Parabolic coordinates ($t, s, z$) provide an orthogonal curvilinear framework where the coordinate surfaces are defined by two families of confocal parabolas that intersect at right angles. By mapping the Cartesian coordinates to quadratic relations of $t$ and $s$, we find that the transformation is governed by a shared scale factor $h_t=h_s=\sqrt{t^2+s^2}$, which simplifies the calculation of differential operators. This system is particularly powerful for solving boundary value problems-such as those found in electrostatics or fluid dynamics-where the physical boundaries are parabolic in shape, as it allows for the separation of variables in the Laplace equation.
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%% Proof and Derivation
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