we re-explain why the modular (algebraicaly) point view plays an excellent frame work to considerations about vector-calculus, multilinear-algebra and differential-geometry phenomena.
So, an -module is a mixed algebraic structure which consists in an abelian group with an action
of a ring which obeys the same rules vector spaces do.
Then if we consider , the ring of differentable functions on an open set of an euclidean space, i.e. all functions
and the set of all vector fields, then there is an obvious action , which is
That is is a -module.
But also is a ring under the sum of vector fields and commutator as a product, so we have an action
which converts to into a -module.