Chapter 4.11, Problem 4P

Verify the chain rule formulas $\frac{\partial F}{\partial x}=\frac{\partial F}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial F}{\partial \theta }\frac{\partial \theta }{\partial x}$, and similar formulas for $\frac{\partial F}{\partial y},\frac{\partial F}{\partial r},\frac{\partial F}{\partial \theta }$, using differentials. For example, write $dF=\frac{\partial F}{\partial r}dr+\frac{\partial F}{\partial \theta }d\theta$ and substitute for dr and $d\theta :$ $dr=\frac{\partial r}{\partial x}dx+\frac{\partial r}{\partial y}dy(\text{ and similarly }d\theta )$

Collect coefficients of dx and dy; these are the values of $\partial F/\partial x$ and $\partial F/\partial y$.