Can anyone help me proofing this? This is not homework.

[yungman]

This is part of a bigger derivation of Green's function. The book just jump to the ending and I cannot understand how to do this. I need someone to give me guidence how to proof this:

$$\frac{x}{r} \frac{\partial u}{\partial x} + \frac{y}{r} \frac{\partial u}{\partial y} +\frac{z}{r} \frac{\partial u}{\partial z} \;=\; \frac{\partial u}{\partial r}$$

Where

$$r=\sqrt{x^2+y^2+z^2}$$

This is not a homework problem.

Thanks

 

[cristo]

Use the chain rule:

$$\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\cdots$$

 

[yungman]

Use the chain rule:

$$\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\cdots$$

I thought of that, but what is $$\frac{\partial x}{\partial r}$$

I know

$$\frac{\partial r}{\partial x}=\frac{x}{r}$$

That's exactly where I got stuck!