For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates.
2.
In orthogonal curvilinear coordinates, separation of variables can still be used, but in some details different from that in Cartesian coordinates.
3.
The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual have inverted units with respect to each other.
4.
There are 11 orthogonal curvilinear coordinate systems ( to within a conformal symmetry ) in which the 3-variable Helmholtz equation can be solved using the separation of variables technique.
5.
There are 17 orthogonal curvilinear coordinate systems ( to within a conformal symmetry ) in which the 3-variable Laplace equation can be solved using the separation of variables technique.
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