Editing Metric tensor (section)

===Raising and lowering indices===
{{See also|Raising and lowering indices}}
In a basis of vector fields {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}}, any smooth tangent vector field {{mvar|X}} can be written in the form

{{NumBlk|:|<math>X =
  v^1[\mathbf{f}]X_1 + v^2 [\mathbf{f}]X_2 + \dots + v^n[\mathbf{f}]X_n =
  \mathbf{f} \begin{bmatrix}v^1[\mathbf{f}] \\ v^2[\mathbf{f}] \\ \vdots \\ v^n[\mathbf{f}]\end{bmatrix} =
  \mathbf{f} v[\mathbf{f}]
</math>|{{EquationRef|7}}}}

for some uniquely determined smooth functions {{math|''v''<sup>1</sup>, ..., ''v''<sup>''n''</sup>}}. Upon changing the basis {{math|'''f'''}} by a nonsingular matrix {{mvar|A}}, the coefficients {{math|''v''<sup>''i''</sup>}} change in such a way that equation ({{EquationNote|7}}) remains true. That is,

:<math>X = \mathbf{fA}v[\mathbf{fA}] = \mathbf{f}v[\mathbf{f}]\,.</math>

Consequently, {{math|''v''['''f'''''A''] {{=}} ''A''<sup>−1</sup>''v''['''f''']}}. In other words, the components of a vector transform ''contravariantly'' (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix {{mvar|A}}. The contravariance of the components of {{math|''v''['''f''']}} is notationally designated by placing the indices of {{math|''v''<sup>''i''</sup>['''f''']}} in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} define the [[dual basis]] to be the [[linear functional]]s {{math|(''θ''<sup>1</sup>['''f'''], ..., ''θ''<sup>''n''</sup>['''f'''])}} such that

:<math>\theta^i[\mathbf{f}](X_j) = \begin{cases} 1 & \mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{cases}</math>

That is, {{math|''θ''<sup>''i''</sup>['''f'''](''X''<sub>''j''</sub>) {{=}} ''δ''<sub>''j''</sub><sup>''i''</sup>}}, the [[Kronecker delta]]. Let

:<math>\theta[\mathbf{f}] = \begin{bmatrix}\theta^1[\mathbf{f}] \\ \theta^2[\mathbf{f}] \\ \vdots \\ \theta^n[\mathbf{f}]\end{bmatrix}.</math>

Under a change of basis {{math|'''f''' ↦ '''f'''''A''}} for a nonsingular matrix {{math|''A''}}, {{math|''θ''['''f''']}} transforms via

:<math>\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].</math>

Any linear functional {{mvar|α}} on tangent vectors can be expanded in terms of the dual basis {{mvar|θ}}

{{NumBlk|:|<math>\begin{align}
  \alpha &= a_1[\mathbf{f}] \theta^1[\mathbf{f}] + a_2[\mathbf{f}] \theta^2[\mathbf{f}] + \cdots + a_n[\mathbf{f}] \theta^n[\mathbf{f}] \\[8pt]
         &= \big\lbrack\begin{array}{cccc}a_1[\mathbf{f}] & a_2[\mathbf{f}] & \dots & a_n[\mathbf{f}]\end{array}\big\rbrack \theta[\mathbf{f}] \\[8pt]
         &= a[\mathbf{f}] \theta[\mathbf{f}]
\end{align}</math>|{{EquationRef|8}}}}

where {{math|''a''['''f''']}} denotes the [[row vector]] {{math|[ ''a''<sub>1</sub>['''f'''] ... ''a''<sub>''n''</sub>['''f'''] ]}}. The components {{math|''a''<sub>''i''</sub>}} transform when the basis {{math|'''f'''}} is replaced by {{math|'''f'''''A''}} in such a way that equation ({{EquationNote|8}}) continues to hold. That is,

