Editing Metric tensor (section)

===Invariance of arclength under coordinate transformations===
{{harvtxt|Ricci-Curbastro|Levi-Civita|1900}} first observed the significance of a system of coefficients {{mvar|E}}, {{mvar|F}}, and {{mvar|G}}, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form ({{EquationNote|1}}) is ''invariant'' under changes in the coordinate system, and that this follows exclusively from the transformation properties of {{mvar|E}}, {{mvar|F}}, and {{mvar|G}}. Indeed, by the chain rule,

:<math>\begin{bmatrix} du \\ dv \end{bmatrix} =
  \begin{bmatrix}
    \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\
    \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
  \end{bmatrix}
  \begin{bmatrix} du' \\ dv' \end{bmatrix}
</math>

so that

:<math>\begin{align}
  ds^2
  &=
    \begin{bmatrix} du & dv \end{bmatrix}
    \begin{bmatrix} E & F \\ F & G \end{bmatrix}
    \begin{bmatrix} du \\ dv \end{bmatrix} \\[6pt]
  &=
    \begin{bmatrix} du' & dv' \end{bmatrix}
    \begin{bmatrix}
      \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt]
      \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
    \end{bmatrix}^\mathsf{T}
    \begin{bmatrix} E & F \\ F & G \end{bmatrix}
    \begin{bmatrix}
      \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt]
      \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
    \end{bmatrix}
    \begin{bmatrix} du' \\ dv' \end{bmatrix} \\[6pt]
  &=
    \begin{bmatrix} du' & dv' \end{bmatrix}
    \begin{bmatrix} E' & F' \\ F' & G' \end{bmatrix}
    \begin{bmatrix} du' \\ dv' \end{bmatrix}\\[6pt]
  &= (ds')^2 \,.
\end{align}</math>