How to visualize differential forms geometrically

Question

I've been attempting to teach myself differential geometry and I have heard that one can visualise them geometrically and that this can sometimes be helpful for an intuitive understanding of them. For example, I've heard that one can visualise a one-form in 3-d as a collection of surfaces and that when one integrates such a one-form over a path it can be thought of loosely as counting the number of these surfaces that this path "pierces" as one "moves" along it. What is the reasoning behind this? Why can one (at least heuristically) think of differential forms in this way?

Answer 1

Yes, that interpretation does make some sense. At a point, the one form is of course a dual vector, which annihilates tangent vectors in every direction but one. The directions it annihilates are tangent to the surface, while the direction it does not is normal to the surface. This means the form is not concerned about movements along the surfaces, but only perpendicular to them. The magnitude of the form can be thought of (strictly heuristically) as being how densely packed the surfaces are.

When you integrate a path, you can break down its movement into movement along the surfaces, which the one-form annihilates so it doesn't contribute to the value, and movement normal to the surfaces, which does contribute according to how densely the surfaces are packed, and so the heuristic interpretation of the integral also holds a sort of sense.