Applied differential geometry

This Is A Self-contained Introductory Textbook On The Calculus Of Differential Forms And Modern Differential Geometry. The Intended Audience Is Physicists, So The Author Emphasises Applications And Geometrical Reasoning In Order To Give Results And Concepts A Precise But Intuitive Meaning Without Getting Bogged Down In Analysis. The Large Number Of Diagrams Helps Elucidate The Fundamental Ideas. Mathematical Topics Covered Include Differentiable Manifolds, Differential Forms And Twisted Forms, The Hodge Star Operator, Exterior Differential Systems And Symplectic Geometry. All Of The Mathematics Is Motivated And Illustrated By Useful Physical Examples. William L. Burke. Includes Index. Bibliography: P. 409-410. Contents......Page 5 Preface page......Page 9 Glossary of notation......Page 13 Introduction......Page 17 I Tensors in linear spaces......Page 27 1 Linear and affine spaces......Page 28 2 Differential calculus......Page 37 3 Tensor algebra......Page 43 4 Alternating products......Page 47 5 Special relativity......Page 53 6 The uses of covariance......Page 60 II Manifolds......Page 67 7 Manifolds......Page 68 8 Tangent vectors and 1-forms......Page 75 9 Lie bracket......Page 84 10 Tensors on manifolds......Page 88 11 Mappings......Page 93 12 Cotangent bundle......Page 100 13 Tangent bundle......Page 106 14 Vector fields and dynamical systems......Page 110 15 Contact bundles......Page 115 16 The geometry of thermodynamics......Page 124 17 Lie groups......Page 131 18 Lie derivative......Page 137 19 Holonomy......Page 148 20 Contact transformations......Page 152 21 Symmetries......Page 157 22 Differential forms......Page 163 23 Exterior calculus......Page 169 24 The * operator......Page 175 25 Metric symmetries......Page 185 26 Normal forms......Page 189 27 Index notation......Page 192 28 Twisted differential forms......Page 199 29 Integration......Page 210 30 Cohomology......Page 218 31 Diffusion equations......Page 223 32 First-order partial differential equations......Page 229 33 Conservation laws......Page 235 34 Calculus of variations......Page 241 35 Constrained variations......Page 249 36 Variations of multiple integrals......Page 255 37 Holonomy and thermodynamics......Page 261 38 Exterior differential systems......Page 264 39 Symmetries and similarity solutions......Page 274 40 Variational principles and conservation laws......Page 280 41 When not to use forms......Page 284 VI Classical electrodynamics......Page 287 42 Electrodynamics and differential forms......Page 288 43 Electrodynamics in spacetime......Page 298 44 Laws of conservation and balance......Page 301 45 Macroscopic electrodynamics......Page 309 46 Electrodynamics of moving bodies......Page 314 VII Dynamics of particles and fields......Page 321 47 Lagrangian mechanics of conservative systems......Page 322 48 Lagrange's equations for general systems......Page 327 49 Lagrangian field theory......Page 330 50 Hamiltonian systems......Page 336 51 Symplectic geometry......Page 341 52 Hamiltonian optics......Page 349 53 Dynamics of wave packets......Page 354 VIII Calculus on fiber bundles......Page 363 54 Connections......Page 365 55 Parallel transport......Page 370 56 Curvature and torsion......Page 374 57 Covariant differentiation......Page 381 58 Metric connections......Page 383 IX Gravitation......Page 387 59 General relativity......Page 388 60 Geodesics......Page 390 61 Geodesic deviation......Page 393 62 Symmetries and conserved quantities......Page 398 63 Schwarzschild orbit problem......Page 403 64 Light deflection......Page 409 65 Gravitational lenses......Page 411 66 Moving frames......Page 418 Bibliography......Page 425 Index......Page 427