(6962,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','[iii/iv]','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two equations and systems of three equations.*4**4*More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.*4*Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through College Algebra (Math 1314).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_A_l_g_/_A_l_g_._a_s_p_x','_m_a_t_h_s_ _-_ _a_l_g_e_b_r_a','_s_3_ _u_p_w_a_r_d_s_ _f_u_l_l','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _A_l_g_e_b_r_a_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_4_*_*_4_*_*_1_*_P_R_E_L_I_M_I_N_A_R_I_E_S_*_2_*_I_n_t_e_g_e_r_ _E_x_p_o_n_e_n_t_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ 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_A_N_D_ _L_O_G_A_R_I_T_H_M_ _F_U_N_C_T_I_O_N_S_*_2_*_E_x_p_o_n_e_n_t_i_a_l_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _w_i_l_l_ _b_e_ _t_a_k_i_n_g_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _o_f_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_*_4_*_L_o_g_a_r_i_t_h_m_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _l_o_g_a_r_i_t_h_m_ _f_u_n_c_t_i_o_n_s_._*_4_*_W_e_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _t_o_ _e_v_a_l_u_a_t_e_ _l_o_g_a_r_i_t_h_m_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _p_r_o_p_e_r_t_i_e_s_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_*_4_*_S_o_l_v_i_n_g_ _E_x_p_o_n_e_n_t_i_a_l_ _E_q_u_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _s_o_l_v_i_n_g_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _e_x_p_o_n_e_n_t_i_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_o_l_v_i_n_g_ _L_o_g_a_r_i_t_h_m_ _E_q_u_a_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_o_l_v_e_ _e_q_u_a_t_i_o_n_s_ _t_h_a_t_ _c_o_n_t_a_i_n_ _l_o_g_a_r_i_t_h_m_s_._*_4_*_A_p_p_l_i_c_a_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_ _o_f_ _e_x_p_o_n_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_ _a_n_d_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _l_o_g_a_r_i_t_h_m_s_._*_1_*_S_Y_S_T_E_M_S_ _O_F_ _E_Q_U_A_T_I_O_N_S_*_2_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_w_o_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _u_s_e_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _t_w_o_ _v_a_r_i_a_b_l_e_s_ _t_o_ _i_n_t_r_o_d_u_c_e_ _t_w_o_ _o_f_ _t_h_e_ _m_a_i_n_ _m_e_t_h_o_d_s_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_._*_4_*_L_i_n_e_a_r_ _S_y_s_t_e_m_s_ _w_i_t_h_ _T_h_r_e_e_ _V_a_r_i_a_b_l_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _w_o_r_k_ _a_ _q_u_i_c_k_ _e_x_a_m_p_l_e_ _t_o_ _s_h_o_w_ _h_o_w_ _t_o_ _u_s_e_ _t_h_e_ _m_e_t_h_o_d_s_ _t_o_ _s_o_l_v_e_ _s_y_s_t_e_m_s_ 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_s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6963,'82.8.167.110','1280483895','http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx','mathematics','[v/vi]','mathematics/algebra','Friday  11:00  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Algebra  - online and e-book (download)*4**4**1*PRELIMINARIES*2*Integer Exponents - In this section we will start looking at exponents and their properties.*4*Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.*4*Real Exponents - This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.*4*Radicals - Here we will define radical notation and relate radicals to rational exponents.  We will also give the properties of radicals.*4*Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.*4*Factoring Polynomials  -This is the most important section of all the preliminaries.*4*Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.*4*Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.*1*SOLVING EQUATIONS AND INEQUALITIES*2*Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.*4*Linear Equations - In this section we will solve linear equations, including equations with rational expressions.*4*Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.*4*Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.*4**4*Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations.*4*We will look at factoring and the square root property in this section.*4**4*Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. *4* We will look at completing the square and quadratic formula in this section.*4**4*Quadratic Equations : A Summary - We*apos*ll give a procedure for determining which method to use in solving quadratic equations in this section.*4*We will also take a quick look at the discriminant.*4**4*Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section*4*only this time they will involve solving a quadratic equation.*4**4*Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.*4*Equations with Radicals - Here we will solve equations with square roots in them.*4**4*Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.*4*Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.*4*Rational Inequalities - Here we will solve inequalities involving rational expressions.*4*Absolute Value Equations - We will officially define absolute value in this section and solve equations that contain absolute value.*4*Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.*1*GRAPHING AND FUNCTIONS*2*Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.*4*Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.*4*Circles - We will look at the equation of a circle and graphing circles in this section.*4**4*The Definition of a Function - We will discuss the definition of a function in this section.*4*We will also introduce the idea of function evaluation.*4**4*Graphing Functions - In this section we will look at the basics of graphing functions.*4*We will also graph some piecewise functions in this section.*4**4*Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.*4*Inverse Functions - We will define and find inverse functions in this section.*1*COMMON GRAPHS*2*Lines, Circles and Piecewise Functions.*4*Parabolas - We*apos*ll be graphing parabolas in this section.*4*Ellipses - In this section we will graph ellipses.*4*Hyperbolas - Here we will be graphing hyperbolas.*4**4*Miscellaneous Functions - In this section we will graph a couple of common functions that don*apos*t really take all that much work to so.*4*We*apos*ll be looking at the constant function, square root, absolute value and a simple cubic function.*4**4*Transformations - We will be looking at shifts and reflections of graphs in this section.*4*Collectively these are often called transformations.*4**4*Symmetry - We will briefly discuss the topic of symmetry in this section.*4*Rational Functions - In this section we will graph some rational functions.  We will also be taking a look at vertical and horizontal asymptotes.*1*POLYNOMIAL FUNCTIONS*2*Dividing Polynomials - We*apos*ll review some of the basics of dividing polynomials in this section.*4*Zeroes/Roots of Polynomials - In this section we*apos*ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.*4*Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.*4*Finding Zeroes of Polynomials - We*apos*ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.*4*Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.*1*EXPONENTIAL AND LOGARITHM FUNCTIONS*2*Exponential Functions - In this section we will introduce exponential functions.*4*We will be taking a look at some of the properties of exponential functions.*4**4*Logarithm Functions - Here we will introduce logarithm functions.*4*We be looking at how to evaluate logarithms as well as the properties of logarithms.*4**4*Solving Exponential Equations - We will be solving equations that contain exponentials in this section.*4*Solving Logarithm Equations - Here we will solve equations that contain logarithms.*4*Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.*1*SYSTEMS OF EQUATIONS*2*Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.*4*Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.*4**4*Augmented Matrices - We will look at the third main method for solving systems in this section.