Fundamentals of Advanced Mathematics, Volume Three, begins with the study of differential and analytic infinite-dimensional manifolds, then progresses into fibered bundles, in particular, tangent and cotangent bundles. In addition, subjects covered include the tensor calculus on manifolds, differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds), an analysis on Lie groups, the Haar measure, the convolution of functions and distributions, and the harmonic analysis over a Lie group. Finally, the theory of connections is (linear connections, principal connections, and Cartan connections) covered, as is the calculus of variations in Lagrangian and Hamiltonian formulations.
This volume is the prerequisite to the analytic and geometric study of nonlinear systems.
1. 1.1 Introduction - p. 1 2. 1.2 Fréchet differential calculus - p. 2 3. 1.3 Other approaches to differential calculus - p. 27 4. 1.4 Smooth partitions of unity - p. 35 5. 1.5 Ordinary differential equations - p. 36 6. 2: Differential and Analytic Manifolds - p. 49 7. Abstract - p. 49 8. 2.1 Introduction - p. 49 9. 2.2 Manifolds: tangent space of a manifold at a point - p. 50 10. 2.3 Tangent linear mappings; submanifolds - p. 65 11. 2.4 Lie groups and their actions - p. 80 12. 3: Fiber Bundles - p. 93 13. Abstract - p. 93 14. 3.1 Introduction - p. 93 15. 3.2 Tangent bundle and cotangent bundle - p. 94 16. 3.3 Fibrations - p. 98 17. 3.4 Vector bundles - p. 108 18. 3.5 Principal bundles - p. 121 19. 4: Tensor Calculus on Manifolds - p. 131 20. Abstract - p. 131 21. 4.1 Introduction - p. 131 22. 4.2 Tensor calculus - p. 132 23. 4.3 Tensor fields - p. 145 24. 4.4 Differential forms - p. 147 25. 4.5 Pseudo-Riemannian manifolds - p. 170 26. 5: Differential and Integral Calculus on Manifolds - p. 173 27. Abstract - p. 173 28. 5.1 Introduction - p. 173 29. 5.2 Currents and differential operators - p. 174 30. 5.3 Manifolds of mappings - p. 183 31. 5.4 Lie derivatives - p. 187 32. 5.5 Exterior differential - p. 195 33. 5.6 Stokes’ formula and applications - p. 200 34. 5.7 Integral curves and manifolds - p. 224 35. 6: Analysis on Lie Groups - p. 245 36. Abstract - p. 245 37. 6.1 Introduction - p. 245 38. 6.2 Convolution - p. 246 39. 6.3 Classification of Lie algebras - p. 256 40. 6.4 Relation between Lie groups and Lie algebras - p. 273 41. 6.5 Harmonic analysis - p. 284 42. 7: Connections - p. 315 43. Abstract - p. 315 44. 7.1 Introduction - p. 315 45. 7.2 Linear connections - p. 317 46. 7.3 Method of moving frames - p. 333 47. 7.4 Riemannian geometry - p. 358 48. References - p. 369 49. Cited Authors - p. 379 50. Index - p. 387
Fundamentals of Advanced Mathematics, Volume Three, begins with the study of differential and analytic infinite-dimensional manifolds, then progresses into fibered bundles, in particular, tangent and cotangent bundles. In addition, subjects covered include the tensor calculus on manifolds, differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds), an analysis on Lie groups, the Haar measure, the convolution of functions and distributions, and the harmonic analysis over a Lie group. Finally, the theory of connections is (linear connections, principal connections, and Cartan connections) covered, as is the calculus of variations in Lagrangian and Hamiltonian formulations.
This volume is the prerequisite to the analytic and geometric study of nonlinear systems.
1. 1.1 Introduction - p. 1 2. 1.2 Fréchet differential calculus - p. 2 3. 1.3 Other approaches to differential calculus - p. 27 4. 1.4 Smooth partitions of unity - p. 35 5. 1.5 Ordinary differential equations - p. 36 6. 2: Differential and Analytic Manifolds - p. 49 7. Abstract - p. 49 8. 2.1 Introduction - p. 49 9. 2.2 Manifolds: tangent space of a manifold at a point - p. 50 10. 2.3 Tangent linear mappings; submanifolds - p. 65 11. 2.4 Lie groups and their actions - p. 80 12. 3: Fiber Bundles - p. 93 13. Abstract - p. 93 14. 3.1 Introduction - p. 93 15. 3.2 Tangent bundle and cotangent bundle - p. 94 16. 3.3 Fibrations - p. 98 17. 3.4 Vector bundles - p. 108 18. 3.5 Principal bundles - p. 121 19. 4: Tensor Calculus on Manifolds - p. 131 20. Abstract - p. 131 21. 4.1 Introduction - p. 131 22. 4.2 Tensor calculus - p. 132 23. 4.3 Tensor fields - p. 145 24. 4.4 Differential forms - p. 147 25. 4.5 Pseudo-Riemannian manifolds - p. 170 26. 5: Differential and Integral Calculus on Manifolds - p. 173 27. Abstract - p. 173 28. 5.1 Introduction - p. 173 29. 5.2 Currents and differential operators - p. 174 30. 5.3 Manifolds of mappings - p. 183 31. 5.4 Lie derivatives - p. 187 32. 5.5 Exterior differential - p. 195 33. 5.6 Stokes’ formula and applications - p. 200 34. 5.7 Integral curves and manifolds - p. 224 35. 6: Analysis on Lie Groups - p. 245 36. Abstract - p. 245 37. 6.1 Introduction - p. 245 38. 6.2 Convolution - p. 246 39. 6.3 Classification of Lie algebras - p. 256 40. 6.4 Relation between Lie groups and Lie algebras - p. 273 41. 6.5 Harmonic analysis - p. 284 42. 7: Connections - p. 315 43. Abstract - p. 315 44. 7.1 Introduction - p. 315 45. 7.2 Linear connections - p. 317 46. 7.3 Method of moving frames - p. 333 47. 7.4 Riemannian geometry - p. 358 48. References - p. 369 49. Cited Authors - p. 379 50. Index - p. 387
