MATH 437 (Differential Forms and Their Applications)
| 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 |
Required: MATH 241; and either MATH 240 or MATH 461.
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 |
| TEXT |
Text(s)
typically used in this course. |
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