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.
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".)
Elementary point-set topology
Manifolds, submanifolds, smooth maps
Inverse Function Theorem
Exterior product, interior product
Vector fields, Tensor fields
Lie derivatives, Exterior derivative,
Applications to Lie groups
Integration on manifolds
Cohomology, de Rham Theorem
Maxwell's Equations and Electrostatics