Quasiconvex function - Revision history

en>SporkBot: Repair duplicate template args

2014-12-23T04:48:55Z

← Older revision		Revision as of 06:48, 23 December 2014
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	~~In [[differential geometry]]~~ and ~~[[general relativity]], the '''Bach tensor''' is~~ a ~~trace-free [[tensor]] of rank 2 which~~ is ~~[[conformally invariant]] in dimension {{nowrap\|1=~~'~~'n'' = 4}}~~.<~~ref~~>Rudolf Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs", ''Mathematische Zeitschrift'', '''9''' (1921) pp. [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002365812&IDDOC=16903 110].<~~/ref~~> ~~Before 1968, it was the only known conformally invariant tensor that is [[algebraically independent]] of the [~~[~~Weyl tensor]].<ref>P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences~~		Greetings! I am Myrtle Shroyer. Doing ceramics is what her family members and her enjoy. Managing people has been his working day occupation for a while. Puerto Rico is exactly where he's been residing for many years and he will by no means transfer.<br><br>Feel free to surf to my webpage; std home test [[http://www.teenvalley.net/blog/17832 Highly recommended Online site]]
	~~Vol. 304, No. 1476 (Apr. 2, 1968), pp.~~ [http://www.~~jstor~~.~~org/pss/2416002 113]–122</ref> In [[abstract index notation\|abstract indices]] the Bach tensor is given by~~
	~~:<math>B_{ab} = P_{cd}{{{W_a}^c}_b}^d+\nabla^c\nabla_aP_{bc}-\nabla^c\nabla_cP_{ab}</math>~~
	~~where ''<math>W</math>'' is the [[Weyl tensor]], and ''<math>P<~~/~~math>'' the [[Schouten tensor]] given in terms of the [[Ricci tensor]] ''<math>R_{ab}<~~/~~math>'' and [[scalar curvature]] ''<math>R</math>'' by~~

	~~:<math>P_{ab}=\frac{1}{n-2}\left(R_{ab}-\frac{R}{2(n-1)}g_{ab}\right).</math>~~

	~~==See also==~~
	*[[Cotton tensor]]
	*[[Obstruction tensor]]

	~~== References ==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	* Arthur L. Besse, ''Einstein Manifolds''. Springer-Verlag, 2007. See Ch.4, §H "Quadratic Functionals".
	* Demetrios Christodoulou, ''Mathematical Problems of General Relativity I''. European Mathematical Society, 2008. Ch.4 §2 "Sketch of the proof of the global stability of Minkowski spacetime".
	* Yvonne Choquet-Bruhat, ''General Relativity and the Einstein Equations''. Oxford University Press, 2011. See Ch.XV §5 "Christodoulou-Klainerman theorem" which notes the Bach tensor is the "dual of the Coton tensor which vanishes for conformally flat metrics".
	* Thomas W. Baumgarte, Stuart L. Shapiro, ''Numerical Relativity: Solving Einstein's Equations on the Computer''. Cambridge University Press, 2010. See Ch.3.

	~~[[Category:Tensors]]~~
	~~[[Category:Tensors in general relativity~~]]


	~~{{differential-geometry-stub}}~~
	~~{{relativity-stub}}~~

115.185.93.198: /* Definition and properties */

2013-10-25T05:48:55Z

Definition and properties

← Older revision		Revision as of 07:48, 25 October 2013
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	~~Nice to meet you~~, ~~my title~~ is ~~Figures Held although I don~~'~~t really like becoming known over the counter~~ [http://~~203~~.~~250~~.78.~~160~~/~~zbxe~~/?~~document_srl=810524&mid~~=~~gallery~~&~~sort_index~~=~~regdate&order_type=desc std testing at home~~] ~~test as like~~ that. ~~One~~ of the ~~extremely best things in the globe for me is to do aerobics~~ and ~~I've been~~ [http://www.~~webmd~~.~~com~~/~~sexual~~-~~conditions~~/~~std-tests performing~~] ~~it for fairly a whilst. I used to be unemployed but now I am a librarian~~ and the ~~salary has been truly satisfying. Puerto Rico is where he~~'~~s always been residing but she needs~~ [~~http~~:/~~/www~~.~~eddysadventurestore~~.~~nl/nieuws/eliminate~~-~~candida~~-~~one~~-~~these-tips at home std test] std test kit to transfer because~~ of ~~her family~~.~~<br><br>My web site~~ :: [~~http~~:~~//calvaryhill.net/xe/board_DsjO50/23648 std home test~~]		In [[differential geometry]] and [[general relativity]], the '''Bach tensor''' is a trace-free [[tensor]] of rank 2 which is [[conformally invariant]] in dimension {{nowrap\|1=''n'' = 4}}.<ref>Rudolf Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs", ''Mathematische Zeitschrift'', '''9''' (1921) pp. [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002365812&IDDOC=16903 110].</ref> Before 1968, it was the only known conformally invariant tensor that is [[algebraically independent]] of the [[Weyl tensor]].<ref>P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
			Vol. 304, No. 1476 (Apr. 2, 1968), pp. [http://www.jstor.org/pss/2416002 113]–122</ref> In [[abstract index notation\|abstract indices]] the Bach tensor is given by
			:<math>B_{ab} = P_{cd}{{{W_a}^c}_b}^d+\nabla^c\nabla_aP_{bc}-\nabla^c\nabla_cP_{ab}</math>
			where ''<math>W</math>'' is the [[Weyl tensor]], and ''<math>P</math>'' the [[Schouten tensor]] given in terms of the [[Ricci tensor]] ''<math>R_{ab}</math>'' and [[scalar curvature]] ''<math>R</math>'' by

			:<math>P_{ab}=\frac{1}{n-2}\left(R_{ab}-\frac{R}{2(n-1)}g_{ab}\right).</math>

			==See also==
			*[[Cotton tensor]]
			*[[Obstruction tensor]]

			== References ==
			{{Reflist}}

			==Further reading==
			* Arthur L. Besse, ''Einstein Manifolds''. Springer-Verlag, 2007. See Ch.4, §H "Quadratic Functionals".
			* Demetrios Christodoulou, ''Mathematical Problems of General Relativity I''. European Mathematical Society, 2008. Ch.4 §2 "Sketch of the proof of the global stability of Minkowski spacetime".
			* Yvonne Choquet-Bruhat, ''General Relativity and the Einstein Equations''. Oxford University Press, 2011. See Ch.XV §5 "Christodoulou-Klainerman theorem" which notes the Bach tensor is the "dual of the Coton tensor which vanishes for conformally flat metrics".
			* Thomas W. Baumgarte, Stuart L. Shapiro, ''Numerical Relativity: Solving Einstein's Equations on the Computer''. Cambridge University Press, 2010. See Ch.3.

			[[Category:Tensors]]
			[[Category:Tensors in general relativity]]


			{{differential-geometry-stub}}
			{{relativity-stub}}

en>Duoduoduo: replacing false informal definition (e.g. the first image in the lede is a counterexample) with correct one

2012-04-16T18:05:34Z

replacing false informal definition (e.g. the first image in the lede is a counterexample) with correct one

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