Differential Forms with Applications to the Physical Sciences (eBook)

Foreword; Preface to the Dover Edition; Preface to the First EditionI. Introduction 1.1 Exterior Differential Forms 1.2 Comparison with TensorsII. 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 ProblemsIII. 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 ProblemsIV. 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 ProblemsV. 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 ProblemsVI. 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 LawVII. 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 TheoremVIII. 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 ProblemsIX. 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 ProblemsX. 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