Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a Jacobian but have been unsuccessful:
$$\int \int \int dr\ d\theta\ d\phi\ r^2 \sin \theta\ f(r,\theta,\phi) = \int \int dr\ d\Omega\ r^2 f(r,\Omega)$$
I believe my confusion is that the solid angle is a surface area in a certain sense, and so I am confused as to how integrating over these surface area values recovers integrating over the full surface area of the sphere. I am also confused because one typically sees a Jacobian determinant employed for such transformations but determinants are defined only for square matrices and the number of variables in these two integrals are not the same.