I am trying to work out a formula and am pretty stuck. I have the start and I know where I want to get... but I cannot work out a couple of intermediate steps. What I have is the following integral:

\int_{\Omega_i}f\left(\vec x, \vec\omega_i, \vec\omega\right)\frac{d^2\Phi_i}{d\Omega_i\,dA}\l  eft(\vec x, \vec\omega_i\right)\,d\Omega_i

Here, \frac{d\Phi_i}{d\Omega_i\,dA} is the flux incident on a surface per unit solid angle, per unit area. The flux varies with position \vec x on the area A of the surface and direction \vec\omega_i on the hemisphere \Omega_i of directions above it. The flux is weighted by the BRDF function f which can also vary with position \vec x and direction \omega_i.

What I would like to do is move the derivative with respect to area A outside the integral, essentially performing differentiation under the integral sign backwards:

\frac{d}{dA}\int_{\Omega_i}f\left(\vec x, \vec\omega_i, \vec\omega\right)\frac{d\Phi_i}{d\Omega_i}\left(\v  ec\omega_i\right)\,d\Omega_i

The things I cannot figure out are:

a) Can I do this at all? The problem is that I have an integral over a two-dimensional space of directions... none of the "easy" explanations out there seem to cover this case.

b) If I can do it, what conditions do the functions have to fulfill? Obviously, the integral should exist in the first place or I would not even be embarking on the journey.

I am thinking that I will have to limit the BRDF to f\left(\vec\omega_i, \vec\omega\right) so that I can write the entire integrand as one large derivative with respect to A, then move the \frac{d}{dA} outside the integral. Am I on the right track here?