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Geometrical Methods of Mathematical Physics - Bernard F. Schutz

Geometrical Methods of Mathematical Physics

Paperback

Published: 5th January 1981
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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.

'Although there are a welter of books where similar material can be found, this book is the most lucid I have come across at this level of exposition. It is eminently suitable for a graduate course (indeed, the more academically able undergraduate should be about to cope with most of it), and the applications should suffice to persuade any physicist or applied mathematician of its importance ... Schutz's book is a triumph ...' The Times Higher Education Supplement

Prefacep. ix
Some basic mathematicsp. 1
The space R[superscript n] and its topologyp. 1
Mappingsp. 5
Real analysisp. 9
Group theoryp. 11
Linear algebrap. 13
The algebra of square matricesp. 16
Bibliographyp. 20
Differentiable manifolds and tensorsp. 23
Definition of a manifoldp. 23
The sphere as a manifoldp. 26
Other examples of manifoldsp. 28
Global considerationsp. 29
Curvesp. 30
Functions on Mp. 30
Vectors and vector fieldsp. 31
Basis vectors and basis vector fieldsp. 34
Fiber bundlesp. 35
Examples of fiber bundlesp. 37
A deeper look at fiber bundlesp. 38
Vector fields and integral curvesp. 42
Exponentiation of the operator d/d[lambda]p. 43
Lie brackets and noncoordinate basesp. 43
When is a basis a coordinate basis?p. 47
One-formsp. 49
Examples of one-formsp. 50
The Dirac delta functionp. 51
The gradient and the pictorial representation of a one-formp. 52
Basis one-forms and components of one-formsp. 55
Index notationp. 56
Tensors and tensor fieldsp. 57
Examples of tensorsp. 58
Components of tensors and the outer productp. 59
Contractionp. 59
Basis transformationsp. 60
Tensor operations on componentsp. 63
Functions and scalarsp. 64
The metric tensor on a vector spacep. 64
The metric tensor field on a manifoldp. 68
Special relativityp. 70
Bibliographyp. 71
Lie derivatives and Lie groupsp. 73
Introduction: how a vector field maps a manifold into itselfp. 73
Lie dragging a functionp. 74
Lie dragging a vector fieldp. 74
Lie derivativesp. 76
Lie derivative of a one-formp. 78
Submanifoldsp. 79
Frobenius' theorem (vector field version)p. 81
Proof of Frobenius' theoremp. 83
An example: the generators of S[superscript 2]p. 85
Invariancep. 86
Killing vector fieldsp. 88
Killing vectors and conserved quantities in particle dynamicsp. 89
Axial symmetryp. 89
Abstract Lie groupsp. 92
Examples of Lie groupsp. 95
Lie algebras and their groupsp. 101
Realizations and representationsp. 105
Spherical symmetry, spherical harmonics and representations of the rotation groupp. 108
Bibliographyp. 112
Differential formsp. 113
The algebra and integral calculus of formsp. 113
Definition of volume -- the geometrical role of differential formsp. 113
Notation and definitions for antisy mmetric tensorsp. 115
Differential formsp. 117
Manipulating differential formsp. 119
Restriction of formsp. 120
Fields of formsp. 120
Handedness and orientabilityp. 121
Volumes and integration on oriented manifoldsp. 121
N-vectors, duals, and the symbol [epsilon][subscript ij...k]p. 125
Tensor densitiesp. 128
Generalized Kronecker deltasp. 130
Determinants and [epsilon][subscript ij...k]p. 131
Metric volume elementsp. 132
The differential calculus of forms and its applicationsp. 134
The exterior derivativep. 134
Notation for derivativesp. 135
Familiar examples of exterior differentiationp. 136
Integrability conditions for partial differential equationsp. 137
Exact formsp. 138
Proof of the local exactness of closed formsp. 140
Lie derivatives of formsp. 142
Lie derivatives and exterior derivatives commutep. 143
Stokes' theoremp. 144
Gauss' theorem and the definition of divergencep. 147
A glance at cohomology theoryp. 150
Differential forms and differential equationsp. 152
Frobenius' theorem (differential forms version)p. 154
Proof of the equivalence of the two versions of Frobenius' theoremp. 157
Conservation lawsp. 158
Vector spherical harmonicsp. 160
Bibliographyp. 161
Applications in physicsp. 163
Thermodynamicsp. 163
Simple systemsp. 163
Maxwell and other mathematical identitiesp. 164
Composite thermodynamic systems: Caratheodory's theoremp. 165
Hamiltonian mechanicsp. 167
Hamiltonian vector fieldsp. 167
Canonical transformationsp. 168
Map between vectors and one-forms provided by [characters not reproducible]p. 169
Poisson bracketp. 170
Many-particle systems: symplectic formsp. 170
Linear dynamical systems: the symplectic inner product and conserved quantitiesp. 171
Fiber bundle structure of the Hamiltonian equationsp. 174
Electromagnetismp. 175
Rewriting Maxwell's equations using differential formsp. 175
Charge and topologyp. 179
The vector potentialp. 180
Plane waves: a simple examplep. 181
Dynamics of a perfect fluidp. 181
Role of Lie derivativesp. 181
The comoving time-derivativep. 182
Equation of motionp. 183
Conservation of vorticityp. 184
Cosmologyp. 186
The cosmological principlep. 186
Lie algebra of maximal symmetryp. 190
The metric of a spherically symmetric three-spacep. 192
Construction of the six Killing vectorsp. 195
Open, closed, and flat universesp. 197
Bibliographyp. 199
Connections for Riemannian manifolds and gauge theoriesp. 201
Introductionp. 201
Parallelism on curved surfacesp. 201
The covariant derivativep. 203
Components: covariant derivatives of the basisp. 205
Torsionp. 207
Geodesicsp. 208
Normal coordinatesp. 210
Riemann tensorp. 210
Geometric interpretation of the Riemann tensorp. 212
Flat spacesp. 214
Compatibility of the connection with volume-measure or the metricp. 215
Metric connectionsp. 216
The affine connection and the equivalence principlep. 218
Connections and gauge theories: the example of electromagnetismp. 219
Bibliographyp. 222
Solutions and hints for selected exercisesp. 224
Notationp. 244
Indexp. 246
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521298872
ISBN-10: 0521298873
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 264
Published: 5th January 1981
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.8
Weight (kg): 0.41