Dilatometer - Revision history

85.127.247.33 at 12:37, 6 September 2014

2014-09-06T12:37:07Z

← Older revision		Revision as of 14:37, 6 September 2014
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	In [[differential geometry]], '''normal coordinates''' at a point ''p'' in a [[differentiable manifold]] equipped with a [[torsion tensor\|symmetric]] [[affine connection]] are a [[local coordinate system]] in a [[neighborhood (mathematics)\|neighborhood]] of ''p'' obtained by applying the [[exponential map]] to the [[tangent space]] at ''p''. In a normal coordinate system, the [[Christoffel symbols]] of the connection vanish at the point ''p'', thus often simplifying local calculations. In normal coordinates associated to the [[Levi-Civita connection]] of a [[Riemannian manifold]], one can additionally arrange that the [[metric tensor]] is the [[Kronecker delta]] at the point ''p'', and that the first [[partial derivative]]s of the metric at ''p'' vanish.		Hi there. Let me begin by introducing the author, her title is Myrtle Cleary. To gather cash is what his family members and him enjoy. Managing individuals is what I do and the wage has been really satisfying. South Dakota is her birth place but she requirements to move simply because of her family members.<br><br>my webpage: [http://www.videokeren.com/blog/71062 at home std testing]

	A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at ''p'' only), and the geodesics through ''p'' are locally linear functions of ''t'' (the affine parameter). This idea was implemented in a fundamental way by [[Albert Einstein]] in the [[general theory of relativity]]: the [[equivalence principle]] uses normal coordinates via [[inertial frame]]s. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or [[Pseudo-Riemannian]] manifold. By contrast, there ~~is no way to define normal coordinates for [[Finsler manifold]]s {{harv\|Busemann\|1955}}~~.

	~~==Geodesic normal coordinates==~~
	~~Geodesic normal coordinates are local coordinates on a manifold with an affine connection afforded~~ by the ~~[[exponential map]]~~

	~~<math> \exp_p : T_{p}M \supset V \rightarrow M </math>~~

	~~and an isomorphism~~

	~~<math> E: \mathbb{R}^n \rightarrow T_{p}M </math>~~

	~~given by any [[basis of a vector space\|basis]] of the tangent space at the fixed basepoint ''p'' ∈ ''M''. If the additional structure of a Riemannian metric is imposed~~, ~~then the basis defined by ''E'' may be required in addition to be [[orthonormal basis\|orthonormal]], and the resulting coordinate system~~ is ~~then known as a '''Riemannian normal coordinate system'''~~.

	~~Normal coordinates exist on a normal neighborhood of a point ''p'' in ''M''. A normal neighborhood ''U'' is a subset of ''M'' such that there~~ is ~~a proper neighborhood ''V'' of the origin in the [[tangent space]] ''T<sub>p</sub>M''~~ and ~~exp<sub>''p''</sub> acts as a [[diffeomorphism]] between ''U'' and ''V''~~. ~~Now let ''U'' be a normal neighborhood of ''p'' in ''M'' then the chart~~ is ~~given by:~~

	~~<math> \varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n </math>~~

	~~The isomorphism ''E'' can be any isomorphism between both vectorspaces, so there are as many charts as different orthonormal bases exist in the domain of ''E''.~~

	~~=== Properties ===~~

	~~The properties of normal coordinates often simplify computations. In the following, assume that ''U'' is a normal neighborhood centered at ''p'' in ''M''~~ and ~~''(x<sup>i</sup>)'' are normal coordinates on ''U''.~~

	* Let ''V'' be some vector from ''T<sub>p</sub>M'' with components ''V<sup>i</sup>'' in local coordinates, and <math> \gamma_V </math> be the ~~[[geodesic]] with starting point ''p'' and velocity vector ''V'', then <math> \gamma_V</math> is represented in normal coordinates by <math> \gamma_V(t) = (tV^1, .~~.~~. , tV^n) </math> as long as it~~ is ~~in ''U''.~~
	* The coordinates of ''p'' are (0, ..., 0)
	* In Riemannian normal coordinates at ''p'' the components of the [[Metric tensor\|Riemannian metric]] ''g'' simplify to ~~<math> \delta_{ij} </math>.~~
	* The [[Christoffel symbols]] vanish at ''p''. In the Riemannian case, so do the first partial derivatives of ~~<math> g_{ij} </math>~~.

