hi.

Work out the outward flux of the field $\displaystyle \vec F=y\hat i+x\hat j + z\hat k$ across the boundary of the volumetric region bounded bty $\displaystyle z^2=x^2+y^2$ and $\displaystyle z=1$. Verify your result using divergence theorem.

that's the question. i tried solve like this;

$\displaystyle \iiint_V \nabla\cdot F dV$

$\displaystyle \nabla\cdot F=1 $ hence volume part of divergence theorem becomes;

$\displaystyle \iiint_V dV$ where our volumetric region is $\displaystyle z^2=x^2+y^2$

after this i couldn't understand, which coordinate system should i work and what should i write as bounds of integrals.

as i look to surface part of the divergence theorem, i used these substitutions;

$\displaystyle \iint_D \left \vec F \cdot \hat n \right d\sigma$ ,,,, $\displaystyle \hat n= \frac{\nabla f}{|\nabla f|}$ ,,,, $\displaystyle d\sigma = \frac{|\nabla f|}{|\nabla f \cdot \hat p|}dA$

where;

$\displaystyle \vec F = y\hat i + x\hat j + z\hat k$ and $\displaystyle f=x^2+y^2-z^2$

these substitutions didn't give me an answer so i need help on both sides of divergence theory.

edit: btw, sorry for grammer mistakes if there is any. my english grammer knowledge isn't good at all.