1. ## Vector Differentiation

Prove that:

$\displaystyle \nabla^2 f(r) = f''(r) + \frac{2}{r}f'(r)$

Where $\displaystyle r^2=x^2+y^2+z^2$

2. Hint:

$\displaystyle \displaystyle \frac{\partial f(r)}{\partial x} = f'(r) \frac{\partial r}{\partial x} = f'(r)\frac{x}{r}$

$\displaystyle \displaystyle \frac{\partial^2 f(r)}{\partial^2 x} = \frac{\partial^2 f'(r)\frac{x}{r}}{\partial x} = f''(r)\frac{x^2}{r^2} + f'\frac{r^2 - x^2}{r^3}$.

3. Originally Posted by snowtea
Hint:

$\displaystyle \displaystyle \frac{\partial f(r)}{\partial x} = f'(r) \frac{\partial r}{\partial x} = f'(r)\frac{x}{r}$

$\displaystyle \displaystyle \frac{\partial^2 f(r)}{\partial^2 x} = \frac{\partial^2 f'(r)\frac{x}{r}}{\partial x} = f''(r)\frac{x^2}{r} + f'\frac{r^2 - x^2}{r^3}$.
Thanks for Hint..
This is correct but I couldn't find the right hand side ..
According to your hint, the first term of the right will be $\displaystyle rf''(r)$..
So, is it correct??

Ok, well, got it..

Thank you very much..

4. Originally Posted by kjchauhan
Thanks for Hint..
This is correct but I couldn't find the right hand side ..
According to your hint, the first term of the right will be $\displaystyle rf''(r)$..
So, is it correct??
I made a typo. It should be $\displaystyle \frac{x^2}{r^2}$. It is now fixed in the previous post.