# Nondifferentiable coordinate transforms

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?

• The transformation is not-differentiable at some points only, or non-differentiable everywhere, at every point? – Maxim Umansky Sep 6 '20 at 14:15
• Nondifferentiable only at points $u(x,y,z)=u_0$ where $u_0$ is a constant. – Tommi Höynälänmaa Sep 6 '20 at 14:51
• 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? – Wolfgang Bangerth Sep 6 '20 at 23:04
• 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. – Maxim Umansky Sep 7 '20 at 2:28
• Ah, but if you exclude the nuclei, then you also exclude the locations where your coordinate transformation is non-smooth. Right? – Wolfgang Bangerth Sep 9 '20 at 17:33