I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f\_1(\mathbf{x}), f\_2(\mathbf{x}),~\dots, f\_n(\mathbf{x})]^{\rm T}$, I am looking for all such functions that:


$\forall~\mathbf{x}\in\mathbb{R}^{n}\quad\&\quad 1\leq i,j\leq n:\quad \big(\nabla f\_i(\mathbf{x})\big)^{\rm T}\nabla f\_j(\mathbf{x}) = \begin{cases}0~&:i\neq j\\g\_{i}(\mathbf{x})&:i=j \end{cases}$


A similar question has been asked [here](https://mathoverflow.net/questions/228964/functions-with-orthogonal-jacobian/228969#228969). As I understood, in Liouville's theorem for conformal maps all the diagonal elements of the Jacobian $\nabla\mathbf{f}(\mathbf{x})$ are the same. Here, however, I am looking for a generalized case where the diagonal elements are not necessarily the same. Do we have something similar to Liouville's theorem for this case?


Thanks.



Functions with a Jacobian whose columns are orthogonal
You are asking about the subject of *orthogonal (coordinate) systems*. There is an extensive literature on this subject, in particular by Darboux when $n=3$, and if you search on "triply orthogonal systems", you will see references to Darboux, Eisenhart, etc. plus many more recent references. There are many classical examples.


The equations are underdetermined when $n=2$, determined when $n=3$, and overdetermined when $n>3$, but they are always involutive and 'integrable' (in the sense of integrable systems).


