• Foreword; Preface to the Dover Edition; Preface to the First EditionI. Introduction1.1 Exterior Differential Forms1.2 Comparison with TensorsII. Exterior algebra2.1 The Space of p-vectors2.2 Determinants2.3 Exterior Products2.4 Linear Transformations2.5 Inner Product Spaces2.6 Inner Products of p-vectors2.7 The Star Operator2.8 ProblemsIII. The Exterior Derivative3.1 Differential Forms3.2 Exterior Derivative3.3 Mappings3.4 Change of coordinates3.5 An Example from Mechanics3.6 Converse of the Poincaré Lemma3.7 An Example3.8 Further Remarks3.9 ProblemsIV. Applications4.1 Moving Frames in E superscript 34.2 Relation between Orthogonal and Skew-symmetric Matrices4.3 The 6-dimensional Frame Space4.4 The Laplacian, Orthogonal Coordinates4.5 Surfaces4.6 Maxwell's Field Equations4.7 ProblemsV. Manifolds and Integration5.1 Introduction5.2 Manifolds5.3 Tangent Vectors5.4 Differential Forms5.5 Euclidean Simplices5.6 Chains and Boundaries5.7 Integration of Forms5.8 Stokes' Theorem5.9 Periods and De Rham's Theorems5.10 Surfaces; Some Examples5.11 Mappings of Chains5.12 ProblemsVI. Applications in Euclidean Space6.1 Volumes in E superscript n6.2 Winding Numbers, Degree of a Mapping6.3 The Hopf Invariant6.4 Linking Numbers, the Gauss Integral, Ampère's LawVII. Applications to Different Equations7.1 Potential Theory7.2 The Heat Equation7.3 The Frobenius Integration Theorem7.4 Applications of the Frobenius Theorem7.5 Systems of Ordinary Equations7.6 The Third Lie TheoremVIII. Applications to Differential Geometry8.1 Surfaces (Continued)8.2 Hypersurfaces8.3 Riemannian Geometry, Local Theory8.4 Riemannian Geometry, Harmonic Integrals8.5 Affine Connection8.6 ProblemsIX. Applications to Group Theory9.1 Lie Groups9.2 Examples of Lie Groups9.3 Matrix Groups9.4 Examples of Matrix Groups9.5 Bi-invariant Forms9.6 ProblemsX. Applications to Physics10.1 Phase and State Space10.2 Hamiltonian Systems10.3 Integral-invariants10.4 Brackets10.5 Contact Transformations10.6 Fluid Mechanics10.7 ProblemsBibliography; Glossary of Notation; Index

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