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