Editing Metric tensor (section)

===Coordinate transformations===
Suppose now that a different parameterization is selected, by allowing {{mvar|u}} and {{mvar|v}} to depend on another pair of variables {{math|''u''′}} and {{math|''v''′}}. Then the analog of ({{EquationNote|2}}) for the new variables is
{{NumBlk|:|<math>
  E' = \vec r_{u'} \cdot \vec r_{u'}, \quad
  F' = \vec r_{u'} \cdot \vec r_{v'}, \quad
  G' = \vec r_{v'} \cdot \vec r_{v'}.
</math>|{{EquationRef|2'}}}}

The [[chain rule]] relates {{math|''E''′}}, {{math|''F''′}}, and {{math|''G''′}} to {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} via the [[matrix (mathematics)|matrix]] equation

{{NumBlk|:|<math>\begin{bmatrix} E' & F' \\ F' & G' \end{bmatrix} =
  \begin{bmatrix}
    \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
    \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
  \end{bmatrix}^\mathsf{T}
  \begin{bmatrix} E & F \\ F & G \end{bmatrix}
  \begin{bmatrix}
    \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
    \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
  \end{bmatrix}
</math>|{{EquationRef|3}}}}

where the superscript T denotes the [[matrix transpose]]. The matrix with the coefficients {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} arranged in this way therefore transforms by the [[Jacobian matrix]] of the coordinate change

:<math>
  J = \begin{bmatrix}
    \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
    \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
\end{bmatrix}\,.</math>

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

:<math>\begin{bmatrix} E & F \\ F & G \end{bmatrix}</math>

with the transformation law ({{EquationNote|3}}) is known as the metric tensor of the surface.