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| Preface to the English Edition Preface to the First German Edition 1 Differentiable Manifolds 1.1 The Concept of a Manifold 1.2 Differentiable Maps 1.3 The Rank 1.4 Submanifolds 1.5 Examples of Manifolds 1.6 Sums, Products, and Quotients of Manifolds 1.7 Will Submanifolds of Euclidean Spaces Do? 1.8 Test 1.9 Exercises 1.10 Hints for the Exercises 2 The Tangent Space 2.1 Tangent Spaces in Euclidean Space 2.2 Three Versions of the Concept of a Tangent Space 2.3 Equivalence of the Three Versions 2.4 Definition of the Tangent Space 2.5 The Differential 2.6 The Tangent Spaces to a Vector Space 2.7 Velocity Vectors of Curves 2.8 Another Look at the Ricci Calculus 2.9 Test 2.10 Exercises 2.11 Hints for the Exercises 3 Differential Forms 3.1 Alternating k-Forms 3.2 The Components of an Alternating k-Form 3.3 Alternating n-Forms and the Determinant 3.4 Differential Forms 3.5 One-Forms 3.6 Test 3.7 Exercises 3.8 Hints for the Exercises 4 The Concept of Orientation 4.1 Introduction 4.2 The TWO Orientations of an n-Dimensional Real Vector Space 4.3 Oriented Manifolds 4.4 Construction of Orientations 4.5 Test 4.6 Exercises 4.7 Hints for the Exercises 5 Integration on Manifolds 5.1 What Are the Right Integrands? 5.2 The Idea behind the Integration Process 5.3 Lebesgue Background Package 5.4 Definition of Integration on Manifolds 5.5 Some Properties of the Integral 5.6 Test …… 6 Manifolds-with-Boundary 7 The Intuitoive Meaning of Stokes's Theorem 8 The Wedge Product and the Defintion of the Cartan Derivative 9 Stokes's Theorem 10 Classical Vector Analysis 11 De Rham Cohomology 12 Differential Forms on Riemannian Manifolds 13 Calculations in coordiates 14 Answers to the Test Questions Bibligraphy Index |
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