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On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which object is measured by $k$-form?

Any help would be very much appreciated. Thanks in advance for your time.

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Can you get hold of a copy of Misner, Thorne and Wheeler's Gravity ? They spend a lot of time explaining this point of view with great pictures. My memory (lost my copy in a postdoctoral move somewhere) is that what is measured is the flux of the $k$-form. – Michael Murray Apr 5 '13 at 13:08

See Theorem 1 in Anders Kock's paper “Differential forms as infinitesimal cochains”, which is devoted precisely to this question. Specifically, the map b in the formula (1) establishes an explicit bijection between differential forms and functions on the space of infinitesimal parallelepipeds (or, equivalently, simplices) that vanish on degenerate parallelepipeds/simplices. The fact that this isomorphism commutes with differentials is precisely the infinitesimal version of Stokes' formula.

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