I am supposed to proof: $$ \frac{1}{r}\frac{d^2}{dr^2} \left(\varphi \vec{r}\right) = \Delta \varphi $$ I was able to solve another similar problem:
$$\frac{1}{r^2}\frac{d}{dr^2} \left(r^2 \frac{d\varphi}{dr} \right) = \Delta \varphi $$" />
but i do have a hard time with the first one. It seems to me that the left side is supposed to be a vector... or not? Could I rewrite it to this form? $$ \frac{1}{r}\frac{d^2}{dr^2} \left(\varphi x, \varphi y, \varphi z\right) = \left(\frac{\partial^2}{\partial x^2}(\varphi) + \frac{\partial^2}{\partial y^2}(\varphi) + \frac{\partial^2}{\partial z^2}(\varphi) \right) $$ But in that case.. isn't left side a vector and the right side a scalar?
Thank you for all your thoughts and insights on where I went wrong...
Edit
Thanks, the other way seems to be right: $$\frac{1}{r}\frac{d^2}{dr^2}\left(\varphi r\right) = \frac{1}{r}\frac{d}{dr}\left(r\frac{d\varphi}{dr} + \varphi\frac{dr}{dr}\right) = \frac{1}{r} \left(\frac{d^2\varphi}{dr^2}r + \frac{dr}{dr} \frac{d\varphi}{dr} + \frac{d\varphi}{dr}\right) = \frac{d^2\varphi}{dr^2} + \frac{1}{r}\frac{d\varphi}{dr} + \frac{1}{r}\frac{d\varphi}{dr}= \frac{d^2\varphi}{dr^2} + \frac{2}{r}\frac{d\varphi}{dr} $$