I know that if I have a random variable $X$ with probability density $f_X(x)$ and define $Y=g(X)$, where $g(x)$ is monotonic and bijective, then I can find the probability density of $Y$ with $$ f_Y(y) = f_X(g^{-1}(y))\left|\frac{d}{dy}g^{-1}(y)\right|. $$ This result can be generalized to non-monotonic functions for $Y=g(X)$ and can be extrapolated to multivariate $X$ and $Y$.
But what I have not seen is what to do if $g(x)$ is an operator instead of a function, for example $y=\frac{d}{dz}x$ or for multivariate variables, $\boldsymbol{y}=\nabla\cdot\boldsymbol{x}$.
In the case that the transformation that takes $x\rightarrow y$ is an operator can I still use the regular change of variables formulas? If not is there an equivalent?