This course is an introduction to differential forms and their applications. The exterior differential calculus of Elie Cartan is one of the most successful and illuminating techniques for calculations. The fundamental theorems of multivariable calculus are united in a general Stokes theorem which holds for smooth manifolds in any number of dimensions. This course develops this theory and technique to perform calculations in analysis and geometry. This course is independent of Math 436, although it overlaps with it. The point of view is more abstract and axiomatic, beginning with the general notion of a topological space, and developing the tools necessary for applying techniques of calculus. Local coordinates are necessary, but coordinate-free concepts are emphasized. Local calculations relate to global topological invariants, as exemplified by the Gauss-Bonnet theorem.
1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).
Recommended: MATH405, MATH403, MATH436, MATH410, or MATH432. (For appropriate "mathematical maturity".)
Level of Rigor
Vector Analysis, by Klaus Janich. Published by Springer-Verlag
Differential Forms and Connections, 1st Edition, by R.W.R. Darling
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Students interested in grad school in MATH should consider this course