A Student's Guide to Vectors and Tensors

Vectors and tensors are among the most powerful problem-solving tools available, with applications ranging from mechanics and electromagnetics to general relativity. Understanding the nature and application of vectors and tensors is critically important to students of physics and engineering. Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language. Written for undergraduate and beginning graduate students, the book provides a thorough grounding in vectors and vector calculus before transitioning through contra and covariant components to tensors and their applications. Matrices and their algebra are reviewed on the book's supporting website, which also features interactive solutions to every problem in the text where students can work through a series of hints or choose to see the entire solution at once. Audio podcasts give students the opportunity to hear important concepts in the book explained by the author.

Daniel Fleisch is Professor in the Department of Physics at Wittenberg University, where he specializes in electromagnetics and space physics. He is the author of A Student's Guide to Maxwell's Equations (Cambridge University Press, 2008)

Preface	p. vii
Acknowledgments	p. x
Vectors	p. 1
Definitions (basic)	p. 1
Cartesian unit vectors	p. 5
Vector components	p. 7
Vector addition and multiplication by a scalar	p. 11
Non-Cartesian unit vectors	p. 14
Basis vectors	p. 20
Chapter 1 problems	p. 23
Vector operations	p. 25
Scalar product	p. 25
Cross product	p. 27
Triple scalar product	p. 30
Triple vector product	p. 32
Partial derivatives	p. 35
Vectors as derivatives	p. 41
Nabla - the del operator	p. 43
Gradient	p. 44
Divergence	p. 46
Curl	p. 50
Laplacian	p. 54
Chapter 2 problems	p. 60
Vector applications	p. 62
Mass on an inclined plane	p. 62
Curvilinear motion	p. 72
The electric field	p. 81
The magnetic field	p. 89
Chapter 3 problems	p. 95
Covariant and contravariant vector components	p. 97
Coordinate-system transformations	p. 97
Basis-vector transformations	p. 105
Basis-vector vs. component transformations	p. 109
Non-orthogonal coordinate systems	p. 110
Dual basis vectors	p. 113
Finding covariant and contravariant components	p. 117
Index notation	p. 122
Quantities that transform contravariantly	p. 124
Quantities that transform covariantly	p. 127
Chapter 4 problems	p. 130
Higher-rank tensors	p. 132
Definitions (advanced)	p. 132
Covariant, contravariant, and mixed tensors	p. 134
Tensor addition and subtraction	p. 135
Tensor multiplication	p. 137
Metric tensor	p. 140
Index raising and lowering	p. 147
Tensor derivatives and Christoffel symbols	p. 148
Covariant differentiation	p. 153
Vectors and one-forms	p. 156
1 Chapter 5 problems	p. 157
Tensor applications	p. 159
The inertia tensor	p. 159
The electromagnetic field tensor	p. 171
The Riemann curvature tensor	p. 183
Chapter 6 problems	p. 192
Further reading	p. 194
Index	p. 195
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A Student's Guide to Vectors and Tensors

Summary

Author Biography

Table of Contents

Supplemental Materials