Editing Metric tensor (section)

===Arc length===
If the variables {{mvar|u}} and {{mvar|v}} are taken to depend on a third variable, {{mvar|t}}, taking values in an [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}}, then {{math|{{vec|''r''}}(''u''(''t''), ''v''(''t''))}} will trace out a [[parametric curve]] in parametric surface {{mvar|M}}. The [[arc length]] of that curve is given by the [[integral]]

: <math> \begin{align}
  s &= \int_a^b\left\|\frac{d}{dt}\vec{r}(u(t),v(t))\right\|\,dt \\[5pt]
    &= \int_a^b \sqrt{u'(t)^2\,\vec{r}_u\cdot\vec{r}_u + 2u'(t)v'(t)\, \vec{r}_u\cdot\vec{r}_v + v'(t)^2\,\vec{r}_v\cdot\vec{r}_v}\, dt \,,
\end{align}</math>

where <math> \left\| \cdot \right\| </math> represents the [[Norm (mathematics)#Euclidean norm|Euclidean norm]]. Here the [[chain rule]] has been applied, and the subscripts denote [[partial derivative]]s:

:<math>\vec{r}_u = \frac{\partial \vec{r}}{\partial u}\,, \quad \vec{r}_v = \frac{\partial \vec{r}}{\partial v}\,.</math>

The integrand is the restriction<ref>More precisely, the integrand is the [[pullback (differential geometry)|pullback]] of this differential to the curve.</ref> to the curve of the square root of the ([[quadratic form|quadratic]]) [[differential (infinitesimal)|differential]]

{{NumBlk|:|<math>(ds)^2 = E \,(du)^2 + 2F \,du\, dv + G\, (dv)^2 ,</math>|{{EquationRef|1}}}}

where

{{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 quantity {{mvar|ds}} in ({{EquationNote|1}}) is called the [[line element]], while {{math|''ds''<sup>2</sup>}} is called the [[first fundamental form]] of {{mvar|M}}. Intuitively, it represents the [[principal part]] of the square of the displacement undergone by {{math|{{vec|''r''}}(''u'', ''v'')}} when {{mvar|u}} is increased by {{mvar|du}} units, and {{mvar|v}} is increased by {{mvar|dv}} units.

Using matrix notation, the first fundamental form becomes
:<math>ds^2 =
  \begin{bmatrix} du & dv \end{bmatrix}
  \begin{bmatrix} E & F \\ F & G \end{bmatrix}
  \begin{bmatrix} du \\ dv \end{bmatrix}
</math>