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I am learning a bit about differential forms: defining differential forms in terms of elementary forms, integrating forms over parametrized domains, etc.

I would like to relate this to my previous knowledge of multivariable calculus. If multivariable calculus is best understood in terms of differential forms, does that mean an arbitrary integrand ending in dxdydz can be expressed nicely in terms of $f(x,y,z)dx\wedge dy\wedge dz+g(x,y,z)dx\wedge dz...$, or some linear combination of k form fields?

For example, how can I write $\int\int e^{xy}dxdy$ in terms of forms?

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$\displaystyle \iint_D e^{xy} dx dy$ is simply $\displaystyle\int_D e^{xy} dx \wedge dy$, with a normal direction in mind though. – Shuhao Cao Apr 15 '13 at 23:22
up vote 2 down vote accepted

$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y)\, \mathrm d x \,\mathrm d y = \int_{\mathbb{R}^2} f(x,y) \,\mathrm dx \wedge \mathrm d y$$ (This all assumes a particular orientation.)

In general, you want to substitute in the preferred volume form for the manifold. (See Wikipedia.)

It's difficult to say very much about this. The moral reason why this works is all to to do with the Hodge dual, which identifies $k$ forms with $n-k$ forms for an $n$-dimensional manifold, given a preferred volume form.

Given a 0 form $f$ (a function), $\star f$ is a 'top form', or an $n$-form, and can therefore be integrated. More generally, therefore, you might write

$$\int_{\mathbb{R}^n} f \,\mathrm {d} V = \int \star f$$

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