structural:reformatting, information:extraction, structural:expansion
Metric in coordinates

A system of n real-valued functions (x,..., x), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U

The metric g has components relative to this frame given by

Relative to a new system of local coordinates, say

the metric tensor will determine a different matrix of coefficients,

This new system of functions is related to the original g (f) by means of the chain rule

so that

Or, in terms of the matrices G = (g) and G = (g),

where Dy denotes the Jacobian matrix of the coordinate change.