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I'm interested in constructing a curvilinear coordinate system with coordinate axes being these two curves: $$x=a, \qquad y^2+x=b,$$ where $a, b$ are parameters that specify the families of the curve, and points on this coordinate can be described by $(a,b)$. Is that something possible? Given two curves, what are some conditions such that they can form a coordinate system?

Sean
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  • Is the map $(x,y)\mapsto (a,b)$ bijective? – Kurt G. Nov 19 '23 at 18:53
  • Thanks for pointing it out. Is bijectivity between $(x,y)$ and $(a,b)$ the only condition required to form a curvilinear coordinate system in $(a,b)$? – Sean Nov 19 '23 at 19:33
  • This follows from the first sentence in the relevant Wikipedia article. Most (if not all) coordinate systems we use are however nicer, namely, differentiable. This is not a problem in your case. Is your system a coordinate system? If not on what subset of $\mathbb R^2$ it is one? – Kurt G. Nov 19 '23 at 19:37
  • The map is bijective on the upper half of the coordinate plane ${(x,y)|x\in \mathbf{R}, y\geq 0}$, so it seems like they form a coordinate system in that space. However, I also heard that the integral curves of two vector fields form a coordinate system iff their Lie bracket vanishes. In this case, the above two curves are the integral curves of the two vector fields $v_1 = \frac{\partial}{\partial y}, v_2=-2y\frac{\partial}{\partial x} + \frac{\partial}{\partial y} $, and their Lie bracket do not vanish. It seems to suggest that they cannot form a coordinate system? – Sean Nov 19 '23 at 21:00
  • I have never heard that the integral curves of two vector fields form a coordinate system iff their Lie bracket vanishes. – Kurt G. Nov 19 '23 at 22:41
  • It seems I found the reason for this confusion and made a few comments there. – Kurt G. Nov 20 '23 at 10:33
  • @KurtG. - FYI, your link refers back to this same thread. I assume it was supposed to point to a different thread? – Paul Sinclair Nov 20 '23 at 14:33
  • @PaulSinclair . Thanks ! I meant this thread. – Kurt G. Nov 20 '23 at 14:47

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