#### 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

MATH 241 and (MATH 240 or MATH 461); or MATH 340 and MATH 341.

Recommended: MATH 403, MATH 405, MATH 410, MATH 432 or MATH 436. (For appropriate "mathematical maturity".)

#### 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