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
Preface | p. ix |
Some basic mathematics | p. 1 |
The space R[superscript n] and its topology | p. 1 |
Mappings | p. 5 |
Real analysis | p. 9 |
Group theory | p. 11 |
Linear algebra | p. 13 |
The algebra of square matrices | p. 16 |
Bibliography | p. 20 |
Differentiable manifolds and tensors | p. 23 |
Definition of a manifold | p. 23 |
The sphere as a manifold | p. 26 |
Other examples of manifolds | p. 28 |
Global considerations | p. 29 |
Curves | p. 30 |
Functions on M | p. 30 |
Vectors and vector fields | p. 31 |
Basis vectors and basis vector fields | p. 34 |
Fiber bundles | p. 35 |
Examples of fiber bundles | p. 37 |
A deeper look at fiber bundles | p. 38 |
Vector fields and integral curves | p. 42 |
Exponentiation of the operator d/d[lambda] | p. 43 |
Lie brackets and noncoordinate bases | p. 43 |
When is a basis a coordinate basis? | p. 47 |
One-forms | p. 49 |
Examples of one-forms | p. 50 |
The Dirac delta function | p. 51 |
The gradient and the pictorial representation of a one-form | p. 52 |
Basis one-forms and components of one-forms | p. 55 |
Index notation | p. 56 |
Tensors and tensor fields | p. 57 |
Examples of tensors | p. 58 |
Components of tensors and the outer product | p. 59 |
Contraction | p. 59 |
Basis transformations | p. 60 |
Tensor operations on components | p. 63 |
Functions and scalars | p. 64 |
The metric tensor on a vector space | p. 64 |
The metric tensor field on a manifold | p. 68 |
Special relativity | p. 70 |
Bibliography | p. 71 |
Lie derivatives and Lie groups | p. 73 |
Introduction: how a vector field maps a manifold into itself | p. 73 |
Lie dragging a function | p. 74 |
Lie dragging a vector field | p. 74 |
Lie derivatives | p. 76 |
Lie derivative of a one-form | p. 78 |
Submanifolds | p. 79 |
Frobenius' theorem (vector field version) | p. 81 |
Proof of Frobenius' theorem | p. 83 |
An example: the generators of S[superscript 2] | p. 85 |
Invariance | p. 86 |
Killing vector fields | p. 88 |
Killing vectors and conserved quantities in particle dynamics | p. 89 |
Axial symmetry | p. 89 |
Abstract Lie groups | p. 92 |
Examples of Lie groups | p. 95 |
Lie algebras and their groups | p. 101 |
Realizations and representations | p. 105 |
Spherical symmetry, spherical harmonics and representations of the rotation group | p. 108 |
Bibliography | p. 112 |
Differential forms | p. 113 |
The algebra and integral calculus of forms | p. 113 |
Definition of volume -- the geometrical role of differential forms | p. 113 |
Notation and definitions for antisy mmetric tensors | p. 115 |
Differential forms | p. 117 |
Manipulating differential forms | p. 119 |
Restriction of forms | p. 120 |
Fields of forms | p. 120 |
Handedness and orientability | p. 121 |
Volumes and integration on oriented manifolds | p. 121 |
N-vectors, duals, and the symbol [epsilon][subscript ij...k] | p. 125 |
Tensor densities | p. 128 |
Generalized Kronecker deltas | p. 130 |
Determinants and [epsilon][subscript ij...k] | p. 131 |
Metric volume elements | p. 132 |
The differential calculus of forms and its applications | p. 134 |
The exterior derivative | p. 134 |
Notation for derivatives | p. 135 |
Familiar examples of exterior differentiation | p. 136 |
Integrability conditions for partial differential equations | p. 137 |
Exact forms | p. 138 |
Proof of the local exactness of closed forms | p. 140 |
Lie derivatives of forms | p. 142 |
Lie derivatives and exterior derivatives commute | p. 143 |
Stokes' theorem | p. 144 |
Gauss' theorem and the definition of divergence | p. 147 |
A glance at cohomology theory | p. 150 |
Differential forms and differential equations | p. 152 |
Frobenius' theorem (differential forms version) | p. 154 |
Proof of the equivalence of the two versions of Frobenius' theorem | p. 157 |
Conservation laws | p. 158 |
Vector spherical harmonics | p. 160 |
Bibliography | p. 161 |
Applications in physics | p. 163 |
Thermodynamics | p. 163 |
Simple systems | p. 163 |
Maxwell and other mathematical identities | p. 164 |
Composite thermodynamic systems: Caratheodory's theorem | p. 165 |
Hamiltonian mechanics | p. 167 |
Hamiltonian vector fields | p. 167 |
Canonical transformations | p. 168 |
Map between vectors and one-forms provided by [characters not reproducible] | p. 169 |
Poisson bracket | p. 170 |
Many-particle systems: symplectic forms | p. 170 |
Linear dynamical systems: the symplectic inner product and conserved quantities | p. 171 |
Fiber bundle structure of the Hamiltonian equations | p. 174 |
Electromagnetism | p. 175 |
Rewriting Maxwell's equations using differential forms | p. 175 |
Charge and topology | p. 179 |
The vector potential | p. 180 |
Plane waves: a simple example | p. 181 |
Dynamics of a perfect fluid | p. 181 |
Role of Lie derivatives | p. 181 |
The comoving time-derivative | p. 182 |
Equation of motion | p. 183 |
Conservation of vorticity | p. 184 |
Cosmology | p. 186 |
The cosmological principle | p. 186 |
Lie algebra of maximal symmetry | p. 190 |
The metric of a spherically symmetric three-space | p. 192 |
Construction of the six Killing vectors | p. 195 |
Open, closed, and flat universes | p. 197 |
Bibliography | p. 199 |
Connections for Riemannian manifolds and gauge theories | p. 201 |
Introduction | p. 201 |
Parallelism on curved surfaces | p. 201 |
The covariant derivative | p. 203 |
Components: covariant derivatives of the basis | p. 205 |
Torsion | p. 207 |
Geodesics | p. 208 |
Normal coordinates | p. 210 |
Riemann tensor | p. 210 |
Geometric interpretation of the Riemann tensor | p. 212 |
Flat spaces | p. 214 |
Compatibility of the connection with volume-measure or the metric | p. 215 |
Metric connections | p. 216 |
The affine connection and the equivalence principle | p. 218 |
Connections and gauge theories: the example of electromagnetism | p. 219 |
Bibliography | p. 222 |
Solutions and hints for selected exercises | p. 224 |
Notation | p. 244 |
Index | p. 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