orthogonal curvilinear coordinate system

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.

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.

They have developed a scheme based on a finite difference approximation of the equations of motion, applied on a boundary-fitted orthogonal curvilinear coordinate system, inside and outside the drop.

When generalized the technique provides a powerful tool fordifferentiation, with respect to any parameter, of not only unit coordinate vectors but any vector that remains fixed in a general orthogonal curvilinear coordinate system.