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:
Here, is the flux incident on a surface per unit solid angle, per unit area. The flux varies with position on the area of the surface and direction on the hemisphere of directions above it. The flux is weighted by the BRDF function which can also vary with position and direction .
What I would like to do is move the derivative with respect to area outside the integral, essentially performing differentiation under the integral sign backwards:
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 so that I can write the entire integrand as one large derivative with respect to , then move the outside the integral. Am I on the right track here?