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Let $M$ be a compact smooth manifold subset of some $\Bbb R^n$ and $f:M\times M\to\Bbb R$ a $C^\infty$ function.

I need to check wether $\bar f(p)=\int_M f(p,q)dq$ is smooth, but I have no clue where to start, or to come up with a contradiction. Thanks.

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Let $\phi: p_0 \in U \subset M \rightarrow \mathbb{R}^k$ a chart.

Then $g: (x,q) \in \mathbb{R}^k \times M \longmapsto f(\phi^{-1}(x),q)$ is smooth.

Thus (with standard theorems) $x \longmapsto \overline{g}(x)=\int_M{g(x,q)\,dq}$ is smooth.

So $\overline{f}(p)=\overline{g}(\phi(p))$, which ends the proof.

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