Editing Line element (section)

===Definition of the line element and arc length===

The [[coordinate]]-independent definition of the square of the line element ''ds'' in an ''n''-[[dimension]]al [[Riemannian manifold|Riemannian]] or [[Pseudo Riemannian manifold]] (in physics usually a [[spacetime manifold|Lorentzian manifold]]) is the  "square of the length"  of an infinitesimal displacement <math>d\mathbf{q}</math><ref name="Kay">Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, {{isbn|0-07-033484-6}}</ref> (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: 
<math display="block"> ds^2 = d\mathbf{q}\cdot d\mathbf{q} = g(d\mathbf{q},d\mathbf{q})</math> 
where ''g'' is the [[metric tensor]], '''·''' denotes [[inner product]], and ''d'''''q''' an [[infinitesimal]] [[Displacement (vector)|displacement]] on  the (pseudo) Riemannian manifold. By parametrizing a curve <math>\mathbf{q}(\lambda)</math>, we can define the [[arc length]] of the curve length of the curve between <math>\mathbf{q}_1=\mathbf{q}(\lambda_1)</math>, and <math>\mathbf{q}_2=\mathbf{q}(\lambda_2)</math> as the [[integral]]:<ref name="SpiegelLipschutzSpellman">Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, {{isbn|978-0-07-161545-7}}</ref>
<math display="block"> s = \int_{\mathbf{q}_1}^{\mathbf{q}_2}\sqrt{ \left|ds^2\right|} = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{ \left|g\left(\frac{d\mathbf{q}}{d\lambda},\frac{d\mathbf{q}}{d\lambda}\right)\right|} = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{ \left|g_{ij}\frac{dq^i}{d\lambda}\frac{dq^j}{d\lambda}\right|}.</math>

To compute a sensible length of curves in pseudo Riemannian manifolds, it is best to assume that the infinitesimal displacements have the same sign everywhere. E.g. in physics the square of a line element along a timeline curve would (in the <math>-+++</math> signature convention) be negative and the negative square root of the square of the line element along the curve would measure the proper time passing for an observer moving along the curve.
From this point of view, the metric also defines in addition to line element the [[surface (topology)|surface]] and [[volume element]]s etc.