Editing Metric tensor (section)

===Inverse metric===
Let {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} be a basis of vector fields, and as above let {{math|''G''['''f''']}} be the matrix of coefficients
:<math>g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right) \,.</math>
One can consider the [[inverse matrix]] {{math|''G''['''f''']<sup>−1</sup>}}, which is identified with the '''inverse metric''' (or ''conjugate'' or ''dual metric''). The inverse metric satisfies a transformation law when the frame {{math|'''f'''}} is changed by a matrix {{mvar|A}} via

{{NumBlk|:|<math>G[\mathbf{f}A]^{-1} = A^{-1}G[\mathbf{f}]^{-1}\left(A^{-1}\right)^\mathsf{T}.</math>|{{EquationRef|5}}}}

The inverse metric transforms ''[[Covariance and contravariance of vectors|contravariantly]]'', or with respect to the inverse of the change of basis matrix {{mvar|A}}. 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 functional]]s.

To see this, suppose that {{mvar|α}} is a covector field. To wit, for each point {{mvar|p}}, {{mvar|α}} determines a function {{math|''α''<sub>''p''</sub>}} defined on tangent vectors at {{mvar|p}} so that the following [[linear transformation|linearity]] condition holds for all tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}, and all real numbers {{mvar|a}} and {{mvar|b}}:

:<math>\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,.</math>

As {{mvar|p}} varies, {{mvar|α}} is assumed to be a [[smooth function]] in the sense that

:<math>p \mapsto \alpha_p \left(X_p\right)</math>

is a smooth function of {{mvar|p}} for any smooth vector field {{mvar|X}}.

Any covector field {{mvar|α}} has components in the basis of vector fields {{math|'''f'''}}. These are determined by

:<math>\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,.</math>

Denote the [[row vector]] of these components by

:<math>\alpha[\mathbf{f}] = \big\lbrack\begin{array}{cccc} \alpha_1 & \alpha_2 & \dots & \alpha_n \end{array}\big\rbrack \,.</math>

Under a change of {{math|'''f'''}} by a matrix {{mvar|A}}, {{math|''α''['''f''']}} changes by the rule

:<math>\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A \,.</math>

That is, the row vector of components {{math|''α''['''f''']}} transforms as a ''covariant'' vector.

For a pair {{mvar|α}} and {{mvar|β}} of covector fields, define the inverse metric applied to these two covectors by

{{NumBlk|:|<math>\tilde{g}(\alpha,\beta) = \alpha[\mathbf{f}]G[\mathbf{f}]^{-1}\beta[\mathbf{f}]^\mathsf{T}.</math>|{{EquationRef|6}}}}

The resulting definition, although it involves the choice of basis {{math|'''f'''}}, does not actually depend on {{math|'''f'''}} in an essential way. Indeed, changing basis to {{math|'''f'''''A''}} gives

:<math>\begin{align}
      &\alpha[\mathbf{f}A] G[\mathbf{f}A]^{-1} \beta[\mathbf{f}A]^\mathsf{T} \\
  ={} &\left(\alpha[\mathbf{f}]A\right) \left(A^{-1}G[\mathbf{f}]^{-1} \left(A^{-1}\right)^\mathsf{T}\right) \left(A^\mathsf{T}\beta[\mathbf{f}]^\mathsf{T}\right) \\
  ={} &\alpha[\mathbf{f}] G[\mathbf{f}]^{-1} \beta[\mathbf{f}]^\mathsf{T}.
\end{align}
</math>

So that the right-hand side of equation ({{EquationNote|6}}) is unaffected by changing the basis {{math|'''f'''}} to any other basis {{math|'''f'''''A''}} whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix {{math|''G''['''f''']}} are denoted by {{math|''g''<sup>''ij''</sup>}}, where the indices {{mvar|i}} and {{mvar|j}} have been raised to indicate the transformation law ({{EquationNote|5}}).