*4*We will look at systems of two 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_s_e_c_t_i_o_n_._*_4_*_E_l_l_i_p_s_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _e_l_l_i_p_s_e_s_._*_4_*_H_y_p_e_r_b_o_l_a_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _g_r_a_p_h_i_n_g_ _h_y_p_e_r_b_o_l_a_s_._*_4_*_*_4_*_M_i_s_c_e_l_l_a_n_e_o_u_s_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _g_r_a_p_h_ _a_ _c_o_u_p_l_e_ _o_f_ _c_o_m_m_o_n_ _f_u_n_c_t_i_o_n_s_ _t_h_a_t_ _d_o_n_*_a_p_o_s_*_t_ _r_e_a_l_l_y_ _t_a_k_e_ _a_l_l_ _t_h_a_t_ _m_u_c_h_ _w_o_r_k_ _t_o_ _s_o_._*_4_*_W_e_*_a_p_o_s_*_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _t_h_e_ _c_o_n_s_t_a_n_t_ _f_u_n_c_t_i_o_n_,_ _s_q_u_a_r_e_ _r_o_o_t_,_ _a_b_s_o_l_u_t_e_ _v_a_l_u_e_ _a_n_d_ _a_ _s_i_m_p_l_e_ _c_u_b_i_c_ _f_u_n_c_t_i_o_n_._*_4_*_*_4_*_T_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _-_ _W_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_h_i_f_t_s_ _a_n_d_ _r_e_f_l_e_c_t_i_o_n_s_ _o_f_ _g_r_a_p_h_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_C_o_l_l_e_c_t_i_v_e_l_y_ _t_h_e_s_e_ _a_r_e_ _o_f_t_e_n_ _c_a_l_l_e_d_ 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_o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_ _w_i_t_h_ _t_h_r_e_e_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_c_e_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _t_h_i_r_d_ _m_a_i_n_ _m_e_t_h_o_d_ _f_o_r_ _s_o_l_v_i_n_g_ _s_y_s_t_e_m_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_y_s_t_e_m_s_ _o_f_ _t_w_o_ _e_q_u_a_t_i_o_n_s_ _a_n_d_ _s_y_s_t_e_m_s_ _o_f_ _t_h_r_e_e_ _e_q_u_a_t_i_o_n_s_._*_4_*_*_4_*_M_o_r_e_ _o_n_ _t_h_e_ _A_u_g_m_e_n_t_e_d_ _M_a_t_r_i_x_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _s_o_m_e_ _s_p_e_c_i_a_l_ _c_a_s_e_s_ _t_o_ _t_h_e_ _s_o_l_u_t_i_o_n_s_ _t_o_ _s_y_s_t_e_m_s_ _a_n_d_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _t_h_e_m_ _u_s_i_n_g_ _t_h_e_ _a_u_g_m_e_n_t_e_d_ _m_a_t_r_i_x_ _m_e_t_h_o_d_._*_4_*_N_o_n_l_i_n_e_a_r_ _S_y_s_t_e_m_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _s_o_l_v_i_n_g_ _n_o_n_l_i_n_e_a_r_ _s_y_s_t_e_m_s_ _o_f_ _e_q_u_a_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_o_l_l_e_g_e_ _A_l_g_e_b_r_a_ _(_M_a_t_h_ _1_3_1_4_)_.','_1','0','_3','0','0'),(6964,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','[11th/12th]','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6965,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','ap','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6966,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','[s5/s6]','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6967,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','higher','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6968,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','adv higher','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6969,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','gcse','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6970,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','a level','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6971,'82.8.167.110','1280490311','http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx','mathematics','[v/vi]','mathematics/calculus','Friday  12:46  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus  - online and e-book (download)*1*REVIEW*2*Review : Functions - Here is a quick review of functions, function notation and a couple of fairly important ideas about functions.*4*Review : Inverse Functions - A quick review of inverse functions and the notation for inverse functions.*4*Review : Trig Functions - A review of trig functions, evaluation of trig functions and the unit circle.*4*Review : Solving Trig Equations - A reminder on how to solve trig equations.*4**4*Review : Solving Trig Equations with Calculators, Part I - The previous section worked problem whose answers were always the *apos*standard*apos* angles.*4*In this section we work some problems whose answers are not *apos*standard*apos* and so a calculator is needed.*4*This section is always covered in my class as most trig equations in the remainder will need a calculator.*4**4*Review : Solving Trig Equations with Calculators, Part II - Even more trig equations requiring a calculator to solve.*4*Review : Exponential Functions - A review of exponential functions.*4*Review : Logarithm Functions - A review of logarithm functions and logarithm properties.*4*Review : Exponential and Logarithm Equations - How to solve exponential and logarithm equations.*4*Review : Common Graphs - This section isn*apos*t much. It*apos*s mostly a collection of graphs of many of the common functions that are liable to be seen in a Calculus class.*1*LIMITS*2*Tangent Lines and Rates of Change - In this section we will take a look at two problems that we will see time and again in this course.*4*These problems will be used to introduce the topic of limits.*4**4*The Limit - Here we will take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us.*4*One-Sided Limits - A brief introduction to one-sided limits.*4*Limit Properties - Properties of limits that we*apos*ll need to use in computing limits.  We will also compute some basic limits in this section*4**4*Computing Limits - Many of the limits we*apos*ll be asked to compute will not be *apos*simple*apos* limits.  In other words, we won*apos*t be able to just apply the properties and be done.*4*In this section we will look at several types of limits that require some work before we can use the limit properties to compute them.*4**4*Infinite Limits - Here we will take a look at limits that have a value of infinity or negative infinity.  We*apos*ll also take a brief look at vertical asymptotes.*4**4*Limits At Infinity, Part I - In this section we*apos*ll look at limits at infinity.  In other words, limits in which the variable gets very large in either the positive or negative sense.*4*We*apos*ll also take a brief look at horizontal asymptotes in this section.  We*apos*ll be concentrating on polynomials and rational expression involving polynomials in this section.*4**4*Limits At Infinity, Part II - We*apos*ll continue to look at limits at infinity in this section, but this time we*apos*ll be looking at exponential, logarithms and inverse tangents.*4*Continuity - In this section we will introduce the concept of continuity and how it relates to limits.  We will also see the Mean Value Theorem in this section.*4*The Definition of the Limit - We will give the exact definition of several of the limits covered in this section.  We*apos*ll also give the exact definition of continuity.*1*DERIVATIVES*2*The Definition of the Derivative - In this section we will be looking at the definition of the derivative. *4*Interpretation of the Derivative - Here we will take a quick look at some interpretations of the derivative.*4*Differentiation Formulas - Here we will start introducing some of the differentiation formulas used in a calculus course.*4**4*Product and Quotient Rule - In this section we will took at differentiating products and quotients of functions.*4**4*Derivatives of Trig Functions - We*apos*ll give the derivatives of the trig functions in this section.*4*Derivatives of Exponential and Logarithm Functions - In this section we will get the derivatives of the exponential and logarithm functions.*4*Derivatives of Inverse Trig Functions - Here we will look at the derivatives of inverse trig functions.*4*Derivatives of Hyperbolic Functions - Here we will look at the derivatives of hyperbolic functions.*4**4*Chain Rule - The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions.*4*Implicit Differentiation - In this section we will be looking at implicit differentiation.  Without this we won*apos*t be able to work some of the applications of derivatives.*4**4*Related Rates - In this section we will look at the lone application to derivatives in this chapter.*4*This topic is here rather than the next chapter because it will help to cement in our minds one of the more important*4*concepts about derivatives and because it requires implicit differentiation.*4**4*Higher Order Derivatives - Here we will introduce the idea of higher order derivatives.*4**4*Logarithmic Differentiation - The topic of logarithmic differentiation is not always presented in a standard calculus course.*4*It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.*1*APPLICATIONS OF DERIVATIVES*2*Rates of Change - The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter.  Namely, rates of change.*4**4*Critical Points - In this section we will define critical points.  Critical points will show up in many of the sections in this chapter so it will be important to understand them.*4*Minimum and Maximum Values - In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.*4*Finding Absolute Extrema - Here is the first application of derivatives that we*apos*ll look at in this chapter.  We will be determining the largest and smallest value of a function on an interval.*4**4*The Shape of a Graph, Part I - We will start looking at the information that the first derivatives can tell us about the graph of a function.*4*We will be looking at increasing/decreasing functions as well as the First Derivative Test.*4**4*The Shape of a Graph, Part II - In this section we will look at the information about the graph of a function that the second derivatives can tell us.*4*We will look at inflection points, concavity, and the Second Derivative Test.*4**4*The Mean Value Theorem - Here we will take a look that the Mean Value Theorem.*4*Optimization Problems - This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possible subject to some constraint.*4*More Optimization Problems - Here are even more optimization problems.*4**4*L*apos*Hospital*apos*s Rule and Indeterminate Forms - This isn*apos*t the first time that we*apos*ve looked at indeterminate forms.*4*In this section we will take a look at L*apos*Hospital*apos*s Rule.  This rule will allow us to compute some limits that we couldn*apos*t do until this section.*4**4*Linear Approximations - Here we will use derivatives to compute a linear approximation to a function. *4*As we will see however, we*apos*ve actually already done this.*4**4*Differentials - We will look at differentials in this section as well as an application for them.*4*Newton*apos*s Method - With this application of derivatives we*apos*ll see how to approximate solutions to an equation.*4*Business Applications - Here we will take a quick look at some applications of derivatives to the business field.