I still don't understand why we need sin\theta here. I think the rd\theta and rd\phi are already the width/height of the rectangle dA?
I am also wondering about the sin theta. I guess that r d\theta * r d\phi is not the true area because A is not actually a rectangle, but a curved region of a sphere that approximates a rectangle; but still, where does the sin theta come from?
you can kind of imagine that when theta goes up, per phi covers smaller and smaller areas...so it's definitely not r*dphi
rd\theta is the height of the rectangle dA, r sin theta d\phi is the width of the rectangle. If you look at the circle that lies in the upper horizontal plane, its radius is r sin theta. The width of the rectangle is an arc on that circle that corresponds to an angle of d\phi.
Why is the differential angle the area projected?.. I'm very confused. ;-(
Is this an easy way to sweep across the surface of a sphere? I'm a little bit confused on how exactly differential solid angles relate to light and radiance.
how many variables are there in this case?
Why is "unit" emphasized? Were we working with the unit sphere originally, or is this true for every sphere?
Are we using r, theta, and phi just to be in spherical coordinates? Would we also use these in practice?
I'm still not entirely sure what irradiance based on solid angle gives us, versus the many other quantities we've already seen. Is this to model that each point source of light may have different radiance along different directions?