# Demonstration of constant radiance for Lambertian Surfaces

I'm approaching to radiometry and I'm struggling with one of the properties of Lambertian surfaces. We know that the radiance is the radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit project area: $$L_{e,\Omega}=\frac{\partial^2 \phi_{e}}{\partial^{\Omega}\partial({Acos\theta})}$$ where $$\Omega$$ is the solid angle, $$\phi_{e}$$ is the radiant power and $$Acos\theta$$ is the projected area.

Since the radiant intensity $$I_{e,\Omega}$$ is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, we can simplify the equation as $$L_{e,\Omega}=\frac{\partial{I_{e,\Omega}}}{\partial(Acos\theta)}$$ Now, I would like to introduce the hypotesis of Lambertian surface. We know that for a Lambertian surface the radiant intensity is proportional to the cosine of the angle between the direction of the viewer and the normal to the surface. If $$I_0$$ is the incident radiant intensity, the Lambert's cosine law states that radiant intensity at a certain angle $$\theta$$ is $$I=I_0cos\theta$$ The problem arises now. How can I mix up these information to demonstrate that the radiance is indipendent from the angle of view? If I just plug in the Lambert's cosine law, I get $$L_{e,\Omega}=\frac{\partial({I_0cos\theta})}{\partial({Acos\theta})}$$ How can I solve this derivative? Thank you in advance!

We have the function $$f(\theta)=I_0\cos\theta$$ for which we wish to find $$\partial_{A\cos\theta}f(\theta)$$. First, rewrite $$A\cos\theta$$ as $$u$$, and then we will express $$f$$ in terms of $$u$$:
\begin{align*}u&=A\cos\theta\\\implies\theta&=\cos^{-1}\frac{u}{A}\\\implies f(u)&=I_0\cos\left(\cos^{-1}\frac{u}{A}\right)\\&=\frac{I_0}{A}u\\\implies\partial_uf(u)&=\boxed{\frac{I_0}{A}}\end{align*} which is indeed independent of $$\theta$$.