*1*INTEGRALS*2*Indefinite Integrals - In this section we will start with the definition of indefinite integral.  *4*This section will be devoted mostly to the definition and properties of indefinite integrals and we won*apos*t be working many examples in this section.*4**4*Computing Indefinite Integrals - In this section we will compute some indefinite integrals and take a look at a quick application of indefinite integrals.*4**4*Substitution Rule for Indefinite Integrals - Here we will look at the Substitution Rule as it applies to indefinite integrals.*4*Many of the integrals that we*apos*ll be doing later on in the course and in later courses will require use of the substitution rule.*4**4*More Substitution Rule - Even more substitution rule problems. *4*Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals.*4**4*Definition of the Definite Integral - We will formally define the definite integral in this section and give many of its properties.*4*We will also take a look at the first part of the Fundamental Theorem of Calculus.*4**4*Computing Definite Integrals - We will take a look at the second part of the Fundamental Theorem of Calculus in this section and start to compute definite integrals.*4*Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals.*1*APPLICATIONS OF INTEGRALS*2*Average Function Value - We can use integrals to determine the average value of a function.*4*Area Between Two Curves - In this section we*apos*ll take a look at determining the area between two curves.*4**4*Volumes of Solids of Revolution / Method of Rings - This is the first of two sections devoted to find the volume of a solid of revolution.*4*In this section we look that the method of rings/disks.*4**4*Volumes of Solids of Revolution / Method of Cylinders - This is the second section devoted to finding the volume of a solid of revolution.*4*Here we will look at the method of cylinders.*4**4*More Volume Problems - In this section we*apos*ll take a look at find the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.*4*Work - The final application we will look at is determining the amount of work required to move an object.*4**1*EXTRAS*2*Proof of Various Limit Properties - In we prove several of the limit properties and facts that were given in various sections of the Limits chapter.*4**4*Proof of Various Derivative Facts/Formulas/Properties - In this section we give the proof for several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.*4*Included are multiple proofs of the Power Rule, Product Rule, Quotient Rule and Chain Rule.*4**4*Proof of Trig Limits - Here we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions.*4*Proofs of Derivative Applications Facts/Formulas - We*apos*ll give proofs of many of the facts that we saw in the Applications of Derivatives chapter.*4**4*Proof of Various Integral Facts/Formulas/Properties - Here we will give the proofs of some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.*4**4*Area and Volume Formulas - Here is the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.*4*Types of Infinity - This is a discussion on the types of infinity and how these affect certain limits.*4*Summation Notation - Here is a quick review of summation notation.*4*Constant of Integration - This is a discussion on a couple of subtleties involving constants of integration that many students don*apos*t think about.*4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus I (Math 2413).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_/_C_a_l_c_I_._a_s_p_x','_m_a_t_h_s_ _c_a_l_c_u_l_u_s','_s_5_ _f_u_l_l_ _u_p_w_a_r_d_s','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6972,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','[11th/12th]','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6973,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','ap','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6974,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','[s5/s6]','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6975,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','higher','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6976,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','adv higher','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6977,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','gcse','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6978,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','a level','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6979,'82.8.167.110','1280506391','http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx','mathematics','[v/vi]','mathematics/calculus','Friday  17:14  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus II  - online and e-book (download)*1*INTEGRATION TECHNIQUES*2*Integration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.*4*Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions.*4**4*Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.*4**4*Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions.*4**4*Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots.*4*Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics.*4*Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.*4*Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral.*4*Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.*4*Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge.*4**4*Approximating Definite Integrals - There are many ways to approximate the value of a definite integral.  We will look at three of them in this section.*1*APPLICATIONS OF INTEGRALS*2*Arc Length - We*apos*ll determine the length of a curve in this section.*4*Surface Area - In this section we*apos*ll determine the surface area of a solid of revolution.*4*Center of Mass - Here we will determine the center of mass or centroid of a thin plate.*4**4*Hydrostatic Pressure and Force - We*apos*ll determine the hydrostatic pressure and force on a vertical plate submerged in water.*4**4*Probability - Here we will look at probability density functions and computing the mean of a probability density function.*1*PARAMETRIC EQUATIONS AND POLAR COORDINATES*2*Parametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)*4*Tangents with Parametric Equations - Finding tangent lines to parametric curves.*4**4*Area with Parametric Equations - Finding the area under a parametric curve.*4*Arc Length with Parametric Equations - Determining the length of a parametric curve.*4**4*Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.*4**4*Polar Coordinates - We*apos*ll introduce polar coordinates in this section. We*apos*ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.*4*Tangents with Polar Coordinates - Finding tangent lines of polar curves.*4*Area with Polar Coordinates - Finding the area enclosed by a polar curve.*4*Arc Length with Polar Coordinates - Determining the length of a polar curve.*4**4*Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.*4*Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters.*1*SEQUENCES AND SERIES*2*Sequences - We will start the chapter off with a brief discussion of sequences.  This section will focus on the basic terminology and convergence of sequences*4**4*More on Sequences - Here we will take a quick look about monotonic and bounded sequences.*4*Series The Basics - In this section we will discuss some of the basics of infinite series.*4*Series  Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.*4*Series  Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.*4**4*Integral Test - Using the Integral Test to determine if a series converges or diverges.*4*Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.*4*Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges.*4**4*Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence.*4*Ratio Test - Using the Ratio Test to determine if a series converges or diverges.*4*Root Test - Using the Root Test to determine if a series converges or diverges.*4**4*Strategy for Series - A set of general guidelines to use when deciding which test to use.*4*Estimating the Value of a Series - Here we will look at estimating the value of an infinite series.*4*Power Series - An introduction to power series and some of the basic concepts.*4*Power Series and Functions - In this section we will start looking at how to find a power series representation of a function.*4**4*Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function.*4*Applications of Series - In this section we will take a quick look at a couple of applications of series.*4*Binomial Series - A brief look at binomial series.*1*VECTORS*2*Vectors The Basics - In this section we will introduce some of the basic concepts about vectors.*4*Vector Arithmetic - Here we will give the basic arithmetic operations for vectors.*4*Dot Product - We will discuss the dot product in this section as well as an application or two.*4*Cross Product - In this section we*apos*ll discuss the cross product and see a quick application.*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes.  When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_/_C_a_l_c_I_I_._a_s_p_x','_m_a_t_h_s','_s_5_ _u_p','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','String Length Greater Than Maximum Allowed (15000).  