The components of the metric in any basis of vector fields, or frame, are given by

The functions form the entries of an symmetric matrix, . If
are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity:

Denoting the matrix by and arranging the components of the vectors and into column vectors and ,

where and denote the transpose of the vectors and , respectively. Under a change of basis of the form

for some invertible matrix , the matrix of components of the metric changes by as well. That is,

or, in terms of the entries of this matrix,

For this reason, the system of quantities is said to transform covariantly with respect to changes in the frame .

Components of the metric
Metric in coordinates
The round metric on a sphere
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