Editing Line element (section)

===Identification of the square of the line element with the metric tensor===
Since <math>d\mathbf{q}</math> is an arbitrary "square of the arc length", <math>ds^2</math> completely defines the metric, and it is therefore usually best to consider the expression for <math>ds^2</math> as a definition of the metric tensor itself, written in a suggestive but non tensorial notation:
<math display="block">ds^2 = g</math> 
This identification of the square of arc length <math>ds^2</math> with the metric is even more easy to see in ''n''-dimensional general [[curvilinear coordinates]] {{nowrap|1='''q''' = (''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>, ..., ''q<sup>n</sup>'')}}, where it is  written as a symmetric rank 2 tensor<ref name="SpiegelLipschutzSpellman"/><ref>An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, {{isbn|0-582-44355-5}}</ref> coinciding with the metric tensor:
<math display="block"> ds^2= g_{ij} dq^i dq^j = g .</math>

Here the [[Ricci calculus|indices]] ''i'' and ''j'' take values 1, 2, 3, ..., ''n'' and [[Einstein summation convention]] is used. Common examples of (pseudo) Riemannian spaces include [[three-dimensional]] [[space]] (no inclusion of [[time]] coordinates), and indeed [[four-dimensional]] [[spacetime]].