Flux of a unit sphere cap so I am stuck with this problem, english is not my first language but I believe it is called flux, feel free to rename the question otherwise;
calculate the flow of the field
$F = \langle x, y, z \rangle$ outwards through the unit sphere $x^2+y^2+z^2=1$ above $z=\dfrac{1}{\sqrt2}$
[http://prntscr.com/b9rs7k] (from mathematica)..
so some people say that you can use the divergence theorem for similar problems to make it real easy, some however say that in some cases it wont work and I cannot tell if this is one of those cases since I am too wet behind the ears =).  
I am unsure of what to do with the z limit here, which I guess translates to what limits to use for $\phi$. I try to think of $\phi$ as the angle available on the z plane it makes no sense to me at the moment..  is it simply $0\le\phi\le\pi$ 
is this even remotely right? (problem being I cant control the answer)
$\begin{eqnarray}
r_\theta\times r_\phi&=&\left|\begin{matrix}i& j& k\\ 
-\sin\theta\sin\phi&\cos\theta\sin\phi& 0\\ \cos\theta\cos\phi& \sin\theta\cos\phi& -\sin\phi
\end{matrix}\right|
\\ \ \\ 
&=&(-\cos\theta\sin^2\phi, -\sin\theta\sin^2\phi, -\sin\phi\cos\phi).
\end{eqnarray}$
Turning it away from centre (right? once again I dont understand fully)  
$\begin{eqnarray}
F(r(\theta,\phi))\cdot(r_\theta\times r_\phi)&=& 
(\cos\theta\sin\phi\cos\phi,\sin\theta\sin\phi\cos\phi,\cos^2\phi) \\
& &\cdot(\cos\theta\sin^2\phi, \sin\theta\sin^2\phi, \sin\phi\cos\phi)
\\ &=&
\cos^2\theta\sin^3\phi\cos\phi+\sin^2\theta\sin^3\phi\cos\phi+\sin\phi\cos^3\phi\\
&=&
\sin\phi\cos\phi(\cos^2\theta\sin^2\phi+\sin^2\theta\sin^2\phi+\cos^2\phi)\\
&=&\sin\phi\cos\phi.
\end{eqnarray}$
$\int\!\!\!\!\int_S F\cdot n\, dS = \int_0^{2\pi}\!\!\int_0^{\pi}\sin\phi\cos\phi\,d\theta d\phi$
This rurns out to be 0  ...   hmm cant be right?
...
or should/could I go with a triple integral
(cant seem to get the root fraction limit to fit properly *mathjax issue)
$\int_0^{2\pi}\int_0^{\pi}\int_{0}^{1}[function] \,dr\,d\theta\,d\phi$   
or is it $\int_0^{2\pi}\int_0^{\pi}\int_{\dfrac{1}{\sqrt2}}^{1}$..  hmm where does the z limit come into play?
Ok, so as I guess you can see I am confused =),  will you masters please help me through this problem?,   
some of this is adapted from
Finding the outward flux through a sphere
 A: This isn't a systematic answer (which would spoil your fun), but a collection of general and specific hints.


*

*You are not integrating over a closed surface, so Gauss's theorem does not directly apply. (You could conceivably "cap off" your surface usefully, but there's an easier strategy for your situation.)

*In your situation, the flux density $F \cdot n$ is best computed in Cartesian coordinates, not by parametrizing: What is the Cartesian expression for the outward unit normal vector on the unit sphere? What happens when you dot with $F$? (At first you get a polynomial in $x$, $y$, $z$, but because you're on the unit sphere this polynomial simplifies. There must be a mistake in your calculation, probably a sign error; I didn't check carefully.)

*The resulting integral can in fact be computed using geometry and a theorem of Archimedes.

*If you prefer to calculate the integral by integrating, however, you're perfectly correct that the condition $z \geq 1/\sqrt{2}$ gives a restriction on $\phi$. To find the limits on $\phi$, recall that $z = \cos\phi$ on the unit sphere.
