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Differential Forms with Applications to the Physical Sciences
Beschreibung:
"To the reader who wishes to obtain a bird's-eye view of the theory of differential forms with applications to other branches of pure mathematics, applied mathematic and physics, I can recommend no better book." — T. J. Willmore, London Mathematical Society Journal.
This excellent text introduces the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory, it is directed primarily to engineers and physical scientists, but it has also been used successfully to introduce modern differential geometry to students in mathematics.
Chapter I introduces exterior differential forms and their comparisons with tensors. The next three chapters take up exterior algebra, the exterior derivative and their applications. Chapter V discusses manifolds and integration, and Chapter VI covers applications in Euclidean space. The last three chapters explore applications to differential equations, differential geometry, and group theory.
"The book is very readable, indeed, enjoyable — and, although addressed to engineers and scientists, should be not at all inaccessible to or inappropriate for ... first year graduate students and bright undergraduates." — F. E. J. Linton, Wesleyan University, American Mathematical Monthly.
I. Introduction
1.1 Exterior Differential Forms
1.2 Comparison with Tensors
II. Exterior algebra
2.1 The Space of p-vectors
2.2 Determinants
2.3 Exterior Products
2.4 Linear Transformations
2.5 Inner Product Spaces
2.6 Inner Products of p-vectors
2.7 The Star Operator
2.8 Problems
III. The Exterior Derivative
3.1 Differential Forms
3.2 Exterior Derivative
3.3 Mappings
3.4 Change of coordinates
3.5 An Example from Mechanics
3.6 Converse of the Poincaré Lemma
3.7 An Example
3.8 Further Remarks
3.9 Problems
IV. Applications
4.1 Moving Frames in E superscript 3
4.2 Relation between Orthogonal and Skew-symmetric Matrices
4.3 The 6-dimensional Frame Space
4.4 The Laplacian, Orthogonal Coordinates
4.5 Surfaces
4.6 Maxwell's Field Equations
4.7 Problems
V. Manifolds and Integration
5.1 Introduction
5.2 Manifolds
5.3 Tangent Vectors
5.4 Differential Forms
5.5 Euclidean Simplices
5.6 Chains and Boundaries
5.7 Integration of Forms
5.8 Stokes' Theorem
5.9 Periods and De Rham's Theorems
5.10 Surfaces; Some Examples
5.11 Mappings of Chains
5.12 Problems
VI. Applications in Euclidean Space
6.1 Volumes in E superscript n
6.2 Winding Numbers, Degree of a Mapping
6.3 The Hopf Invariant
6.4 Linking Numbers, the Gauss Integral, Ampère's Law
VII. Applications to Different Equations
7.1 Potential Theory
7.2 The Heat Equation
7.3 The Frobenius Integration Theorem
7.4 Applications of the Frobenius Theorem
7.5 Systems of Ordinary Equations
7.6 The Third Lie Theorem
VIII. Applications to Differential Geometry
8.1 Surfaces (Continued)
8.2 Hypersurfaces
8.3 Riemannian Geometry, Local Theory
8.4 Riemannian Geometry, Harmonic Integrals
8.5 Affine Connection
8.6 Problems
IX. Applications to Group Theory
9.1 Lie Groups
9.2 Examples of Lie Groups
9.3 Matrix Groups
9.4 Examples of Matrix Groups
9.5 Bi-invariant Forms
9.6 Problems
X. Applications to Physics
10.1 Phase and State Space
10.2 Hamiltonian Systems
10.3 Integral-invariants
10.4 Brackets
10.5 Contact Transformations
10.6 Fluid Mechanics
10.7 Problems
Bibliography; Glossary of Notation; Index