  We introduce $\Psi \mathrm{ec}$, a discretization of Cartan's exterior
calculus of differential forms using wavelets. Our construction consists of
differential $r$-form wavelets with flexible directional localization that
provide tight frames for the spaces $\Omega^r(\mathbb{R}^n)$ of forms in
$\mathbb{R}^2$ and $\mathbb{R}^3$. By construction, the wavelets satisfy the de
Rahm co-chain complex, the Hodge decomposition, and that the $k$-dimensional
integral of an $r$-form is an $(r-k)$-form. They also verify Stokes' theorem
for differential forms, with the most efficient finite dimensional
approximation attained using directionally localized, curvelet- or
ridgelet-like forms. The construction of $\Psi \mathrm{ec}$ builds on the
geometric simplicity of the exterior calculus in the Fourier domain. We
establish this structure by extending existing results on the Fourier transform
of differential forms to a frequency description of the exterior calculus,
including, for example, a Plancherel theorem for forms and a description of the
symbols of all important operators.