	~~==Polar coordinates==~~
	On a Riemannian manifold, a normal coordinate system at ''p'' facilitates the introduction of a system of [[spherical coordinates]], known as '''polar coordinates'''. These are the coordinates on ''M'' obtained by introducing the standard spherical coordinate system on the Euclidean space ''T''<~~sub~~>~~''p''~~<~~/sub~~>''M''. That is, one introduces on ''T''<sub>''p''</sub>''M'' the standard spherical coordinate system (''r'',φ) where ''r'' ≥ 0 is the radial parameter and φ = (φ<sub>1</sub>,...,φ<sub>''n''−1</sub>) is a parameterization of the [[~~N sphere\|(''n''−1)-sphere]]. Composition of (''r'',φ) with the inverse of the exponential map at ''p'' is a polar coordinate system.~~

	~~Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant~~: geometrically it represents the geodesic distance to ''p'' of nearby points. [[Gauss's lemma (Riemannian geometry)\|Gauss's lemma]] asserts that the [[gradient]] of ''r'' is simply the [[partial derivative]] <math>\partial/\partial r</~~math>. That is,~~
	~~:<math>\langle df, dr\rangle = \frac{\partial f}{\partial r}<~~/~~math>~~
	~~for any smooth function ''&fnof;''~~. ~~As a result, the metric in polar coordinates assumes a [[block diagonal]] form~~
	~~:<math>g = \begin{bmatrix}~~
	~~1&0&\cdots\ 0\\~~
	~~0&&\\~~
	~~\vdots &&g_{\phi\phi}(r,\phi)\\~~
	~~0&&~~
	~~\end{bmatrix}~~.</~~math>~~

	~~==References==~~
	* {{Citation \| last1=Busemann \| first1=Herbert \| title=On normal coordinates in Finsler spaces \| id={{MathSciNet \| id = 0071075}} \| year=1955 \| journal=[[Mathematische Annalen]] \| issn=0025-5831 \| volume=129 \| pages=417–423 \| doi=10.1007/~~BF01362381}}.~~
	* {{citation \| last1=Kobayashi\|first1=Shoshichi\|last2=Nomizu\|first2=Katsumi \| title = [[Foundations of Differential Geometry]]\|volume=Vol. 1\| publisher=[[Wiley Interscience]] \| year=1996\|edition=New\|isbn=0-471-15733-3}}.
	* Chern, S. S.; Chen, W. H.; Lam, K. S.; ''Lectures on Differential Geometry'', World Scientific, 2000

	~~==See also==~~
	*[[Fermi coordinates]]
	*[[Local reference frame]]

	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Coordinate systems in differential geometry]~~]

en>Monkbot: /* Applications */Fix CS1 deprecated date parameter errors

2014-01-16T19:48:23Z

Applications: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 21:48, 16 January 2014
Line 1:		Line 1:
	~~Marvella~~ [~~http://fankut~~.~~com/index.php?do=/profile~~-~~2572/info/ over~~ the ~~counter std test~~] is ~~what you can contact her but it~~'s ~~not~~ the ~~most feminine name out~~ there. ~~Minnesota has usually~~ ~~home std test been his house but his spouse wants them~~ to ~~move. What I love performing~~ is ~~performing ceramics but I haven~~'~~t produced~~ a ~~dime with it~~. ~~Managing people~~ is ~~what I do~~ and [~~http~~://~~independentmumbaiescort~~.~~com~~/~~anamika~~/~~blogs~~/~~post/4358 over~~ the ~~counter std test~~] ~~salary has been really home [http:~~//~~Www~~.~~Womansday~~.~~com~~/~~sex-relationships/sex-tips/everything-you-need-to-know-about-lubricants-101625 std test~~] ~~[http:~~/~~/www~~.~~Huffingtonpost~~.~~com~~/~~2013~~/03/29/~~celebrities~~-with~~-stds_n_2979493~~.~~html kit fulfilling~~].<br><br>~~my weblog: home std test kit~~; [~~http~~://~~support~~.~~kaponline~~.~~com/entries/46228144~~-~~Endocrine~~-~~Illnesses~~-~~Canines-Part-1 Recommended Web page~~],		In [[differential geometry]], '''normal coordinates''' at a point ''p'' in a [[differentiable manifold]] equipped with a [[torsion tensor\|symmetric]] [[affine connection]] are a [[local coordinate system]] in a [[neighborhood (mathematics)\|neighborhood]] of ''p'' obtained by applying the [[exponential map]] to the [[tangent space]] at ''p''. In a normal coordinate system, the [[Christoffel symbols]] of the connection vanish at the point ''p'', thus often simplifying local calculations. In normal coordinates associated to the [[Levi-Civita connection]] of a [[Riemannian manifold]], one can additionally arrange that the [[metric tensor]] is the [[Kronecker delta]] at the point ''p'', and that the first [[partial derivative]]s of the metric at ''p'' vanish.

			A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at ''p'' only), and the geodesics through ''p'' are locally linear functions of ''t'' (the affine parameter). This idea was implemented in a fundamental way by [[Albert Einstein]] in the [[general theory of relativity]]: the [[equivalence principle]] uses normal coordinates via [[inertial frame]]s. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or [[Pseudo-Riemannian]] manifold. By contrast, there is no way to define normal coordinates for [[Finsler manifold]]s {{harv\|Busemann\|1955}}.

