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| {{about|lines in mathematics|Long Interspersed Nuclear Elements in DNA|Retrotransposon#LINEs}}
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| In [[geometry]], the '''line element''' or '''length element''' can most generally be thought of as the change in a [[position vector]] in an [[affine space]] expressing the change of the [[arc length]]. An easy way of visualizing this relationship is by parameterizing the given [[curve]] by [[Frenet–Serret formulas]]. As such, a ''line element'' is then naturally a function of the metric, and can be related to the [[Riemann curvature tensor|curvature tensor]]. It is usually denoted by ''s'' or ''{{ell}}'', and [[differential (infinitesimal)|differential]]s of this are then written ''ds'' or ''d{{ell}}''.
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| Line elements are used in [[physics]], especially in theories of [[gravitation]] (most notably [[general relativity]]) where [[spacetime]] is modelled as a curved [[manifold]] with a metric. For example, if a [[mass]]ive object causes some [[curvature]] in spacetime, the [[trajectory]] of an object with negligible mass over that curvature would follow the line element according to the [[geodesic equation]].<ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| ==General formulation==
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| {{for|notation used|Ricci calculus|Einstein notation}}
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| ===Definition using metric===
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| The [[coordinate]]-independent definition of the square of the line element d''s'' in an ''n''-[[dimension]]al [[metric space]] is:<ref>Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, ISBN 0-07-033484-6</ref>
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| :<math> ds^2 = d\bold{q}\cdot d\bold{q} = g(d\bold{q},d\bold{q}) </math>
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| where ''g'' is the [[metric tensor]], '''·''' denotes [[inner product]], and ''d'''''q''' an [[infinitesimal]] [[Displacement (vector)|displacement]] in the metric space.
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| In ''n''-dimensional general [[curvilinear coordinates]] '''q''' = (''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>...''q<sup>n</sup>''), the square of arc length is:<ref>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><ref>An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, ISBN 0-582-44355-5</ref>
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| :<math> ds^2= \sum g_{ij}dq^i dq^j </math> | |
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| where the [[Ricci calculus|indices]] ''i'' and ''j'' take values 1, 2, 3 ... ''n''. Common examples of metric spaces include [[three-dimensional]] [[space]] (no inclusion of [[time]] coordinates), and indeed [[four-dimensional]] [[spacetime]]. The metric is the origin of the line element, in addition to the [[surface]] and [[volume element]]s etc.
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| ===Total arc length===
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| By parameterizing a curve with a [[parameter]] λ, so that '''q'''(λ), the [[arc length]] of the curve between the points '''q'''(λ<sub>1</sub>) and '''q'''(λ<sub>2</sub>) is the [[integral]]:<ref>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>
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| :<math> s = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{ g_{ij}\frac{dq^i}{d\lambda}\frac{dq^j}{d\lambda}} </math>
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| ==Line elements in Euclidean space==
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| {{main|Euclidean space}}
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| [[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 (yellow).]]
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| Following are examples of how the line elements are found from the metric.
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| ===Cartesian coordinates===
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| The simplest line element is in [[Cartesian coordinates]] - in which case the metric is just the [[Kronecker delta]]:
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| :<math>g_{ij} = \delta_{ij}</math>
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| (here ''i, j'' = 1, 2, 3 for space) or in [[matrix (mathematics)|matrix]] form (''i'' denotes row, ''j'' denotes column):
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| :<math>[g_{ij}] = \begin{pmatrix}
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| 1 & 0 & 0\\
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| 0 & 1 & 0\\
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| 0 & 0 & 1
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| \end{pmatrix}</math>
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| The general curvilinear coordinates reduce to Cartesian coordinates:
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| :<math>(q^1,q^2,q^3) = (x, y, z)\,\Rightarrow\,d\bold{r}=(dx,dy,dz)</math>
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| so
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| :<math> ds^2 = \sum g_{ij}dq^idq^j = dx^2 +dy^2 +dz^2 </math>
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| ===Orthogonal curvilinear coordinates===
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| For all [[orthogonal coordinates]] the metric is given by:<ref>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>
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| :<math>[g_{ij}] = \begin{pmatrix}
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| h_1^2 & 0 & 0\\
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| 0 & h_2^2 & 0\\
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| 0 & 0 & h_3^2
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| \end{pmatrix}</math>
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| where
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| :<math>h_i = \left|\frac{\partial\bold{r}}{\partial q^i}\right|</math>
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| for ''i'' = 1, 2, 3 are [[curvilinear coordinates#Orthogonal curvilinear coordinates in 3d|scale factor]]s, so the square of the line element is:
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| :<math>ds^2 = h_1^2(q^1)^2 + h_2^2(q^2)^2 + h_3^2(q^3)^2 </math>
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| Some examples of line elements in these coordinates are below.<ref>Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, ISBN 0-07-033484-6</ref>
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| :{| class="wikitable"
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| |-
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| ! Coordinate system
