Understanding the BRDF Equation | pressku.com

I americium studying nan BRDF equation. From Wikipedia, nan equation is arsenic follows:

I person 1 main question:

• Where does nan $\cos{\theta_i}$ successful nan denominator travel from?
  The denominator implies that $dE_i(\omega_i) = L_i(\omega_i)\cos{\theta_i}d\omega_i$. Now, nan matter beneath nan equation states that $L$ measures nan powerfulness received connected a projected area. But this should mean $L_i$ itself includes a cosine facet already; then, why do we request a $\cos{\theta_i}$ erstwhile equating $dE_i(\omega_i)$ to $L_i(\omega_i)$? Isn't nan cosine facet already included from $L_i(\omega_i)$?

I americium besides confused astir Wikipedia's reasoning for why $f_r$ is defined successful position of 2 differentials alternatively than nan undifferentiated quantities:

What does "irradiating ray different than $d E_i(\omega_i)$, which are of nary liking for $f_r(\omega_i, \omega_r)$, mightiness illuminate nan surface" mean? How does nan magnitude of ray considered alteration erstwhile utilizing $d E_i(\omega_i)$ successful spot of $E_i(\omega_i)$? And really precisely would this "unintentionally affect" $L_r$?