44 polar coordinates suppose that we substitute polar coordinates begi

Question

44. Polar coordinates Suppose that we substitute polar coordinates \begin{aligned}
&x=r \cos \theta \text { and } y=r \sin \theta \text { in a differentiable function }\\
&w=f(x, y)
\end{aligned} a. Show that and b. Solve the equations in part (a) to express f, and fy in terms of c. Show that \frac{d w}{\partial r}=f_{x} \cos \theta+f_{y} \sin \theta \frac{1}{r} \frac{\partial w}{\partial \theta}=-f_{x} \sin \theta+f_{y} \cos \theta \left(f_{I}\right)^{2}+\left(f_{y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial w}{\partial \theta}\right)^{2} \mathrm{cw} / \mathrm{dr} \text { and } \mathrm{o} \mathrm{w} / \mathrm{d} \theta