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	Einstein notation - Wikipedia, the free encyclopedia
		In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae.
		According to this convention, when an index variable appears twice in a single term, once in an upper and once in a lower position, it implies that we are summing over all of its possible values.
		In Einstein notation, an index that is repeated twice in an equation implies a summation, and the summation symbol need not be included.
	en.wikipedia.org /wiki/Einstein_notation (611 words)

	PlanetMath: Einstein summation convention (Site not responding. Last check: 2007-10-08)
		The Einstein summation convention imply that when an index occurs more than once in the same expression, the expression is implicitly summed over all possible values for that index.
		Therefore, in order to use the summation convention, it must be clear from the context over what range indices should be summed.
		This is version 1 of Einstein summation convention, born on 2003-03-27.
	planetmath.org /encyclopedia/EinsteinSummationConvention.html (277 words)

	Einstein notation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Sometimes (as in (A generalization of special relativity to include gravity (based on the principle of equivalence)) general relativity), the index is required to appear once as a superscript and once as a subscript; in other applications, all indices are subscripts.
		The innovation of the Einstein notation is the recognition that an index that is repeated twice in an equation implies a summation, and the summation symbol need not be included.
		The value of the Einstein convention is that it applies to other vector spaces built from V using the (Click link for more info and facts about tensor product) tensor product and ((geometry) the interchangeability of the roles of points and planes in the theorems of projective geometry) duality.
	www.absoluteastronomy.com /encyclopedia/e/ei/einstein_notation.htm (1687 words)

	Einstein Summation Conventions (Site not responding. Last check: 2007-10-08)
		Einstein's summation convention is a way of dealing with tensors in a compact and consistent way.
		The idea here is to use indices to describe a generic element and apply tensor algebra to calculate the required quantities.
		Another typical application is the use of the Einstein's summations conventions when evaluating cross products.
	web.mit.edu /8.21/www/notes/notes/node6.html (198 words)

	PlanetMath: summation (Site not responding. Last check: 2007-10-08)
		It must be noted that, although the running variable usually takes integer values, the summation function needs not, and it can lie on any algebraic structure where a sum is defined.
		We now give formulas for evaluating many common summations, which can be combined using the mentioned properties to evaluate a wide range of sums.
		This is version 7 of summation, born on 2004-10-12, modified 2005-03-17.
	planetmath.org /encyclopedia/Sum.html (1061 words)

	Ricci curvature - Wikipedia, the free encyclopedia
		For instance, Einstein manifolds do not have to have constant curvature in dimensions 4 and up.
		An explicit expression for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols.
		Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
	en.wikipedia.org /wiki/Ricci_tensor (651 words)

	MAT 392 Lecture Notes -- Summation and Matrix Notation (Site not responding. Last check: 2007-10-08)
		An innovation due to Einstein, not as famous as E=mc^2 but more widely used, is to decree that whenever an index appears as both a superscript and a subscript we are to sum over all possible values.
		All of the preceding conventions are fine for abstract algebraic calculations, but sooner or later we must replace symbols like a and x by tables of numbers.
		The convention I have used so far, and will continue to use, is that the upper index gives the row number while the lower index gives the column number.
	www.math.princeton.edu /~stalker/392s00/notation.html (538 words)

	Tensor/Old: Definition and Links by Encyclopedian.com - All about Tensor/Old
		They were resurrected from obscurity by Einstein in order to formulate General relativity in such a way that the physical laws that were described were independent of the coordinate system chosen.
		Throughout we will use the Einstein summation convention in which repeated indices imply a sum over components.
		The reader should already be familiar with vector spaces and their properties.
	www.encyclopedian.com /te/Tensor---Old.html (483 words)

	Stress-energy tensor - Wikipedia, the free encyclopedia
		Please note that throughout we will assume the use of the Einstein summation convention also x
		There is in fact no way to define a global energy-momentum vector in a general curved spacetime.
		In General Relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as
	en.wikipedia.org /wiki/Stress-energy_tensor (753 words)

	General Relativity (Site not responding. Last check: 2007-10-08)
		and to use the Einstein summation convention that a sum is understood over repeated indices.
		We introduce the convention that an index that has a superscript in the denominator in a derivative, as in Eq.
		The curvature tensor is applied in the Einstein equation of the gravitational field, and is used to describe the motion of particles in curved spacetime.
	www.nikhef.nl /~henkjan/astro/node13.html (1621 words)

