Calculus Problem: Showing Partial Derivatives Equal Each Other

[Mathman23]

Hi

Given a function

z = f(x,y), where x = r * cos(\phi) and y = r * sin (\phi)

First I show that

$$\frac{\partial z}{\partial r} = \frac{\partial z}{\partial x} cos(\phi) + \frac{\partial z}{\partial y} sin (\phi)$$

and

$$\frac{\partial z}{\partial \phi} = - \frac{\partial z}{\partial x} r \cdot sin(\phi) + \frac{\partial z}{\partial y} r \cdot sin(\phi)$$

Finally I need to show that

$$(\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2 = (\frac{\partial z}{\partial r})^2 + \frac{1}{r^2} (\frac{\partial z}{\partial \phi}) ^2$$

How do I approach this part of the problem?

Sincerley

Fred

 

[arildno]

Hmm..square both previous equations and add them together, mayhap?

 

[Mathman23]

This is my solution for (b) please look at them at see if I made a mistake:

solving (b)


$$(\frac{\partial z}{\partial x}^2) ^2 + \frac{1}{r^2} (\frac{\partial z}{\partial \phi})^2 = (\frac{\partial z}{\partial x})^2 \cdot cos ^2 (\phi) + (\frac{\partial z}{\partial y})^2 \cdot sin(\phi) + 2 \frac{\partial z}{\partial x} \cdot \frac{\partial z}{\partial y} sin(\phi) \cdot cos(\phi) + (\frac{\partial z}{\partial x}) ^2 \cdot \frac{r^2 \cdot sin^2 (\phi)}{r^2} + (\frac{\partial z}{\partial y})^2 \cdot \frac{r^2 \cdot cos^2 (\phi)}{r^2} - 2 \frac{\partial z}{\partial x} \cdot \frac{\partial z}{\partial y} \cdot \frac{r^2 sin(\phi) \cdot cos(\phi)}{r^2} =$$

$$= (\frac{\partial z}{\partial x})^2 \cdot (cos^2(\phi) + sin^2 (\phi) \cdot (\frac{\partial z}{\partial y})^2 sin^2 (\phi) = (\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2$$


Sincerely Yours
Fred