Let $M$ be a smooth manifold of dimension $n$, let $u \in \mathcal{D}'(M)$ be a distribution, and let $V \subset M$ be an open subset. Let $\iota: V \to M$ denote the open inclusion. Define the restricted distribution $u|_V \in \mathcal{D}'(V)$ by
for every $\phi \in C_c^\infty(V)$, where $\widetilde{\phi} \in C_c^\infty(M)$ is the extension of $\phi$ by $0$ outside $V$. For each $x \in V$, identify $T_x^*V$ with $T_x^*M$ by the inverse of the fiberwise linear isomorphism