It's a practice problem for my calc 3 class.
Given: z=f(x,y)=g(r, theta), where x=7cos(theta), y=7sin(theta) Find: (a) dz/dr; (b) dz/d(theta); (c) d^2(z)/dr/d(theta)
(a) and (b) are easy; I'm having trouble simplifying part (c) according to my solution manual (see the picture below).
Many thanks!!!
This is simply an application of chain rule. $z$ is naturally a function of $x$ and $y$ but $x$ and $y$ are a function of $r$ and $\theta$. For simplicity, let $h = \dfrac{\partial z}{\partial x}$, then you want to compute $\dfrac{\partial h}{\partial\theta}$. Well $h$ is a function of $x$ and $y$, so chain rule gives us
$$\frac{\partial h}{\partial \theta} = \frac{\partial h}{\partial x}\frac{\partial x}{\partial\theta} + \frac{\partial h}{\partial y}\frac{\partial y}{\partial \theta}.$$
Substituting back $\dfrac{\partial z}{\partial x}$ for $h$ gives that
$$\frac{\partial}{\partial\theta}\frac{\partial z}{\partial x} = \frac{\partial^2 z}{\partial x^2}\frac{\partial x}{\partial\theta}+\frac{\partial^2 z}{\partial x\partial y}\frac{\partial y}{\partial\theta}.$$