Lectures on Vector Calculus - CSUSB

Lectures on Vector Calculus

Paul Renteln

Department of Physics California State University San Bernardino, CA 92407 March, 2009; Revised March, 2011

c Paul Renteln, 2009, 2011

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Contents

1 Vector Algebra and Index Notation

1

1.1 Orthonormality and the Kronecker Delta . . . . . . . . . . . . 1

1.2 Vector Components and Dummy Indices . . . . . . . . . . . . 4

1.3 Vector Algebra I: Dot Product . . . . . . . . . . . . . . . . . . 8

1.4 The Einstein Summation Convention . . . . . . . . . . . . . . 10

1.5 Dot Products and Lengths . . . . . . . . . . . . . . . . . . . . 11

1.6 Dot Products and Angles . . . . . . . . . . . . . . . . . . . . . 12

1.7 Angles, Rotations, and Matrices . . . . . . . . . . . . . . . . . 13

1.8 Vector Algebra II: Cross Products and the Levi Civita Symbol 18

1.9 Products of Epsilon Symbols . . . . . . . . . . . . . . . . . . . 23

1.10 Determinants and Epsilon Symbols . . . . . . . . . . . . . . . 27

1.11 Vector Algebra III: Tensor Product . . . . . . . . . . . . . . . 28

1.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Vector Calculus I

32

2.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 The Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Vector Calculus with Indices . . . . . . . . . . . . . . . . . . . 43

2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Vector Calculus II: Other Coordinate Systems

48

3.1 Change of Variables from Cartesian to Spherical Polar . . . . 48

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3.2 Vector Fields and Derivations . . . . . . . . . . . . . . . . . . 49 3.3 Derivatives of Unit Vectors . . . . . . . . . . . . . . . . . . . . 53 3.4 Vector Components in a Non-Cartesian Basis . . . . . . . . . 54 3.5 Vector Operators in Spherical Coordinates . . . . . . . . . . . 54 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Vector Calculus III: Integration

57

4.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Volume Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Integral Theorems

70

5.1 Green's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Stokes' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Gauss' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 The Generalized Stokes' Theorem . . . . . . . . . . . . . . . . 74

5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A Permutations

76

B Determinants

77

B.1 The Determinant as a Multilinear Map . . . . . . . . . . . . . 79

B.2 Cofactors and the Adjugate . . . . . . . . . . . . . . . . . . . 82

B.3 The Determinant as Multiplicative Homomorphism . . . . . . 86

B.4 Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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List of Figures

1 Active versus passive rotations in the plane . . . . . . . . . . . 13 2 Two vectors spanning a parallelogram . . . . . . . . . . . . . . 20 3 Three vectors spanning a parallelepiped . . . . . . . . . . . . . 20 4 Reflection through a plane . . . . . . . . . . . . . . . . . . . . 31 5 An observer moving along a curve through a scalar field . . . . 33 6 Some level surfaces of a scalar field . . . . . . . . . . . . . . 35 7 Gradients and level surfaces . . . . . . . . . . . . . . . . . . . 36 8 A hyperbola meets some level surfaces of d . . . . . . . . . . . 37 9 Spherical polar coordinates and corresponding unit vectors . . 49 10 A parameterized surface . . . . . . . . . . . . . . . . . . . . . 65

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