Geometrical Methods of Mathematical Physics

ISBN-10: 0521298873

ISBN-13: 9780521298872

Edition: 1980

Authors: Bernard Schutz

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Description:

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.
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Book details

List price: $54.99
Copyright year: 1980
Publisher: Cambridge University Press
Publication date: 1/28/1980
Binding: Paperback
Pages: 264
Size: 6.00" wide x 9.25" long x 0.75" tall
Weight: 0.792
Language: English

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