Editing Line element (section)

===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>