Tensor calculus for physics: a concise guide
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| Format: | Book |
| Language: | English |
| Published: |
Baltimore
Johns Hopkins Univ. Press
2015
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | X, 227 S. Ill., graph. Darst. |
| ISBN: | 142141564X 1421415658 1421415666 9781421415642 9781421415659 |
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Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts—these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry.
Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the inertia tensor, and in electricity and magnetism, they encounter the polarization tensor. However, this piecemeal approach can set students up for misconceptions when they have to learn about tensors in more advanced physics and mathematics studies (e.g., while enrolled in a graduate-level general relativity course or when studying non-Euclidean geometries in a higher mathematics class).
Dwight E. Neuenschwander s Tensor Calculus for Physics is a bottom-up approach that emphasizes motivations before providing definitions. Using a clear, step-by-step approach, the book strives to embed the logic of tensors in contexts that demonstrate why that logic is worth pursuing. It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.
Dwight E. Neuenschwander is a professor of physics at Southern Naz-arene University. He is a columnist for the Observer, the magazine of the Society for Physics Students, and the author of Emmy Noether s Wonderful Theorem, also published by Johns Hopkins, and Howto Involve Undergraduates in Research: A Field Guide for Faculty.
Contents
n
Preface xi Acknowledgments xiii
Chapter 1. Tensors Need Context 1
1.1 Why Aren’t Tensors Defined by What They Are? 1
1.2 Euclidean Vectors, without Coordinates 3
1.3 Derivatives of Euclidean Vectors with Respect to a Scalar
1.4 The Euclidean Gradient 6
1.5 Euclidean Vectors, with Coordinates 7
1.6 Euclidean Vector Operations with and without Coordinates
1.7 Transformation Coefficients as Partial Derivatives 18
1.8 What Is a Theory of Relativity? 20
1.9 Vectors Represented as Matrices 23
1.10 Discussion Questions and Exercises 28
Chapter 2. Two-Index Tensors 33
2.1 The Electric Susceptibility Tensor 33
2.2 The Inertia Tensor 34
2.3 The Electric Quadrupole Tensor 37
CONTENTS
viii
2.4 The Electromagnetic Stress Tensor 39
2.5 Transformations of Two-Index Tensors 42
2.6 Finding Eigenvectors and Eigenvalues 46
2.7 Two-Index Tensor Components as Products of Vector Components 50
2.8 More Than Two Indices 51
2.9 Integration Measures and Tensor Densities 51
2.10 Discussion Questions and Exercises 53
Chapter 3. The Metric Tensor 63
3.1 The Distinction between Distance and Coordinate Displacement 63
3.2 Relative Motion 65
3.3 Upper and Lower Indices 72
3.4 Converting between Vectors and Duals 77
3.5 Contravariant, Covariant, and “Ordinary7 Vectors 79
3.6 Tensor Algebra 83
3.7 Tensor Densities Revisited 84
3.8 Discussion Questions and Exercises 90
Chapter 4. Derivatives of Tensors 9T
4.1 Signs of Trouble 97
4.2 The Affine Connection 99
4.3 The Newtonian Limit 101
4.4 Transformation of the Affine Connection 103
4.5 The Covariant Derivative 105
4.0 Relation of the Affine Connection to the Metric Tensor
107
CONTENTS
ix
4.7 Divergence, Curl, and Laplacian with Covariant Derivatives
4.8 Disccussion Questions and Exercises 113
Chapter 5. Curvature 119
5.1 What Is Curvature? 119
5.2 The Riemann Tensor 122
5.3 Measuring Curvature 125
5.4 Linearity in the Second Derivative 128
5.5 Discussion Questions and Exercises 131
Chapter 6. Covariance Applications 137
6.1 Covariant Electrodynamics 137
6.2 General Covariance and Gravitation 143
6.3 Discussion Questions and Exercises 148
Chapter 7. Tensors and Manifolds 155
7.1 Tangent Spaces, Charts, and Manifolds 157
7.2 Metrics on Manifolds and Their Tangent Spaces 161
7.3 Dual Basis Vectors 163
7.4 Derivatives of Basis Vectors and the Affine Connection 167
7.5 Discussion Questions and Exercises 171
Chapter 8. Getting Acquainted with Differential Forms
8.1 Tensors as Multilinear Forms 176
109
175
8.2 1-Forms and Their Extensions
180
X
CONTENTS
8.3 Exterior Products and Differential Forms; 190
8.4 The Exterior Derivative 195
8.5 An Application to Physics: Max well’s Equations
8.6 Integrals of Differential Forms 200
8.7 Discussion Question^ and Exercises 203
Appendix A: Common Coordinate Systems Appendix B: Theorem of Alternatives 211
Appendix C: Abstract Vector Spaces 213
Bibliography 215
198
209
Index
221
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| spelling | Neuenschwander, Dwight E. Verfasser aut Tensor calculus for physics a concise guide Dwight E. Neuenschwander Baltimore Johns Hopkins Univ. Press 2015 X, 227 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Tensorrechnung (DE-588)4192487-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Tensorrechnung (DE-588)4192487-3 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4214-1566-6 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027625068&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027625068&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Tensor calculus for physics a concise guide |
| title_auth | Tensor calculus for physics a concise guide |
| title_exact_search | Tensor calculus for physics a concise guide |
| title_full | Tensor calculus for physics a concise guide Dwight E. Neuenschwander |
| title_fullStr | Tensor calculus for physics a concise guide Dwight E. Neuenschwander |
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| title_short | Tensor calculus for physics |
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