A system of real-valued functions , giving a local coordinate system on an open set in , determines a basis of vector fields on 

The metric 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 by means of the chain rule

so that

Or, in terms of the matrices and ,

where denotes the Jacobian matrix of the coordinate change.

Metric in coordinates
The energy, variational principles and geodesics
Area
Inverse metric