Multilinear antisymmetric forms and differential forms on manifolds
We discussed the module of differential 1-forms dual to the module of smooth vector fields on a manifold. Differential 1-forms are generated by differentials of smooth functions and as such can be pulled back by smooth maps.
The “raison d’être” of differential 1-forms is to be integrated over smooth curves in the manifold, the result being dependent only on the orientation of the curve and not on its specific parametrization.
At the second hour we discussed the notion of forms of higher degree, which required to introduce the Grassman algebra on the dual space to an abstract finite-dimensional linear space . The Grassmann (exterior) algebra is a mathematical miracle that was discovered by a quest for unusual and unknown, with only slight “motivations” from outside.
The day ended up with the definition of the differential -forms and their functoriality (i.e., in what direction and how they are carried by smooth maps between manifolds).
The lecture notes are available here.