I was reading a paper where the property of light known as Illuminance, for a specific setup (as in the figure)
is given with the following formula:
The description below the formula says:
The illuminance $E$ on the road surface is given by the formula... where $d\Phi$ is the luminous flux (lm), $dA$ is the area of the road surface (m2), $d\omega$ is the solid angle (sr), $I(\alpha, \beta)$ is the luminous intensity (cd), $\alpha$ and $\beta$ is the horizontal and vertical angle (in relation to the headlamp axis), respectively, $r$ is the distance between the light source and the small area $dA$, and $\theta$ is the angle between the road surface normal and the incident direction
Now I want to abstract for a moment and look at the formula purely mathematically, ignoring the light terminology.
Can someone explain me the presence of $d$ in the formula? What does it represent mathematically? Something related to the derivatives? Why?
Can you please provide a "simple English" explanation of the formula?
As usual in physics, here the symbol $d$ is used to indicate a little (infinitesimal) quantity.
The quotient $\dfrac{d \Phi}{dA}$ indicate the luminous flux between the area element $dA$ per unit area. This is a function of the solid angle $d \omega$, so that the chain rule is used in the second step of the equality.
For the last step it's used the fact that, as illustrated in the figure: $$ dA=\frac{dA'}{\cos \theta}=\frac{r^2d \omega}{\cos \theta}. $$
as defined in the referenced wiki page, the illuminance is the total luminous flux incident on a surface, per unit area, so, for an infinitesimal area $dA$ it is $E= \frac{d \Phi}{dA}$ where $d\Phi$ is the flux.
From the figure we see that $d \omega$ is the solid angle that subtends the area $dA'$ orthogonal the the radius $r$ so that $dA'=r^2d \omega$ and, since this area in inclined by $\theta$ with respect to the area $A$, we have $dA=\dfrac{dA'}{\cos \theta}$.
Finally $I(\alpha,\beta)$ is the luminous intensity, i.e. , by definition, the luminous flux emitted by a light source in a particular direction per unit solid angle, that is: $$ I(\alpha,\beta)=\frac{d\Phi}{d \omega} $$