:<math>\alpha = a[\mathbf{f}A]\theta[\mathbf{f}A] = a[\mathbf{f}]\theta[\mathbf{f}]</math>

whence, because {{math|''θ''['''f'''''A''] {{=}} ''A''<sup>−1</sup>''θ''['''f''']}}, it follows that {{math|1=''a''['''f'''''A''] {{=}} ''a''['''f''']''A''}}. That is, the components {{mvar|a}} transform ''covariantly'' (by the matrix {{mvar|A}} rather than its inverse). The covariance of the components of {{math|''a''['''f''']}} is notationally designated by placing the indices of {{math|''a''<sub>''i''</sub>['''f''']}} in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding {{math|''X''<sub>''p''</sub>}} fixed, the function

:<math>g_p(X_p, -) : Y_p \mapsto g_p(X_p, Y_p)</math>

of tangent vector {{math|''Y''<sub>''p''</sub>}} defines a [[linear functional]] on the tangent space at {{mvar|p}}. This operation takes a vector {{math|''X''<sub>''p''</sub>}} at a point {{mvar|p}} and produces a covector {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, −)}}. In a basis of vector fields {{math|'''f'''}}, if a vector field {{mvar|X}} has components {{math|''v''['''f''']}}, then the components of the covector field {{math|''g''(''X'', −)}} in the dual basis are given by the entries of the row vector
:<math>a[\mathbf{f}] = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}].</math>

Under a change of basis {{math|'''f''' ↦ '''f'''''A''}}, the right-hand side of this equation transforms via
:<math>
  v[\mathbf{f}A]^\mathsf{T} G[\mathbf{f}A] =
    v[\mathbf{f}]^\mathsf{T} \left(A^{-1}\right)^\mathsf{T} A^\mathsf{T} G[\mathbf{f}]A =
    v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]A
</math>

so that {{math|''a''['''f'''''A''] {{=}} ''a''['''f''']''A''}}: {{mvar|a}} transforms covariantly. The operation of associating to the (contravariant) components of a vector field {{math|''v''['''f'''] {{=}} [ ''v''<sup>1</sup>['''f'''] ''v''<sup>2</sup>['''f'''] ... ''v''<sup>''n''</sup>['''f'''] ]}}<sup>T</sup> the (covariant) components of the covector field {{math|''a''['''f'''] {{=}} [ ''a''<sub>1</sub>['''f'''] ''a''<sub>2</sub>['''f'''] … ''a''<sub>''n''</sub>['''f'''] ]}}, where
:<math>a_i[\mathbf{f}] = \sum_{k=1}^n v^k[\mathbf{f}]g_{ki}[\mathbf{f}]</math>

is called '''lowering the index'''.

To ''raise the index'', one applies the same construction but with the inverse metric instead of the metric. If {{math|''a''['''f'''] {{=}} [ ''a''<sub>1</sub>['''f'''] ''a''<sub>2</sub>['''f'''] ... ''a''<sub>''n''</sub>['''f'''] ]}} are the components of a covector in the dual basis {{math|''θ''['''f''']}}, then the column vector
{{NumBlk|:|<math>v[\mathbf{f}] = G^{-1}[\mathbf{f}]a[\mathbf{f}]^\mathsf{T}</math>|{{EquationRef|9}}}}
has components which transform contravariantly:
:<math>v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].</math>

Consequently, the quantity {{math|''X'' {{=}} '''f'''''v''['''f''']}} does not depend on the choice of basis {{math|'''f'''}} in an essential way, and thus defines a vector field on {{mvar|M}}. The operation ({{EquationNote|9}}) associating to the (covariant) components of a covector {{math|''a''['''f''']}} the (contravariant) components of a vector {{math|''v''['''f''']}} given is called '''raising the index'''. In components, ({{EquationNote|9}}) is
:<math>v^i[\mathbf{f}] = \sum_{k=1}^n g^{ik}[\mathbf{f}] a_k[\mathbf{f}].</math>