			==Geodesic normal coordinates==
			Geodesic normal coordinates are local coordinates on a manifold with an affine connection afforded by the [[exponential map]]

			<math> \exp_p : T_{p}M \supset V \rightarrow M </math>

			and an isomorphism

			<math> E: \mathbb{R}^n \rightarrow T_{p}M </math>

			given by any [[basis of a vector space\|basis]] of the tangent space at the fixed basepoint ''p'' ∈ ''M''. If the additional structure of a Riemannian metric is imposed, then the basis defined by ''E'' may be required in addition to be [[orthonormal basis\|orthonormal]], and the resulting coordinate system is then known as a '''Riemannian normal coordinate system'''.

			Normal coordinates exist on a normal neighborhood of a point ''p'' in ''M''. A normal neighborhood ''U'' is a subset of ''M'' such that there is a proper neighborhood ''V'' of the origin in the [[tangent space]] ''T<sub>p</sub>M'' and exp<sub>''p''</sub> acts as a [[diffeomorphism]] between ''U'' and ''V''. Now let ''U'' be a normal neighborhood of ''p'' in ''M'' then the chart is given by:

			<math> \varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n </math>

			The isomorphism ''E'' can be any isomorphism between both vectorspaces, so there are as many charts as different orthonormal bases exist in the domain of ''E''.

			=== Properties ===

			The properties of normal coordinates often simplify computations. In the following, assume that ''U'' is a normal neighborhood centered at ''p'' in ''M'' and ''(x<sup>i</sup>)'' are normal coordinates on ''U''.

			* Let ''V'' be some vector from ''T<sub>p</sub>M'' with components ''V<sup>i</sup>'' in local coordinates, and <math> \gamma_V </math> be the [[geodesic]] with starting point ''p'' and velocity vector ''V'', then <math> \gamma_V</math> is represented in normal coordinates by <math> \gamma_V(t) = (tV^1, ... , tV^n) </math> as long as it is in ''U''.
			* The coordinates of ''p'' are (0, ..., 0)
			* In Riemannian normal coordinates at ''p'' the components of the [[Metric tensor\|Riemannian metric]] ''g'' simplify to <math> \delta_{ij} </math>.
			* The [[Christoffel symbols]] vanish at ''p''. In the Riemannian case, so do the first partial derivatives of <math> g_{ij} </math>.

			==Polar coordinates==
			On a Riemannian manifold, a normal coordinate system at ''p'' facilitates the introduction of a system of [[spherical coordinates]], known as '''polar coordinates'''. These are the coordinates on ''M'' obtained by introducing the standard spherical coordinate system on the Euclidean space ''T''<sub>''p''</sub>''M''. That is, one introduces on ''T''<sub>''p''</sub>''M'' the standard spherical coordinate system (''r'',φ) where ''r'' ≥ 0 is the radial parameter and φ = (φ<sub>1</sub>,...,φ<sub>''n''−1</sub>) is a parameterization of the [[N sphere\|(''n''−1)-sphere]]. Composition of (''r'',φ) with the inverse of the exponential map at ''p'' is a polar coordinate system.

			Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to ''p'' of nearby points. [[Gauss's lemma (Riemannian geometry)\|Gauss's lemma]] asserts that the [[gradient]] of ''r'' is simply the [[partial derivative]] <math>\partial/\partial r</math>. That is,
			:<math>\langle df, dr\rangle = \frac{\partial f}{\partial r}</math>
			for any smooth function ''&fnof;''. As a result, the metric in polar coordinates assumes a [[block diagonal]] form
			:<math>g = \begin{bmatrix}
			1&0&\cdots\ 0\\
			0&&\\
			\vdots &&g_{\phi\phi}(r,\phi)\\
			0&&
			\end{bmatrix}.</math>

			==References==
			* {{Citation \| last1=Busemann \| first1=Herbert \| title=On normal coordinates in Finsler spaces \| id={{MathSciNet \| id = 0071075}} \| year=1955 \| journal=[[Mathematische Annalen]] \| issn=0025-5831 \| volume=129 \| pages=417–423 \| doi=10.1007/BF01362381}}.
			* {{citation \| last1=Kobayashi\|first1=Shoshichi\|last2=Nomizu\|first2=Katsumi \| title = [[Foundations of Differential Geometry]]\|volume=Vol. 1\| publisher=[[Wiley Interscience]] \| year=1996\|edition=New\|isbn=0-471-15733-3}}.
			* Chern, S. S.; Chen, W. H.; Lam, K. S.; ''Lectures on Differential Geometry'', World Scientific, 2000

			==See also==
			*[[Fermi coordinates]]
			*[[Local reference frame]]

			[[Category:Riemannian geometry]]
			[[Category:Coordinate systems in differential geometry]]

en>PhnomPencil: /* Applications */Fixing links to disambiguation pages -- Sinter using AWB

2012-04-16T21:20:44Z

Applications: Fixing links to disambiguation pages -- Sinter using AWB

New page

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