Let $F:\Bbb R^m \times \Bbb R^n \rightarrow \Bbb R^n$, $F(x,y)=0$, and let $y=G(x)$.
If we have that $F(x,G(x))=0$, then by the chain rule we can get $D_x f +D_y f \ D_xG$ where $D_x$ Denotes the Jacobi matrix (with respect to the subscript).
Could someone explain this to me? I guess I am getting confused because, to me, $F(x, G(x))$ is not $F$ composed with $G$, so I don't quite see how to apply the chain rule. Or do we assume that $G$ is a $2\times 1$ vector $\left(x, G(x)\right)$?
Thank you.
If you want to see a direct composition, define a function $\Gamma \colon \def\R{\mathbf R}\R^m \to \R^{n} \times \R^{m}$ by $$ \Gamma(x) := \bigl(x, G(x)\bigr) $$ That is, we denote the "$2\times 1$-vector" $(x, G(x))$ (in fact, its an $n+m$-vector) by $\Gamma(x)$. Then applying the chain rule to $F\circ \Gamma(x)$, gives $$ D F\bigl(\Gamma(x)\bigr) D\Gamma(x) = \bigl(D_xF, D_y F) \cdot \binom{{\rm Id}}{D_x G} = D_x F + D_y F\cdot D_x G $$