Please Re-enter.','_1','0','_3','0','0'),(6980,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','[11th/12th]','mathematics/calculus','Friday  17:28  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  In particular we will look at line integrals with respect to arc length.*4*Line Integrals  Part II - We will continue looking at line integrals in this section.  Here we will be looking at line integrals with respect to x, y, and/or z.*4*Line Integrals of Vector Fields - Here we will look at a third type of line integrals, line integrals of vector fields.*4*Fundamental Theorem for Line Integrals - In this section we will look at a version of the fundamental theorem of calculus for line integrals of vector fields.*4*Conservative Vector Fields - Here we will take a somewhat detailed look at conservative vector fields and how to find potential functions.*4*Green*apos*s Theorem - We will give Green*apos*s Theorem in this section as well as an interesting application of Green*apos*s Theorem.*4*Curl and Divergence - In this section we will introduce the concepts of the curl and the divergence of a vector field.  We will also give two vector forms of Green*apos*s Theorem.*1*SURFACE INTEGRALS*2*Parametric Surfaces - In this section we will take a look at the basics of representing a surface with parametric equations.  We will also take a look at a couple of applications.*4*Surface Integrals - Here we will introduce the topic of surface integrals.  We will be working with surface integrals of functions in this section.*4*Surface Integrals of Vector Fields - We will look at surface integrals of vector fields in this section.*4*Stokes*apos* Theorem - We will look at Stokes*apos* Theorem in this section.*4*Divergence Theorem - Here we will take a look at the Divergence Theorem.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus III (Math 2415).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_I_/_C_a_l_c_I_I_I_._a_s_p_x','_m_a_t_h_s','_s_5','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _C_a_l_c_u_l_u_s_ _I_I_I_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_1_*_T_H_R_E_E_ _D_I_M_E_N_S_I_O_N_A_L_ _S_P_A_C_E_*_2_*_T_h_i_s_ _i_s_ _t_h_e_ _o_n_l_y_ _c_h_a_p_t_e_r_ _t_h_a_t_ _e_x_i_s_t_s_ _i_n_ _t_w_o_ _p_l_a_c_e_s_ _i_n_ _m_y_ _n_o_t_e_s_._ _W_h_e_n_ _I_ _o_r_i_g_i_n_a_l_l_y_ _w_r_o_t_e_ _t_h_e_s_e_ _n_o_t_e_s_ _a_l_l_ _o_f_ _t_h_e_s_e_ _t_o_p_i_c_s_ _w_e_r_e_ _c_o_v_e_r_e_d_ _i_n_ _C_a_l_c_u_l_u_s_ _I_I_ _h_o_w_e_v_e_r_,_*_4_*_w_e_ _h_a_v_e_ _s_i_n_c_e_ _m_o_v_e_d_ _s_e_v_e_r_a_l_ _o_f_ _t_h_e_m_ _i_n_t_o_ _C_a_l_c_u_l_u_s_ _I_I_I_._ _ _S_o_,_ _r_a_t_h_e_r_ _t_h_a_n_ _s_p_l_i_t_ _t_h_e_ _c_h_a_p_t_e_r_ _u_p_ _I_ _h_a_v_e_ _k_e_p_t_ _i_t_ _i_n_ _t_h_e_ _C_a_l_c_u_l_u_s_ _I_I_ _n_o_t_e_s_ _a_n_d_ _a_l_s_o_ _p_u_t_ _a_ _c_o_p_y_ _i_n_ _t_h_e_ _C_a_l_c_u_l_u_s_ _I_I_I_ _n_o_t_e_s_._*_4_*_*_4_*_T_h_e_ _3_-_D_ _C_o_o_r_d_i_n_a_t_e_ _S_y_s_t_e_m_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_s_ _a_n_d_ _n_o_t_a_t_i_o_n_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _L_i_n_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_v_e_l_o_p_ _t_h_e_ _v_a_r_i_o_u_s_ _f_o_r_m_s_ _f_o_r_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _l_i_n_e_s_ _i_n_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _s_p_a_c_e_._*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _P_l_a_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_v_e_l_o_p_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _p_l_a_n_e_._*_4_*_Q_u_a_d_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_o_m_e_ _e_x_a_m_p_l_e_s_ _o_f_ _q_u_a_d_r_i_c_ _s_u_r_f_a_c_e_s_._*_4_*_F_u_n_c_t_i_o_n_s_ _o_f_ _S_e_v_e_r_a_l_ _V_a_r_i_a_b_l_e_s_ _-_ _A_ _q_u_i_c_k_ _r_e_v_i_e_w_ _o_f_ _s_o_m_e_ _i_m_p_o_r_t_a_n_t_ _t_o_p_i_c_s_ _a_b_o_u_t_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _W_e_ _c_o_n_c_e_n_t_r_a_t_e_ _p_r_i_m_a_r_i_l_y_ _o_n_ _c_u_r_v_e_s_ _i_n_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _s_p_a_c_e_._ _ _W_e_ _w_i_l_l_ _h_o_w_e_v_e_r_,_ _t_o_u_c_h_ _b_r_i_e_f_l_y_ _o_n_ _s_u_r_f_a_c_e_s_ _a_s_ _w_e_l_l_._*_4_*_C_a_l_c_u_l_u_s_ _w_i_t_h_ _V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _l_i_m_i_t_s_,_ _d_e_r_i_v_a_t_i_v_e_s_,_ _a_n_d_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_s_._*_4_*_T_a_n_g_e_n_t_,_ _N_o_r_m_a_l_ _a_n_d_ _B_i_n_o_r_m_a_l_ _V_e_c_t_o_r_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_a_n_g_e_n_t_,_ _n_o_r_m_a_l_ _a_n_d_ _b_i_n_o_r_m_a_l_ _v_e_c_t_o_r_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_A_r_c_ _L_e_n_g_t_h_ _w_i_t_h_ _V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _f_i_n_d_ _t_h_e_ _a_r_c_ _l_e_n_g_t_h_ _o_f_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_u_r_v_a_t_u_r_e_ _-_ _W_e_ _w_i_l_l_ _d_e_t_e_r_m_i_n_e_ _t_h_e_ _c_u_r_v_a_t_u_r_e_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_V_e_l_o_c_i_t_y_ _a_n_d_ _A_c_c_e_l_e_r_a_t_i_o_n_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _a_ _s_t_a_n_d_a_r_d_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_e_r_i_v_a_t_i_v_e_s_._ _ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _v_e_l_o_c_i_t_y_ _a_n_d_ _a_c_c_e_l_e_r_a_t_i_o_n_ _o_f_ _a_n_ _o_b_j_e_c_t_ _w_h_o_s_e_ _p_o_s_i_t_i_o_n_ _f_u_n_c_t_i_o_n_ _i_s_ _g_i_v_e_n_ _b_y_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _T_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _a_n_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_4_*_S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._ _ _T_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _y_e_t_ _a_n_o_t_h_e_r_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_1_*_P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_L_i_m_i_t_s_ _-_ _T_a_k_i_n_g_ _l_i_m_i_t_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _s_t_a_n_d_a_r_d_ _n_o_t_a_t_i_o_n_s_ _a_n_d_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_._*_4_*_I_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _i_m_p_o_r_t_a_n_t_ _i_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_4_*_*_4_*_H_i_g_h_e_r_ _O_r_d_e_r_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _h_i_g_h_e_r_ _o_r_d_e_r_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_f_f_e_r_e_n_t_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _e_x_t_e_n_d_ _t_h_e_ _i_d_e_a_ _o_f_ _d_i_f_f_e_r_e_n_t_i_a_l_s_ _t_o_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_C_h_a_i_n_ _R_u_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _c_h_a_i_n_ _r_u_l_e_ _f_o_r_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_D_i_r_e_c_t_i_o_n_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _s_e_e_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_ _a_n_d_ _s_e_e_ _a_ _c_o_u_p_l_e_ _o_f_ _n_i_c_e_ _f_a_c_t_s_ _p_e_r_t_a_i_n_i_n_g_ _t_o_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_1_*_A_P_P_L_I_C_A_T_I_O_N_S_ _O_F_ _P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _L_i_n_e_a_r_ _A_p_p_r_o_x_i_m_a_t_i_o_n_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _t_o_ _s_u_r_f_a_c_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_._*_4_*_G_r_a_d_i_e_n_t_ _V_e_c_t_o_r_,_ _T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _N_o_r_m_a_l_ _L_i_n_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_h_e_ _g_r_a_d_i_e_n_t_ _v_e_c_t_o_r_ _c_a_n_ _b_e_ _u_s_e_d_ _t_o_ _f_i_n_d_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _a_n_d_ _n_o_r_m_a_l_ _l_i_n_e_s_ _t_o_ _a_ _s_u_r_f_a_c_e_._*_4_*_R_e_l_a_t_i_v_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_e_e_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _r_e_l_a_t_i_v_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_._*_4_*_A_b_s_o_l_u_t_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _W_e_ _w_i_l_l_ _f_i_n_d_ _a_b_s_o_l_u_t_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _o_v_e_r_ _a_ _g_i_v_e_n_ _r_e_g_i_o_n_._*_4_*_L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_o_ _u_s_e_ _L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _t_o_ _f_i_n_d_ _t_h_e_ _a_b_s_o_l_u_t_e_ _e_x_t_r_e_m_a_ _f_o_r_ _a_ _f_u_n_c_t_i_o_n_ _s_u_b_j_e_c_t_ _t_o_ _a_ _g_i_v_e_n_ _c_o_n_s_t_r_a_i_n_t_._*_1_*_M_U_L_T_I_P_L_E_ _I_N_T_E_G_R_A_L_S_*_2_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_I_t_e_r_a_t_e_d_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _w_e_ _a_c_t_u_a_l_l_y_ _c_o_m_p_u_t_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _o_v_e_r_ _G_e_n_e_r_a_l_ _R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ 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_f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_a_l_c_u_l_u_s_ _I_I_I_ _(_M_a_t_h_ _2_4_1_5_)_.','_1','0','_3','0','0'),(6981,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','ap','mathematics/calculus','Friday  17:28  30/07/10','1','0','0','1','1','0','0','0','0','0','4','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  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We will also give two vector forms of Green*apos*s Theorem.*1*SURFACE INTEGRALS*2*Parametric Surfaces - In this section we will take a look at the basics of representing a surface with parametric equations.  We will also take a look at a couple of applications.*4*Surface Integrals - Here we will introduce the topic of surface integrals.  