Unit 3 Functions of Several Variables
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1 Unit 3 Functions of Several Variables In this unit, we consider several simple examples of multi-variable functions, quadratic surfaces and projections, level curves and surfaces, partial derivatives of multi-variable functions, open and closed sets, and limits and continuities of functions of several variables. Note: Unit 3 is based on Chapter 14 of the textbook, Salas and Hille s Calculus: Several Variables, 7th ed., revised by Garret J. Etgen (New York: Wiley, 1995). All assigned readings and exercises are from that textbook, unless otherwise indicated. Objectives Detailed objectives are given in each of the sections listed below. 1. Elementary Examples 2. Quadratic Surfaces 3. Level Curves and Level Surfaces 4. Partial Derivatives 5. Open and Closed Sets 6. Limits, Continuity and the Equality of Mixed Partials Objective 1 a. find the domain and range of functions of two or three variables. b. form vector difference quotients for functions of several variables, and calculate the limit as the denominator goes to zero. c. express various geometrical forms as functions of several variables. Read Section 14.1, pages Mathematics 365 / Study Guide 17
2 Exercises Complete problems 1-11, 14, 17, 22, 25, 27, 31, 33 and 35 on pages real valued function of two variables domain of a real valued function of two variables range of a real valued function of two variables open unit disk real valued function of three variables domain of a real valued function of three variables range of a real valued function of three variables unit sphere open unit ball bounded function Before you proceed to Objective 2, make certain that you can meet each of the sub-objectives listed under Objective 1. Objective 2 a. identify a quadratic surface given its formula, find its special properties (if any), and sketch the quadratic surface. b. provide an equation for a given quadratic surface. c. rotate a given curve to generate a surface. d. determine the projection of a given curve onto a plane. Read Section 14.2, pages Calculus Several Variables
3 Exercises Complete all of the odd-numbered problems on page 909. For each of the quadratic surfaces listed below, make certain that you can provide an equation, and identify any special properties (i.e., intercepts, traces, sections, centre, symmetry and boundedness or unboundedness), and sketch the curve. ellipsoid hyperboloid of one sheet hyperboloid of two sheets quadratic cone elliptic paraboloid hyperbolic paraboloid parabolic cylinder elliptic cylinder hyperbolic cylinder semi-axes of an ellipsoid ellipsoid of revolution hyperboloid of revolution double circular cone nappes of a cone vertex elliptic paraboloid paraboloid of revolution minimax saddle point cylinder Mathematics 365 / Study Guide 19
4 generators of a cylinder right circular cylinder projection of a curve onto a plane Before you proceed to Objective 3, make certain that you can meet each of the sub-objectives listed under Objective 2. Objective 3 a. identify the level curve defined by a function, and sketch the curve for given values in the range of the function. b. identify and sketch selected level surfaces, given their equations. c. find an equation for a level curve containing a given point. d. find an equation for a level surface containing a given point. Read Section 14.3, pages Exercises Complete all of the odd-numbered problems on pages level curve level surface Before you proceed to Objective 4, make certain that you can meet each of the sub-objectives listed under Objective Calculus Several Variables
5 Objective 4 a. calculate the partial derivatives of a given multi-variable function. b. determine the value of a partial derivative of a known function at a given point. c. find partial derivatives by determining the value as h 0 of their difference equations. d. use partial derivatives to solve vector parametrization and tangent problems. e. demonstrate various conclusions relating to partial derivatives and partial differential equations. Read Section 14.4, pages , and the brief section on partial differential equations that appears under problem 57 on page 927. Exercises Complete all of the odd-numbered problems 1-39, problems and problem 58 on pages partial derivative of a function of two variables partial derivative of a function of three variables Cauchy-Riemann equations ordinary differential equation partial differential equation Before you proceed to Objective 5, make certain that you can meet each of the sub-objectives listed under Objective 4. Mathematics 365 / Study Guide 21
6 Objective 5 state whether a given set is open, closed or neither, specify the interior and boundary of the set, and sketch the set. Read Section 14.5, pages Exercises Complete all of the odd-numbered problems on page 931. point concept open interval neighbourhood of a point interior of a set boundary of a set open set closed set deleted neighbourhood Before you proceed to Objective 6, make certain that you can meet Objective 5. Objective 6 a. calculate the second-order partial derivatives of a given equation. b. determine whether a given function is harmonic. c. determine whether a given function is a solution to the wave equation. d. demonstrate various conclusions relating to second-order partial derivatives. 22 Calculus Several Variables
7 Read Section 13.6, pages , the section titled Laplace s Equation under problem 21 on page 939, and the section titled The Wave Equation, under problem 27 on page 940. Exercises Complete problems 1-5 on page 887. limit of a function of several variables continuous multi-variable function continuous rational function continuity of composite functions second-order partial mixed partial Laplace s equation harmonic function wave equation Before you complete the second tutor-marked assignment and write the midterm examination, make certain that you can meet each of the subobjectives listed under Objective 6. Midterm Examination Before you begin the second tutor marked assignment, contact the Office of the Registrar to request the midterm examination. Please see your Student Manual for further information. Mathematics 365 / Study Guide 23
8 24 Calculus Several Variables
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