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Exterior Analysis

Book Description

Exterior analysis uses differential forms (a mathematical technique) to analyze curves, surfaces, and structures. Exterior Analysis is a first-of-its-kind resource that uses applications of differential forms, offering a mathematical approach to solve problems in defining a precise measurement to ensure structural integrity.

The book provides methods to study different types of equations and offers detailed explanations of fundamental theories and techniques to obtain concrete solutions to determine symmetry. It is a useful tool for structural, mechanical and electrical engineers, as well as physicists and mathematicians.



  • Provides a thorough explanation of how to apply differential equations to solve real-world engineering problems
  • Helps researchers in mathematics, science, and engineering develop skills needed to implement mathematical techniques in their research
  • Includes physical applications and methods used to solve practical problems to determine symmetry

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface
  6. Chapter I. Exterior Algebra
    1. 1.1 Scope of the Chapter
    2. 1.2 Linear Vector Spaces
    3. 1.3 Multilinear Functionals
    4. 1.4 Alternating k-Linear Functionals
    5. 1.5 Exterior Algebra
    6. 1.6 Rank of an Exterior Form
    7. I Exercises
  7. Chapter II. Differentiable Manifolds
    1. 2.1 Scope of the Chapter
    2. 2.2 Differentiable Manifolds
    3. 2.3 Differentiable Mappings
    4. 2.4 Submanifolds
    5. 2.5 Differentiable Curves
    6. 2.6 Vectors. Tangent Spaces
    7. 2.7 Differential of a Map Between Manifolds
    8. 2.8 Vector Fields. Tangent Bundle
    9. 2.9 Flows Over Manifolds
    10. 2.10 Lie Derivative
    11. 2.11 Distributions. The Frobenius Theorem
    12. II Exercises
  8. Chapter III. Lie Groups
    1. 3.1 Scope of the Chapter
    2. 3.2 Lie Groups
    3. 3.3 Lie Algebras
    4. 3.4 Lie Group Homomorphisms
    5. 3.5 One-Parameter Subgroups
    6. 3.6 Adjoint Representation
    7. 3.7 Lie Transformation Groups
    8. Exercises
  9. Chapter IV. Tensor Fields on Manifolds
    1. 4.1 Scope of the Chapter
    2. 4.2 Cotangent Bundle
    3. 4.3 Tensor Fields
    4. IV Exercises
  10. Chapter V. Exterior Differential Forms
    1. 5.1 Scope of the Chapter
    2. 5.2 Exterior Differential Forms
    3. 5.3 Some Algebraic Properties
    4. 5.4 Interior Product
    5. 5.5 Bases Induced by the Volume Form
    6. 5.6 Ideals of the Exterior Algebra Λ(M)
    7. 5.7 Exterior Forms Under Mappings
    8. 5.8 Exterior Derivative
    9. 5.9 Riemannian Manifolds. Hodge Dual
    10. 5.10 Closed Ideals
    11. 5.11 Lie Derivatives of Exterior Forms
    12. 5.12 Isovector Fields of Ideals
    13. 5.13 Exterior Systems and Their Solutions
    14. 5.14 Forms Defined on a Lie Group
    15. V Exercises
  11. Chapter VI. Homotopy Operator
    1. 6.1 Scope of the Chapter
    2. 6.2 Star-Shaped Regions
    3. 6.3 Homotopy Operator
    4. 6.4 Exact and Antiexact Forms
    5. 6.5 Change of Centre
    6. 6.6 Canonical Forms of 1-Forms, Closed 2- Forms
    7. 6.7 An Exterior Differential Equation
    8. 6.8 A System of Exterior Differential Equations
    9. VI Exercises
  12. Chapter VII. Linear Connections
    1. 7.1 Scope of the Chapter
    2. 7.2 Connections on Manifolds
    3. 7.3 Cartan Connection
    4. 7.4 Levi-Civita Connection
    5. 7.5 Differential Operators
    6. VII Exercises
  13. Chapter VIII. Integration of Exterior Forms
    1. 8.1 Scope of the Chapter
    2. 8.2 Orientable Manifolds
    3. 8.3 Integration of Forms in the Euclidean Space
    4. 8.4 Simplices and Chains
    5. 8.5 Integration of Forms on Manifolds
    6. 8.6 The Stokes Theorem
    7. 8.7 Conservation Laws
    8. 8.8 The Cohomology of De Rham
    9. 8.9 Harmonic Forms. Theory of Hodge-De Rham
    10. 8.10 Poincare Duality
    11. VIII Exercises
  14. Chapter IX. Partial Differential Equations
    1. 9.1 Scope of the Chapter
    2. 9.2 Ideals Formed by Differential Equations
    3. 9.3 Isovector Fields of the Contact Ideal
    4. 9.4 Isovector Fields of Balance Ideals
    5. 9.5 Similarity Solutions
    6. 9.6 The Method of Generalised Characteristics
    7. 9.7 Horizontal Ideals and Their Solutions
    8. 9.8 Equivalence Transformations
    9. IX Exercises
  15. Chapter X. Calculus of Variations
    1. 10.1 Scope of the Chapter
    2. 10.2 Stationary Functionals
    3. 10.3 Euler-Lagrange Equations
    4. 10.4 Noetherian Vector Fields
    5. 10.5 Variational Problem for a General Action Functional
    6. X Exercises
  16. Chapter XI. Some Physical Applications
    1. 11.1 Scope of the Chapter
    2. 11.2 Conservative Mechanics
    3. 11.3 Poisson Bracket of 1-Forms and Smooth Functions
    4. 11.4 Canonical Transformations
    5. 11.5 Non-Conservative Mechanics
    6. 11.6 Electromagnetism
    7. 11.7 Thermodynamics
    8. XI Exercises
  17. References
  18. Index of Symbols
  19. Name Index
  20. Subject Index