Klaus Janich. Vector Analysis. Translated by Leslie Kay. With 108 Illustrations. Springer

Transcription

1 Klaus Janich Vector Analysis Translated by Leslie Kay With 108 Illustrations Springer

2 Preface to the English Edition Preface to the First German Edition Differentiable Manifolds The Concept of a Manifold Differentiable Maps The Rank Submanifolds Examples of Manifolds Sums, Products, and Quotients of Manifolds Will Submanifolds of Euclidean Spaces Do? Test Exercises Hints for the Exercises,. 22 The Tangent Space Tangent Spaces in Euclidean Space Three Versions of the Concept of a Tangent Space 27 v vii IX

3 2.3 Equivalence of the Three Versions Definition of the Tangent Space The Differential The Tangent Spaces to a Vector Space Velocity Vectors of Curves Another Look at the Ricci Calculus Test Exercises Hints for the Exercises 48 3 Differential Forms Alternating Jc-Forms The Components of an Alternating fc-form Alternating n-forms and the Determinant Differential Forms One-Forms Test Exercises Hints for the Exercises 62 4 The Concept of Orientation Introduction The TVo Orientations of an n-dimensional Real Vector Space Oriented Manifolds Construction of Orientations Test Exercises Hints for the Exercises 77 5 Integration on Manifolds * What Are the Right Integrands? ' The Idea behind the Integration Process Lebesgue Background Package Definition of Integration on Manifolds Some Properties of the Integral Test 96

4 5.7 Exercises Hints for the Exercises 99 6 Manifolds-with-Boundary Introduction Differentiability in the Half-Space The Boundary Behavior of Diffeomorphisms The Concept of Manifolds-with-Boundary Submanifolds Construction of Manifolds-with-Boundary Tangent Spaces to the Boundary The Orientation Convention Test Ill 6.10 Exercises Hints for the Exercises The Intuitive Meaning of Stokes's Theorem Comparison of the Responses to Cells and Spans The Net Flux of an n-form through an n-cell Source Strength and the Cartan Derivative Stokes's Theorem The de Rham Complex Simplicial Complexes Thede Rham Theorem The Wedge Product and the Definition of the Cartan Derivative The Wedge Product of Alternating Forms.< A Characterization of the Wedge Product The Defining Theorem for the Cartan Derivative Proof for a Chart Domain Proof for the Whole Manifold The Naturality of the Cartan Derivative

5 8.7 The de Rham Complex Test Exercises Hints for the Exercises Stokes's Theorem The Theorem Proof for the Half-Space Proof for a Chart Domain The General Case Partitions of Unity Integration via Partitions of Unity Test Exercises Hints for the Exercises Classical Vector Analysis Introduction The Translation Isomorphisms Gradient, Curl, and Divergence Line and Area Elements The Classical Integral Theorems The Mean-Value Property of Harmonic Functions The Area Element in the Coordinates of the Surface The Area Element of the Graph of a Function of TWo Variables The Concept of the Integral in Classical Vector Analysis Test, Exercises Hints for the Exercises De Rham Cohomology Definition of the de Rham Functor A Few Properties 197

6 xili 11.3 Homotopy Invariance: Looking for the Idea of the Proof Carrying Out the Proof The Poincar6 Lemma The Hairy Ball Theorem Test Exercises Hints for the Exercises Differential Forms on Riemannian Manifolds Semi-Riemannian Manifolds The Scalar Product of Alternatingfc-Forms The Star Operator The Coderivative Harmonic Forms and the Hodge Theorem Poincar6 Duality Test Exercises Hints for the Exercises Calculations in Coordinates The Star Operator and the Coderivative in Three-Dimensional Euclidean Space Forms and Dual Forms on Manifolds without a Metric Three Principles of the Ricci Calculus on Manifolds without a Metric Tensor Fields Raising and Lowering Indices in the Ricci Calculus, The Invariant Meaning of Raising and Lowering Indices Scalar Products of Tensors in the Ricci Calculus The Wedge Product and the Star Operator in the Ricci Calculus 255

7 13.9 The Divergence and the Laplacian in the Ricci Calculus Concluding Remarks Test Exercises Hints for the Exercises Answers to the Test Questions 269 Bibliography 273 Index 275