Preface |
Some basic mathematics / 1: |
The space R[superscript n] and its topology / 1.1: |
Mappings / 1.2: |
Real analysis / 1.3: |
Group theory / 1.4: |
Linear algebra / 1.5: |
The algebra of square matrices / 1.6: |
Bibliography / 1.7: |
Differentiable manifolds and tensors / 2: |
Definition of a manifold / 2.1: |
The sphere as a manifold / 2.2: |
Other examples of manifolds / 2.3: |
Global considerations / 2.4: |
Curves / 2.5: |
Functions on M / 2.6: |
Vectors and vector fields / 2.7: |
Basis vectors and basis vector fields / 2.8: |
Fiber bundles / 2.9: |
Examples of fiber bundles / 2.10: |
A deeper look at fiber bundles / 2.11: |
Vector fields and integral curves / 2.12: |
Exponentiation of the operator d/d[lambda] / 2.13: |
Lie brackets and noncoordinate bases / 2.14: |
When is a basis a coordinate basis? / 2.15: |
One-forms / 2.16: |
Examples of one-forms / 2.17: |
The Dirac delta function / 2.18: |
The gradient and the pictorial representation of a one-form / 2.19: |
Basis one-forms and components of one-forms / 2.20: |
Index notation / 2.21: |
Tensors and tensor fields / 2.22: |
Examples of tensors / 2.23: |
Components of tensors and the outer product / 2.24: |
Contraction / 2.25: |
Basis transformations / 2.26: |
Tensor operations on components / 2.27: |
Functions and scalars / 2.28: |
The metric tensor on a vector space / 2.29: |
The metric tensor field on a manifold / 2.30: |
Special relativity / 2.31: |
Lie derivatives and Lie groups / 2.32: |
Introduction: how a vector field maps a manifold into itself / 3.1: |
Lie dragging a function / 3.2: |
Lie dragging a vector field / 3.3: |
Lie derivatives / 3.4: |
Lie derivative of a one-form / 3.5: |
Submanifolds / 3.6: |
Frobenius' theorem (vector field version) / 3.7: |
Proof of Frobenius' theorem / 3.8: |
An example: the generators of S[superscript 2] / 3.9: |
Invariance / 3.10: |
Killing vector fields / 3.11: |
Killing vectors and conserved quantities in particle dynamics / 3.12: |
Axial symmetry / 3.13: |
Abstract Lie groups / 3.14: |
Examples of Lie groups / 3.15: |
Lie algebras and their groups / 3.16: |
Realizations and representations / 3.17: |
Spherical symmetry, spherical harmonics and representations of the rotation group / 3.18: |
Differential forms / 3.19: |
The algebra and integral calculus of forms / A: |
Definition of volume -- the geometrical role of differential forms / 4.1: |
Notation and definitions for antisy mmetric tensors / 4.2: |
Manipulating differential forms / 4.3: |
Restriction of forms / 4.5: |
Fields of forms / 4.6: |
Handedness and orientability / 4.7: |
Volumes and integration on oriented manifolds / 4.8: |
N-vectors, duals, and the symbol [epsilon][subscript ij...k] / 4.9: |
Tensor densities / 4.10: |
Generalized Kronecker deltas / 4.11: |
Determinants and [epsilon][subscript ij...k] / 4.12: |
Metric volume elements / 4.13: |
The differential calculus of forms and its applications / B: |
The exterior derivative / 4.14: |
Notation for derivatives / 4.15: |
Familiar examples of exterior differentiation / 4.16: |
Integrability conditions for partial differential equations / 4.17: |
Exact forms / 4.18: |
Proof of the local exactness of closed forms / 4.19: |
Lie derivatives of forms / 4.20: |
Lie derivatives and exterior derivatives commute / 4.21: |
Stokes' theorem / 4.22: |
Gauss' theorem and the definition of divergence / 4.23: |
A glance at cohomology theory / 4.24: |
Differential forms and differential equations / 4.25: |
Frobenius' theorem (differential forms version) / 4.26: |
Proof of the equivalence of the two versions of Frobenius' theorem / 4.27: |
Conservation laws / 4.28: |
Vector spherical harmonics / 4.29: |
Applications in physics / 4.30: |
Thermodynamics |
Simple systems / 5.1: |
Maxwell and other mathematical identities / 5.2: |
Composite thermodynamic systems: Caratheodory's theorem / 5.3: |
Hamiltonian mechanics |
Hamiltonian vector fields / 5.4: |
Canonical transformations / 5.5: |
Map between vectors and one-forms provided by [characters not reproducible] / 5.6: |
Poisson bracket / 5.7: |
Many-particle systems: symplectic forms / 5.8: |
Linear dynamical systems: the symplectic inner product and conserved quantities / 5.9: |
Fiber bundle structure of the Hamiltonian equations / 5.10: |
Electromagnetism / C: |
Rewriting Maxwell's equations using differential forms / 5.11: |
Charge and topology / 5.12: |
The vector potential / 5.13: |
Plane waves: a simple example / 5.14: |
Dynamics of a perfect fluid / D: |
Role of Lie derivatives / 5.15: |
The comoving time-derivative / 5.16: |
Equation of motion / 5.17: |
Conservation of vorticity / 5.18: |
Cosmology / E: |
The cosmological principle / 5.19: |
Lie algebra of maximal symmetry / 5.20: |
The metric of a spherically symmetric three-space / 5.21: |
Construction of the six Killing vectors / 5.22: |
Open, closed, and flat universes / 5.23: |
Connections for Riemannian manifolds and gauge theories / 5.24: |
Introduction / 6.1: |
Parallelism on curved surfaces / 6.2: |
The covariant derivative / 6.3: |
Components: covariant derivatives of the basis / 6.4: |
Torsion / 6.5: |
Geodesics / 6.6: |
Normal coordinates / 6.7: |
Riemann tensor / 6.8: |
Geometric interpretation of the Riemann tensor / 6.9: |
Flat spaces / 6.10: |
Compatibility of the connection with volume-measure or the metric / 6.11: |
Metric connections / 6.12: |
The affine connection and the equivalence principle / 6.13: |
Connections and gauge theories: the example of electromagnetism / 6.14: |
Solutions and hints for selected exercises / 6.15: |
Notation |
Index |
Appendix |
Preface |
Some basic mathematics / 1: |
The space R[superscript n] and its topology / 1.1: |