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In the theory of generalized (curvilinear) coordinates, usually one defines coordinates first, e.g. $(u^1,u^2,u^3)$, then a position vector $\mathbf{r}(u^1,u^2,u^3)$, and finally the basis vectors, $$ e_i = \frac{\partial \mathbf{r}}{\partial u^i} $$ and the reciprocal (dual) basis vectors, $$ e^i = \nabla u^i $$ Suppose instead that you know a set of basis vectors at every point in $\mathbb{R}^3$, e.g. you are given $e_i$ (or $e^i$). Is it possible to compute the coordinates $(u^1,u^2,u^3)$ given these basis vectors?

As a simple/specific example, suppose you are given the spherical basis vectors $\hat{r},\hat{\theta},\hat{\phi}$ in their Cartesian components everywhere in $\mathbb{R}^3$. Could you then conclude that $r,\theta,\phi$ are the corresponding coordinates? How would you go about doing this? It seems to me that you would need to compute the integral curves of each basis vector and then use those curves to determine the coordinates, but its unclear exactly how to do this.

As a follow-up question, suppose you are given basis vector fields $e^i$ which are not conservative -- e.g. there is no function $u^i$ such that $e^i = \nabla u^i$. Does this mean there are no coordinates corresponding to the given $e^i$?

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I think the best way to think about this is that even though you are talking about $\mathbb R^3$ and not an arbitrary differentiable manifold, the basis vectors actually live not in $\mathbb R^3$ but in the tangent bundle $T\mathbb R^3$. In other words, the basis vectors at a point $P$ live in the tangent space at $P$ while the basis vectors at a different point $Q$ live in the tangent space at $Q$. These tangent spaces are different spaces though they are all isomorphic to $\mathbb R^3$. Therefore, if you want to do something globally, you need an affine connection which connects nearby tangent spaces. The point is that you need to know what happens to the $e_i$ when you transport it parallelly from one tangent space to another.

One nice thing about $\mathbb R^3$ is its metric which allows a unique affine connection (the Levi-Civita connection) to be constructed. But to use that, you need the metric at each point and not just the basis vectors. With the metric, you can compute the Christoffel symbols for Euclidean space quite easily.

Once the connection is in place, you can in principle choose some point to be the origin and transport the basis vectors at that point to any other point along a smooth curve to construct the global coordinates.

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  • $\begingroup$ Metrics and connections are fine and all, but this isn’t OP’s issue. Op seems to have accidentally stumbled upon the Frobenius integrability problem, since they’re asking how to recover coordinates from a (local) basis of vector fields. $\endgroup$ Commented Aug 21 at 8:54

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