Is knowing unit basis vector enough to specify a coordinate system?

Question

In orthonormal curvilinear coordinate system, we define unit basis vector as $$\mathbf{\hat{e}}_u = \frac{1}{h_u} \frac{d\mathbf{r}}{du},$$ where $h_u = |\frac{d\mathbf{r}}{du}|$. Suppose we only know $\mathbf{e}_u$ in relation to Cartesian coordinates, are we able to construct coordinate system from there? Using polar coordinates $(s,\phi)$ as an example, given only the information $$\mathbf{\hat{s}}=\frac{x\mathbf{\hat{x}} + y \mathbf{\hat{y}}}{\sqrt{x^2+y^2}}, \qquad \mathbf{\hat{\phi}}=\frac{-y\mathbf{\hat{x}}+x\mathbf{\hat{y}}}{\sqrt{x^2+y^2}}, $$ arę we able to construct a coordinate system with $\mathbf{\hat{s}}$ and $\mathbf{\hat{\phi}}$ as basis unit vector? More generally, suppose I want to construct a new coordinate system $(u,v)$ but is only given $$ \mathbf{\hat{u}}=f(x,y)\mathbf{\hat{x}} + g(x,y)\mathbf{\hat{y}}, \qquad \mathbf{\hat{v}}=g(x,y)\mathbf{\hat{x}} - f(x,y)\mathbf{\hat{y}} $$ as the unit basis vector. Is there a way to find $$\mathbf{\hat{u}}=f'(u,v)\mathbf{\hat{x}} + g'(u,v)\mathbf{\hat{y}}, \qquad \mathbf{\hat{v}}=f'(u,v)\mathbf{\hat{x}} - g'(u,v)\mathbf{\hat{y}} ,$$ which gives a relationship between $x=f(u,v), y=g(u,v)$. This will enable the computation of norm $h_u$ of tangent vector $\frac{d\mathbf{r}}{du}$. Then, we are able to specify the differential in $(u,v)$ as $$ d\mathbf{r}= \mathbf{\hat{u}} h_u du + \mathbf{\hat{v}} h_v dv .$$

Answer 1

A vector field is a coordinate basis (i.e. defines a coordinate system) if and only if their pairwise Lie brackets are zero. Intuitively, the vanishing of the Lie brackets means that the coordinate lines (i.e. the integral curves of the basis vector fields) are well-defined in the sense that they form closed grids locally. It ensures that traveling along a basis vector $\mathbf{e}_i$ and then along a different basis vector $\mathbf{e}_j$ is the same as first traveling along $\mathbf{e}_j$ and then along $\mathbf{e}_i$. Otherwise, the same coordinates will be assigned to different points and your coordinate system will be ill-defined.