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Suppose that we have coordinates $u=u(x,y)$ and $v=v(x,y)$ in $\mathbb{R}^2$ so that $v$ is not differentiable when $u(x,y)=u_0$ where $u_0$ is a constant. Can we solve a differential equation, such as the Schrödinger equation, using these coordinates? Can we compute integrals involving the Jacobian determinant in these coordinates?

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  • $\begingroup$ The transformation is not-differentiable at some points only, or non-differentiable everywhere, at every point? $\endgroup$ – Maxim Umansky Sep 6 '20 at 14:15
  • $\begingroup$ Nondifferentiable only at points $u(x,y,z)=u_0$ where $u_0$ is a constant. $\endgroup$ – Tommi Höynälänmaa Sep 6 '20 at 14:51
  • $\begingroup$ What's the domain on which you pose the differential equation? Is it a subset of $(x,y,z)$ space, or a subset of $(u,v,z)$ space? $\endgroup$ – Wolfgang Bangerth Sep 6 '20 at 23:04
  • $\begingroup$ For solution of a differential equation there should not be any major problem. In the worst case, a matching condition would have to be formulated for the locations where the coordinate transformation is not differentiable. For the Jacobian, would be ok too, if the singularity is integrable. $\endgroup$ – Maxim Umansky Sep 7 '20 at 2:28
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    $\begingroup$ Ah, but if you exclude the nuclei, then you also exclude the locations where your coordinate transformation is non-smooth. Right? $\endgroup$ – Wolfgang Bangerth Sep 9 '20 at 17:33

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