Let $X$ and $Y$ be smooth second-countable Hausdorff manifolds of dimensions $n$ and $m$, respectively, and let $F: X \to Y$ be a smooth submersion. For each $x \in X$, let $dF_x: T_xX \to T_{F(x)}Y$ denote the differential and let $dF_x^*: T^*_{F(x)}Y \to T_x^*X$ denote its cotangent pullback. Let $u \in \mathcal{D}'(X; |\Omega_X|)$ be a distributional density on $X$, and assume that the restricted map

\begin{align*} F|_{\operatorname{supp} u}: \operatorname{supp} u \to Y \end{align*}

is proper. For every $\phi \in C_c^\infty(Y)$, let $\chi \in C_c^\infty(X)$ be any function satisfying $\chi = 1$ on an open neighbourhood of the compact set

\begin{align*} \operatorname{supp} u \cap F^{-1}(\operatorname{supp}\phi). \end{align*}

Define

\begin{align*} (F_*u)(\phi) := u\bigl(\chi(\phi \circ F)\bigr). \end{align*}

Then this definition is independent of the choice of $\chi$, and it defines a distributional density $F_*u \in \mathcal{D}'(Y; |\Omega_Y|)$. Moreover, if

\begin{align*} C_F(\operatorname{WF}(u)) := \{(F(x), \eta) \in T^*Y \setminus \{0\} : x \in X,\ \eta \in T^*_{F(x)}Y \setminus \{0\},\ (x, dF_x^*\eta) \in \operatorname{WF}(u)\}, \end{align*}

then

\begin{align*} \operatorname{WF}(F_*u) \subset C_F(\operatorname{WF}(u)). \end{align*}