Differential Forms with Applications to the Physical Sciences

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 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, Amp