In a physics course we could define the scattering cross section written $\frac{dN}{d\Omega}$ counting the number of particles going through a certain area of the sphere over a certain amount of time. Is it possible to give mathematical definition of the derivative with respect to solid angle?
Let $f: S^2 \to \mathbb{R}$ be some sort of real-valued function on the sphere, we could try to define the "derivative" at a point by:
$$ f(x_0) = \lim_{|\epsilon| \to 0} \frac{1}{\pi \epsilon^2} \int_{B_\epsilon(x_0) \cap S^2} f(x) \,d\Omega $$
Here $\Omega = \frac{A}{r^2}$ could be measured in steradians. We could try to recover the value at a point using the fundamental theorem of calculus.
This is just like differentiating under the integral sign with the same kinds of mistakes. Does this limit necessarily exist? I could imagine a very chaotic function real-valued function $f$ defined on $S^2$ and we could integrate over a sequence of regions $\Omega_n \to \{ x_0\}$ and yet the limit doesn't exist.
Lots of discussions on Math.SE of sphere integrals where the formulas are well-defined. Surface Integral over a sphere