Radiance is sort of an abstract quantity, but I think it's not so bad if you think about it in terms of it's dimensions. When rendering, we like to think of light in terms of geometrical optics. So, instead of light being waves with a continuous spread of energy, they are discrete rays that shoot straight from the surface you're rendering to the pixel element on the screen. This takes some doing, however, because in reality, light is
a continuous wave (classically-speaking -- no intention of modeling things at the quantum level here).
So how do you turn a continuous quantity like an EM wave into a discrete ray? By analogy, consider mass. As a human, you have a certain mass. However, that mass is not distributed evenly in your body. Some tissue is more dense than others. For instance, muscle is more dense than bone. Lets say you knew about the mass density function for your body. That is, if someone gives you a coordinate (x,y,z) that is inside your body, you can plug it into the function and the result will be the mass density at that coordinate. How would you calculate the total mass of your body with this function? Well, you would split up the volume into a bunch of tiny cubes, and you sample the density function in the center (say) of those cubes, and then multiply that density by the volume of the cube to get the mass of that cube, then add up the masses of all the tiny cubes. The tinier the cubes, the more of them you'll have to use, but this will make your mass calculation more accurate. Where integral calculus comes into play is that it tells you the mass you get in the limiting case where the cubes are infinitely tiny and there are infinitely many of them. In my opinion, it's easier to reason about it as "a zillion tiny small cubes" and just remember that the only difference with integral calculus is that you get an exact answer rather than an approximation.
So consider a surface that you're rendering. It is reflecting a certain amount of light that it has received from a light source. We want to think of the light in terms of energy, unlike the mass example. The surface as a whole is reflecting a certain amount of energy every second, which we call the energy flux (measured in Watts, also known as Joules/sec). However, we don't really care what the entire surface is doing. We just want the energy density along a specific ray. So, let's break the surface down into tiny little area elements (squares) and figure out how much flux is coming from each tiny area element. We only care about the area element that is under our pixel. That gives us a flux density per unit area, which is called Irradiance (or Exitance, depending on the situation). So now we know the energy flux density being emitted from the area under our pixel. But wait! Not all of that energy is moving toward
our pixel. That little surface element is emitting energy in all directions. We only want to know how much energy is moving in the specific direction of our pixel. So, we need to further break down that Irradiance according to direction, to find out how much of that Irradiance is being emitted along each direction (a.k.a. infinitesimal solid angle). This gives us an energy density with respect to time, area, and solid angle, known as Radiance.
Edited by CDProp, 29 October 2014 - 12:59 PM.