Find Flux Vector

**Jimmy_W** · September 22nd 2009, 05:42 AM

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 $\vec{J} = 0$

**Jimmy_W** · October 4th 2009, 02:08 AM

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?

**MarkW** · October 4th 2009, 02:26 AM

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?

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