| Preface | p. ix |
| Some basic mathematics | p. 1 |
| The space R[superscript n] and its topology | p. 1 |
| Mappings | p. 5 |
| Real analysis | p. 9 |
| Group theory | p. 11 |
| Linear algebra | p. 13 |
| The algebra of square matrices | p. 16 |
| Bibliography | p. 20 |
| Differentiable manifolds and tensors | p. 23 |
| Definition of a manifold | p. 23 |
| The sphere as a manifold | p. 26 |
| Other examples of manifolds | p. 28 |
| Global considerations | p. 29 |
| Curves | p. 30 |
| Functions on M | p. 30 |
| Vectors and vector fields | p. 31 |
| Basis vectors and basis vector fields | p. 34 |
| Fiber bundles | p. 35 |
| Examples of fiber bundles | p. 37 |
| A deeper look at fiber bundles | p. 38 |
| Vector fields and integral curves | p. 42 |
| Exponentiation of the operator d/d[lambda] | p. 43 |
| Lie brackets and noncoordinate bases | p. 43 |
| When is a basis a coordinate basis? | p. 47 |
| One-forms | p. 49 |
| Examples of one-forms | p. 50 |
| The Dirac delta function | p. 51 |
| The gradient and the pictorial representation of a one-form | p. 52 |
| Basis one-forms and components of one-forms | p. 55 |
| Index notation | p. 56 |
| Tensors and tensor fields | p. 57 |
| Examples of tensors | p. 58 |
| Components of tensors and the outer product | p. 59 |
| Contraction | p. 59 |
| Basis transformations | p. 60 |
| Tensor operations on components | p. 63 |
| Functions and scalars | p. 64 |
| The metric tensor on a vector space | p. 64 |
| The metric tensor field on a manifold | p. 68 |
| Special relativity | p. 70 |
| Bibliography | p. 71 |
| Lie derivatives and Lie groups | p. 73 |
| Introduction: how a vector field maps a manifold into itself | p. 73 |
| Lie dragging a function | p. 74 |
| Lie dragging a vector field | p. 74 |
| Lie derivatives | p. 76 |
| Lie derivative of a one-form | p. 78 |
| Submanifolds | p. 79 |
| Frobenius' theorem (vector field version) | p. 81 |
| Proof of Frobenius' theorem | p. 83 |
| An example: the generators of S[superscript 2] | p. 85 |
| Invariance | p. 86 |
| Killing vector fields | p. 88 |
| Killing vectors and conserved quantities in particle dynamics | p. 89 |
| Axial symmetry | p. 89 |
| Abstract Lie groups | p. 92 |
| Examples of Lie groups | p. 95 |
| Lie algebras and their groups | p. 101 |
| Realizations and representations | p. 105 |
| Spherical symmetry, spherical harmonics and representations of the rotation group | p. 108 |
| Bibliography | p. 112 |
| Differential forms | p. 113 |
| The algebra and integral calculus of forms | p. 113 |
| Definition of volume -- the geometrical role of differential forms | p. 113 |
| Notation and definitions for antisy mmetric tensors | p. 115 |
| Differential forms | p. 117 |
| Manipulating differential forms | p. 119 |
| Restriction of forms | p. 120 |
| Fields of forms | p. 120 |
| Handedness and orientability | p. 121 |
| Volumes and integration on oriented manifolds | p. 121 |
| N-vectors, duals, and the symbol [epsilon][subscript ij...k] | p. 125 |
| Tensor densities | p. 128 |
| Generalized Kronecker deltas | p. 130 |
| Determinants and [epsilon][subscript ij...k] | p. 131 |
| Metric volume elements | p. 132 |
| The differential calculus of forms and its applications | p. 134 |
| The exterior derivative | p. 134 |
| Notation for derivatives | p. 135 |
| Familiar examples of exterior differentiation | p. 136 |
| Integrability conditions for partial differential equations | p. 137 |
| Exact forms | p. 138 |
| Proof of the local exactness of closed forms | p. 140 |
| Lie derivatives of forms | p. 142 |
| Lie derivatives and exterior derivatives commute | p. 143 |
| Stokes' theorem | p. 144 |
| Gauss' theorem and the definition of divergence | p. 147 |
| A glance at cohomology theory | p. 150 |
| Differential forms and differential equations | p. 152 |
| Frobenius' theorem (differential forms version) | p. 154 |
| Proof of the equivalence of the two versions of Frobenius' theorem | p. 157 |
| Conservation laws | p. 158 |
| Vector spherical harmonics | p. 160 |
| Bibliography | p. 161 |
| Applications in physics | p. 163 |
| Thermodynamics | p. 163 |
| Simple systems | p. 163 |
| Maxwell and other mathematical identities | p. 164 |
| Composite thermodynamic systems: Caratheodory's theorem | p. 165 |
| Hamiltonian mechanics | p. 167 |
| Hamiltonian vector fields | p. 167 |
| Canonical transformations | p. 168 |
| Map between vectors and one-forms provided by [characters not reproducible] | p. 169 |
| Poisson bracket | p. 170 |
| Many-particle systems: symplectic forms | p. 170 |
| Linear dynamical systems: the symplectic inner product and conserved quantities | p. 171 |
| Fiber bundle structure of the Hamiltonian equations | p. 174 |
| Electromagnetism | p. 175 |
| Rewriting Maxwell's equations using differential forms | p. 175 |
| Charge and topology | p. 179 |
| The vector potential | p. 180 |
| Plane waves: a simple example | p. 181 |
| Dynamics of a perfect fluid | p. 181 |
| Role of Lie derivatives | p. 181 |
| The comoving time-derivative | p. 182 |
| Equation of motion | p. 183 |
| Conservation of vorticity | p. 184 |
| Cosmology | p. 186 |
| The cosmological principle | p. 186 |
| Lie algebra of maximal symmetry | p. 190 |
| The metric of a spherically symmetric three-space | p. 192 |
| Construction of the six Killing vectors | p. 195 |
| Open, closed, and flat universes | p. 197 |
| Bibliography | p. 199 |
| Connections for Riemannian manifolds and gauge theories | p. 201 |
| Introduction | p. 201 |
| Parallelism on curved surfaces | p. 201 |
| The covariant derivative | p. 203 |
| Components: covariant derivatives of the basis | p. 205 |
| Torsion | p. 207 |
| Geodesics | p. 208 |
| Normal coordinates | p. 210 |
| Riemann tensor | p. 210 |
| Geometric interpretation of the Riemann tensor | p. 212 |
| Flat spaces | p. 214 |
| Compatibility of the connection with volume-measure or the metric | p. 215 |
| Metric connections | p. 216 |
| The affine connection and the equivalence principle | p. 218 |
| Connections and gauge theories: the example of electromagnetism | p. 219 |
| Bibliography | p. 222 |
| Solutions and hints for selected exercises | p. 224 |
| Notation | p. 244 |
| Index | p. 246 |
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