	Orthogonal Transformations (Site not responding. Last check: 2007-10-08)
		If we change notation then Einstein's summation convention can be used, i.e.
		Einstein's Summation Convention: If an index appears twice in a term, once as a superscript and once as a subscript, then summation is implied over the range that the indices are allowed to take on.
		The summation convention can now be used to express A
	www.geocities.com /physics_world/ma/orthog_trans.htm (515 words)

	9.4.1 Some rules of tensor analysis on manifolds
		Einstein summation convention: Repeated indices are summed, unless otherwise noted.
		Sometimes it is useful to write the transformation matrix in traditional matrix form.
		The convention is that the index which is placed a bit closer to the
	www.gfdl.gov /%7Esmg/MOM/web/guide_parent/s2node107.html (738 words)

	Chiral Mechanics
		The usual Einstein summation convention for repeated indices is used and the comma denotes differentiation with respect to spatial coordinates.
		In terms of the macrostrain and conventional notation for the elastic constants, these may be written as in the original article.
		These are the constitutive equations for a micropolar solid which is isotropic with respect to coordinate rotations but not with respect to inversions.
	silver.neep.wisc.edu /~lakes/chiral.Cmp.html (2477 words)

	outline1.html
		Einstein's equation describes how the flow of energy and momentum through spacetime affects the metric.
		As always, we follow the Einstein summation convention and sum over repeated indices when one is up and the other is down.
		The remaining nine components of the matrix describe the flow of the x, y, and z components of momentum in the x, y, and z directions.
	math.ucr.edu /home/baez/gr/outline2.html (3217 words)

	PlanetMath: Levi-Civita permutation symbol (Site not responding. Last check: 2007-10-08)
		See the entry on the generalized Kronecker symbol for details.
		When using the Levi-Civita permutation symbol and the generalized Kronecker delta symbol, the Einstein summation convention is usually employed.
		Using equation 1, we have for equation 2
	planetmath.org /encyclopedia/LeviCivitaPermutationSymbol3.html (315 words)

	Einstein notation: Definition and Links by Encyclopedian.com - All about Einstein notation
		Einstein notation: Definition and Links by Encyclopedian.com - All about Einstein notation
		Sometimes, the index is required to appear once as a superscript and once as a subscript; in other applications, all indices are subscripts.
		For example, V V, the tensor product of V with itself, has a basis consisting of tensors of the form e
	www.encyclopedian.com /ei/Einstein-notation.html (1259 words)

	Summation Convention (Site not responding. Last check: 2007-10-08)
		Einstein introduced his summation convention saying that when the same script appeared just twice in a term, once as a superscript and once as a subscript, then a summation sign for that script running over the four dimensions was to be imagined before the term.
		In the context of a curved manifold with a metric tensor, adherence to this index pattern avoids many bugs.
		The upper and lower rule ensured that only vectors from a space and its dual were “multiplied”.
	www.cap-lore.com /MathPhys/Tumble/sum.html (139 words)

	[No title]
		In the second expressionFORM made an assumption which is called the `Einstein summation convention': indices that occur twice inside the same term are considered to be summed over.
		This name convention has the great benefit of allowing FORM to work with single objects while it is clear for the user that this object stands in reality for something much more complex.
		This is a summation function which has a slightly different rule for the elements of the sum.
	thproxy.jinr.ru /file-archive/doc/other/form1_doc.txt (13112 words)

	ModPhy1
		The first step in quantifying a Riemannian space is to select as system of coordinates that span the space.
		In the following equations, we will use Einstein’s summation convention such that whenever an index is repeated in a term, once as a subscript and once as a superscript, it signifies that that term should be summed over all possible values of the index.
		The Einstein tensor is a symmetrical second rank tensor whose divergence is identically zero.
	physics.tamuk.edu /~hewett/ModPhy1/Unit1/GeneralRelativity/EinsteinGravity/EinsteinT/EinsteinT.html (724 words)

	qg5.1
		The summation convention applies when you have 2 copies of the same index: i and i, not i and j.
		Also, the summation convention only applies when one of the indices is up and the other is down.
		And that's the answer, with the Einstein summation convention telling us to sum over j.".
	math.ucr.edu /home/baez/qg-fall2000/qg5.1.html (1545 words)

