I'm learning about differential forms (while also reviewing my long-forgotten multivariable calculus), and in particular trying to get an inuitive understanding of the exterior derivative. One helpful suggestion has been to treat the exterior derivative of a k-form $\omega$ as (the limit of) a normalized integral of $\omega$ over the oriented boundary of a small (k+1)-parallelotope.
So for a 0-form $f$, we get:
$$df(v)\big|_p = \lim_{t \to 0}\dfrac{1}{t} \left( f(p+tv) - f(p)\right)$$
For a 1-form, intuitively we get curl (we're integrating around a parallelogram). But from reading Prof. Tao's intro to differential forms, it seems like what we should get is some kind of flux (edit: I just mean something flow-like). If we're working in Euclidean 3-space, should I view the two as something like Hodge duals of each other? And what if we're working in an arbitrary space? Should I think of an exact 2-form as curl-like or flux-like?
Any other help in thinking about all of this in an intuitive-but-formal way would be appreciated.
Thanks to Ted Shifrin for clearing this up.
Suppose we identify a vector field $V$ with its dual, the 1-form $\omega$. This is apparently a common thing to do. Then what is $d\omega$? It is a 2-form such that $d\omega(v_1, v_2)$ gives the (normalized) circulation of $V$ around the (infinitesimal) parallelogram defined by $(v_1, v_2)$. This works in general.
But a flow only makes sense across a hypersurface. In $\mathbb{R}^n$, it must be represented by an $(n-1)$-form $\tau$, measuring the flow of the vector field corresponding to $\star\tau$. In $\mathbb{R}^3$, therefore, we can choose to interpret the 2-form $d\omega$ as measuring the flow of the field $\star d\omega$ (what we normally call curl $V$). But we cannot think of a 2-form as measuring a flow in general.
More generally, it seems useful to keep in mind that a $k$-form simply assigns a scalar to a $k$-volume. This can have more than one interpretation depending on context.