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MATH 100 Spring 2006-07 Introduction to Multivariable Calculus Lecture Notes Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology May 4, 2007 (MATH100)notes100-ch5.pdf downloaded by twuac from at 2014-02-26 20:43:36. Academic use within HKUST only.
ii (MATH100)notes100-ch5.pdf downloaded by twuac from at 2014-02-26 20:43:36. Academic use within HKUST only.
Contents Table of Contents iii 1 Vectors and Geometry of Space 1 1.1 Three-Dimensional Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Equations of Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Vector-Valued Functions 33 2.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Calculus with Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Tangent, Normal and Binormal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Arc Length in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Partial Derivatives 49 3.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6 Applications of Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4 Multiple Integrals 115 4.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Double Integrals Over Non-rectangular Regions . . . . . . . . . . . . . . . . . . . . . 125 4.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.5 Triple Integrals in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 155 4.6 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 158 5 Integration in Vector Fields 163 5.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.4 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 iii (MATH100)notes100-ch5.pdf downloaded by twuac from at 2014-02-26 20:43:36. Academic use within HKUST only.
CONTENTS 5.5 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.6 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 iv (MATH100)notes100-ch5.pdf downloaded by twuac from at 2014-02-26 20:43:36. Academic use within HKUST only.
Chapter 5 Integration in Vector Fields In this chapter we combine ideas from the preceding four chapters: space curves, vector functions, partial derivatives, and multiple integrals. The result is the development of line integrals, vector fields, and surface integrals giving powerful mathematical tools for science and engineering. Line integrals are used to find the work done by a force in moving an object along a path, and to find the mass of a wire having a varying density. Surface integrals are used to find the rate at which a fluid flows across a surface. Finally, we conclude with three major theorems, Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem. These theorems connect the new tools (i.e., the new integrals) and offer insight into their mathematical calculations and physical applications. 5.1 Vector Fields We start this chapter with the definition of a vector field as they will be a major component of this chapter. Let us start with the formal definition of a vector field.

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