could someone, please, help me to understand how to interpret this formula $L=\frac{d^2\Phi}{dA dw}$ ($\Phi$ - radiant flux, $A$ - unit area, $w$ unit solid angle) as a radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area.
I have a basic knowledge of multivariable calculus.
I know that $\Phi$ is a radiant flux which is "radiant energy emitted, reflected, transmitted, or received per unit time". Starting from this we're taking a partial derivative first let's say with respect to $A$ which gives us some function that indicates how our radiant flux $\Phi$ will change with change in area. Then we take another derivative from our result and now I'm trying to interpret this: $\frac{\frac{d\Phi}{dA}}{dw}$. I see it as a change in our "change with area" function with respect to $w$ and visualising it like we're fixating $A$ to some constant value $a$ in this function $S(A,w)=\frac{d\Phi}{dA}$ (let the name be $S$) and then we're evaluating how our $S$ changes with respect to $w$ at each $(a,w_i)$ point.
So, the resulting function gives us, by my interpretation, how change in $\Phi$ with respect to $A$ changes with respect to $w$, and as you can tell, it's not radiant flux per unit solid angle per unit projected area.
What am I missing in my reasoning ?
My interpretation of differentials is what have confused me, I think. It helped me to simplify the radiance formula $L(A,w)=\frac{d^2\Phi}{dA dw}$ by removing differentials, so that the amount of energy that light source gives per particular angle is constant, so it spreads it equally in every direction. And also I've treated it like any piece of the source area gives the same amount of energy (energy / area = same constant at every source point). This way dividing amount of energy by particular angle we get amount of energy in particular direciton, and then if we divide by the area, we get amount of energy per particular direction per particular point of the source ($L(A,w)=\frac{\frac{\Delta\Phi}{\Delta w}}{\Delta A}$). Which is exactly what wikipedia tells.
And to go backwards to differentials (our initial radiance formula) we consider non-equal spread of energy per direction ($dw$) and non-equal spread of energy per source area piece ($dA$).
(I hope I got this right, still getting confused when see differentials)