We will be working with surface integrals of functions in this section.*4*Surface Integrals of Vector Fields - We will look at surface integrals of vector fields in this section.*4*Stokes*apos* Theorem - We will look at Stokes*apos* Theorem in this section.*4*Divergence Theorem - Here we will take a look at the Divergence Theorem.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus III (Math 2415).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_I_/_C_a_l_c_I_I_I_._a_s_p_x','_m_a_t_h_s','_s_5','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _C_a_l_c_u_l_u_s_ _I_I_I_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_1_*_T_H_R_E_E_ _D_I_M_E_N_S_I_O_N_A_L_ _S_P_A_C_E_*_2_*_T_h_i_s_ _i_s_ _t_h_e_ _o_n_l_y_ _c_h_a_p_t_e_r_ _t_h_a_t_ _e_x_i_s_t_s_ _i_n_ _t_w_o_ _p_l_a_c_e_s_ _i_n_ _m_y_ 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_R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ 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_f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ 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INSERT INTO `full_link_ok` VALUES (6982,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','[s5/s6]','mathematics/calculus','Friday  17:29  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  In particular we will look at line integrals with respect to arc length.*4*Line Integrals  Part II - We will continue looking at line integrals in this section.  Here we will be looking at line integrals with respect to x, y, and/or z.*4*Line Integrals of Vector Fields - Here we will look at a third type of line integrals, line integrals of vector fields.*4*Fundamental Theorem for Line Integrals - In this section we will look at a version of the fundamental theorem of calculus for line integrals of vector fields.*4*Conservative Vector Fields - Here we will take a somewhat detailed look at conservative vector fields and how to find potential functions.*4*Green*apos*s Theorem - We will give Green*apos*s Theorem in this section as well as an interesting application of Green*apos*s Theorem.*4*Curl and Divergence - In this section we will introduce the concepts of the curl and the divergence of a vector field.  We will also give two vector forms of Green*apos*s Theorem.*1*SURFACE INTEGRALS*2*Parametric Surfaces - In this section we will take a look at the basics of representing a surface with parametric equations.  We will also take a look at a couple of applications.*4*Surface Integrals - Here we will introduce the topic of surface integrals.  We will be working with surface integrals of functions in this section.*4*Surface Integrals of Vector Fields - We will look at surface integrals of vector fields in this section.*4*Stokes*apos* Theorem - We will look at Stokes*apos* Theorem in this section.*4*Divergence Theorem - Here we will take a look at the Divergence Theorem.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus III (Math 2415).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_I_/_C_a_l_c_I_I_I_._a_s_p_x','_m_a_t_h_s','_s_5','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _C_a_l_c_u_l_u_s_ _I_I_I_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_1_*_T_H_R_E_E_ _D_I_M_E_N_S_I_O_N_A_L_ _S_P_A_C_E_*_2_*_T_h_i_s_ _i_s_ _t_h_e_ _o_n_l_y_ _c_h_a_p_t_e_r_ _t_h_a_t_ _e_x_i_s_t_s_ _i_n_ _t_w_o_ _p_l_a_c_e_s_ _i_n_ _m_y_ 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_f_u_n_d_a_m_e_n_t_a_l_ _t_h_e_o_r_e_m_ _o_f_ _c_a_l_c_u_l_u_s_ _f_o_r_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_._*_4_*_C_o_n_s_e_r_v_a_t_i_v_e_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _s_o_m_e_w_h_a_t_ _d_e_t_a_i_l_e_d_ _l_o_o_k_ _a_t_ _c_o_n_s_e_r_v_a_t_i_v_e_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _a_n_d_ _h_o_w_ _t_o_ _f_i_n_d_ _p_o_t_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _g_i_v_e_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _i_n_t_e_r_e_s_t_i_n_g_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_4_*_C_u_r_l_ _a_n_d_ _D_i_v_e_r_g_e_n_c_e_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_s_ _o_f_ _t_h_e_ _c_u_r_l_ _a_n_d_ _t_h_e_ _d_i_v_e_r_g_e_n_c_e_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_w_o_ _v_e_c_t_o_r_ _f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_a_l_c_u_l_u_s_ _I_I_I_ _(_M_a_t_h_ _2_4_1_5_)_.','_1','0','_3','0','0'),(6983,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','higher','mathematics/calculus','Friday  17:29  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  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_s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _a_ _s_t_a_n_d_a_r_d_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_e_r_i_v_a_t_i_v_e_s_._ _ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _v_e_l_o_c_i_t_y_ _a_n_d_ _a_c_c_e_l_e_r_a_t_i_o_n_ _o_f_ _a_n_ _o_b_j_e_c_t_ _w_h_o_s_e_ _p_o_s_i_t_i_o_n_ _f_u_n_c_t_i_o_n_ _i_s_ _g_i_v_e_n_ _b_y_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _T_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _a_n_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_4_*_S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._ _ _T_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _y_e_t_ _a_n_o_t_h_e_r_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_1_*_P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_L_i_m_i_t_s_ _-_ _T_a_k_i_n_g_ _l_i_m_i_t_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _s_t_a_n_d_a_r_d_ _n_o_t_a_t_i_o_n_s_ _a_n_d_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_._*_4_*_I_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _i_m_p_o_r_t_a_n_t_ _i_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_4_*_*_4_*_H_i_g_h_e_r_ _O_r_d_e_r_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _h_i_g_h_e_r_ _o_r_d_e_r_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_f_f_e_r_e_n_t_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _e_x_t_e_n_d_ _t_h_e_ _i_d_e_a_ _o_f_ _d_i_f_f_e_r_e_n_t_i_a_l_s_ _t_o_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_C_h_a_i_n_ _R_u_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _c_h_a_i_n_ _r_u_l_e_ _f_o_r_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_D_i_r_e_c_t_i_o_n_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _s_e_e_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_ _a_n_d_ _s_e_e_ _a_ _c_o_u_p_l_e_ _o_f_ _n_i_c_e_ _f_a_c_t_s_ _p_e_r_t_a_i_n_i_n_g_ _t_o_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_1_*_A_P_P_L_I_C_A_T_I_O_N_S_ _O_F_ _P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _L_i_n_e_a_r_ _A_p_p_r_o_x_i_m_a_t_i_o_n_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _t_o_ _s_u_r_f_a_c_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_._*_4_*_G_r_a_d_i_e_n_t_ _V_e_c_t_o_r_,_ _T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _N_o_r_m_a_l_ _L_i_n_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_h_e_ _g_r_a_d_i_e_n_t_ _v_e_c_t_o_r_ _c_a_n_ _b_e_ _u_s_e_d_ _t_o_ _f_i_n_d_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _a_n_d_ _n_o_r_m_a_l_ _l_i_n_e_s_ _t_o_ _a_ _s_u_r_f_a_c_e_._*_4_*_R_e_l_a_t_i_v_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_e_e_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _r_e_l_a_t_i_v_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_._*_4_*_A_b_s_o_l_u_t_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _W_e_ _w_i_l_l_ _f_i_n_d_ _a_b_s_o_l_u_t_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _o_v_e_r_ _a_ _g_i_v_e_n_ _r_e_g_i_o_n_._*_4_*_L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_o_ _u_s_e_ _L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _t_o_ _f_i_n_d_ _t_h_e_ _a_b_s_o_l_u_t_e_ _e_x_t_r_e_m_a_ _f_o_r_ _a_ _f_u_n_c_t_i_o_n_ _s_u_b_j_e_c_t_ _t_o_ _a_ _g_i_v_e_n_ _c_o_n_s_t_r_a_i_n_t_._*_1_*_M_U_L_T_I_P_L_E_ _I_N_T_E_G_R_A_L_S_*_2_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_I_t_e_r_a_t_e_d_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _w_e_ _a_c_t_u_a_l_l_y_ _c_o_m_p_u_t_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _o_v_e_r_ _G_e_n_e_r_a_l_ _R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_._ _ _I_n_ _p_a_r_t_i_c_u_l_a_r_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _r_e_s_p_e_c_t_ _t_o_ _a_r_c_ _l_e_n_g_t_h_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _c_o_n_t_i_n_u_e_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _r_e_s_p_e_c_t_ _t_o_ _x_,_ _y_,_ _a_n_d_/_o_r_ _z_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _t_h_i_r_d_ _t_y_p_e_ _o_f_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_,_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_._*_4_*_F_u_n_d_a_m_e_n_t_a_l_ _T_h_e_o_r_e_m_ _f_o_r_ _L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _v_e_r_s_i_o_n_ _o_f_ _t_h_e_ _f_u_n_d_a_m_e_n_t_a_l_ _t_h_e_o_r_e_m_ _o_f_ _c_a_l_c_u_l_u_s_ _f_o_r_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_._*_4_*_C_o_n_s_e_r_v_a_t_i_v_e_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _s_o_m_e_w_h_a_t_ _d_e_t_a_i_l_e_d_ _l_o_o_k_ _a_t_ _c_o_n_s_e_r_v_a_t_i_v_e_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _a_n_d_ _h_o_w_ _t_o_ _f_i_n_d_ _p_o_t_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _g_i_v_e_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _i_n_t_e_r_e_s_t_i_n_g_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_4_*_C_u_r_l_ _a_n_d_ _D_i_v_e_r_g_e_n_c_e_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_s_ _o_f_ _t_h_e_ _c_u_r_l_ _a_n_d_ _t_h_e_ _d_i_v_e_r_g_e_n_c_e_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_w_o_ _v_e_c_t_o_r_ _f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_a_l_c_u_l_u_s_ _I_I_I_ _(_M_a_t_h_ _2_4_1_5_)_.','_1','0','_3','0','0'),(6984,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','adv higher','mathematics/calculus','Friday  17:29  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. 