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- I’m informally learning the basics of differential forms from various sources. If anyone has book recommendations please share!

So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector?

In particular I’m thinking of this variation of the divergence theorem:

$$\int_{\partial V} \rho d\vec A = \int_V (\nabla \rho )dV$$

Which seems to fit so perfectly into the generalized stokes theorem:

$$\int_{\partial V} \rho = \int_V d\rho$$

Except that it seems to go against what a couple sources said; 1-forms (like ##d\rho = \nabla \rho##) are supposed to be integrated over lines, not volumes.

Another random question; can physical things be given an absolute classification as a k-form? E.g. is pressure always a 0-form? Or can it also be viewed as a 3-form (on ##R^3##)?

[EDITED; I first typed div(rho) instead of grad(rho) by accident]

In particular I’m thinking of this variation of the divergence theorem:

$$\int_{\partial V} \rho d\vec A = \int_V (\nabla \rho )dV$$

Which seems to fit so perfectly into the generalized stokes theorem:

$$\int_{\partial V} \rho = \int_V d\rho$$

Except that it seems to go against what a couple sources said; 1-forms (like ##d\rho = \nabla \rho##) are supposed to be integrated over lines, not volumes.

Another random question; can physical things be given an absolute classification as a k-form? E.g. is pressure always a 0-form? Or can it also be viewed as a 3-form (on ##R^3##)?

[EDITED; I first typed div(rho) instead of grad(rho) by accident]

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