Is there a geometric interpretation/visualization of the exterior derivative, at least in three dimensions ?

Suppose we have a 1-form on a 3-dimensional basis {dx

^{1}, dx

^{2},dx

^{3}} :

with a set of real-valued coefficients

*f*. The exterior derivative is then, by definition, the 2-form

A 1-form in three dimensions is an oriented line segment, a 2-form an oriented surface element. So, is there an intuitive way to "visualize" the exterior derivative operation, in a geometric sense ? For example, I can roughly visualise the Hodge Star as giving us the elements which are

*orthogonal* to whatever the operator acts on in the particular coordinate basis, at least up to a sign. I know this isn't precise or rigorous, but it does help me to grasp the intuitive meaning, and it even works in more than three dimensions. Does something similar exist for the exterior derivative ? Can it be visualised roughly as "wedging" with a linearly independent basis element in a way that orientation is preserved, thereby increasing the degree by one ? This would explain how, for example, the exterior derivative turns an oriented line segment into an oriented surface element.

I am fine with the abstract definitions of the operators, and its connections to the usual div/grad/curl, but it would be very helpful to have some way to visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations. Guitarist has shown us the algebraic part, which I am fine with; my problem is that, just by looking at

*dF=0* and

*d*F=uJ *it is very hard to visualise what this actually implies in a geometric sense.