I have a very long cylindrical mass. Now say I have some cylindrical Gaussian surface $\Omega$ centered around this cylinder, visualized in purple below (ignore everything else in the diagram),
for which I apply Gauss's law of gravity $$\oint_\Omega\mathbf g\,\cdot\,\text{d}\mathbf S = -4\pi GM$$
The gravitational field $\mathbf g$ is obviously constant over and perpendicular to the curved part, while parallel to the two circular areas. My textbook simply says that based on this, we can make $\mathbf g$ a scalar, "drop" the dot product, and pull it out of the integral $$g\oint_\Omega\text{d}S = g\cdot 2\pi d\ell = -4\pi GM$$
It also says that since the two circular faces are parallel to $\mathbf g$, they do not contribute to the remaining integral. Then, we can solve for $g$ and make it a vector again in cylindrical coordinates by attaching a radial unit vector $$g = -\frac{2GM}{d\ell} \iff \mathbf g = -\frac{2GM}{d\ell}\, \hat{\mathbf r}$$
This seems a bit sketchy, so I was wondering, is there a more rigorous way of treating this? I would imagine we have to take the magnitudes of both $\mathbf g$ and each surface differential d$\mathbf S$ with a cosine factor for the dot product???
And what about the end portion where we just make $g$ a vector again, is this like mathematically rigorous?
Much thanks!