Applied Exterior Calculus (1985)

Contents......Page 11
Part One. Vectors and Forms......Page 17
1-1. Number Space, Points, and Coordinates......Page 18
1-2. Composition of Mappings......Page 20
1-3. Some Useful Theorems......Page 24
1-4. Systems of Autonomous First-Order Differential Equations......Page 27
1-5. Vector Spaces, Dual Spaces, and Algebras......Page 31
2-1. The Tangent to a Curve at a Point......Page 36
2-2. The Tangent Space T(E_n) and Vector Fields on E_n......Page 41
2-3. Operator Representations of Vector Fields......Page 44
2-4. Orbits......Page 49
2-5. Linear and Quasilinear First-Order Partial Differential Equations......Page 53
2-6. The Natural Lie Algebra of T(E_n)......Page 66
2-7. Lie Subalgebra of T(E_n) and Systems of Linear First-Order Partial Differential Equations......Page 71
2-8. Behavior under Mappings......Page 80
3-1. The Dual Space T^*(E_n)......Page 92
3-2. The Exterior or "Veck" Product......Page 98
3-3. Algebraic Results......Page 103
3-4. Inner Multiplication......Page 108
3-5. Top Down Generation of Bases......Page 115
3-6. Ideals of \Lambda(E_n)......Page 119
3-7. Behavior under Mappings......Page 124
4-1. The Exterior Derivative......Page 137
4-2. Closed Ideals and a Confluence of Ideas......Page 146
4-3. The Frobenius Theorem......Page 151
4-4. The Darboux Class of a 1-Form and the Darboux Theorem......Page 156
4-5. Finite Deformations and Lie Derivatives......Page 166
4-6. Ideals and Isovector Fields......Page 175
4-7. Integration of Exterior Forms, Stokes' Theorem, and the Divergence Theorem......Page 182
5-1. Perspective......Page 189
5-2. Starshaped Regions......Page 190
5-3. The Homotopy Operator H......Page 191
5-4. The Exact Part of a Form and the Vector Subspace of Exact Forms......Page 195
5-5. The Module of Antiexact Forms......Page 196
5-6. Representations......Page 199
5-7. Change of Center......Page 202
5-8. Behavior under Mappings......Page 204
5-9. An Introductory Problem......Page 206
5-10. Representations for 2-Forms That Are Not Closed......Page 212
5-11. Differential Systems of Degree k and Class r......Page 214
5-12. Integration of the Connection Equations......Page 217
5-13. The Attitude Matrix of a Differential System......Page 219
5-14. Integration of the Curvature Equations......Page 223
5-15. Integration of a Differential System......Page 225
5-16. Equivalent Differential Systems......Page 227
5-17. Horizontal and Vertical Ideals of a Distribution......Page 230
5-18. n-Forms and Integration......Page 236
5-19. The Adjoint of a Linear Operator on \Lambda(E_n)......Page 239
References for Further Study......Page 247
Part Two. Applications to Mathematics......Page 248
6-1. The Graph Space of Solutions to Partial Differential Equations......Page 249
6-2. Kinematic Space and the Contact 1-Forms......Page 251
6-3. The Contact Ideal and Its Isovectors......Page 252
6-4. Explicit Characterization of TC(K)......Page 257
6-5. Second-Order Partial Differential Equations and Balance Forms......Page 262
6-6. Isovectors of the Balance Ideal and Generation of Solutions......Page 265
6-7. Examples......Page 267
6-8. Similarity Variables and Similarity Solutions......Page 279
6-9. Inverse Isovector Methods......Page 285
6-10. Dimension Reduction in the Calculation of Isovectors......Page 291
6-11. Contact Forms of Higher Order......Page 295
7-1. Formulation of the Problem......Page 301
7-2. Finite Variations, Stationarity, and the Euler-Lagrange Equations......Page 306
7-3. Properties of Euler-Lagrange Forms and Stationarizing Maps......Page 313
7-4. Noetherian Vector Fields and Their Associated Currents......Page 319
7-5. Boundary Conditions and the Null Class......Page 325
7-6. Problems with Boundary Integrals and Differential Constraints......Page 332
References for Further Study......Page 340
Part Three. Applications to Physics......Page 342
8-1. Formulation of the Problem......Page 343
8-2. The Work Function for Thermostatically Conservative Forces......Page 345
8-3. Internal Energy, Heat Addition, and Irreversibility......Page 348
8-4. Homogeneous Systems with Internal Degrees of Freedom......Page 355
8-5. Homogeneous Systems with Nonconservative Forces......Page 361
8-6. Nonequilibrium Thermodynamics......Page 367
8-7. Reformulation as a Well Posed Mathematical Problem......Page 369
9-1. The Four-Dimensional Base Space......Page 373
9-2. Electrodynamics with Free Magnetic Charge and Current......Page 377
9-3. General Solutions of Maxwell's Equations......Page 379
9-4. The Ray Operator and Explicit Representations......Page 381
9-5. Properties of the Field Equations and Their Solutions......Page 384
9-6. Constitutive Relations and Wave Properties......Page 389
9-7. Variational Formulation of the Field Equations......Page 394
9-8. Field Induced Momentum and Force Densities......Page 398
10-1. The Origin of Gauge Theories......Page 406
10-2. The Minimal Replacement Construct......Page 408
10-4. Gauge Connection 1-Forms and the Gauge Covariant Exterior Derivative......Page 411
10-5. Gauge Covariant Differentiation of Group Associated Quantities......Page 417
10-6. Derivation of the Field Equations......Page 423
10-7. Transformation Properties and Gauge Covariance......Page 428
10-8. Integrability Conditions and Current Conservation......Page 430
10-9. Gauge Group Orbits and the Antiexact Gauge......Page 434
10-10. Direct Integration of the Field Equations......Page 437
10-11. Momentum-Energy Complexes and Interaction Forces......Page 441
10-12. General Symmetry Groups of the Action n-Form......Page 444
10-13. Operator-Valued Connection 1-Forms......Page 446
10-14. Properties of D and Operator-Valued Curvature Forms......Page 449
10-15. Lie Connections......Page 452
10-16. The Minimal Replacement Construct......Page 454
10-17. Variations and the Field Equations......Page 460
10-18. Gauge Theory for the Poincare Group......Page 466
References for Further Study......Page 471
Appendix......Page 473
Index......Page 476