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.

 

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Filed under algebra, cucei math, differential geometry, geometry, mathematics, multilinear algebra, what is math

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