18.952 - Theory Of Differential Forms (Spring 2018)

Instructor: Victor Guillemin

Office: Room 2-270

Email: vwg [at] math . mit .edu

Lectures: M W F, 11am, 2-143

Office Hours: M W F, 2pm - 3pm

COURSE DESCRIPTION

Multilinear algebra: tensors and exterior forms. Differential forms on $ \mathbf{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 $ \mathbf{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