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| Preface | |
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| Some basic mathematics | |
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| The space R[superscript n] and its topology | |
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| Mappings | |
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| Real analysis | |
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| Group theory | |
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| Linear algebra | |
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| The algebra of square matrices | |
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| Bibliography | |
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| Differentiable manifolds and tensors | |
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| Definition of a manifold | |
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| The sphere as a manifold | |
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| Other examples of manifolds | |
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| Global considerations | |
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| Curves | |
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| Functions on M | |
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| Vectors and vector fields | |
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| Basis vectors and basis vector fields | |
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| Fiber bundles | |
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| Examples of fiber bundles | |
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| A deeper look at fiber bundles | |
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| Vector fields and integral curves | |
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| Exponentiation of the operator d/d[lambda] | |
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| Lie brackets and noncoordinate bases | |
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| When is a basis a coordinate basis? | |
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| One-forms | |
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| Examples of one-forms | |
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| The Dirac delta function | |
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| The gradient and the pictorial representation of a one-form | |
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| Basis one-forms and components of one-forms | |
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| Index notation | |
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| Tensors and tensor fields | |
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| Examples of tensors | |
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| Components of tensors and the outer product | |
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| Contraction | |
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| Basis transformations | |
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| Tensor operations on components | |
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| Functions and scalars | |
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| The metric tensor on a vector space | |
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| The metric tensor field on a manifold | |
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| Special relativity | |
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| Bibliography | |
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| Lie derivatives and Lie groups | |
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| Introduction: how a vector field maps a manifold into itself | |
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| Lie dragging a function | |
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| Lie dragging a vector field | |
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| Lie derivatives | |
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| Lie derivative of a one-form | |
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| Submanifolds | |
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| Frobenius' theorem (vector field version) | |
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| Proof of Frobenius' theorem | |
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| An example: the generators of S[superscript 2] | |
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| Invariance | |
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| Killing vector fields | |
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| Killing vectors and conserved quantities in particle dynamics | |
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| Axial symmetry | |
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| Abstract Lie groups | |
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| Examples of Lie groups | |
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| Lie algebras and their groups | |
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| Realizations and representations | |
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| Spherical symmetry, spherical harmonics and representations of the rotation group | |
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| Bibliography | |
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| Differential forms | |
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| The algebra and integral calculus of forms | |
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| Definition of volume -- the geometrical role of differential forms | |
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| Notation and definitions for antisy mmetric tensors | |
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| Differential forms | |
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| Manipulating differential forms | |
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| Restriction of forms | |
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| Fields of forms | |
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| Handedness and orientability | |
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| Volumes and integration on oriented manifolds | |
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| N-vectors, duals, and the symbol [epsilon][subscript ij...k] | |
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| Tensor densities | |
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| Generalized Kronecker deltas | |
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| Determinants and [epsilon][subscript ij...k] | |
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| Metric volume elements | |
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| The differential calculus of forms and its applications | |
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| The exterior derivative | |
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| Notation for derivatives | |
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| Familiar examples of exterior differentiation | |
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| Integrability conditions for partial differential equations | |
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| Exact forms | |
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| Proof of the local exactness of closed forms | |
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| Lie derivatives of forms | |
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| Lie derivatives and exterior derivatives commute | |
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| Stokes' theorem | |
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| Gauss' theorem and the definition of divergence | |
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| A glance at cohomology theory | |
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| Differential forms and differential equations | |
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| Frobenius' theorem (differential forms version) | |
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| Proof of the equivalence of the two versions of Frobenius' theorem | |
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| Conservation laws | |
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| Vector spherical harmonics | |
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| Bibliography | |
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| Applications in physics | |
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| Thermodynamics | |
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| Simple systems | |
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| Maxwell and other mathematical identities | |
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| Composite thermodynamic systems: Caratheodory's theorem | |
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| Hamiltonian mechanics | |
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| Hamiltonian vector fields | |
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| Canonical transformations | |
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| Map between vectors and one-forms provided by [characters not reproducible] | |
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| Poisson bracket | |
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| Many-particle systems: symplectic forms | |
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| Linear dynamical systems: the symplectic inner product and conserved quantities | |
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| Fiber bundle structure of the Hamiltonian equations | |
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| Electromagnetism | |
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| Rewriting Maxwell's equations using differential forms | |
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| Charge and topology | |
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| The vector potential | |
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| Plane waves: a simple example | |
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| Dynamics of a perfect fluid | |
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| Role of Lie derivatives | |
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| The comoving time-derivative | |
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| Equation of motion | |
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| Conservation of vorticity | |
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| Cosmology | |
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| The cosmological principle | |
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| Lie algebra of maximal symmetry | |
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| The metric of a spherically symmetric three-space | |
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| Construction of the six Killing vectors | |
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| Open, closed, and flat universes | |
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| Bibliography | |
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| Connections for Riemannian manifolds and gauge theories | |
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| Introduction | |
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| Parallelism on curved surfaces | |
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| The covariant derivative | |
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| Components: covariant derivatives of the basis | |
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| Torsion | |
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| Geodesics | |
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| Normal coordinates | |
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| Riemann tensor | |
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| Geometric interpretation of the Riemann tensor | |
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| Flat spaces | |
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| Compatibility of the connection with volume-measure or the metric | |
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| Metric connections | |
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| The affine connection and the equivalence principle | |
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| Connections and gauge theories: the example of electromagnetism | |
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| Bibliography | |
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| Solutions and hints for selected exercises | |
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| Notation | |
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| Index | |