Is there a geometric interpretation/visualization of the exterior derivative, at least in the case of three dimensions ?(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a 1-form on a 3-dimensional basis {dx^{1}, dx^{2},dx^{3}} :

[tex]\displaystyle{\omega =f_{i}dx^{i}}[/tex]

with a set of real-valued coefficientsf. The exterior derivative is then, by definition, the 2-form

[tex]\displaystyle{d\omega =\sum_{i,j}\frac{\partial f_{j}}{\partial x^{i}}dx^{i}\wedge dx^{j}}[/tex]

Intuitively, 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 ? Can it be visualised roughly as "wedging" with an orthogonal 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 intuitively visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations; my problem is that, just by looking atdF=0andd*F=uJit is very hard to visualise what this actually implies in a geometric sense.

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# Visualising the exterior derivative ?

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