differentials of smoothly varying family of maps

Question

Let $F:N\times M \to M'$ be a smooth map, which we interpret as a "smooth" family of maps $M \to M'$, parametrized by N, so we have a map $F(y,\cdot):M \to M' \: \forall y \in N$.
Show that the family of the corresponding differentials, that is, the map:
$\tilde{F}:N \times TM \to TM', (y,v) \mapsto d(F(y,\cdot))(v)$ is also smooth.

Unfortunately my ideas didn't work out. Thanks in advance for any help.

Answer 1

As already pointed out by @Thorgott, The differential of a smooth map is again smooth.
Therefore, taking the differential of the map F we get $dF:T(N\times M)\to TM'$. There is a natural, linear diffeomorphism $\Phi:TN \times TM \to T(N \times M)$. So we can instead regard $dF$ as a map $dF: TN \times TM \to TM'$. Since $dF$ is linear, for tangent vectors $(p,v_1) \in TN$ and $(q,v_2)$ in $TM$ we have $dF_{(p,q)}(v_1+v_2) = d(F(\cdot,q))(v_1) + d(F(p,\cdot))(v_2)$. Which one could also understand by taking $d(F(\cdot,q))$ to be the first $m' \times n$ -block of the Jacobian and $d(F(p,\cdot))$ the latter $m' \times m$ - block. But then by restricting the map $dF:TN \times TM \to TM'$ to $N \times \{0\}$ (which is an embedded submanifold of N), we obtain the smoothness of the map $(p,(q,v_2)) \to d(F(p,\cdot))(v_2)$.

Answer 2

Following @Deane's suggestion, I will also try to write out the map explicitly in local coordinates.
Let $y \in N$ and $v \in TM$ be arbitrary. Assume that $v$ is a tangent vector based at some $p \in M$. Then I can choose charts:
$1.\: \phi : U \to \mathbb{R}^m$ around $p$ in M.
$2. \: \psi: V \to \mathbb{R}^n$ around $y$ in N.
$3. \: \sigma: W \to \mathbb{R}^n$ around $F(p)$ in M'.
such that $F_{\tilde{y}}(U) \subset W \: \forall \tilde{y} \in V.$ This is possible, since F is continuous in both variables.
I now construct charts for $N \times TM$ and $TM'$. On $N \times TM$ we can take the chart $(\psi \times \tilde{\phi}, V \times TU)$ around $(y,v)$. Where $\tilde{\phi}:TU \to \phi(U) \times \mathbb{R}^n, \: \sum_{i = 1}^{n}c_i\cdot \frac{\partial}{\partial{x^i}}|_{p} \mapsto (\phi^1(p),...,\phi^m(p)).$ And similarly a chart $(\tilde{\sigma},TW)$. In these coords the map assumes the form:
$\tilde{\sigma} \circ \tilde{F} \circ (\psi^{-1} \times \tilde{\phi}^{-1}): \psi(V) \times \phi(U) \times \mathbb{R}^n \to \sigma(W)\times \mathbb{R}^{m'}$
$(\psi(v),\phi(u),c) \mapsto (\sigma(F(u)), \sum_{j = 1}^{m}\frac{\partial{F}^{1}}{\partial{x}^{j}}\cdot c_j,...,\sum_{j = 1}^{m}\frac{\partial{F}^{m}}{\partial{x}^{j}}\cdot c_j$) which is smooth, since F is smooth. I hope that is all correct.