Quick question on partial derivatives.

Question

Can one have the following: $$\frac{\partial F}{\partial v}=\frac{\partial F}{\partial x}\left(\frac{\partial x}{\partial v}\right)?$$

Answer 1

In general if $F=F(x,y)$ but after we have $x=x(v,w)$ and $y=y(v,w)$ then $$\frac{\partial F}{\partial v}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial v},$$ and $$\frac{\partial F}{\partial w}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial w}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial w},$$ are available.