Differential Forms : Theory and Practice

[BOOK DESCRIPTION]

Differential forms are a powerful mathematical technique to help students, researchers, and engineers solve problems in geometry and analysis, and their applications. They both unify and simplify results in concrete settings, and allow them to be clearly and effectively generalized to more abstract settings. Differential Forms has gained high recognition in the mathematical and scientific community as a powerful computational tool in solving research problems and simplifying very abstract problems. Differential Forms, 2nd Edition, is a solid resource for students and professionals needing a general understanding of the mathematical theory and to be able to apply that theory into practice. * Provides a solid theoretical basis of how to develop and apply differential forms to real research problems* Includes computational methods to enable the reader to effectively use differential forms* Introduces theoretical concepts in an accessible manner

[TABLE OF CONTENTS]

Preface                                            ix
1 Differential Forms in Rn, I                      1  (50)
  1.0 Euclidean spaces, tangent spaces, and        1  (4)
  tangent vector fields
  1.1 The algebra of differential forms            5  (6)
  1.2 Exterior differentiation                     11 (22)
  1.3 The fundamental correspondence               33 (9)
  1.4 The Converse of Poincares Lemma, I           42 (5)
  1.5 Exercises                                    47 (4)
2 Differential Forms in Rn, II                     51 (50)
  2.1 1-Forms                                      51 (6)
  2.2 k-Forms                                      57 (21)
  2.3 Orientation and signed volume                78 (9)
  2.4 The Converse of PoincareLemma, II          87 (11)
  2.5 Exercises                                    98 (3)
3 Push-forwards and Pull-backs in Rn               101(40)
  3.1 Tangent vectors                              101(3)
  3.2 Points, tangent vectors, and push-forwards   104(5)
  3.3 Differential forms and pull-backs            109(14)
  3.4 Pull-backs, products, and exterior           123(6)
  derivatives
  3.5 Smooth homotopies and the Converse of        129(10)
  Poincar駸 Lemma, III
  3.6 Exercises                                    139(2)
4 Smooth Manifolds                                 141(66)
  4.1 The notion of a smooth manifold              144(16)
  4.2 Tangent vectors and differential forms       160(6)
  4.3 Further constructions                        166(5)
  4.4 Orientations of manifolds-intuitive          171(13)
  discussion
  4.5 Orientations of manifolds-careful            184(11)
  development
  4.6 Partitions of unity                          195(5)
  4.7 Smooth homotopies and the Converse of        200(3)
  Poincares Lemma in general
  4.8 Exercises                                    203(4)
5 Vector Bundles and the Global Point of View      207(20)
  5.1 The definition of a vector bundle            208(7)
  5.2 The dual bundle, and related bundles         215(7)
  5.3 The tangent bundle of a smooth manifold,     222(2)
  and related bundles
  5.4 Exercises                                    224(3)
6 Integration of Differential Forms                227(82)
  6.1 Definite integrals in Rn                     228(5)
  6.2 Definition of the integral in general        233(8)
  6.3 The integral of a 0-form over a point        241(2)
  6.4 The integral of a 1-form over a curve        243(32)
  6.5 The integral of a 2-form over a surface      275(23)
  6.6 The integral of a 3-form over a solid body   298(1)
  6.7 Chains and integration on chains             299(3)
  6.8 Exercises                                    302(7)
7 The Generalized Stokes's Theorem                 309(52)
  7.1 Statement of the theorem                     309(3)
  7.2 The fundamental theorem of calculus and      312(4)
  its analog for line integrals
  7.3 Cap independence                             316(4)
  7.4 Green's and Stokes's theorems                320(11)
  7.5 Gauss's theorem                              331(6)
  7.6 Proof of the GST                             337(11)
  7.7 The converse of the GST                      348(8)
  7.8 Exercises                                    356(5)
8 de Rham Cohomology                               361(32)
  8.1 Linear and homological algebra               361(10)
  constructions
  8.2 Definition and basic properties              371(8)
  8.3 Computations of cohomology groups            379(6)
  8.4 Cohomology with compact supports             385(4)
  8.5 Exercises                                    389(4)
Index                                              393