Lets say i have a continiously differentialbe function $f(x, y, z)$ that is nonnegative everywhere and has a finite volume integral over all space that yields 1, $$ \int^\infty_{-\infty }dx\int^\infty_{-\infty }dy\int^\infty_{-\infty }dz f(x,y,z) = 1. $$ Now i'd like to find the levelsurface defined by an isovalue $k$, such that the integral inside the volume bounded by that surface yields a specific value $0<l<1$.

My thought is that an isovalue should fix an isosurface which fixes a volume which fixes a volume integral value and thus there should be a direct mapping from isovalue to volume integral value. Can i find this map in a closed form or is this in general impossible ? What i would like to have would be a function of the form $$ U(k) = l $$ where $$ l = \int_{V_k}fdV $$ and $V_k$ is the volume inside the isosurface defined by isovalue $k$.

The functions $$f(x,y,z)$$ could be the magnitude square of atomic orbitals for example. The question stems from the attempt to plot an isovalue contur that contains exactly 95% of the probability to find an electron. I could find such contours/isovalues by trial and error but i wondered if there is a more systematic approach to find the isovalue.


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