# Find Flux Vector

• Sep 22nd 2009, 05:42 AM
Jimmy_W
Find Flux Vector
For $\displaystyle \Phi (x,y,z) = x^{xy} \sin(x+z) + y \ln (z)$

find the flux vector $\displaystyle \vec{J} = \triangledown \Phi$

and prove that curl $\displaystyle \vec{J} = 0$
• Oct 4th 2009, 02:08 AM
Jimmy_W
Is this correct?

$\displaystyle \triangledown \Phi = \frac{\partial \Phi}{\partial x} i + \frac{\partial \Phi}{\partial y} j + \frac{\partial \Phi}{\partial z} k$

where

$\displaystyle \frac{\partial \Phi}{\partial x} = e^{xy} \cos(x+z) + \sin(x+z) y e^{xy}$

$\displaystyle \frac{\partial \Phi}{\partial y} = \sin(x+z) x e^{xy} + \ln(z)$

and

$\displaystyle \frac{\partial \Phi}{\partial z} = e^{xy} \cos(x+z) + \frac{y}{x}$

and

$\displaystyle \triangledown \Phi = (e^{xy} \cos(x+z) + \sin(x+z) y e^{xy}) i + (\sin(x+z) x e^{xy} + \ln(z) ) j + (e^{xy} \cos(x+z) + \frac{y}{x})k$

Is there anything more I need to do with this?
• Oct 4th 2009, 02:26 AM
MarkW
In the k component it should be y/z, not y/x.

Other than that, all you need to do is calculate the curl and it will cancel out nicely.

Do you know how to find the general flux vector in the second part of question 1?