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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

Contents
Table of Contents iii
1 Vectors and Geometry of Space 1
1.1 Three-DimensionalCoordinateSystems ......................... 1

1.2 Vectors ........................................... 5

1.3 TheDotProduct ...................................... 10

1.4 TheCrossProduct ..................................... 13

1.5 EquationsofLines ..................................... 18

1.6 EquationsofPlanes .................................... 22

1.7 QuadricSurfaces ...................................... 29

2 Vector-Valued Functions 33
2.1 VectorFunctions ...................................... 33

2.2 CalculuswithVectorFunctions .............................. 39

2.3 Tangent,NormalandBinormalVectors ......................... 42

2.4 ArcLengthinSpace .................................... 46

3 Partial Derivatives 49
3.1 FunctionsofSeveralVariables............................... 49

3.2 LimitsandContinuity ................................... 53

3.3 PartialDerivatives ..................................... 58

3.4 TheChainRule....................................... 73

3.5 DirectionalDerivatives ................................... 81

3.6 ApplicationsofPartialDerivatives ............................ 89

4 Multiple Integrals 115
4.1 DoubleIntegrals ...................................... 115

4.2 DoubleIntegralsOverNon-rectangularRegions . . . . . . . . . . . . . . . . . . . . . 125
4.3 DoubleIntegralsinPolarCoordinates .......................... 136

4.4 TripleIntegrals ....................................... 148

4.5 TripleIntegralsinCylindricalCoordinates........................ 155

4.6 TripleIntegralsinSphericalCoordinates ......................... 158

5 Integration in Vector Fields 163
5.1 VectorFields ........................................ 163

5.2 LineIntegrals ........................................ 168

5.3 GreensTheorem ...................................... 190

5.4 SurfaceIntegrals ...................................... 201

CONTENTS
5.5 DivergenceTheorem .................................... 220

5.6 StokesTheorem ...................................... 223

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 .elds, and surface integrals giving powerful mathematical tools for science and engineering. Line integrals are used to .nd the work done by a force in moving an object along a path, and to .nd the mass of a wire having a varying density. Surface integrals are used to .nd the rate at which a .uid .ows across a surface. Finally, we conclude with three major theorems, Greens Theorem,