$S\subseteq \mathbb{R}^N, U\subseteq \mathbb{R}^k, \phi\in C^1(U,\mathbb{R}^N) \ \text{ s.t.} \ \phi(U)=S.$
$J_\psi$ is the Jacobian matrix. Let $V\subseteq\mathbb{R}^k$ be open, and let $\psi:V\to S$ a bijection so that $h:=\psi^{-1}\circ\phi\in C^1(U,V)$ and $h^{-1}=\phi^{-1}\circ\psi\in C^1(V,U)$
I'm trying to get to the end statement that $J_\psi(h(x))J_h(x)=J_\phi(x)$ for all $x\in U$.
I'm thinking along the lines of chain rules, but the notation is strange as it is. I think that $J_\phi(x)$ would be the values of $\phi$ with respect to the partial derivatives $x_i$, but it's sending me in circles.
Ideas?