Jason (jcreed) wrote,

Lately I have been staying up relaly late either reading about math on Wikipedia, or discussing it with my housemate Andreas. Neal mentioned "Differential Forms" earlier today while giving me a ride home from the weekly hotdog outing.

From the wikipedia article on them, there occurs the pair of sentences:
In differential geometry, a differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of a manifold. At any point p on a manifold, a k-form gives a multilinear map from the k-th cartesian power of the tangent space at p to R.
I astonished myself by (after some poking around for definitions and contemplation) understanding both sentences' content, though I'm still not sure why the second is true. I guess I would expect the domain to be the k-th exterior power of the tangent bundle at p, not the cartesian power. Anyway, I've still got the definition of integration over k-forms to understand before I can really know what the über-general form of Stokes Theorem is saying. Although it looks a fuckton like an adjunction to my late-night- and category-theory-addled brain. Like, boundary is (left/right) adjoint to derivative, assuming integration can be thought of as, er, hom-set formation. If you squint really hard, see.
Tags: categories, math, wiki

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