Description

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.

Prerequisites

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

Advanced

Sample Textbooks

Vector Analysis, by Klaus Janich. Published by Springer-Verlag

Differential Forms and Connections, 1st Edition,  by R.W.R. Darling

Applications

If you like this course, you might also consider the following courses

Additional Notes

Students interested in grad school in MATH should consider this course 

Topics

Essential background

Elementary point-set topology

Manifolds, submanifolds, smooth maps

Tangent Spaces

Inverse Function Theorem

Exterior algebra

Exterior product, interior product

Graded derivations

Exterior Calculus

Vector fields, Tensor fields

Lie derivatives, Exterior derivative,

Applications to Lie groups

Integration on manifolds

Stokes Theorem

Cohomology, de Rham Theorem

Harmonic theory

Gauss-Bonnet-Theorem

Maxwell's Equations and Electrostatics