Exterior Analysis

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