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Disclaimer: physicist learning differential geometry here. Possible I get a lot wrong, please bear with me.

In basic differential geometry it seems there is an emphasis on the exterior algebra of covectors. for example $dx\wedge dy \wedge dz$ is an object that can be integrated over and is also an element of $\Lambda^3(V^*)$. It seems like we can equivalently define $\partial_x \wedge \partial_y \wedge \partial_z$ which is an element of $\Lambda^3(V)$, but I don't see this done as much, especially in physics literature.

I would also think elements of $\Lambda^3(V)$ are somehow dual to elements in $\Lambda^3(V^*)$ such that you could get away with talking about one or the other as dictated by preference.

If what I say is correct (it may not be), then why the preference for exterior algebra of covectors but not vectors?

Jagerber48
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    My Prof used the slogan "vector fields can't be pushed forward by the differential, but differential forms can be pulled back". A lot of constructions require the pullback operation of differential forms (duals to vector fields) and you can't always find the "vector fields version" of them. – FreeFunctor Apr 10 '22 at 17:07
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    On an $n$-manifold it's $n$-forms that can be integrated invariantly, i.e., independently of coordinates. In the context of physics, one often has a metric in the background, in which case one can anyway identify elements of $T_p M$ and $(T_p M)^* = T_p^* M$. – Travis Willse Apr 10 '22 at 22:04
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    I occasionally use multivectors (exterior products of vectors) as well, but the true power is not just the exterior algebra of dual vectors but the interplay with differential and integral calculus. You can differentiate and integrate (over submanifolds of appropriate dimension) differential forms and you have Stokes's Theorem. There's nothing quite analogous with multivectors. – Ted Shifrin Apr 10 '22 at 22:05
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    To emphasize, the superficial algebraic duality between differential forms and multivectors is broken by mappings, which are single-valued (each point of the domain maps to a unique point of the codomain) but need not be bijective (each point of the codomain comes from exactly one point of the domain), and the pointwise behavior of vectors (which push forward) and covectors (which pull back). – Andrew D. Hwang Apr 10 '22 at 22:53

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