The module explains Stokes' formula
∫∂S ω = ∫S ∂ω
where:
ω is a differential p-form in n-variables;
S is a "nice" (n+1)-dimensional region in Euclidean space ℝn+1;
∂S denotes the boundary of the region S;
∂ω denotes the exterior derivative of the form ω
It's a good topic for a university in the West of Ireland since George Gabriel Stokes was born and brought up in Skreen, County Sligo. The formula is best explained using the differential forms language of Élie Cartan. The module covers
- the case p=0 (8 lectures),
- the case p=1 (8 lectures),
- the case p≥ 2 (8 lectures),
and explains how Stokes' formula can be applied to
- the usual problems of multivariate calculus,
- financial mathematics,
- electromagnetism.