The Chain Rule - finding differentials

Question

I have been given:

• $V=f(x,y,z)$, with $x= r\cos\theta$, $y= r\sin\theta$ , and $z=t$.
• And asked, find $dV/dr$, $dV/d\theta$ and $dV/dt$

Would $$ \frac{dV}{dr} = \frac{dV}{dx} * \frac{dx}{dr} + \frac{dV}{dy} * \frac{dy}{dr} $$ ? If so, what is $dV/dx$ if I have just been given $V=f(x,y,z)$?

I've obtained:

$$V_{r}=V_{x}\cos(\theta )+V_{y}\sin(\theta )$$

I am unsure how to display my final answer.

Answer 1

First: using the same name for different things is a bad habit that causes confusion. Use different names for different things. In your case: $$V(r,\theta,t) = f(r\cos\theta,r\sin\theta,t).$$ Then: $$ \frac{\partial V}{\partial r} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial r} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial r} = \cdots $$