Exercise 13

Let $z = u^2 - v^2$ and suppose that $$\begin{aligned} u & = e^x\cos(y) \\ v & = e^x\sin(y) \end{aligned}$$

1. Find the values of $u$ and $v$ that correspond to $x=0$ and $y=2\pi/3$.

2. Use the Chain Rule to find the general partial derivatives $$\begin{gathered} \frac{\partial z}{\partial x} \ \mbox{and} \ \frac{\partial z}{\partial y} \end{gathered}$$ and then determine both $\frac{\partial z}{\partial x}\restrict{(0, \frac{2\pi}{3})}{}$ and $\frac{\partial z}{\partial y}\restrict{(0, \frac{2\pi}{3})}{}$