Transport theory basics: can't understand solid angles

Question

I don't understand something in transport theory:

$$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$

This is the number of particles flowing across a differential surface element in the direction $\vec{w}$. I understand that I need something like

$$\text{total number of particles} = \text{densityofparticles} \times \text{volumeconsidered}$$

so the density towards a direction $* \cos\theta * dA$ makes sense to me.. but I don't understand why isn't there a 1/3 term in the equation.. aren't we interested in the area where the solid angle subtends so it should be a pyramid?

Since my comprehension of solid angles isn't perfect either, I need someone to explain this to me please

Answer 1

You have the wrong model, I believe. Solid angle isn't needed here. You want to think about a parallelepiped (box) of material that flows across the patch of surface. So the height of the box is $\cos\theta dw$ (if I guess what your notation means) and its base is $dA$.