Lambert's cosine law

Remarks

Proof: seen from a direction at angle $\theta$, the element has projected area $dA \cos\theta$, which gives clause 1. For clause 2, integrate over the hemisphere with $d\Omega = \sin\theta \, d\theta \, d\phi$: $$M = \int_0^{2\pi} \int_0^{\pi/2} L \cos\theta \sin\theta \, d\theta \, d\phi = 2\pi L \cdot \tfrac12 = \pi L.$$ The factor is $\pi$, not $2\pi$, because of the cosine.

Example: a black body is such a surface, so its exitance is $\pi$ times its radiance, which turns Planck's radiance into the Stefan–Boltzmann exitance. Matte paper and fresh snow are nearly such surfaces, while a mirror or a glossy surface is not, and for them the cosine law fails.

Going deeper

For a student

Surfaces that obey the law are called Lambertian, or ideal diffuse. The law is the assumption behind view factors and radiosity methods, both in heat transfer and in computer graphics.