Editing Line element (section)

==Line elements in Euclidean space==
{{main|Euclidean space}}

[[File:Line element.svg|thumb|Vector line element d'''r''' (green) in [[three-dimensional|3d]] Euclidean space, where λ is a [[parametric equation|parameter]] of the space curve (light green).]]

Following are examples of how the line elements are found from the metric.

===Cartesian coordinates===

The simplest line element is in [[Cartesian coordinates]] - in which case the metric is just the [[Kronecker delta]]:
<math display="block">g_{ij} = \delta_{ij}</math>
(here ''i, j'' = 1, 2, 3 for space) or in [[matrix (mathematics)|matrix]] form (''i'' denotes row, ''j'' denotes column):
<math display="block">[g_{ij}] = \begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix}</math>

The general curvilinear coordinates reduce to Cartesian coordinates:
<math display="block">(q^1,q^2,q^3) = (x, y, z)\,\Rightarrow\,d\mathbf{r}=(dx,dy,dz)</math>
so
<math display="block"> ds^2 = g_{ij} dq^i dq^j = dx^2 +dy^2 +dz^2 </math>

===Orthogonal curvilinear coordinates===

For all [[orthogonal coordinates]] the metric is given by:<ref name="SpiegelLipschutzSpellman"/>
<math display="block">[g_{ij}] = \begin{pmatrix}
h_1^2 & 0 & 0\\
0 & h_2^2 & 0\\
0 & 0 & h_3^2
\end{pmatrix}</math>
where
<math display="block">h_i = \left|\frac{\partial\mathbf{r}}{\partial q^i}\right|</math>

for ''i'' = 1, 2, 3 are [[curvilinear coordinates#Orthogonal curvilinear coordinates in 3d|scale factor]]s, so the square of the line element is:
<math display="block">ds^2 = h_1^2(dq^1)^2 + h_2^2(dq^2)^2 + h_3^2(dq^3)^2 </math>

Some examples of line elements in these coordinates are below.<ref name="Kay"/>

{| class="wikitable"
|-
! Coordinate system
! {{math|(''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>)}}
! Metric
! Line element
|-
|[[Cartesian coordinate system|Cartesian]]
|{{math|(''x'', ''y'', ''z'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}</math>
|<math> ds^2 = dx^2 + dy^2 + dz^2 </math>
|-
|[[Polar coordinate system|Plane polars]]
|{{math|(''r'', ''θ'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 \\
0 & r^2 \\
\end{pmatrix}</math>
|<math> ds^2= dr^2 +r^2 d \theta^2</math>
|-
|[[Spherical coordinate system|Spherical polar]]s
|{{math|(''r'', ''θ'', ''φ'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & r^2\sin^2\theta \\
\end{pmatrix}</math>
|<math> ds^2=dr^2+r^2 d \theta\ ^2+ r^2 \sin^2 \theta d \varphi^2 </math>
|-
|[[Cylindrical polar coordinates|Cylindrical polar]]s
|{{math|(''r'', ''φ'', ''z'')}}
|<math>[g_{ij}] = \begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}</math>
|<math> ds^2=dr^2+ r^2 d \varphi^2 +dz^2 </math>
|}