Ψec: A local spectral exterior calculus

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