Differential operators in arbitrary coordinate systems?

[lordkelvin]

Hi, physics undergraduate here. I don't know much about differential geometry yet, but I'm curious about this idea:

Say I encounter a boundary value problem, and I'm not sure what coordinate system would be 'easiest' to solve the problem in. Is there some way to put the differential operator in terms of an unknown metric tensor, then impose some conditions stemming from the boundary values of the problem onto the arbitrary metric tensor in order to select some 'best' coordinate system?

Say I wanted to find the eigenmodes of a parralelogram-shaped drumhead. I'm basically curious if there is some way for me to have mathematics tell me I'd be best off using skew coordinates. Same thing with spherical coordinates on a round drumhead.

 

[Chestermiller]

Often, it is best to choose a coordinate system such that the coordinates on the boundaries are constants. This makes it much easier to apply the boundary conditions.