Vector Analysis

by Janich, Klaus; Kay, L.; Kay, Leslie D.; Axler, S.; Gehring, F. W.; Ribet, K. A.

• ISBN13: 9780387986494
• ISBN10: 0387986499
• Format: Hardcover
• Copyright: 2001-04-01
• Publisher: Springer Nature

Summary

'The present book is a marvelous introduction in the modern theory of manifolds and differential forms. The undergraduate student can closely examine tangent spaces, basic concepts of differential forms, integration on manifolds, Stokes theorem, de Rham- cohomology theorem, differential forms on Riema-nnian manifolds, elements of the theory of differential equations on manifolds (Laplace-Beltrami operators). Every chapter contains useful exercises for the students.' ' ZENTRALBLATT MATH

Table of Contents

Preface to the English Edition	v
Preface to the First German Edition	vii
Differentiable Manifolds	1	(24)
The Concept of a Manifold	1	(2)
Differentiable Maps	3	(2)
The Rank	5	(2)
Submanifolds	7	(2)
Examples of Manifolds	9	(3)
Sums, Products, and Quotients of Manifolds	12	(5)
Will Submanifolds of Euclidean Spaces Do?	17	(1)
Test	18	(3)
Exercises	21	(1)
Hints for the Exercises	22	(3)
The Tangent Space	25	(24)
Tangent Spaces in Euclidean Space	25	(2)
Three Versions of the Concept of a Tangent Space	27	(5)
Equivalence of the Three Versions	32	(4)
Definition of the Tangent Space	36	(1)
The Differential	37	(3)
The Tangent Spaces to a Vector Space	40	(1)
Velocity Vectors of Curves	41	(1)
Another Look at the Ricci Calculus	42	(3)
Test	45	(2)
Exercises	47	(1)
Hints for the Exercises	48	(1)
Differential Forms	49	(16)
Alternating k-Forms	49	(2)
The Components of an Alternating k-Form	51	(3)
Alternating n-Forms and the Determinant	54	(1)
Differential Forms	55	(2)
One-Forms	57	(2)
Test	59	(2)
Exercises	61	(1)
Hints for the Exercises	62	(3)
The Concept of Orientation	65	(14)
Introduction	65	(2)
The Two Orientations of an n-Dimensional Real Vector Space	67	(3)
Oriented Manifolds	70	(1)
Construction of Orientations	71	(3)
Test	74	(2)
Exercises	76	(1)
Hints for the Exercises	77	(2)
Integration on Manifolds	79	(22)
What Are the Right Integrands?	79	(4)
The Idea behind the Integration Process	83	(2)
Lebesgue Background Package	85	(3)
Definition of Integration on Manifolds	88	(5)
Some Properties of the Integral	93	(3)
Test	96	(2)
Exercises	98	(1)
Hints for the Exercises	99	(2)
Manifolds-with-Boundary	101	(16)
Introduction	101	(1)
Differentiability in the Half-Space	102	(1)
The Boundary Behavior of Diffeomorphisms	103	(2)
The Concept of Manifolds-with-Boundary	105	(1)
Submanifolds	106	(1)
Construction of Manifolds-with-Boundary	107	(2)
Tangent Spaces to the Boundary	109	(1)
The Orientation Convention	110	(1)
Test	111	(3)
Exercises	114	(1)
Hints for the Exercises	115	(2)
The Intuitive Meaning of Stokes's Theorem	117	(16)
Comparison of the Responses to Cells and Spans	117	(1)
The Net Flux of an n-Form through an n-Cell	118	(3)
Source Strength and the Cartan Derivative	121	(1)
Stokes's Theorem	122	(1)
The de Rham Complex	123	(1)
Simplicial Complexes	124	(3)
The de Rham Theorem	127	(6)
The Wedge Product and the Definition of the Cartan Derivative	133	(18)
The Wedge Product of Alternating Forms	133	(2)
A Characterization of the Wedge Product	135	(2)
The Defining Theorem for the Cartan Derivative	137	(2)
Proof for a Chart Domain	139	(1)
Proof for the Whole Manifold	140	(3)
The Naturality of the Cartan Derivative	143	(1)
The de Rham Complex	144	(1)
Test	145	(3)
Exercises	148	(1)
Hints for the Exercises	148	(3)
Stokes's Theorem	151	(16)
The Theorem	151	(1)
Proof for the Half-Space	152	(2)
Proof for a Chart Domain	154	(1)
The General Case	155	(1)
Partitions of Unity	156	(2)
Integration via Partitions of Unity	158	(2)
Test	160	(3)
Exercises	163	(1)
Hints for the Exercises	163	(4)
Classical Vector Analysis	167	(28)
Introduction	167	(1)
The Translation Isomorphisms	168	(2)
Gradient, Curl, and Divergence	170	(3)
Line and Area Elements	173	(2)
The Classical Integral Theorems	175	(3)
The Mean-Value Property of Harmonic Functions	178	(2)
The Area Element in the Coordinates of the Surface	180	(5)
The Area Element of the Graph of a Function of Two Variables	185	(1)
The Concept of the Integral in Classical Vector Analysis	186	(3)
Test	189	(2)
Exercises	191	(1)
Hints for the Exercises	192	(3)
De Rham Cohomology	195	(20)
Definition of the de Rham Functor	195	(2)
A Few Properties	197	(2)
Homotopy Invariance: Looking for the Idea of the Proof	199	(2)
Carrying Out the Proof	201	(2)
The Poincare Lemma	203	(3)
The Hairy Ball Theorem	206	(3)
Test	209	(2)
Exercises	211	(1)
Hints for the Exercises	212	(3)
Differential Forms on Riemannian Manifolds	215	(24)
Semi-Riemannian Manifolds	215	(3)
The Scalar Product of Alternating k-Forms	218	(3)
The Star Operator	221	(4)
The Coderivative	225	(2)
Harmonic Forms and the Hodge Theorem	227	(3)
Poincare Duality	230	(2)
Test	232	(3)
Exercises	235	(1)
Hints for the Exercises	236	(3)
Calculations in Coordinates	239	(30)
The Star Operator and the Coderivative in Three-Dimensional Euclidean Space	239	(2)
Forms and Dual Forms on Manifolds without a Metric	241	(2)
Three Principles of the Ricci Calculus on Manifolds without a Metric	243	(3)
Tensor Fields	246	(3)
Raising and Lowering Indices in the Ricci Calculus	249	(2)
The Invariant Meaning of Raising and Lowering Indices	251	(2)
Scalar Products of Tensors in the Ricci Calculus	253	(2)
The Wedge Product and the Star Operator in the Ricci Calculus	255	(2)
The Divergence and the Laplacian in the Ricci Calculus	257	(2)
Concluding Remarks	259	(2)
Test	261	(3)
Exercises	264	(2)
Hints for the Exercises	266	(3)
Answers to the Test Questions	269	(4)
Bibliography	273	(2)
Index	275