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.

Author Notes

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)

Reviews 1

Choice Review

This book is an excellent resource for science and engineering students who can use it as a quick reference while studying topics such as physics, statics, dynamics, electromagnetism, and fluid mechanics. Fleisch (physics, Wittenberg Univ.; A Student's Guide to Maxwell's Equations, 2008) manages to cover in a modest volume a wide range of essential topics on vectors and tensors from a practical point of view. In less than 200 pages, he reviews vectors and their algebraic and differential operators with applications to curvilinear motion and electric and magnetic fields before making a transition to tensors. Undergraduates can fully utilize the content on vectors, while graduate students can brush up on tensors with a quick turnaround and relative ease. The author is commended for his effective and elucidating style, with graphical explanations and without mathematical long-windedness. This book is an ideal corequisite to courses like dynamics, fluid mechanics, and electromagnetic field theory. Reading specific sections in this book a priori not only serves as just-in-time preparation, but also empowers students to tackle subjects that require a good grasp of vector algebra, vector differential operators, vector transformation, and tensors. Summing Up: Highly recommended. Lower-division undergraduates through professionals/practitioners. R. N. Laoulache University of Massachusetts Dartmouth

Preface	p. vii
Acknowledgments	p. x
1 Vectors	p. 1
1.1 Definitions (basic)	p. 1
1.2 Cartesian unit vectors	p. 5
1.3 Vector components	p. 7
1.4 Vector addition and multiplication by a scalar	p. 11
1.5 Non-Cartesian unit vectors	p. 14
1.6 Basis vectors	p. 20
1.7 Chapter 1 problems	p. 23
2 Vector operations	p. 25
2.1 Scalar product	p. 25
2.2 Cross product	p. 27
2.3 Triple scalar product	p. 30
2.4 Triple vector product	p. 32
2.5 Partial derivatives	p. 35
2.6 Vectors as derivatives	p. 41
2.7 Nabla - the del operator	p. 43
2.8 Gradient	p. 44
2.9 Divergence	p. 46
2.10 Curl	p. 50
2.11 Laplacian	p. 54
2.12 Chapter 2 problems	p. 60
3 Vector applications	p. 62
3.1 Mass on an inclined plane	p. 62
3.2 Curvilinear motion	p. 72
3.3 The electric field	p. 81
3.4 The magnetic field	p. 89
3.5 Chapter 3 problems	p. 95
4 Covariant and contravariant vector components	p. 97
4.1 Coordinate-system transformations	p. 97
4.2 Basis-vector transformations	p. 105
4.3 Basis-vector vs. component transformations	p. 109
4.4 Non-orthogonal coordinate systems	p. 110
4.5 Dual basis vectors	p. 113
4.6 Finding covariant and contravariant components	p. 117
4.7 Index notation	p. 122
4.8 Quantities that transform contravariantly	p. 124
4.9 Quantities that transform covariantly	p. 127
4.10 Chapter 4 problems	p. 130
5 Higher-rank tensors	p. 132
5.1 Definitions (advanced)	p. 132
5.2 Covariant, contravariant, and mixed tensors	p. 134
5.3 Tensor addition and subtraction	p. 135
5.4 Tensor multiplication	p. 137
5.5 Metric tensor	p. 140
5.6 Index raising and lowering	p. 147
5.7 Tensor derivatives and Christoffel symbols	p. 148
5.8 Covariant differentiation	p. 153
5.9 Vectors and one-forms	p. 156
5.10 1 Chapter 5 problems	p. 157
6 Tensor applications	p. 159
6.1 The inertia tensor	p. 159
6.2 The electromagnetic field tensor	p. 171
6.3 The Riemann curvature tensor	p. 183
6.4 Chapter 6 problems	p. 192
Further reading	p. 194
Index	p. 195

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