TITLE: How to define a differential form on a fractal?
QUESTION [30 upvotes]: It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one can write (and solve) the heat equation on the fractal.
The question is, can one run a fluid flow through a fractal set? In other words, is there
a proper way to write the Navier-Stokes equations on a fractal? In order to do this, it seems that  we need a "correct" notion of divergence at least.
More generally, is there a "correct" way to define a differential form on a fractal?

REPLY [6 votes]: There is an example of Hassler Whitney that is a lot of fun.

Whitney, Hassler, "A function not constant on a connected set of critical points." 
  Duke Math. J. 1 (1935), no. 4, 514–517. 

He constructs a curve $C$ (this is your fractal) in the plane...



and a $C^1$ function $f : \mathbb R^2 \to \mathbb R$ such that $f$ is not constant
on the curve, yet the gradient of $f$ vanishes at every point of the curve.
(Time-travelers may attend my lecture at the undergraduate math club...)



The relevance here: if you want
$$
\int_C df = f(b) - f(a) ,
$$
then that "differential" $df$ must involve more than just the partial derivatives of $f$ at the points of $C$...