Differential Forms with Applications to the Physical Sciences
Published on 28. March 2003
Book
Paperback/Softback
240 pages
978-0-486-66169-8 (ISBN)
Description
A graduate-level text introducing 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. Directed primarily to graduate-level engineers and physical scientists, it has also been used successfully to introduce modern differential geometry to graduate students in mathematics. Includes 45 illustrations. Index.
More details
Language
English
Place of publication
United States
Target group
College/higher education
Professional and scholarly
Dimensions
Height: 218 mm
Width: 136 mm
Thickness: 17 mm
Weight
259 gr
ISBN-13
978-0-486-66169-8 (9780486661698)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Foreword; Preface to the Dover Edition; Preface to the First Edition
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 Poincare 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, Ampere'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
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 Poincare 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, Ampere'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