the importance of a modular view

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 $R$-module is a mixed algebraic structure which consists in an abelian group with an action $R\times M\to M$

of a ring $R$ which obeys the same rules vector spaces do.

Then if we consider $R=C^{\infty}(\Omega)$, the ring of differentable functions on an open set of an euclidean space, i.e. all functions $\Omega\to \mathbb{R}$

and ${\cal{X}}(\Omega)$ the set of all vector fields, then there is an obvious action $C^{\infty}( \Omega)\times{\cal{X}}(\Omega)\to{\cal{X}}(\Omega)$, which is $(f,X)\mapsto (fX^s)\partial_s$

That is ${\cal{X}}(\Omega)$ is a $C^{\infty}(\Omega)$-module.

But also ${\cal{X}}(\Omega)$ is a ring under the sum of vector fields and commutator $[X,Y]$ as a product, so we have an action ${\cal{X}}(\Omega)\times C^{\infty}(\Omega)\to C^{\infty}(\Omega)$ $(X,f)\mapsto Xf=X^s\partial_sf$

which converts to $C^{\infty}(\Omega)$ into a ${\cal{X}}(\Omega)$-module.