	Re: Einstein and gauge theory (Site not responding. Last check: 2007-10-08)
		Einstein had Marcel Grossman, a friend from college, and a professional mathematician who was also a differential geometer!
		by the time Einstein was ready to use it, the Italian school (Levi-Civita and Ricci) had taken Riemann's invention quite a long way.
		Of course, Bianchi hadn't found his identities yet, so Einstein used physical principles to apply the needed constraints.
	www.lns.cornell.edu /spr/1999-11/msg0019701.html (222 words)

	General Relativity and Spacetime (Site not responding. Last check: 2007-10-08)
		According to Einsteins principle of general covariance, it is possible - and sometimes nescessary - to formulate the physical laws in equations, which takes the same form whatever coordinates
		This step from a description in inertial frames to a general covariant description also means, that we cannot use t as an absolute parameter any more.
		In the last expression the convenient Einstein summation convention is introduced, it says: sum over each repeated index (eg.
	www.astro.ku.dk /%7Ecramer/RelViz/text/geom_web/node2.html (900 words)

	Week 50 of 2003 (Site not responding. Last check: 2007-10-08)
		However, Einstein used it to explain the most wonderful theory of the universal: Relativity.
		A rotation in normal 3D space, Einstein summation convention,
		In the last expression, we used the Einstein summation convention: "if an expression contains the same index twice, it is implied that we should sum over this index".
	home.jtan.com /~febdian/febdian.net/diary_03wk50.htm (706 words)

	summation - OneLook Dictionary Search
		Summation : Online Plain Text English Dictionary [home, info]
		Phrases that include summation: summation by parts, summation gallop, einstein summation, einstein summation convention, poisson summation formula, more...
		Words similar to summation: addition, plus, rundown, summational, summing up, more...
	www.onelook.com /?w=summation (278 words)

	Astron. Astrophys. 346, 713-720 (1999)
		We use the Einstein summation convention, where needed, and a semicolon for the covariant derivative.
		The summation is limited to rays which reach infinity.
		To do so it is sufficient to neglect in the summation all the rays which do not enter the region of interest and multiply the result by the correction factor
	aa.springer.de /papers/9346002/2300713/sc2.htm (1623 words)

	PERTTI LOUNESTO SHRINE (Site not responding. Last check: 2007-10-08)
		Albert Einstein (1879-1955) did not contribute anything to mathematics (unless we count the Einstein summation convention).
		But in comparison with physicists of his time, Einstein's knowledge of mathematics was superior, excellent.
		More importantly, Einstein was a revolutionary visionary, a creative physicist, among physicists, one of the greatest.
	www.tiki-lounge.com /~raf/lounesto/LounestoShrine.html (1319 words)

	Re: Einstein summation convention (Site not responding. Last check: 2007-10-08)
		Baez - > > I read in sci.physics.research the post titled: > Question re origin of Einstein summation convention > > And it got me wondering why the Einstein summation > convention works in the first place.
		A Hybrid Mathematican's/Physicist's Notation (was: Einstein summation convention)
		Next by thread: A Hybrid Mathematican's/Physicist's Notation (was: Einstein summation convention)
	www.lns.cornell.edu /spr/2003-05/msg0051367.html (185 words)

	Green Paper #1
		The last definition uses the Einstein summation convention.
		The convention states that if an index is repeated (in this case i appears twice) it is summed over.
		Another convention is to name quantities where the index is low as covariant (co-is-low) and where the index is high as contravariant.
	www.pa.msu.edu /courses/2000fall/PHY491/gp2/GreenPaper2.htm (476 words)

	Re: Einstein and gauge theory (Site not responding. Last check: 2007-10-08)
		I was wondering from a few months, how much the environment where Einstein was submerged was responsible for his inventions and how much Einstein himself has shaped the surrounding people into the great scientists they are by the questions he was asking and the answers he was seeking?
		by the time > Einstein was ready to use it, the Italian school (Levi-Civita and Ricci) had > taken Riemann's invention quite a long way.
		Of course, Bianchi hadn't found > his identities yet, so Einstein used physical principles to apply the needed > constraints.
	www.lns.cornell.edu /spr/1999-12/msg0019736.html (195 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		where the Einstein summation convention, according to which one sums over repeated upper and lower indices, has been used.
		These connection coefficients are also known as the Christoffel symbols and define a covariant derivative of tensors of arbitrary rank by
		, where again the Einstein summation convention for repeated indices has been used.
	gravity.phys.psu.edu /~sperhake/Research/GRBasics/GRbasics.html (470 words)

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