Exterior Derivative: A Visual Explanation

[KarateMan]

I would like to ask you about exterior derivative.

I have found the exterior derivative very difficult to visualize. Does it have anything to do with the ordinary derivative of a scalar function? What I mean is that the ordinary differentiation is the rate of change of the scalar function with respect to the variable. So the exterior derivative is also the rate of change of something? Or it is entirely something else? It simply doesn't seem to me that way. How can I visualize taking exterior derivative of a 1-form is a 2-form?

 

[Hurkyl]

Forms are functions. You don't visualize them: you visualize what they do.


A k-form is simply a way of measuring k-dimensional surfaces in your manifold.

By Stokes' theorem, \int_S d\omega = \int_{\delta S} \omega.

This tells you exactly what the exterior derivative does: dw measures a region by applying w to the boundary.

If you want to picture it locally, then just imagine little tiny regions. For example, let's work in 3-space.

Let w be the 2-form that measures how much of your surface is perpendicular to a certain vector field F. By picturing little tiny spheres, what do you think dw is?

Hint below:
















Note that

\int_S \omega = \\int_S \vec{F} \cdot \hat{n} \, dA

where n is the unit normal to your surface.

 

[mathwonk]

i think d of a one form measures by how much the field asociated to the one form fails to be conservative, i.e. how far the one form fails to be a gradient (locally).

this is a derivative of sorts. by looking at the stokes foprmula in hurkyls note, and dividing both siodes by the area of the small surface S, you are getting the ratio of the path integral on the right, to the area of the surface. hence taking the limit as the surface shrinks to a point, gives the rate of change of the path integral around discs centered at that point, i.e. the "curl" of then vector field there, wrt area.


this is the old fashioned point of view, as in books like L.Hopf, Partial differential equations of physics,