Given the vector field $X(x,y,z)=(1,y,x)$, I am asked to compute the flow of this vector field through the northern hemisphere $(z\geq 0)$ of the elipsoid with equation $$\frac{x^2}{4}+\frac{y^2}{4}+z^2=1$$ with the orientation determined by the normal vector who's third component is positive over this part of the surface.
I tried to apply the Divergence Theorem on this surface but since the divergence of this vector field is not zero $$\text{div}(X)=1$$ my integral will be difficult to compute since $$\int_{R}\text{div}(X)\cdot dV=\int_{R}dV=Vol(R)=\int_{\partial R}\langle F,N\rangle dS=\int_{S}\langle F,N\rangle dS+\int_{D}\langle F,(0,0,-1)\rangle dS$$ So, $$\int_{S}\langle F,N\rangle dS=Vol(R)-\int_{D}\langle F,(0,0,-1)\rangle dS$$ where $R$ is the region enclosed $R:=\{(x,y,z)\in\mathbb{R}^3|\frac{x^2}{4}+\frac{y^2}{4}+z^2\leq1,z\geq0\}$, $S$ corresponds to our surface $S:=\{(x,y,z)\in\mathbb{R^3}|\frac{x^2}{4}+\frac{y^2}{4}+z^2=1,z\geq0\}$ and $D$ corresponds to the disc or cover closes our surface $D:=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2=4\}$.
I think that I am complicating myself when it comes to computing this flow. I think there is also a way to compute this using parametrizations (brute-force). But I get some weird integrals and I shold get a simple value for this calculation, like $4\pi$ or something like it. Any help will be much appreciated.
Jacobian transformation can be used map ellipsoid to a sphere. $$let\;x\;=2u$$ $$let\;y\;=2v$$ $$let\;z\;=w$$ Where u,v,w is a different coordinate system. $$\frac{\partial{(x,y,z)}}{\partial{(u,v,w)}}=Det\begin{vmatrix} X_{u} & X_{v} & X_{w}\\ Y_{u} & Y_{v} & Y_{w}\\ Z_{u} & Z_{v} & Z_{w} \end{vmatrix}$$ $$=Det\begin{vmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{vmatrix}=4$$ using gauss's divergence theorem $${\subset\!\supset}\llap{\iint}_S\left(\vec{F}\cdot \hat{n}\right)da=\iiint_V(\nabla.F)dv$$ $$V=\iiint\;Det\begin{vmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{vmatrix}dudvdw$$ $$V=\iiint4r^2sin{\phi}d{\theta}d{\phi}dr$$ For $$0\leq\phi\leq\pi$$ $$0\leq\theta\leq2\pi$$ $$0\leq\ r\leq 1$$ Since $$u^2+v^2+w^2=1,\;r=1$$ $$u=rsin{\phi}cos{\theta}$$ $$v=rsin{\phi}sin{\theta}$$ $$w=rcos{\phi}$$ $$since\;\nabla.F=1$$ We just have to compute $$\iiint_Vdv=\frac{16\pi}{3}$$ For half sphere volume is divided by 2.