Let $M$ and $N$ be smooth manifolds and let $F:M\rightarrow N$ be a smooth map. Show that that the function $M\rightarrow M\left(m\times n,\mathbb{R}\right)$ which maps each point $p\in M$ to $dF_{p}$ (the derivative of $F$ at $p$) is smooth. By $ M\left(m\times n,\mathbb{R}\right)$, I mean that space of all $m\times n $ matrices considered as a smooth manifold with its standard smooth structure, where $m$ is the dimension of $M$ and $n$ is the dimension of $N$.

  • $\begingroup$ Maybe you could work in local coordinates? Do you know what the map $p \mapsto dF_p$ looks like in local coordinates? $\endgroup$ – Kenny Wong Apr 23 '17 at 10:05
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    $\begingroup$ it is not at all clear how the map $dF_p$ is defined, since there is no natural isomorphism from the linear maps from $T_p M$ to $T_{f(p)} N$ to a set of matrices. $\endgroup$ – Thomas Apr 23 '17 at 10:22
  • $\begingroup$ @Thomas Nonetheless, if $p \mapsto dF_p$ is smooth w.r.t. one choice of local coordinates, then it is smooth w.r.t. all others. I would presume that the setter of the question had this in mind? Maybe a better question would be to prove that $dF$ is a smooth map between the total spaces of the respective tangent bundles? $\endgroup$ – Kenny Wong Apr 23 '17 at 10:28
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    $\begingroup$ @KennyWong If the map is not well defined it does not make sense to ask whether it is smooth. $\endgroup$ – Thomas Apr 23 '17 at 11:19
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    $\begingroup$ @Eigenfield This is what people do when they describe this using vector bundles. Note, however, that your space $M$ now depends on $p$, and this is the variable with respect to which you want to differentiate. What you need before asking for smoothness is a stringent definition of what you are working with. I'd suggest that you grab any recent textbook on differential geometry and look up how to do this consistently. $\endgroup$ – Thomas Apr 23 '17 at 12:28

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