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How do you visualize the exterior derivative of differential forms?

I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I imagine two vectors, constituting a parallelogram.

So I think provided I can imagine a field of oriented line segments, with exterior derivative I should imagine an appropriate field of oriented areas.

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1 Answer 1

I usually think of differential forms as things "dual" to lines, surfaces, etc.

Here I picture forms in a 3-dimensional space. The generalization is obvious, with little care.

A foliation of the space (think of the sedimentary rocks) is always a 1-form. Not all 1-forms are foliations, but they can always be written as sums of foliations.

A line integral of a 1-form is simply "how many layers the line crosses". With signs.

A stream of "flux lines", that cross a surface, is a 2-form in R³. Not all 2-forms are streams of lines, but they can be sums of streams of lines. The surface integral is, again, the number of "intersections".

A 3-form in R³ is simply a "cloud of points". Volume integration is "how many points are within a certain region".

The exterior derivative is the boundary of those objects.

Think of the foliation/1-form. If a layer breaks, its boundary is a line. Many layers that break form a stream of lines, that is a 2-form.

You see that if you take a closed loop, the integral of the 1-form along such loop is precisely the number of layers that the loop crossed without crossing back. If the loop is a keyring, the integral is the number of keys. The value of this integral is the number of layers that were "born" or "dead" within the loop. Or, the number of stream-lines, of the exterior derivative, that crossed an area enclosed by the loop! This is exactly Stokes' theorem: the integral of a 1-form around a closed loop is equal to the integral of its derivative on an area enclosed by such loop.

Ultimately, Stokes' theorem is a "conservation of intersections".

This works for any order, and any dimension.

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