Geometrical Methods of Mathematical Physics

Bernard Schutz - Geometrical methods of mathematical physics
CONTENTS
PREFACE
1 - Some basic mathematics
1.1 - The space R^n and its topology
1.2 - Mappings
1.3 - Real analysis
1.4 - Group theory
1.5 - Linear algebra
1.6 - The algebra of square matrices
1.7 - Bibliography
2 - Differentiable manifolds and tensors
2.1 - Definition of a manifold
2.2 - The sphere as a manifold
2.3 - Other examples of manifolds
2.4 - Global considerations
2.5 - Curves
2.6 - Functions on M
2.7 - Vectors and vector fields
2.8 - Basis vectors and basis vector fields
2.9 - Fiber bundles
2.10 - Examples of fiber bundles
2.11 - A deeper look at fiber bundles
2.12 - Vector fields and integral curves
2.13 - Exponentiation of the operator d/dλ
2.14 - Lie brackets and noncoordinate bases
2.15 - When is a basis a coordinate basis?
2.16 - One-forms
2.17 - Examples of one-forms
2.18 - The Dirac delta function
2.19 - The gradient and the pictorial representation of a one-form
2.20 - Basis one-forms and components of one-forms
2.21 - Index notation
2.22 - Tensors and tensor fields
2.23 - Examples of tensors
2.24 - Components of tensors and the outer product
2.25 - Contraction
2.26 - Basis transformations
2.27 - Tensor operations on components
2.28 - Functions and scalars
2.29 - The metric tensor on a vector space
2.30 - The metric tensor field on a manifold
2.31 - Special relativity
2.32 - Bibliography
3 - Lie derivatives and Lie groups
3.1 - Introduction: how a vector field maps a manifold into itself
3.2 - Lie dragging a function
3.3 - Lie dragging a vector field
3.4 - Lie derivatives
3.5 - Lie derivative of a one-form
3.6 - Submanifolds
3.7 - Frobenius’ theorem (vector field version)
3.8 - Proof of Frobenius’ theorem
3.9 - An example: the generators of S²
3.10 - Invariance
3.11 - Killing vector fields
3.12 - Killing vectors and conserved quantities in particle dynamics
3.13 - Axial symmetry
3.14 - Abstract Lie groups
3.15 - Examples of Lie groups
3.16 - Lie algebras and their groups
3.17 - Realizations and representations
3.18 - Spherical symmetry, spherical harmonics and representations of the rotation group
3.19 - Bibliography
4 - Differential forms
A - The algebra and integral calculus of forms
4.1 - Definition of volume — the geometrical role of differential forms
4.2 - Notation and definitions for antisymmetric tensors
4.3 - Differential forms
4.4 - Manipulating differential forms
4.5 - Restriction of forms
4.6 - Fields of forms
4.7 - Handedness and orientability
4.8 - Volumes and integration on oriented manifolds
4.9 - N-vectors, duals, and the symbol ϵij...k
4.10 - Tensor densities
4.11 - Generalized Kronecker deltas
4.12 - Determinants and ϵij...k
4.13 - Metric volume elements
B - The differential calculus of forms and its applications
4.14 - The exterior derivative
4.15 - Notation for derivatives
4.16 - Familiar examples of exterior differentiation
4.17 - Integrability conditions for partial differential equations
4.18 - Exact forms
4.19 - Proof of the local exactness of closed forms
4.20 - Lie derivatives of forms
4.21 - Lie derivatives and exterior derivatives commute
4.22 - Stokes’ theorem
4.23 - Gauss’ theorem and the definition of divergence
4.24 - A glance at cohomology theory
4.25 - Differential forms and differential equations
4.26 - Frobenius’ theorem (differential forms version)
4.27 - Proof of the equivalence of the two versions of Frobenius’ theorem
4.28 - Conservation laws
4.29 - Vector spherical harmonics
4.30 - Bibliography
5 - Applications in physics
A - Thermodynamics
5.1 - Simple systems
5.2 - Maxwell and other mathematical identities
5.3 - Composite thermodynamic systems: Caratheodory’s theorem
B - Hamiltonian mechanics
5.4 - Hamiltonian vector fields
5.5 - Canonical transformations
5.6 - Map between vectors and one-forms provided by ῶ
5.7 - Poisson bracket
5.8 - Many-particle systems: symplectic forms
5.9 - Linear dynamical systems: the symplectic inner product and conserved quantities
5.10 - Fiber bundle structure of the Hamiltonian equations
C - Electromagnetism
5.11 - Rewriting Maxwell’s equations using differential forms
5.12 - Charge and topology
5.13 - The vector potential
5.14 - Plane waves: a simple example
D - Dynamics of a perfect fluid
5.15 - Role of Lie derivatives
5.16 - The comoving time-derivative
5.17 - Equation of motion
5.18 - Conservation of vorticity
E - Cosmology
5.19 - The cosmological principle
5.20 - Lie algebra of maximal symmetry
5.21 - The metric of a spherically symmetric three-space
5.22 - Construction of the six Killing vectors
5.23 - Open, closed, and flat universes
5.24 - Bibliography
6 - Connections for Riemannian manifolds and gauge theories
6.1 - Introduction
6.2 - Parallelism on curved surfaces
6.3 - The covariant derivative
6.4 - Components: covariant derivatives of the basis
6.5 - Torsion
6.6 - Geodesics
6.7 - Normal coordinates
6.8 - Riemann tensor
6.9 - Geometric interpretation of the Riemann tensor
6.10 - Flat spaces
6.11 - Compatibility of the connection with volume-measure or the metric
6.12 - Metric connections
6.13 - The affine connection and the equivalence principle
6.14 - Connections and gauge theories: the example of electromagnetism
6.15- Bibliography
Appendix: solutions and hints for selected exercises
Notation
Index