How can I find the integral $\int_{S} 2~dydz + 1~dzdx - (3x)~dxdy$, where $S$ is the following surface?
$$x^2 + 2y^2 + 3z^2 + xyze^{(x+y+z)\sin(x^2 -y+z)} = 1$$
$$0\leq x,y,z$$
I think that I must use Gauss's law, but I am not sure if the surface is closed. I thought if it isn't I shall close it with a sphere with radius $\epsilon$.
Then I simply calculate the flux through the ball and say:
$$\int_{S}F\cdot\vec{n} = \int_{Ball+S}F\cdot\vec{n} - \int_{Ball}F\cdot\vec{n} = 0 \:(\text{Gauss's law}) - \int_{Ball}F\cdot\vec{n}$$
If I can't use Gauss's law, what should I do?
Your surface is not closed. Its traces on the $OXY$, $OXZ$ and $OYZ$ planes are quarters of ellipses. You can make it closed by adding the coordinate planes. Your total flux through the closed surface is zero by Gauss' Theorem, so the flux through your surface is the opposite of the flux through the elliptical pieces of coordinate planes that you added. The latter is easily found as double integrals.