Exterior Analysis

Exterior Analysis

Using Applications of Differential Forms

1st Edition - August 29, 2013

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  • Author: Erdogan Suhubi
  • eBook ISBN: 9780124159280
  • Hardcover ISBN: 9780124159020

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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.

Key Features

  • 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

Readership

Physical Scientists, Engineers, Applied Mathematicians

Table of Contents

  • Preface

    Chapter I. Exterior Algebra

    1.1 Scope of the Chapter

    1.2 Linear Vector Spaces

    1.3 Multilinear Functionals

    1.4 Alternating k-Linear Functionals

    1.5 Exterior Algebra

    1.6 Rank of an Exterior Form

    I Exercises

    Chapter II. Differentiable Manifolds

    2.1 Scope of the Chapter

    2.2 Differentiable Manifolds

    2.3 Differentiable Mappings

    2.4 Submanifolds

    2.5 Differentiable Curves

    2.6 Vectors. Tangent Spaces

    2.7 Differential of a Map Between Manifolds

    2.8 Vector Fields. Tangent Bundle

    2.9 Flows Over Manifolds

    2.10 Lie Derivative

    2.11 Distributions. The Frobenius Theorem

    II Exercises

    Chapter III. Lie Groups

    3.1 Scope of the Chapter

    3.2 Lie Groups

    3.3 Lie Algebras

    3.4 Lie Group Homomorphisms

    3.5 One-Parameter Subgroups

    3.6 Adjoint Representation

    3.7 Lie Transformation Groups

    Exercises

    Chapter IV. Tensor Fields on Manifolds

    4.1 Scope of the Chapter

    4.2 Cotangent Bundle

    4.3 Tensor Fields

    IV Exercises

    Chapter V. Exterior Differential Forms

    5.1 Scope of the Chapter

    5.2 Exterior Differential Forms

    5.3 Some Algebraic Properties

    5.4 Interior Product

    5.5 Bases Induced by the Volume Form

    5.6 Ideals of the Exterior Algebra Λ(M)

    5.7 Exterior Forms Under Mappings

    5.8 Exterior Derivative

    5.9 Riemannian Manifolds. Hodge Dual

    5.10 Closed Ideals

    5.11 Lie Derivatives of Exterior Forms

    5.12 Isovector Fields of Ideals

    5.13 Exterior Systems and Their Solutions

    5.14 Forms Defined on a Lie Group

    V Exercises

    Chapter VI. Homotopy Operator

    6.1 Scope of the Chapter

    6.2 Star-Shaped Regions

    6.3 Homotopy Operator

    6.4 Exact and Antiexact Forms

    6.5 Change of Centre

    6.6 Canonical Forms of 1-Forms, Closed 2- Forms

    6.7 An Exterior Differential Equation

    6.8 A System of Exterior Differential Equations

    VI Exercises

    Chapter VII. Linear Connections

    7.1 Scope of the Chapter

    7.2 Connections on Manifolds

    7.3 Cartan Connection

    7.4 Levi-Civita Connection

    7.5 Differential Operators

    VII Exercises

    Chapter VIII. Integration of Exterior Forms

    8.1 Scope of the Chapter

    8.2 Orientable Manifolds

    8.3 Integration of Forms in the Euclidean Space

    8.4 Simplices and Chains

    8.5 Integration of Forms on Manifolds

    8.6 The Stokes Theorem

    8.7 Conservation Laws

    8.8 The Cohomology of De Rham

    8.9 Harmonic Forms. Theory of Hodge-De Rham

    8.10 Poincare Duality

    VIII Exercises

    Chapter IX. Partial Differential Equations

    9.1 Scope of the Chapter

    9.2 Ideals Formed by Differential Equations

    9.3 Isovector Fields of the Contact Ideal

    9.4 Isovector Fields of Balance Ideals

    9.5 Similarity Solutions

    9.6 The Method of Generalised Characteristics

    9.7 Horizontal Ideals and Their Solutions

    9.8 Equivalence Transformations

    IX Exercises

    Chapter X. Calculus of Variations

    10.1 Scope of the Chapter

    10.2 Stationary Functionals

    10.3 Euler-Lagrange Equations

    10.4 Noetherian Vector Fields

    10.5 Variational Problem for a General Action Functional

    X Exercises

    Chapter XI. Some Physical Applications

    11.1 Scope of the Chapter

    11.2 Conservative Mechanics

    11.3 Poisson Bracket of 1-Forms and Smooth Functions

    11.4 Canonical Transformations

    11.5 Non-Conservative Mechanics

    11.6 Electromagnetism

    11.7 Thermodynamics

    XI Exercises

    References

    Index of Symbols

    Name Index

    Subject Index

Product details

  • No. of pages: 779
  • Language: English
  • Copyright: © Academic Press 2013
  • Published: August 29, 2013
  • Imprint: Academic Press
  • eBook ISBN: 9780124159280
  • Hardcover ISBN: 9780124159020

About the Author

Erdogan Suhubi

Affiliations and Expertise

Yeditepe University, Department of Mathematics, Kayisdagi, Turkey

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