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We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  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_t_o_u_c_h_ _b_r_i_e_f_l_y_ _o_n_ _s_u_r_f_a_c_e_s_ _a_s_ _w_e_l_l_._*_4_*_C_a_l_c_u_l_u_s_ _w_i_t_h_ _V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _l_i_m_i_t_s_,_ _d_e_r_i_v_a_t_i_v_e_s_,_ _a_n_d_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_s_._*_4_*_T_a_n_g_e_n_t_,_ _N_o_r_m_a_l_ _a_n_d_ _B_i_n_o_r_m_a_l_ _V_e_c_t_o_r_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_a_n_g_e_n_t_,_ _n_o_r_m_a_l_ _a_n_d_ _b_i_n_o_r_m_a_l_ _v_e_c_t_o_r_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_A_r_c_ _L_e_n_g_t_h_ _w_i_t_h_ _V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _f_i_n_d_ _t_h_e_ _a_r_c_ _l_e_n_g_t_h_ _o_f_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_u_r_v_a_t_u_r_e_ _-_ _W_e_ _w_i_l_l_ _d_e_t_e_r_m_i_n_e_ _t_h_e_ _c_u_r_v_a_t_u_r_e_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_V_e_l_o_c_i_t_y_ _a_n_d_ _A_c_c_e_l_e_r_a_t_i_o_n_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _a_ _s_t_a_n_d_a_r_d_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_e_r_i_v_a_t_i_v_e_s_._ _ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _v_e_l_o_c_i_t_y_ _a_n_d_ _a_c_c_e_l_e_r_a_t_i_o_n_ _o_f_ _a_n_ _o_b_j_e_c_t_ _w_h_o_s_e_ _p_o_s_i_t_i_o_n_ _f_u_n_c_t_i_o_n_ _i_s_ _g_i_v_e_n_ _b_y_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _T_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _a_n_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_4_*_S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._ _ _T_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _y_e_t_ _a_n_o_t_h_e_r_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_1_*_P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_L_i_m_i_t_s_ _-_ _T_a_k_i_n_g_ _l_i_m_i_t_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _s_t_a_n_d_a_r_d_ _n_o_t_a_t_i_o_n_s_ _a_n_d_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_._*_4_*_I_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _i_m_p_o_r_t_a_n_t_ _i_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_4_*_*_4_*_H_i_g_h_e_r_ _O_r_d_e_r_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _h_i_g_h_e_r_ _o_r_d_e_r_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_f_f_e_r_e_n_t_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _e_x_t_e_n_d_ _t_h_e_ _i_d_e_a_ _o_f_ _d_i_f_f_e_r_e_n_t_i_a_l_s_ _t_o_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_C_h_a_i_n_ _R_u_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _c_h_a_i_n_ _r_u_l_e_ _f_o_r_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_D_i_r_e_c_t_i_o_n_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _s_e_e_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_ _a_n_d_ _s_e_e_ _a_ _c_o_u_p_l_e_ _o_f_ _n_i_c_e_ _f_a_c_t_s_ _p_e_r_t_a_i_n_i_n_g_ _t_o_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_1_*_A_P_P_L_I_C_A_T_I_O_N_S_ _O_F_ _P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _L_i_n_e_a_r_ _A_p_p_r_o_x_i_m_a_t_i_o_n_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _t_o_ _s_u_r_f_a_c_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_._*_4_*_G_r_a_d_i_e_n_t_ _V_e_c_t_o_r_,_ _T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _N_o_r_m_a_l_ _L_i_n_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_h_e_ _g_r_a_d_i_e_n_t_ _v_e_c_t_o_r_ _c_a_n_ _b_e_ _u_s_e_d_ _t_o_ _f_i_n_d_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _a_n_d_ _n_o_r_m_a_l_ _l_i_n_e_s_ _t_o_ _a_ _s_u_r_f_a_c_e_._*_4_*_R_e_l_a_t_i_v_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_e_e_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _r_e_l_a_t_i_v_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_._*_4_*_A_b_s_o_l_u_t_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _W_e_ _w_i_l_l_ _f_i_n_d_ _a_b_s_o_l_u_t_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _o_v_e_r_ _a_ _g_i_v_e_n_ _r_e_g_i_o_n_._*_4_*_L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_o_ _u_s_e_ _L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _t_o_ _f_i_n_d_ _t_h_e_ _a_b_s_o_l_u_t_e_ _e_x_t_r_e_m_a_ _f_o_r_ _a_ _f_u_n_c_t_i_o_n_ _s_u_b_j_e_c_t_ _t_o_ _a_ _g_i_v_e_n_ _c_o_n_s_t_r_a_i_n_t_._*_1_*_M_U_L_T_I_P_L_E_ _I_N_T_E_G_R_A_L_S_*_2_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_I_t_e_r_a_t_e_d_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _w_e_ _a_c_t_u_a_l_l_y_ _c_o_m_p_u_t_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _o_v_e_r_ _G_e_n_e_r_a_l_ _R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_._ _ _I_n_ _p_a_r_t_i_c_u_l_a_r_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _r_e_s_p_e_c_t_ _t_o_ _a_r_c_ _l_e_n_g_t_h_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_I_ _-_ _W_e_ _w_i_l_l_ _c_o_n_t_i_n_u_e_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _H_e_r_e_ _w_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _r_e_s_p_e_c_t_ _t_o_ _x_,_ _y_,_ _a_n_d_/_o_r_ _z_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _t_h_i_r_d_ _t_y_p_e_ _o_f_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_,_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_._*_4_*_F_u_n_d_a_m_e_n_t_a_l_ _T_h_e_o_r_e_m_ _f_o_r_ _L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _a_ _v_e_r_s_i_o_n_ _o_f_ _t_h_e_ 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_f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_a_l_c_u_l_u_s_ _I_I_I_ _(_M_a_t_h_ _2_4_1_5_)_.','_1','0','_3','0','0'),(6985,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','gcse','mathematics/calculus','Friday  17:29  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  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_n_o_t_e_s_._ _W_h_e_n_ _I_ _o_r_i_g_i_n_a_l_l_y_ _w_r_o_t_e_ _t_h_e_s_e_ _n_o_t_e_s_ _a_l_l_ _o_f_ _t_h_e_s_e_ _t_o_p_i_c_s_ _w_e_r_e_ _c_o_v_e_r_e_d_ _i_n_ _C_a_l_c_u_l_u_s_ _I_I_ _h_o_w_e_v_e_r_,_*_4_*_w_e_ _h_a_v_e_ _s_i_n_c_e_ _m_o_v_e_d_ _s_e_v_e_r_a_l_ _o_f_ _t_h_e_m_ _i_n_t_o_ _C_a_l_c_u_l_u_s_ _I_I_I_._ _ _S_o_,_ _r_a_t_h_e_r_ _t_h_a_n_ _s_p_l_i_t_ _t_h_e_ _c_h_a_p_t_e_r_ _u_p_ _I_ _h_a_v_e_ _k_e_p_t_ _i_t_ _i_n_ _t_h_e_ _C_a_l_c_u_l_u_s_ _I_I_ _n_o_t_e_s_ _a_n_d_ _a_l_s_o_ _p_u_t_ _a_ _c_o_p_y_ _i_n_ _t_h_e_ _C_a_l_c_u_l_u_s_ _I_I_I_ _n_o_t_e_s_._*_4_*_*_4_*_T_h_e_ _3_-_D_ _C_o_o_r_d_i_n_a_t_e_ _S_y_s_t_e_m_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_s_ _a_n_d_ _n_o_t_a_t_i_o_n_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _L_i_n_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_v_e_l_o_p_ _t_h_e_ _v_a_r_i_o_u_s_ _f_o_r_m_s_ _f_o_r_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _l_i_n_e_s_ _i_n_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _s_p_a_c_e_._*_4_*_E_q_u_a_t_i_o_n_s_ _o_f_ _P_l_a_n_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_v_e_l_o_p_ _t_h_e_ _e_q_u_a_t_i_o_n_ _o_f_ _a_ _p_l_a_n_e_._*_4_*_Q_u_a_d_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _b_e_ _l_o_o_k_i_n_g_ _a_t_ _s_o_m_e_ _e_x_a_m_p_l_e_s_ _o_f_ _q_u_a_d_r_i_c_ _s_u_r_f_a_c_e_s_._*_4_*_F_u_n_c_t_i_o_n_s_ _o_f_ _S_e_v_e_r_a_l_ _V_a_r_i_a_b_l_e_s_ _-_ _A_ _q_u_i_c_k_ _r_e_v_i_e_w_ _o_f_ _s_o_m_e_ _i_m_p_o_r_t_a_n_t_ _t_o_p_i_c_s_ _a_b_o_u_t_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _W_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _W_e_ _c_o_n_c_e_n_t_r_a_t_e_ _p_r_i_m_a_r_i_l_y_ _o_n_ _c_u_r_v_e_s_ _i_n_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _s_p_a_c_e_._ _ _W_e_ _w_i_l_l_ _h_o_w_e_v_e_r_,_ _t_o_u_c_h_ _b_r_i_e_f_l_y_ _o_n_ _s_u_r_f_a_c_e_s_ _a_s_ _w_e_l_l_._*_4_*_C_a_l_c_u_l_u_s_ _w_i_t_h_ _V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _q_u_i_c_k_ _l_o_o_k_ _a_t_ _l_i_m_i_t_s_,_ _d_e_r_i_v_a_t_i_v_e_s_,_ _a_n_d_ _i_n_t_e_g_r_a_l_s_ _w_i_t_h_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_s_._*_4_*_T_a_n_g_e_n_t_,_ _N_o_r_m_a_l_ _a_n_d_ _B_i_n_o_r_m_a_l_ _V_e_c_t_o_r_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_a_n_g_e_n_t_,_ _n_o_r_m_a_l_ _a_n_d_ _b_i_n_o_r_m_a_l_ _v_e_c_t_o_r_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_A_r_c_ _L_e_n_g_t_h_ _w_i_t_h_ _V_e_c_t_o_r_ _F_u_n_c_t_i_o_n_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _f_i_n_d_ _t_h_e_ _a_r_c_ _l_e_n_g_t_h_ _o_f_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_u_r_v_a_t_u_r_e_ _-_ _W_e_ _w_i_l_l_ _d_e_t_e_r_m_i_n_e_ _t_h_e_ _c_u_r_v_a_t_u_r_e_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_*_4_*_V_e_l_o_c_i_t_y_ _a_n_d_ _A_c_c_e_l_e_r_a_t_i_o_n_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _r_e_v_i_s_i_t_ _a_ _s_t_a_n_d_a_r_d_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_e_r_i_v_a_t_i_v_e_s_._ _ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _v_e_l_o_c_i_t_y_ _a_n_d_ _a_c_c_e_l_e_r_a_t_i_o_n_ _o_f_ _a_n_ _o_b_j_e_c_t_ _w_h_o_s_e_ _p_o_s_i_t_i_o_n_ _f_u_n_c_t_i_o_n_ _i_s_ _g_i_v_e_n_ _b_y_ _a_ _v_e_c_t_o_r_ _f_u_n_c_t_i_o_n_._*_4_*_C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _T_h_e_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _a_n_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_4_*_S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._ _ _T_h_e_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _i_s_ _y_e_t_ _a_n_o_t_h_e_r_ _a_l_t_e_r_n_a_t_e_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_ _f_o_r_ _t_h_e_ _t_h_r_e_e_ _d_i_m_e_n_s_i_o_n_a_l_ _c_o_o_r_d_i_n_a_t_e_ _s_y_s_t_e_m_._*_1_*_P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_L_i_m_i_t_s_ _-_ _T_a_k_i_n_g_ _l_i_m_i_t_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _i_d_e_a_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _a_s_ _w_e_l_l_ _a_s_ _t_h_e_ _s_t_a_n_d_a_r_d_ _n_o_t_a_t_i_o_n_s_ _a_n_d_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_._*_4_*_I_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _i_m_p_o_r_t_a_n_t_ _i_n_t_e_r_p_r_e_t_a_t_i_o_n_s_ _o_f_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_4_*_*_4_*_H_i_g_h_e_r_ _O_r_d_e_r_ _P_a_r_t_i_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _h_i_g_h_e_r_ _o_r_d_e_r_ _p_a_r_t_i_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_f_f_e_r_e_n_t_i_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _e_x_t_e_n_d_ _t_h_e_ _i_d_e_a_ _o_f_ _d_i_f_f_e_r_e_n_t_i_a_l_s_ _t_o_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_C_h_a_i_n_ _R_u_l_e_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _t_h_e_ _c_h_a_i_n_ _r_u_l_e_ _f_o_r_ _f_u_n_c_t_i_o_n_s_ _o_f_ _s_e_v_e_r_a_l_ _v_a_r_i_a_b_l_e_s_._*_4_*_*_4_*_D_i_r_e_c_t_i_o_n_a_l_ _D_e_r_i_v_a_t_i_v_e_s_ _-_ _W_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _s_e_e_ _h_o_w_ _t_o_ _c_o_m_p_u_t_e_ _t_h_e_m_ _a_n_d_ _s_e_e_ _a_ _c_o_u_p_l_e_ _o_f_ _n_i_c_e_ _f_a_c_t_s_ _p_e_r_t_a_i_n_i_n_g_ _t_o_ _d_i_r_e_c_t_i_o_n_a_l_ _d_e_r_i_v_a_t_i_v_e_s_._*_1_*_A_P_P_L_I_C_A_T_I_O_N_S_ _O_F_ _P_A_R_T_I_A_L_ _D_E_R_I_V_A_T_I_V_E_S_*_2_*_T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _L_i_n_e_a_r_ _A_p_p_r_o_x_i_m_a_t_i_o_n_s_ _-_ _W_e_*_a_p_o_s_*_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _t_o_ _s_u_r_f_a_c_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_._*_4_*_G_r_a_d_i_e_n_t_ _V_e_c_t_o_r_,_ _T_a_n_g_e_n_t_ _P_l_a_n_e_s_ _a_n_d_ _N_o_r_m_a_l_ _L_i_n_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_h_e_ _g_r_a_d_i_e_n_t_ _v_e_c_t_o_r_ _c_a_n_ _b_e_ _u_s_e_d_ _t_o_ _f_i_n_d_ _t_a_n_g_e_n_t_ _p_l_a_n_e_s_ _a_n_d_ _n_o_r_m_a_l_ _l_i_n_e_s_ _t_o_ _a_ _s_u_r_f_a_c_e_._*_4_*_R_e_l_a_t_i_v_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _s_e_e_ _h_o_w_ _t_o_ _i_d_e_n_t_i_f_y_ _r_e_l_a_t_i_v_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_._*_4_*_A_b_s_o_l_u_t_e_ _M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _W_e_ _w_i_l_l_ _f_i_n_d_ _a_b_s_o_l_u_t_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _o_v_e_r_ _a_ _g_i_v_e_n_ _r_e_g_i_o_n_._*_4_*_L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_o_ _u_s_e_ _L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _t_o_ _f_i_n_d_ _t_h_e_ _a_b_s_o_l_u_t_e_ _e_x_t_r_e_m_a_ _f_o_r_ _a_ _f_u_n_c_t_i_o_n_ _s_u_b_j_e_c_t_ _t_o_ _a_ _g_i_v_e_n_ _c_o_n_s_t_r_a_i_n_t_._*_1_*_M_U_L_T_I_P_L_E_ _I_N_T_E_G_R_A_L_S_*_2_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_I_t_e_r_a_t_e_d_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _w_e_ _a_c_t_u_a_l_l_y_ _c_o_m_p_u_t_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _o_v_e_r_ _G_e_n_e_r_a_l_ _R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ 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_f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_a_l_c_u_l_u_s_ _I_I_I_ _(_M_a_t_h_ _2_4_1_5_)_.','_1','0','_3','0','0'),(6986,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','a level','mathematics/calculus','Friday  17:29  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  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We will also give two vector forms of Green*apos*s Theorem.*1*SURFACE INTEGRALS*2*Parametric Surfaces - In this section we will take a look at the basics of representing a surface with parametric equations.  We will also take a look at a couple of applications.*4*Surface Integrals - Here we will introduce the topic of surface integrals.  We will be working with surface integrals of functions in this section.*4*Surface Integrals of Vector Fields - We will look at surface integrals of vector fields in this section.*4*Stokes*apos* Theorem - We will look at Stokes*apos* Theorem in this section.*4*Divergence Theorem - Here we will take a look at the Divergence Theorem.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus III (Math 2415).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_I_/_C_a_l_c_I_I_I_._a_s_p_x','_m_a_t_h_s','_s_5','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _C_a_l_c_u_l_u_s_ _I_I_I_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_1_*_T_H_R_E_E_ _D_I_M_E_N_S_I_O_N_A_L_ _S_P_A_C_E_*_2_*_T_h_i_s_ _i_s_ _t_h_e_ _o_n_l_y_ _c_h_a_p_t_e_r_ _t_h_a_t_ _e_x_i_s_t_s_ _i_n_ _t_w_o_ _p_l_a_c_e_s_ _i_n_ _m_y_ 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_R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ 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_f_u_n_d_a_m_e_n_t_a_l_ _t_h_e_o_r_e_m_ _o_f_ _c_a_l_c_u_l_u_s_ _f_o_r_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_._*_4_*_C_o_n_s_e_r_v_a_t_i_v_e_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _s_o_m_e_w_h_a_t_ _d_e_t_a_i_l_e_d_ _l_o_o_k_ _a_t_ _c_o_n_s_e_r_v_a_t_i_v_e_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _a_n_d_ _h_o_w_ _t_o_ _f_i_n_d_ _p_o_t_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _g_i_v_e_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _i_n_t_e_r_e_s_t_i_n_g_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_4_*_C_u_r_l_ _a_n_d_ _D_i_v_e_r_g_e_n_c_e_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_s_ _o_f_ _t_h_e_ _c_u_r_l_ _a_n_d_ _t_h_e_ _d_i_v_e_r_g_e_n_c_e_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_w_o_ _v_e_c_t_o_r_ _f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _S_t_o_k_e_s_*_a_p_o_s_*_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _D_i_v_e_r_g_e_n_c_e_ _T_h_e_o_r_e_m_._*_4_*_*_4_*_*_4_*_ _E_x_t_r_e_m_e_l_y_ _u_s_e_f_u_l_ _f_o_r_ _t_e_a_c_h_e_r_s_ _o_f_ _M_a_t_h_s_/_M_a_t_h_e_m_a_t_i_c_s_ _w_o_r_k_i_n_g_ _t_h_r_o_u_g_h_ _C_a_l_c_u_l_u_s_ _I_I_I_ _(_M_a_t_h_ _2_4_1_5_)_.','_1','0','_3','0','0'),(6987,'82.8.167.110','1280507292','http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx','mathematics','[v/vi]','mathematics/calculus','Friday  17:29  30/07/10','1','0','0','1','1','0','0','0','0','0','6','1','anon','Excellent collection of guides/resources for Calculus III  - online and e-book (download)*1*THREE DIMENSIONAL SPACE*2*This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however,*4*we have since moved several of them into Calculus III.  So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.*4**4*The 3-D Coordinate System - We will introduce the concepts and notation for the three dimensional coordinate system in this section.*4**4*Equations of Lines - In this section we will develop the various forms for the equation of lines in three dimensional space.*4*Equations of Planes - Here we will develop the equation of a plane.*4*Quadric Surfaces - In this section we will be looking at some examples of quadric surfaces.*4*Functions of Several Variables - A quick review of some important topics about functions of several variables.*4**4*Vector Functions - We introduce the concept of vector functions in this section.  We concentrate primarily on curves in three dimensional space.  We will however, touch briefly on surfaces as well.*4*Calculus with Vector Functions - Here we will take a quick look at limits, derivatives, and integrals with vector functions.*4*Tangent, Normal and Binormal Vectors - We will define the tangent, normal and binormal vectors in this section.*4*Arc Length with Vector Functions - In this section we will find the arc length of a vector function.*4*Curvature - We will determine the curvature of a function in this section.*4**4*Velocity and Acceleration - In this section we will revisit a standard application of derivatives.  We will look at the velocity and acceleration of an object whose position function is given by a vector function.*4*Cylindrical Coordinates - We will define the cylindrical coordinate system in this section.  The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.*4*Spherical Coordinates - In this section we will define the spherical coordinate system.  The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.*1*PARTIAL DERIVATIVES*2*Limits - Taking limits of functions of several variables.*4*Partial Derivatives - In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.*4*Interpretations of Partial Derivatives - Here we will take a look at a couple of important interpretations of partial derivatives.*4**4*Higher Order Partial Derivatives - We will take a look at higher order partial derivatives in this section.*4*Differentials - In this section we extend the idea of differentials to functions of several variables.