This is a list of all courses offered by the Math Department.  Not all courses are offered each year.  What is provided is a general description of the courses and the prerequisites.  The actual content may vary.

<- Return to Course List


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".)


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
Maxwell's Equations and Electrostatics