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| ! (q<sup>1</sup>, q<sup>2</sup>, q<sup>3</sup>)
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| ! Metric
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| ! Line element
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| |-
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| |[[Polar coordinate system|Plane polars]]
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| |(''r'', θ)
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| |<math>[g_{ij}] = \begin{pmatrix}
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| 1 & 0 \\
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| 0 & r^2 \\
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| \end{pmatrix}</math>
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| |<math> ds^2= dr^2 +r^2 d \theta\ ^2</math>
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| |-
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| |[[Spherical coordinate system|Spherical polar]]s
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| |(''r'', θ, φ)
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| |<math>[g_{ij}] = \begin{pmatrix}
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| 1 & 0 & 0 \\
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| 0 & r^2 & 0 \\
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| 0 & 0 & r^2\sin^2\theta \\
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| \end{pmatrix}</math>
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| |<math> ds^2=dr^2+r^2 d \theta\ ^2+ r^2 \sin^2 \theta\ d \phi\ ^2 </math>
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| |-
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| |[[Cylindrical polar coordinates|Cylindrical polar]]s
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| |(''r'', θ, ''z'')
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| |<math>[g_{ij}] = \begin{pmatrix}
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| 1 & 0 & 0 \\
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| 0 & r^2 & 0 \\
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| 0 & 0 & 1 \\
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| \end{pmatrix}</math>
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| |<math> ds^2=dr^2+ r^2 d \theta\ ^2 +dz^2 </math>
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| |-
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| |}
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| ===General curvilinear coordinates===
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| In general curvilinear coordinates, the metric has elements given by:<ref>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>
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| :<math>g_{ij} = \frac{\partial \bold{r}}{\partial q^i}\cdot\frac{\partial \bold{r}}{\partial q^j}</math>
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| so the square of the line element is
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| :<math>ds^2 = g_{ij}dq^idq^j = \frac{\partial \bold{r}}{\partial q^i}\cdot\frac{\partial \bold{r}}{\partial q^j}dq^idq^j</math>
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| ==Line elements in 4d spacetime==
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| ===Minkowskian spacetime===
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| The [[Minkowski metric]] is:<ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0</ref><ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| :<math>[g_{ij}] = \begin{pmatrix}
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| \pm 1 & 0 & 0 & 0 \\
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| 0 & \mp 1 & 0 & 0 \\
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| 0 & 0 & \mp 1 & 0 \\
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| 0 & 0 & 0 & \mp 1 \\
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| \end{pmatrix}</math>
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| where one sign or the other is chosen, both conventions are used. This applies only for [[flat spacetime]]. The coordinates are given by the [[4-position]]:
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| :<math>\bold{x} = (x^0,x^1,x^2,x^3) = (ct,\bold{r}) \,\Rightarrow,\, d\bold{x} = (cdt,d\bold{r})</math>
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| so the line element is:
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| :<math>ds^2 = \pm c^2dt^2 \mp d\bold{r}\cdot d\bold{r}</math>
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| ===General spacetime===
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| The coordinate-independent definition of the square of the line element d''s'' in [[Spacetime#Spacetime intervals|spacetime]] is:<ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| :<math> ds^2 = d\bold{x}\cdot d\bold{x} = g(d\bold{x},d\bold{x}) </math>
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| In terms of coordinates:
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| :<math> ds^2= g_{\alpha\beta}dx^\alpha dx^\beta </math>
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| where for this case the indices α and β run over 0, 1, 2, 3 for spacetime.
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| This is the '''invariant interval''' - the measure of separation between two arbitrarily close [[Event (relativity)|events]] in [[spacetime]]. In [[special relativity]] it is invariant under [[Lorentz transformation]]s; in [[general relativity]] it is invariant under arbitrary [[inverse function|invertible]] [[differentiable]] [[coordinate transformations]].
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| ==See also==
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| *[[Covariance and contravariance of vectors]]
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| *[[First fundamental form]]
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| *[[List of integration and measure theory topics]]
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| *[[Metric tensor]]
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| *[[Ricci calculus]]
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| *[[Raising and lowering indices]]
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| ==References==
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| {{reflist}}
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| [[Category:Affine geometry]]
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| [[Category:Riemannian geometry]]
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| [[Category:Special relativity]]
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| [[Category:General relativity]]
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| [[da:Linjeelement]]
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| [[de:Linienelement]]
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