# Homework Help: Laplacian(Nabla x v)

1. May 17, 2008

### cscott

1. The problem statement, all variables and given/known data

In cylindrical:

Get $$\frac{1}{\rho} \frac{d}{dp} \left( \rho\frac{d^2 v}{d\rho^2}\right) - \frac{1}{\rho^2}\frac{dv}{dp} = 0$$

Out of $$\nabla^2\left(\nabla \times \vec{v}\right) = 0$$

where $$\vec{v} = v(\rho)\hat{z}$$

3. The attempt at a solution

I get $$\nabla \times \vec{v} = -\frac{dv}{dp}\hat{\theta}$$

But how do I apply the Laplacian? I can't even get that out of Maple.

Last edited: May 17, 2008
2. May 17, 2008

### cscott

Last edited: May 17, 2008
3. May 17, 2008

### lzkelley

Its a pain, but doable. You only have a theta direction, as a function of radius - so it'll simplify. Plug and chug.
http://en.wikipedia.org/wiki/Laplacian

4. May 19, 2008

### cscott

So I should be applying $$\nable^2 f = \frac{1}{\rho} \frac{d}{d\rho}\left(\rho\frac{df}{d\rho}\right) + \frac{1}{\rho^2}\frac{d^2f}{d\theta^2} + \frac{d^2f}{dz^2}$$ (all should be partials)

to simply $$f = -\frac{dv}{d\rho}\hat{\theta}$$?

and use the derivatives of the unit vectors here http://mathworld.wolfram.com/CylindricalCoordinates.html?

Last edited: May 19, 2008
5. May 19, 2008

### cscott

I got $$\left[ \frac{1}{\rho^2}\frac{dv}{dp} - \frac{1}{\rho} \frac{d}{dp} \left( \rho\frac{d^2 v}{d\rho^2}\right) \right] \hat{\theta} = 0$$

This making sense anyone? Thanks

6. May 19, 2008

### lzkelley

Thats exactly what you were looking for right?

7. May 19, 2008

### cscott

Yeah. I guess applying the Laplacian to a vector messed me up a bit.

I also wonder why they don't give the answer in vector form...