# Thread: Find Flux Vector

1. ## Find Flux Vector

For $\Phi (x,y,z) = x^{xy} \sin(x+z) + y \ln (z)$

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

and prove that curl $\vec{J} = 0$

2. Is this correct?

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

where

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

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

and

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

and

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

Is there anything more I need to do with this?

3. 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?