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I am trying to solve the PDE from the Ginzburg-Landau equations using the FEM, but as or right now, I have only been focusing on the equations in Cartesian coordinates as I derive the weak form (here). Ultimately, I want to solve the PDE in cylindrical coordinates, and the $\phi$-derivatives are obviously going to be different (with their extra factors of $r$). In general, does the weak form of a PDE depend on the coordinate system? Or does it simply depend on the equation? That some weak forms may be the same regardless of the choice of coordinate system? I think that with the way I am deriving it, I'm not really assuming that I'm working in a specific coordinate system, so would the only difference come up in how I implement the weak form into the FEM solver?

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