Vector field flow over surface in 3D

[skrat]

Homework Statement

Calculate the flow of $\vec{F}=(y^2,x^2,x^2y^2)$ over surface $S$ defined as $x^2+y^2+z^2=R^2$ for $z \geq 0$ with normal pointed away from the origin.

Homework Equations

The Attempt at a Solution

The easiest was is probably with Gaussian law. I would be really happy if somebody could correct me if I am wrong and answer my question below:

Gaussian law: $\int \int _O\vec{F}d\vec{S}+\int \int _S\vec{F}d\vec{S}=\int \int \int_{Body} \nabla\vec{F}dV$ where I used notation $O$ for the circle.

Now $\nabla\vec{F}= 0$ therefore $\int \int _O\vec{F}d\vec{S}+\int \int _S\vec{F}d\vec{S}=0$ so all that remains is to calculate the floe through surface $O$.

Using polar coordinates $x=r \cos \varphi$ and $y= r \sin \varphi$ for $z=0$. Than $r_{\varphi } \times r_{r}=(0,0,-r)$

$\int \int _O\vec{F}d\vec{S}=-\int_{0}^{2\pi }\int_{0}^{R}r^{5} \cos^2 \varphi \sin^2 \varphi d\varphi dr$

That should be $-\frac{\pi R^6}{96}$.

Question here: I am a bit confused weather I should use the other sign here $r_{\varphi } \times r_{r}=(0,0,-r)$ or is this the right one?

 

[vanhees71]

Gauß's Law is for all the surface normal vectors pointing away from the enclosed volume. So your idea is correct.