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Differential Forms, 2nd Edition

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

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. Preface
  7. 1: Differential Forms in , I
    1. 1.0 Euclidean spaces, tangent spaces, and tangent vector fields
    2. 1.1 The algebra of differential forms
    3. 1.2 Exterior differentiation
    4. 1.3 The fundamental correspondence
    5. 1.4 The Converse of Poincaré’s Lemma, I
    6. 1.5 Exercises
  8. 2: Differential Forms in , II
    1. 2.1 -forms
    2. 2.2 -Forms
    3. 2.3 Orientation and signed volume
    4. 2.4 The converse of Poincaré’s Lemma, II
    5. 2.5 Exercises
  9. 3: Push-forwards and Pull-backs in
    1. 3.1 Tangent vectors
    2. 3.2 Points, tangent vectors, and push-forwards
    3. 3.3 Differential forms and pull-backs
    4. 3.4 Pull-backs, products, and exterior derivatives
    5. 3.5 Smooth homotopies and the Converse of Poincaré’s Lemma, III
    6. 3.6 Exercises
  10. 4: Smooth Manifolds
    1. 4.1 The notion of a smooth manifold
    2. 4.2 Tangent vectors and differential forms
    3. 4.3 Further constructions
    4. 4.4 Orientations of manifolds—intuitive discussion
    5. 4.5 Orientations of manifolds—careful development
    6. 4.6 Partitions of unity
    7. 4.7 Smooth homotopies and the Converse of Poincaré’s Lemma in general
    8. 4.8 Exercises
  11. 5: Vector Bundles and the Global Point of View
    1. 5.1 The definition of a vector bundle
    2. 5.2 The dual bundle, and related bundles
    3. 5.3 The tangent bundle of a smooth manifold, and related bundles
    4. 5.4 Exercises
  12. 6: Integration of Differential Forms
    1. 6.1 Definite integrals in
    2. 6.2 Definition of the integral in general
    3. 6.3 The integral of a -form over a point
    4. 6.4 The integral of a -form over a curve
    5. 6.5 The integral of a -form over a surface
    6. 6.6 The integral of a -form over a solid body
    7. 6.7 Chains and integration on chains
    8. 6.8 Exercises
  13. 7: The Generalized Stokes’s Theorem
    1. 7.1 Statement of the theorem
    2. 7.2 The fundamental theorem of calculus and its analog for line integrals
    3. 7.3 Cap independence
    4. 7.4 Green’s and Stokes’s theorems
    5. 7.5 Gauss’s theorem
    6. 7.6 Proof of the GST
    7. 7.7 The converse of the GST
    8. 7.8 Exercises
  14. 8: de Rham Cohomology
    1. 8.1 Linear and homological algebra constructions
    2. 8.2 Definition and basic properties
    3. 8.3 Computations of cohomology groups
    4. 8.4 Cohomology with compact supports
    5. 8.5 Exercises
  15. Index