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Differential Forms

Book Description

Differential forms are utilized as a mathematical technique to help students, researchers, and engineers analyze and interpret problems where abstract spaces and structures are concerned, and when questions of shape, size, and relative positions are involved. 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 through mathematical analysis on a computer. Differential Forms, 2nd Edition, is a solid resource for students and professionals needing a solid general understanding of the mathematical theory and be able to apply that theory into practice. Useful applications are offered to investigate a wide range of problems such as engineers doing risk analysis, measuring computer output flow or testing complex systems. They can also be used to determine the physics in mechanical and/or structural design to ensure stability and structural integrity. The book offers many recent examples of computations and research applications across the fields of applied mathematics, engineering, and physics.

  • The only reference that provides a solid theoretical basis of how to develop and apply differential forms to real research problems
  • Includes computational methods for graphical results essential for math modeling
  • Presents common industry techniques in detail for a deeper understanding of mathematical applications
  • 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