Coordinates and Metric Tensors

At point P referred to the curvilinear coordinates, there exists a set of base vectors IM ), defined as 

In general, curvilinear coordinates are not mutually orthogonal that is, 

The metric tensor g- is a covariant second-order tensor, and it is a symmetric tensor 

For instance, it can be easily shown that the cylindrical coordinates and spherical coordinates are orthogonal curvilinear coordinate systems. 

The contravariant components g i of the metric tensor are obtained by (Hawkins 

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