I struggle to understand how integration over direction takes place in this paper (open access). I will give two examples. The first one is this: $$C\int \int δ(1-s\cdot s')L(r,s')\,ds' = 2πCL(r,s)$$ Here C is a constant, s is a non-changing unit vector, s' is the integrated unit vector and L is a function of position r and direction s'. The integration takes place in 3D space.
I suppose that the left term can be seen as an integral over the entire solid angle with $s\cdot s'=cosθ$ in spherical coordinates. Then $$C\int \int δ(1-s\cdot s')L(r,s')\,ds' = C\int_{0}^{2π} dφ \int_{0}^{π} δ(1-cosθ)L(r,s')\,sinθdθ$$
Shouldn't the integral over θ yield zero due to the delta function? According to a previous question it should actually be equal to zero. If not what am I misunderstanding? I suspect that I may take the wrong coordinate system for the integration (in this case a spherical coordinate system with unit vector s along the z direction so that $s\cdot s' = cosθ$). However I fail to see how this would be the problem.
The second example is this:
$$C\int \int δ(1-s\cdot s')δ(1-s'\cdot z)\,ds' = 0$$
In this case z is a unit vector normal to the boundary of the examined system. I have no clue how to handle this product of the two delta functions or why this integral yields zero.
For anyone who might look for those integrals in the cited paper, work out the derivation of equations 3, 4 and 5. An important tip is that $μ'_s = μ_s(1-g)$. All the variables appart from the ones presented here can be considered as constants, if I am not wrong.