Editing Metric tensor (section)

==Introduction==
[[Carl Friedrich Gauss]] in his 1827 ''[[#CITEREFGauss1827|Disquisitiones generales circa superficies curvas]]'' (''General investigations of curved surfaces'') considered a surface [[parametric surface|parametrically]], with the [[Cartesian coordinates]] {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} of points on the surface depending on two auxiliary variables {{mvar|u}} and {{mvar|v}}. Thus a parametric surface is (in today's terms) a [[vector-valued function]]

:<math>\vec{r}(u,\,v) = \bigl( x(u,\,v),\, y(u,\,v),\, z(u,\,v) \bigr)</math>

depending on an [[ordered pair]] of real variables {{math|(''u'', ''v'')}}, and defined in an [[open set]] {{mvar|D}} in the {{mvar|uv}}-plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.

One natural such invariant quantity is the [[arclength|length of a curve]] drawn along the surface. Another is the [[angle]] between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the [[area]] of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.

The metric tensor is <math display=inline> \begin{bmatrix} E & F \\ F & G \end{bmatrix} </math> in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.
<!-- Since what? -->

===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>

===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.

===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>

===Length and angle===
Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of [[tangent vector]]s to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the [[dot product]] of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface {{mvar|M}} can be written in the form

:<math>\mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v</math>

for suitable real numbers {{math|''p''<sub>1</sub>}} and {{math|''p''<sub>2</sub>}}. If two tangent vectors are given:

:<math>\begin{align}
  \mathbf{a} &= a_1\vec{r}_u + a_2\vec{r}_v \\
  \mathbf{b} &= b_1\vec{r}_u + b_2\vec{r}_v
\end{align}</math>

then using the [[bilinear form|bilinearity]] of the dot product,

:<math>\begin{align}
  \mathbf{a} \cdot \mathbf{b}
    &= a_1 b_1 \vec{r}_u\cdot\vec{r}_u + a_1b_2 \vec{r}_u\cdot\vec{r}_v + a_2b_1 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v \\[8pt]
    &= a_1 b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G. \\[8pt]
    &= \begin{bmatrix} a_1 & a_2 \end{bmatrix}
         \begin{bmatrix} E & F \\ F & G \end{bmatrix}
         \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \,.
\end{align}</math>

This is plainly a function of the four variables {{math|''a''<sub>1</sub>}}, {{math|''b''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}}, and {{math|''b''<sub>2</sub>}}. It is more profitably viewed, however, as a function that takes a pair of arguments {{math|'''a''' {{=}} [''a''<sub>1</sub> ''a''<sub>2</sub>]}} and {{math|'''b''' {{=}} [''b''<sub>1</sub> ''b''<sub>2</sub>]}} which are vectors in the {{mvar|uv}}-plane. That is, put

:<math>g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G \,.</math>

This is a [[symmetric function]] in {{math|'''a'''}} and {{math|'''b'''}}, meaning that

:<math>g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a})\,.</math>

It is also [[bilinear form|bilinear]], meaning that it is [[linear functional|linear]] in each variable {{math|'''a'''}} and {{math|'''b'''}} separately. That is,

:<math>\begin{align}
  g\left(\lambda\mathbf{a} + \mu\mathbf{a}', \mathbf{b}\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}', \mathbf{b}\right),\quad\text{and} \\
  g\left(\mathbf{a}, \lambda\mathbf{b} + \mu\mathbf{b}'\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}, \mathbf{b}'\right)
\end{align}</math>

for any vectors {{math|'''a'''}}, {{math|'''a'''′}}, {{math|'''b'''}}, and {{math|'''b'''′}} in the {{mvar|uv}} plane, and any real numbers {{mvar|μ}} and {{mvar|λ}}.

In particular, the length of a tangent vector {{math|'''a'''}} is given by

:<math> \left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a}, \mathbf{a})}</math>

and the angle {{mvar|θ}} between two vectors {{math|'''a'''}} and {{math|'''b'''}} is calculated by

:<math>\cos(\theta) = \frac{g(\mathbf{a}, \mathbf{b})}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } \,.</math>

===Area===
The [[surface area]] is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface {{mvar|M}} is parameterized by the function {{math|{{vec|''r''}}(''u'', ''v'')}} over the domain {{mvar|D}} in the {{mvar|uv}}-plane, then the surface area of {{mvar|M}} is given by the integral

:<math>\iint_D \left|\vec{r}_u \times \vec{r}_v\right|\,du\,dv</math>

where {{math|×}} denotes the [[cross product]], and the absolute value denotes the length of a vector in Euclidean space. By [[Lagrange's identity]] for the cross product, the integral can be written

:<math>\begin{align}
       &\iint_D \sqrt{\left(\vec{r}_u\cdot\vec{r}_u\right) \left(\vec{r}_v\cdot\vec{r}_v\right) - \left(\vec{r}_u\cdot\vec{r}_v\right)^2}\,du\,dv \\[5pt]
   ={} &\iint_D \sqrt{EG - F^2}\,du\,dv\\[5pt]
   ={} &\iint_D \sqrt{\det \begin{bmatrix} E & F \\ F & G \end{bmatrix}}\, du\, dv
\end{align}</math>

where {{math|det}} is the [[determinant]].