Suppose now that a different parameterization is selected, by allowing and to depend on another pair of variables and . Then the analog of () for the new variables is
The chain rule relates , , and to , , and via the matrix equation

where the superscript T denotes the matrix transpose. The matrix with the coefficients , , and arranged in this way therefore transforms by the Jacobian matrix of the coordinate change

A matrix which transforms in this way is one kind of what is called a tensor. The matrix

with the transformation law () is known as the metric tensor of the surface.

The energy, variational principles and geodesics
Canonical measure and volume form
Signature of a metric
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