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Is it appropriate to think of surface integrals and volume integrals that come up in physics as integrals over a region (connected subset) of a 2D and a 3D manifold, where we add up the contributions of "infintesimal element" of this integral, where these elements are a differential 2-form and 3-form, respectively? If this is a correct intuitive picture, then I do not understand the relevance of antisymmetry of differential forms with regards to integrals, since 2-forms and 3-forms are antisymmetric (0,2) and (0,3) type tensors.

This is especially unclear for simple volume integrals. If this basically means summing up contributions of dx dy dz cubes, it would be more natural to have symmetry rather than the antisymmetry of the underlying 3-forms. What am I missing?

bkocsis
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    To give a simple example, if you tried deriving $dxdy=rdrd\theta$ from$$x=r\cos\theta,,y=r\sin\theta\implies dx=\cos\theta dr-r\sin\theta d\theta,,y=\sin\theta dr+r\cos\theta d\theta$$you'd need to assume infinitestimals anticommute. In fact, any transformation of $n$-tuple integration elements gets its Jacobian determinant that way. – J.G. Nov 29 '23 at 22:08
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    The reason we can integrate forms is because the coordinate transformation formula for the forms matches the coordinate transformation formula for the integral volume element, which is the $\det DF$, where $F$ is the coordinate transformation. – Mason Nov 29 '23 at 22:08
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    It is also related to the interesting question if in a change of variable formula we should take the Jacobian determinant or its absolute value. The answer boils down to: if we want a oriented volume or not. – Kurt G. Nov 30 '23 at 07:08

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