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Let $i_{q0} : M\rightarrow M\times N$, $i_{q0}(p) = (p, q0)$ be a mapping between smooth manifolds. I need some hints to show that it is $C^{\infty}$.

I have so far... Let $(U,\phi)$ and $(V,\psi)$ be charts about $p$ and $i_{q0}$, and let $r^{i}$ be the $ith$ coordinate function on Euclidean space. Then we need to show that $\frac{\partial (r^{i}\circ \psi \circ i_{q0} \circ \phi^{-1})}{\partial r^{j}}$ exists and is continuous at $\phi(p)$ and that we can keep taking partial derivatives.

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    The map $i_{q_0}$ of your question is simply the inclusion map $(x^1,\dots,x^n)\to (x^1,\dots,x^n,0,\dots,0)$ in (an appropriate choice of) local coordinates where $M$ is a smooth $n$-manifold and $N$ is a smooth $m$-manifold (where $m$ is the number of zeros in $(x^1,\dots,x^n,0,\dots,0)$). Clearly, this (inclusion) map is smooth.2012-02-16

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