Let be a basis of vector fields, and as above let be the matrix of coefficients
One can consider the inverse matrix , which is identified with the inverse metric (or "conjugate" or "dual metric"). The inverse metric satisfies a transformation law when the frame is changed by a matrix via

The inverse metric transforms "contravariantly", or with respect to the inverse of the change of basis matrix . Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals.

To see this, suppose that is a covector field. To wit, for each point , determines a function defined on tangent vectors at so that the following linearity condition holds for all tangent vectors and , and all real numbers and :

As varies, is assumed to be a smooth function in the sense that

is a smooth function of for any smooth vector field .

Any covector field has components in the basis of vector fields . These are determined by

Denote the row vector of these components by

Under a change of by a matrix , changes by the rule

That is, the row vector of components transforms as a "covariant" vector.

For a pair and of covector fields, define the inverse metric applied to these two covectors by

The resulting definition, although it involves the choice of basis , does not actually depend on in an essential way. Indeed, changing basis to gives

So that the right-hand side of equation () is unaffected by changing the basis to any other basis whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix are denoted by , where the indices and have been raised to indicate the transformation law ().

Components of the metric
The energy, variational principles and geodesics
Inverse metric
Canonical measure and volume form