COURSE DESCRIPTION

Multilinear algebra: tensors and exterior forms. Differential forms on R^n: exterior differentiation, the pull-back operation and the Poincaré lemma. Applications to physics: Maxwell’s equations from the differential form perspective. Integration of forms on open sets of R^n. The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes’ theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology.

Prerequisites: 18.101; 18.700 or 18.701

Text Book: The primary text for the course will be the notes prepared by Professor Guillemin and posted here on the web. Some useful secondary references include Spivak's Calculus on Manifolds, Munkres's Analysis on Manifolds, and Guillemin and Pollack's Differential Topology.