Ebook: Manifolds and Differential Forms

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