Preface
These are the lecture notes for Math 3210 (formerly named Math 321), Manifolds and Differential Forms, as taught at Cornell University since the Fall of 2001. The course covers manifolds and differential forms for an audience of undergraduates who have taken a typical calculus sequence at a North American university, including basic linear algebra and multivariable calculus up to the integral theorems of Green, Gauss and Stokes. With a view to the fact that vector spaces are nowadays a standard item on the undergraduate menu, the text is not restricted to curves and surfaces in three-dimensional space, but treats manifolds of arbitrary dimension. Some prerequisites are briefly reviewed within the text and in appendices. The selection of material is similar to that in Spivak’s book [Spi71] and in Flanders’ book [Fla89], but the treatment is at a more elementary and informal level appropriate for sophomores and juniors.
A large portion of the text consists of problem sets placed at the end of each chapter. The exercises range from easy substitution drills to fairly involved but, I hope, interesting computations, as well as more theoretical or conceptual problems. More than once the text makes use of results obtained in the exercises.
Because of its transitional nature between calculus and analysis, a text of this kind has to walk a thin line between mathematical informality and rigour. I have tended to err on the side of caution by providing fairly detailed definitions and proofs. In class, depending on the aptitudes and preferences of the audience and also on the available time, one can skip over many of the details without too much loss of continuity. At any rate, most of the exercises do not require a great deal of formal logical skill and throughout I have tried to minimize the use of point-set topology.
These notes, occasionally revised and updated, are available at http://www.math.cornell.edu/~sjamaar/manifolds/.
Corrections, suggestions and comments sent to
[email protected] will be received gratefully.
Ithaca, New York, December 2017
Author(s): Reyer Sjamaar
Year: 2017
Language: English
Tags: Differential Geometry; Differential Topology; Manifolds; Differential Forms
Preface
Chapter 1. Introduction
1.1. Manifolds
1.2. Equations
1.3. Parametrizations
1.4. Configuration spaces
Exercises
Chapter 2. Differential forms on Euclidean space
2.1. Elementary properties
2.2. The exterior derivative
2.3. Closed and exact forms
2.4. The Hodge star operator
2.5. div, grad and curl
Exercises
Chapter 3. Pulling back forms
3.1. Determinants
3.2. Pulling back forms
Exercises
Chapter 4. Integration of 1-forms
4.1. Definition and elementary properties of the integral
4.2. Integration of exact 1-forms
4.3. Angle functions and the winding number
Exercises
Chapter 5. Integration and Stokes' theorem
5.1. Integration of forms over chains
5.2. The boundary of a chain
5.3. Cycles and boundaries
5.4. Stokes' theorem
Exercises
Chapter 6. Manifolds
6.1. The definition
6.2. The regular value theorem
Exercises
Chapter 7. Differential forms on manifolds
7.1. First definition
7.2. Second definition
Exercises
Chapter 8. Volume forms
8.1. n-Dimensional volume in RN
8.2. Orientations
8.3. Volume forms
Exercises
Chapter 9. Integration and Stokes' theorem for manifolds
9.1. Manifolds with boundary
9.2. Integration over orientable manifolds
9.3. Gauss and Stokes
Exercises
Chapter 10. Applications to topology
10.1. Brouwer's fixed point theorem
10.2. Homotopy
10.3. Closed and exact forms re-examined
Exercises
Appendix A. Sets and functions
A.1. Glossary
A.2. General topology of Euclidean space
Exercises
Appendix B. Calculus review
B.1. The fundamental theorem of calculus
B.2. Derivatives
B.3. The chain rule
B.4. The implicit function theorem
B.5. The substitution formula for integrals
Exercises
Appendix C. The Greek alphabet
Bibliography
Notation Index
Index
