Context: I am unhappy with the definition of Irradiance and would like to better formalize it. From Wikipedia:
Irradiance of a surface, denoted $E_e$ is defined as:
$${\displaystyle E_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},}$$
where
- $Φ_e$ is the radiant flux received;
- $A$ is the area.
Generally, let $M$ be an $m$-dimensional Riemannian manifold and consider the set of oriented embedded submanifolds of a fixed dimension $n \le m$.
We can define a sort-of-measure, $\mu$ on them, by mapping $N \subseteq M$ to its volume $\mu(N)$. My question is, does this "sort-of-measure" concept have any formalism around it? Maybe sheaf?
Specifically, I have $M = \mathbb{R}^3$ and have two of these sort-of-measures: $$\mu(N)=\text{surface area of } N$$ $$\nu(N)=\text{energy entering } N \text{ per unit time}$$
This latter quantity doesn't come from a Riemannian metric I don't think.
Now irradiance can be defined as their "Radon-Nikodym derivative" $\frac{d\nu}{d\mu}(x,n)$. The derivative depends not only on the point, but on the surface normal as well: $$ \nu(N) = \int_N \frac{d\nu}{d\mu}(x,n) d\mu(N, x)$$
I don't know of a better formalism than this, but it would be nice to get a global object that is automatically independent of parametrization, for example a 2-form on $\mathbb{R}^3$. Sadly I don't think there is such a form for these pseudo-measures.