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# De Rham map

The *de Rham map* establishes a relation between continuous differential forms over region $D \subset {\bf R}^n$ which is a realization of a cubical complex $K$ (or cell complex) and discrete differential forms, i.e., cochains, over $K$:
$$R:\Omega ^k (D) \rightarrow C^k(K),k=0,1,...$$
defined via integration, as follows:
$$R (\omega )(c) =\int _c \omega,$$
where $c$ is a chain.

This map commutes with the exterior derivative: $$dR=Rd$$ by Stokes' theorem. In other words it is a chain map.

This map establishes an isomorphism between the de Rham cohomology and cubical cohomology.

The inverse link is given by the *Whitney map*:
$$W:C^k(K) \rightarrow \Omega ^k (D),k=0,1,...$$
defined via "interpolation".