I am facing some trouble with understanding the solution to a line integral. Follows the integral:
$$ \oint{d \Omega}= \int_0^{2\pi}{dl} \int_0^\pi \sin \theta ~ d\theta = 4\pi$$
I can easily understand how the 2 integrals multiplied to each other equal $ 4 \pi$ but I cannot understand why the line integral is equal to the 2 other integrals multiplied.
For some context this came up in an astrophysics class, but that is all the context I have, professor gave this to assess where we are in maths, and I'm trying to figure this out but I can't.
Thank you.
This is not a path integral, but rather a surface integral. The circle on the integral denotes that it is a closed surface, not a closed path. The notation $d\Omega$ refers to the area element of the unit sphere, or equivalently an element of solid angle, so this integral gives the surface area of the unit sphere, which is $4\pi.$
The breakdown into two integrals comes from the usual spherical coordinates on the sphere, where $\theta$ is the polar angle and $l$ (odd choice of notation) is the azimuthal angle.