Why is the exterior derivative called exterior derivative

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ satisfies the Leibniz rule? I feel am looking for a way to grasp it as I understand most derivation operators - as measuring some "infinitesimal change" in some sense. Am I completely misguided?

• It generalizes the gradient of a function. And Stokes' Theorem generalizes the fundamental theorem of calculus. – geodude Jun 3 '14 at 11:59
• "Exterior" refers to the exterior algebra on which $d$ is defined, and "differential" refers to the fact that it reduces to the ordinary differential of scalar fields when restricted to $0$ forms. – Giuseppe Negro Jun 3 '14 at 12:04
• Thanks to both of the commentators above. Is there any sort of explanation via viewing the derivative as measuring some "infinitesimal change", or is it completely out of context in exterior calculus? – olamundo Jun 3 '14 at 16:09
• @olamundo: If you do not add the @...: construct, people are not notified of your comments and so they might never come back. Anyway, the exterior derivative is a sort of infinitesimal operation. See for example Bachman's notes on differential forms. – Giuseppe Negro Jun 9 '14 at 17:21
• @GiuseppeNegro Thanks! Those notes look exactly like what I was looking for. If you want you can post this comment as an answer and I'll accept. – olamundo Jun 10 '14 at 21:12

The exterior derivative is a sort of infinitesimal operation. See for example Bachman's notes on differential forms. (See also comments to the question, above).

EDIT: Updated link to Bachman's notes. They are now on the arXiv.

• Those notes have since disappeared, or been moved, and the link is dead. – izœc Jul 23 '15 at 21:19

It is precisely an infinitesimal change. But an infinitesimal difference in what?

A differential 1 form measures the flux across an infinitesimal line situated at some point. By extension a differerential 0 form evaluated at a point in space measures the flux across an infinitesimal point (an infinitiseimAl point is the same as a finite point is the same As a point.)

Consider a smooth curve. Zoom in on a point and look at its neighborhood. The exterior derivative of a 0 form measures the difference between the zero forms evaluated at points infinitesimally apart. As the points draw nearer and nearer, the difference becomes more and more a differential. This physically measures the difference in flow that exits one point and enters the other. If one were to add all these infinitesimal differences across a curve, the interior regions cancel leaving the exterior difference behind. For when the flux exits one end, it immediately enters the next.

This is called the exterior derivative because the integral of the "exterior derivative" leaves the difference of the exterior behind. Loosely the word "integral" cancels "derivative" leaving " exterior" behind...hence its name.

This is naturally extended to differential two forms.

Consider a smooth surface. Zoom in at a point and look at the neighborhood. It looks like an element of a plane which when parametrized leads to a small parrallellogram.

a side note: A differential 2 form evaluated at a point measures the flux across the infinitesimal parallelogram (which is defined by two vectors infinitesimal in length). The skew symmetry comes from the linearity in flux (consider w(x,y) + w(y,x). The sum is w(x+y,x+y) which must be zero since the area of the parallelogram is zero...no passage is admitted since none can be admitted.

Back to topic: Considering this surface and having zoomed in, an exterior derivative of a differential 1 form represents the differences in flux across the lines of the infinitesimal parallelogram. This is different than the two form. The two form would measure the flux of air as it slices through the paper. This "flux a across the lines" represents the flux of water as it travels along the paper.

To understand this draw the parellogram. Suppose we've parametrized our surface to r and s. Then one point of the parellogram is (r,s). The two vectors that make up the parellogram are formed by varying r and s infinitesimally. Let's call them dr and ds. There are the two other vectors parallel to these on the other side. I'll call them drr and dss respectively. First, Find the differences in the flux from dss and ds. This is the differences in one forms from dss and ds. Next, Find the differences in the flux from dr to drr (notice this direction of the difference is polar to the first. This small detail is crucial. I'll ask you why its this way? Its because you'll run into a small issue a bit later when integrating across the surface) Then just add these differences up and that's it. That's the exterior derivative of 1-form...the net flux across the lines of a parallelogram.

As it happens the exterior derivative of 1 form is a 2 form.

Adding them all up across the whole surface leads to again a cancellation of the interiors leaving only the integral across the contour behind. Notice that's an integral of a differential 1 form.

This was a summary of the true inner workings of the calculus, I think. All of it can be made more vigorous and generalized, but this is the spirit of it, I think.

If you look at the definition and how it reduces to the (total) differential in case of manifold=$R^n$ then I think the name is actually a misnomer and it should be called the exterior differential.

Actually the notation suggests that because it uses lower case d (as for regular differentials) instead of upper case d (one of the notations for a derivative).