Given a function $f: \mathbb{R}^3 \to \mathbb{R}^3 $ that maps a vector field $A$ into a vector field $B$ and that is non-injective, meaning that elements in the codomain can potentially be hit twice.
What would be an analytical solution of calculating the average vector in the codomain?
Pardon my inaccurate mathematical lingo, I am a physicist, not a mathematician. I will try to give a specific example:
- Assume we have a vector $a_1 \in A$ that is positioned at {$1,1,1$} and points in any direction
- Assume we have another vector $a_2 \in A$ that is positioned at {$5,5,5$} and points in any direction
- Now assume that $f$ maps both $a_1$ and $a_2$ to the position {$10,11,12$} in $B$. What would be the average vector at this position? (I am looking for a general solution, of course)
I know how to solve this numerically: For all coordinates in $B$, I would simply add up all vectors at that coordinate and then divide by the amount of vectors at that coordinate. However, if there are infinite elements that could map into the same coordinate in $B$, this is not possible. That's why I am looking for an analytical solution.
Side note: Intuitively, I would assume that something like a density function formalism should exist, which describes how often elements in the codomain are hit.
Many thanks!