Find the integral $\int_{S} 2~dydz + dzdx + -3dxdy$ where $S$ is the surface $x^2 + y^2 + z^2 +xyz = 1$ , $0 \leq x,y,z$.
choose the direction of the normal as you like.
i am having hard time parametrizing the equation . how can i parametrize it to calculate the flux .
Although you can parametrize the surface (for example, $z=\frac{1}{2} \left(\sqrt{x^2 y^2-4 x^2-4 y^2+4}-x y\right)$), I believe, it's not the way the problem is intended to be solved.
First, you want to show that the surface projects onto one of the basis planes uniquely (without folds). For example, if you consider the surface equation as an equation on $z$, there is only one positive root.
Now we can think of our integral. We divide the surface into small patches. Expression $dx\,dy$ is an area of a projection of a small patch on plane $xy$. Since unique projection of each patch, sum of projection areas of all patches is the area of the projection $A_{xy}$:
$$ \int_S\left(2dy\,dz+dz\,dx-3dx\,dy\right)=2\int_S dy\,dz+\int_S dz\,dx-3\int_Sdx\,dy = 2A_{yz}+A_{xz}-3A_{xy}. $$
But due to symmetry, areas of all three projections are the same, and the asnwer is $0$.
P.S. If there were other numbers, we needed to calculate the area of the projections, which is easy, since it's a quarter-circle.