*4*Chain Rule - Here we will look at the chain rule for functions of several variables.*4**4*Directional Derivatives - We will introduce the concept of directional derivatives in this section.  We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.*1*APPLICATIONS OF PARTIAL DERIVATIVES*2*Tangent Planes and Linear Approximations - We*apos*ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.*4*Gradient Vector, Tangent Planes and Normal Lines - In this section we*apos*ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.*4*Relative Minimums and Maximums - Here we will see how to identify relative minimums and maximums.*4*Absolute Minimums and Maximums - We will find absolute minimums and maximums of a function over a given region.*4*Lagrange Multipliers  In this section we*apos*ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.*1*MULTIPLE INTEGRALS*2*Double Integrals - We will define the double integral in this section.*4*Iterated Integrals - In this section we will start looking at how we actually compute double integrals.*4*Double Integrals over General Regions - Here we will look at some general double integrals.*4*Double Integrals in Polar Coordinates - In this section we will take a look at evaluating double integrals using polar coordinates.*4*Triple Integrals - Here we will define the triple integral as well as how we evaluate them.*4*Triple Integrals in Cylindrical Coordinates - We will evaluate triple integrals using cylindrical coordinates in this section.*4*Triple Integrals in Spherical Coordinates - In this section we will evaluate triple integrals using spherical coordinates.*4*Change of Variables - In this section we will look at change of variables for double and triple integrals.*4*Surface Area - Here we look at the one real application of double integrals that we*apos*re going to look at in this material.*4*Area and Volume Revisited - We summarize the area and volume formulas from this chapter.*1*LINE INTEGRALS*2*Vector Fields - In this section we introduce the concept of a vector field.*4*Line Integrals  Part I - Here we will start looking at line integrals.  In particular we will look at line integrals with respect to arc length.*4*Line Integrals  Part II - We will continue looking at line integrals in this section.  Here we will be looking at line integrals with respect to x, y, and/or z.*4*Line Integrals of Vector Fields - Here we will look at a third type of line integrals, line integrals of vector fields.*4*Fundamental Theorem for Line Integrals - In this section we will look at a version of the fundamental theorem of calculus for line integrals of vector fields.*4*Conservative Vector Fields - Here we will take a somewhat detailed look at conservative vector fields and how to find potential functions.*4*Green*apos*s Theorem - We will give Green*apos*s Theorem in this section as well as an interesting application of Green*apos*s Theorem.*4*Curl and Divergence - In this section we will introduce the concepts of the curl and the divergence of a vector field.  We will also give two vector forms of Green*apos*s Theorem.*1*SURFACE INTEGRALS*2*Parametric Surfaces - In this section we will take a look at the basics of representing a surface with parametric equations.  We will also take a look at a couple of applications.*4*Surface Integrals - Here we will introduce the topic of surface integrals.  We will be working with surface integrals of functions in this section.*4*Surface Integrals of Vector Fields - We will look at surface integrals of vector fields in this section.*4*Stokes*apos* Theorem - We will look at Stokes*apos* Theorem in this section.*4*Divergence Theorem - Here we will take a look at the Divergence Theorem.*4**4**4* Extremely useful for teachers of Maths/Mathematics working through Calculus III (Math 2415).','1','0','3','_h_t_t_p_:_/_/_t_u_t_o_r_i_a_l_._m_a_t_h_._l_a_m_a_r_._e_d_u_/_C_l_a_s_s_e_s_/_C_a_l_c_I_I_I_/_C_a_l_c_I_I_I_._a_s_p_x','_m_a_t_h_s','_s_5','_1','','0','_1','_1','0','0','0','0','0','_7','_1','','_E_x_c_e_l_l_e_n_t_ _c_o_l_l_e_c_t_i_o_n_ _o_f_ _g_u_i_d_e_s_/_r_e_s_o_u_r_c_e_s_ _f_o_r_ _C_a_l_c_u_l_u_s_ _I_I_I_ _ _-_ _o_n_l_i_n_e_ _a_n_d_ _e_-_b_o_o_k_ _(_d_o_w_n_l_o_a_d_)_*_1_*_T_H_R_E_E_ _D_I_M_E_N_S_I_O_N_A_L_ _S_P_A_C_E_*_2_*_T_h_i_s_ _i_s_ _t_h_e_ _o_n_l_y_ _c_h_a_p_t_e_r_ _t_h_a_t_ _e_x_i_s_t_s_ _i_n_ _t_w_o_ _p_l_a_c_e_s_ _i_n_ _m_y_ 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_M_i_n_i_m_u_m_s_ _a_n_d_ _M_a_x_i_m_u_m_s_ _-_ _W_e_ _w_i_l_l_ _f_i_n_d_ _a_b_s_o_l_u_t_e_ _m_i_n_i_m_u_m_s_ _a_n_d_ _m_a_x_i_m_u_m_s_ _o_f_ _a_ _f_u_n_c_t_i_o_n_ _o_v_e_r_ _a_ _g_i_v_e_n_ _r_e_g_i_o_n_._*_4_*_L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_*_a_p_o_s_*_l_l_ _s_e_e_ _h_o_w_ _t_o_ _u_s_e_ _L_a_g_r_a_n_g_e_ _M_u_l_t_i_p_l_i_e_r_s_ _t_o_ _f_i_n_d_ _t_h_e_ _a_b_s_o_l_u_t_e_ _e_x_t_r_e_m_a_ _f_o_r_ _a_ _f_u_n_c_t_i_o_n_ _s_u_b_j_e_c_t_ _t_o_ _a_ _g_i_v_e_n_ _c_o_n_s_t_r_a_i_n_t_._*_1_*_M_U_L_T_I_P_L_E_ _I_N_T_E_G_R_A_L_S_*_2_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _W_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_I_t_e_r_a_t_e_d_ _I_n_t_e_g_r_a_l_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _s_t_a_r_t_ _l_o_o_k_i_n_g_ _a_t_ _h_o_w_ _w_e_ _a_c_t_u_a_l_l_y_ _c_o_m_p_u_t_e_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _o_v_e_r_ _G_e_n_e_r_a_l_ _R_e_g_i_o_n_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_o_m_e_ _g_e_n_e_r_a_l_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_D_o_u_b_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _P_o_l_a_r_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _e_v_a_l_u_a_t_i_n_g_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _p_o_l_a_r_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _d_e_f_i_n_e_ _t_h_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_ _a_s_ _w_e_l_l_ _a_s_ _h_o_w_ _w_e_ _e_v_a_l_u_a_t_e_ _t_h_e_m_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _C_y_l_i_n_d_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _W_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _c_y_l_i_n_d_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_T_r_i_p_l_e_ _I_n_t_e_g_r_a_l_s_ _i_n_ _S_p_h_e_r_i_c_a_l_ _C_o_o_r_d_i_n_a_t_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _e_v_a_l_u_a_t_e_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_ _u_s_i_n_g_ _s_p_h_e_r_i_c_a_l_ _c_o_o_r_d_i_n_a_t_e_s_._*_4_*_C_h_a_n_g_e_ _o_f_ _V_a_r_i_a_b_l_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _c_h_a_n_g_e_ _o_f_ _v_a_r_i_a_b_l_e_s_ _f_o_r_ _d_o_u_b_l_e_ _a_n_d_ _t_r_i_p_l_e_ _i_n_t_e_g_r_a_l_s_._*_4_*_S_u_r_f_a_c_e_ _A_r_e_a_ _-_ _H_e_r_e_ _w_e_ _l_o_o_k_ _a_t_ _t_h_e_ _o_n_e_ _r_e_a_l_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _d_o_u_b_l_e_ _i_n_t_e_g_r_a_l_s_ _t_h_a_t_ _w_e_*_a_p_o_s_*_r_e_ _g_o_i_n_g_ _t_o_ _l_o_o_k_ _a_t_ _i_n_ _t_h_i_s_ _m_a_t_e_r_i_a_l_._*_4_*_A_r_e_a_ _a_n_d_ _V_o_l_u_m_e_ _R_e_v_i_s_i_t_e_d_ _-_ _W_e_ _s_u_m_m_a_r_i_z_e_ _t_h_e_ _a_r_e_a_ _a_n_d_ _v_o_l_u_m_e_ _f_o_r_m_u_l_a_s_ _f_r_o_m_ _t_h_i_s_ _c_h_a_p_t_e_r_._*_1_*_L_I_N_E_ _I_N_T_E_G_R_A_L_S_*_2_*_V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._*_4_*_L_i_n_e_ _I_n_t_e_g_r_a_l_s_ _ _P_a_r_t_ _I_ _-_ 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_f_u_n_d_a_m_e_n_t_a_l_ _t_h_e_o_r_e_m_ _o_f_ _c_a_l_c_u_l_u_s_ _f_o_r_ _l_i_n_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_._*_4_*_C_o_n_s_e_r_v_a_t_i_v_e_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _s_o_m_e_w_h_a_t_ _d_e_t_a_i_l_e_d_ _l_o_o_k_ _a_t_ _c_o_n_s_e_r_v_a_t_i_v_e_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _a_n_d_ _h_o_w_ _t_o_ _f_i_n_d_ _p_o_t_e_n_t_i_a_l_ _f_u_n_c_t_i_o_n_s_._*_4_*_G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _-_ _W_e_ _w_i_l_l_ _g_i_v_e_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _a_s_ _w_e_l_l_ _a_s_ _a_n_ _i_n_t_e_r_e_s_t_i_n_g_ _a_p_p_l_i_c_a_t_i_o_n_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_4_*_C_u_r_l_ _a_n_d_ _D_i_v_e_r_g_e_n_c_e_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _c_o_n_c_e_p_t_s_ _o_f_ _t_h_e_ _c_u_r_l_ _a_n_d_ _t_h_e_ _d_i_v_e_r_g_e_n_c_e_ _o_f_ _a_ _v_e_c_t_o_r_ _f_i_e_l_d_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _g_i_v_e_ _t_w_o_ _v_e_c_t_o_r_ _f_o_r_m_s_ _o_f_ _G_r_e_e_n_*_a_p_o_s_*_s_ _T_h_e_o_r_e_m_._*_1_*_S_U_R_F_A_C_E_ _I_N_T_E_G_R_A_L_S_*_2_*_P_a_r_a_m_e_t_r_i_c_ _S_u_r_f_a_c_e_s_ _-_ _I_n_ _t_h_i_s_ _s_e_c_t_i_o_n_ _w_e_ _w_i_l_l_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _t_h_e_ _b_a_s_i_c_s_ _o_f_ _r_e_p_r_e_s_e_n_t_i_n_g_ _a_ _s_u_r_f_a_c_e_ _w_i_t_h_ _p_a_r_a_m_e_t_r_i_c_ _e_q_u_a_t_i_o_n_s_._ _ _W_e_ _w_i_l_l_ _a_l_s_o_ _t_a_k_e_ _a_ _l_o_o_k_ _a_t_ _a_ _c_o_u_p_l_e_ _o_f_ _a_p_p_l_i_c_a_t_i_o_n_s_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _-_ _H_e_r_e_ _w_e_ _w_i_l_l_ _i_n_t_r_o_d_u_c_e_ _t_h_e_ _t_o_p_i_c_ _o_f_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_._ _ _W_e_ _w_i_l_l_ _b_e_ _w_o_r_k_i_n_g_ _w_i_t_h_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _f_u_n_c_t_i_o_n_s_ _i_n_ _t_h_i_s_ _s_e_c_t_i_o_n_._*_4_*_S_u_r_f_a_c_e_ _I_n_t_e_g_r_a_l_s_ _o_f_ _V_e_c_t_o_r_ _F_i_e_l_d_s_ _-_ _W_e_ _w_i_l_l_ _l_o_o_k_ _a_t_ _s_u_r_f_a_c_e_ _i_n_t_e_g_r_a_l_s_ _o_f_ _v_e_c_t_o_r_ _f_i_e_l_d_s_ _i_n_ _t_h_i_s_ 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