Curvilinear coordinate systems metrics

The dependence on the curvilinear coordinate system has been made more explicit by replacing the elements by the equivalent covariant metric tensor... 

The metrics (or scale factors) for a large number of orthogonal curvilinear coordinate systems can be found in the appendix by Happel and Brenner (1973) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff International, Leyden, The Netherlands, 1973). 

Figure 15 Basis vectors and functions may be (a) orthogonal, (b) nonorthogonal, or even (c) curvilinear like (a) (x,y), (b) (/, ),and(c) (xi,X2) in this illustration. The metric gjiv = (which can be identified with the overlap matrix in quantum chemistry) describes uniquely the kind of coordinate system (a-c) spanned by the basis and provides a measure for distances, volumes, etc.

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