How Do You Calculate the Laplacian in Polar Coordinates?

[davon806]

Homework Statement

I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached

Homework Equations

The Attempt at a Solution

My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist( If you sum A,B,C and D up). Thanks

 

[kuruman]

I don't think you should be messing with unit vectors and their derivatives. The Laplacian is a scalar operator. It looks like you are doing it in two dimensions. Start with $$\nabla ^2=\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$$
Observe that according to the chain rule
$$\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}$$
and one more time for the second partial derivative. Repeat with y and then add the two results.

This is the "brute force" method and involves tedious algebra. It is the kind of thing that one has to do once in one's life to satisfy oneself that one can do it.