A Characterization of Whitney forms

We give a characterization of Whitney forms on an $n$-simplex $\sigma$ and prove that for every real valued simplicial $k$-cochain $c$ on $\sigma$, the form $Wc$ is the unique differential $k$-form $\phi$ on $\sigma$ with affine coefficients that pulls back to a constant form of degree $k$ on every $k$-face $\tau$ of $\sigma$ and satisfies ${\int }_{\tau }\phi =<c,\tau >$.