I have an experiment where I'm measuring some physical quantity Q as a function of 3 variables which I can physically control (x,y,z). I'm collecting many samples of Q and (x,y,z) and then I can create an interpolation function given these samples.
One of my coworkers raised the possibility that the function should actually be a 2D function with the variables (x*y,z). In other words, that if I take all of the points with equal x*y and z, the function Q(x,y,z) would have the same value.
I'm looking for a good way to test this assumption and possibly visualize this result.
Assume that $(x,y)$ is in the first quadrant. A differential condition for the property in question is the following: $$xf_x-yf_y\equiv0\ .\tag{1}$$ This condition expresses that (for each fixed $z$) the $(x,y)$-gradient of $f$ is parallel to the gradient of the function $g(x,y):=x\,y$. It is easy to check that for an $f$ satisfying $(1)$ the function $$\phi(t):=f(t x_0, y_0/t,z_0)\qquad(t>0)$$ is constant, hence $f$ is constant along the level lines of $g$.