Translated by L. Kay
Classical vector analysis deals with vector fields; the gradient, divergence, and curl operators; line, surface, and volume integrals; and the integral theorems of Gauss, Stokes, and Green. Modern vector analysis distills these into the Cartan calculus and a general form of Stokes' theorem. This essentially modern text carefully develops vector analysis on manifolds and reinterprets it from the classical viewpoint (and with the classical notation) for three-dimensional Euclidean space, then goes on to introduce de Rham cohomology and Hodge theory. The material is accessible to an undergraduate student with calculus, linear algebra, and some topology as prerequisites. The many figures, exercises with detailed hints, and tests with answers make this book particularly suitable for anyone studying the subject independently.
Author(s): Klaus Jänich
Series: Springer Undergraduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 2001
Language: English
Pages: 289
Front Matter....Pages i-xiv
Differentiable Manifolds....Pages 1-24
The Tangent Space....Pages 25-48
Differential Forms....Pages 49-64
The Concept of Orientation....Pages 65-78
Integration on Manifolds....Pages 79-100
Manifolds-with-Boundary....Pages 101-115
The Intuitive Meaning of Stokes’s Theorem....Pages 117-131
The Wedge Product and the Definition of the Cartan Derivative....Pages 133-149
Stokes’s Theorem....Pages 151-165
Classical Vector Analysis....Pages 167-193
De Rham Cohomology....Pages 195-213
Differential Forms on Riemannian Manifolds....Pages 215-237
Calculations in Coordinates....Pages 239-268
Answers to the Test Questions....Pages 269-271
Back Matter....Pages 273-283
