# Difficulty with the Laplacian

1. Jun 23, 2015

### erb12c

1. The problem statement, all variables and given/known data
Given: |r|=√(x^2+y^2+z^2) r=xi+yj+zk

(i)Find the partial derivative with respect to x of |r|.
(ii) Find the Laplacian of |r|.

2. Relevant equations

3. The attempt at a solution
For (i) I got x/|r|
but then for (ii) I got 2/r which I don't think is correct

2. Jun 23, 2015

### Zondrina

If $| \vec r(x, y, z) | = \sqrt{x^2 + y^2 + z^2}$, then:

$$| \vec r(x, y, z) |_x = \frac{\partial}{\partial x} (x^2 + y^2 + z^2)^{\frac{1}{2}} = \frac{1}{2} (x^2 + y^2 + z^2)^{- \frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^2 + y^2 + z^2)$$

What is the definition of the Laplacian?

3. Jun 23, 2015

### RUber

Part i) is the warm up for part ii).
What did you do to get x/|r|?

If $\frac{\partial}{\partial x } |r| = \frac{x}{|r|}$, then what is $\frac{\partial}{\partial x } \frac{x}{|r|}$?

I think 2/|r| is right.

4. Jun 23, 2015

### Zondrina

If you clean up the computation in the second post:

$$\frac{1}{2} (x^2 + y^2 + z^2)^{- \frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^2 + y^2 + z^2) = \frac{x}{\sqrt{x^2 + y^2 + z^2}} = \frac{x}{| \vec r |}$$

It is, but it would be nice if the OP showed some of the work.