Question on Radiance equation The radiance equation is
$$
L = \frac{d}{dA} \frac{2(\phi)}{dW cos(\theta)} (watt/srm^2)
$$
where $\phi$ is the flux.
I am thinking, should not be the cosine term on the numerator instead of the denominator? Having the cosine in the denominator will make L goes to infinity if $\theta = 90$ , which does not make sense to me. My understanding is that if $\theta = 90$ (i.e. flux direction is perpendicular to the surface normal), then $L$ should equal to zero (and not infinity). 
 A: Generally, the radiative transfer equation is given by
$$
\frac{1}{c}\frac{\partial I_{\nu}(\vec{r},\vec{n},t)}{\partial t} +
\vec{n}\cdot\frac{I_{\nu}(\vec{r},\vec{n},t)}{\partial \vec{r}} =
\rho(\vec{r},t) \kappa_{\nu} (\vec{r},t)\left\{-I_{\nu}(\vec{r},\vec{n},t)+S_{\nu}(\vec{r},\vec{n},t)\right\}.
$$
To monochromatic intensity is defined by noting that $I_{\nu}(\vec{r},\vec{n},t)\cos\!\Theta \,d\nu\,df d\Omega\, dt$ at the position $\vec{r}$ gives the radiative energy
in the frequency interval $\nu,\nu+d\nu$ transported through the surface element $df$ into the solid
angle element $d\Omega$ enclosing the direction $\vec{n}$ during a time increment $dt$,
see the appended figure.

Your definition looks a bit strange, there is probably a time differential $dt$ missing in the denominator and the $2\phi$ should be the energy differential $dE$. However, $\cos\theta$ in your definition of the radiance correctly appears in the denominator. Looking at my figure, if $\Theta = 90^{\circ}$ the energy propagates along the area element $df$ instead of though it, so the radiance is indeed ill defined in this case.
A: The formula correctly reflects the definition of radiance, so it's certainly correct in this sense. Is such a definition useful? I guess so - the value defined in this way gives some idea of "brightness" of the source, and "brightness" does tend to infinity when the angle tends to 90 deg - the same flux comes from a very small "perceived" area.
A: I believe the confusion comes from the fact that we speak of radiant flux per PROJECTED area per unit solid angle. We are not looking really at the radiant flux of a small surface element around P or x (depending on which convention you use) but how much flux goes through a projection of that surface in the direction of incidence. And if you consider 1 unit of projected area, as the angle $\theta$ increases, the area of the surface element corresponding to the projection of $dA^\perp$ on the horizontal plane increases (and goes to infinity in the limit of $\theta$ approaching 90 degrees. Thus what you measure here in way is that amount of "light" that is incident or emerging from this very large area back into a very small "projected" area. Looked that way, you can hopefully better understand why it goes to infinity?
http://scratchapixel.com/old/lessons/3d-basic-lessons/lesson-15-introduction-to-shading-and-radiometry/introduction-to-radiometry-2/
