To go from the first line to the second, we make use of spherical coordinates. As in the picture, you can think of theta as tracing from the north pole to the base, and you can think of phi as going in a circle around the base. Using some trig, you can see that at an angle of theta, sin(theta)dthetadphi gives a small patch of area on the sphere. So, this replaces dw. Then, we must trace a quarter circle with theta and a full circle with phi to hit every point on the hemisphere, which gives us the integration limits.
StephenYangjz
I find all these definitions of quantities pretty cohesive. The irradiance of light is per unit solid angle, (also per beam of light), so it's independent of the area. The flux, however, integrates through the whole area. The iradiance here of uniform hemispherical light, on the other hand, integrates through all the angles and represents the total energy.
To go from the first line to the second, we make use of spherical coordinates. As in the picture, you can think of theta as tracing from the north pole to the base, and you can think of phi as going in a circle around the base. Using some trig, you can see that at an angle of theta, sin(theta)dthetadphi gives a small patch of area on the sphere. So, this replaces dw. Then, we must trace a quarter circle with theta and a full circle with phi to hit every point on the hemisphere, which gives us the integration limits.
I find all these definitions of quantities pretty cohesive. The irradiance of light is per unit solid angle, (also per beam of light), so it's independent of the area. The flux, however, integrates through the whole area. The iradiance here of uniform hemispherical light, on the other hand, integrates through all the angles and represents the total energy.