If z = f(x,y), where x = rcos([itex]\theta[/itex]) and y = rsin([itex]\theta[/itex]), find [itex]\frac{\partial z}{\partial r}[/itex], [itex]\frac{\partial z}{\partial\theta}[/itex], and [itex]\frac{\partial^2 z}{\partial r\partial\theta}[/itex](adsbygoogle = window.adsbygoogle || []).push({});

Here's what I've done:

(a)

[tex]\frac{\partial z}{\partial r} = \frac{dz}{dx} \frac{\partial x}{\partial r} + \frac{dz}{dy} \frac{\partial y}{\partial r} = \frac{dz}{dx} \cos{\theta} + \frac{dz}{dy} \sin{\theta}[/tex]

(b)

[tex]\frac{\partial z}{\partial\theta} = \frac{dz}{dx} \frac{\partial x}{\partial\theta} + \frac{dz}{dy} \frac{\partial y}{\partial\theta} = -\frac{dz}{dx} r\sin{\theta} + \frac{dz}{dy} r\cos{\theta}[/tex]

My question is, for parts a and b, is this correct or must something also be done with the dz/dx and dz/dy, and for part c, I don't know how to do it. Can someone help please?

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# Homework Help: Partial differentiation

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