wiktionary productProduct may refer to Business Product business , an item that ideally satisfies a market s want or need Product breakdown structure Product project management , a deliverable or set of deliverables that contribute to a business solution Nanoproduct , a product that has been enhanced with nanotechnology Sciences Product biology , something manufactured by an enzyme from its substrate Product chemistry , a substance found when a chemical reaction ends Product mathematics , the result of multiplying Product category theory , a generalisation of products Arts and entertainment Product Brand X album Product Brand X album , 1979 Product De Press album Product De Press album , 1982 Product, a three CD compilation set by Buzzcocks .the .product, a notable 64K demo by the demogroup Farbrausch disambiguation bs Proizvod vor ca Producte cs Produkt da Produkt de Produkt et Toode es Producto eu Produktu argipena fr Produit gl Produto ia Producto disambiguation it Prodotto lt Produktas reik m s nl Product no Produkt pl Produkt pt Produto ro Produs ru simple Product sk Produkt sr sv Produkt th tr r n uk ur ... more details
Unreferenced date August 2008 In mathematics , the tensor bundle of a manifold is the direct sum of vector bundles direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection mathematics connection is needed. Category Vector bundles geometry stub ru ... more details
Unreferenced date December 2009 In tensor analysis , a mixed tensor is a tensor which is neither strictly Covariance and contravariance of vectors covariant nor strictly Covariance and contravariance of vectors contravariant at least one of the indices of a mixed tensor will be a subscript covariant and at least one of the indices will be a superscript contravariant . A mixed tensor of type math begin pmatrix M N end pmatrix math , also written type M , N , with both M 0 and N 0, is a tensor which has M contravariant indices and N covariant indices. Such tensor can be defined as a linear operator linear function which maps an M N tuple of M one form s and N Vector geometry vector s to a scalar ... other by the covariance contravariance of their indices. A given contravariant index of a tensor can be lowered using the metric tensor g sub sub , and a given covariant index can be raised using the inverse metric tensor g sup sup . Thus, g sub sub could be called the index lowering operator and g sup sup the index raising operator . Generally, the covariant metric tensor, contracted with a tensor of type M , N , yields a tensor of type math M 1,N 1 math , whereas its contravariant inverse, contracted with a tensor of type math M,N math , yields a tensor of type math M 1,N 1 math . Examples As an example, a mixed tensor of type 1, 2 can be obtained by raising an index of a covariant tensor of type 0, 3 , math T alpha beta tau T alpha beta gamma , g gamma tau math , where math T alpha beta tau math is the same tensor as math T alpha beta gamma math , because math T alpha beta ... Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding ... of the metric tensor will be equal to the Kronecker delta, which will also be mixed. See also Covariance and contravariance of vectors Tensor intrinsic definition Two point tensor DEFAULTSORT Mixed Tensor Category Tensors ... more details
A Killing tensor , named after Wilhelm Killing , is a tensor , known in the theory of general relativity , math K math that satisfies math nabla alpha K beta gamma 0 , math where the parentheses on the indices refer to the symmetric tensor symmetric part . This is a generalization of a Killing vector. While Killing vectors are associated with continuous symmetries more precisely, differentiable , and hence very common, the concept of Killing tensor arises much less frequently. The Kerr metric Kerr solution is the most famous example of a semi Riemannian manifold manifold possessing a Killing tensor. See also Killing form Killing vector field Wilhelm Killing Category Riemannian geometry ... more details
External links date October 2009 Tensor software is a class of mathematical software designed for manipulation and calculation with tensor s. Standalone open source software http www.aei.mpg.de peekas ... encountered in field theory. It has extensive functionality for tensor polynomial simplification including ... Maxima is a free open source computer algebra system which can be used for tensor algebra calculations ... without defining all components of the tensor explicitly . It comes with three tensor packages itensor for abstract indicial tensor manipulation, ctensor for component defined tensors, and atensor for algebraic tensor manipulation. http maxima.sourceforge.net docs manual en maxima 27.html SEC90 ... is a tensor analysis system written for the Mathematica system. It provides more than 250 ... is a tensor package written for the Mathematica system. It provides many functions relevant for General ... lee Ricci Ricci is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free. http baldufa.upc.es xjaen ttc index.htm TTC Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds. http www.inp.demokritos.gr sbonano RGTC EDC and RGTC Exterior Differential Calculus and Riemannian Geometry & Tensor Calculus are free Mathematica packages for tensor calculus especially designed but not only for general relativity ... tensor calculus package for Mathematica. http www.xact.es xAct Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions. http library.wolfram.com ... connection and the basic tensors of General Relativity from a given metric tensor. http ... geometry for Maple Software for use with Matlab http csmr.ca.sandia.gov tgkolda TensorToolbox Tensor ... analysis. Libraries http www.oonumerics.org FTensor FTensor is a high performance tensor library written ... tensor library implemented in C used in Dynare . The library allows for folded unfolded, dense ... more details
A by product is a secondary product derived from a manufacturing process or chemical reaction. It is not the primary product or service being produced. A by product can be useful and marketable or it can be considered waste. International Energy Agency IEA offers the following definition for the purpose of life cycle assessment ref http www.ieabioenergy task38.org systemdefining biomitre technical manual.pdf BIOMITRE Technical Manual, Horne, R. E. and Matthews, R., November 2004 ref ... main products, co products which involve similar revenues to the main product , by products which result in smaller revenues , and waste products which provide little or no revenue . Major by products Animal sources blood meal from slaughterhouse operations poultry by product meal clean parts of the carcass of slaughtered poultry, such as necks, feet, undeveloped eggs, and intestines chrome shavings from a stage of leather manufacture collagen and gelatin from the boiled skin and other parts of slaughtered livestock feather s from poultry processing feather meal from poultry processing fetal pig s lanolin from the cleaning of wool leather hides and skins from slaughterhouse operations processed via the leathermaking process manure from animal husbandry meat and bone meal from the rendering food processing rendering of animal bones and offal poultry litter swept from the floors of chicken coops whey from cheese manufacturing Vegetation acidulated soap stock from the refining of vegetable oil bagasse the fibrous residue remaining after sugarcane or sorghum stalks are crushed to extract their juice black liquor from the production of Pulp paper cellulose pulp using the Kraft process bran and cereal germ germ from the milling of whole grain s into refined grains brewer s yeast from ethanol fermentation ... from wastewater treatment waste heat from electricity production and usage See also By product synergy ... ja no Biprodukt pt Subproduto simple By product fi Sivutuote sv Biprodukt th ... more details
citation style date February 2011 Infobox musical artist name The Product image Theproductandlogo.jpg background group or band origin Detroit, Michigan genre Alternative Rock website URL productrock.com years active Start date 2009 Present current members B.J. Perry br R.J. Perry br Rich Bennett br Charlie Jewell br past members Jerome Reilly The Product is an Alternative Rock band based out of Detroit, Michigan . The Product formed in early 2009 with the original line up of B.J. Perry, R.J. Perry, Rich Bennett and Jerome Reilly. Several months later the band parted ways with Jerome, and picked up former band mate and high school friend Charlie Jewell. They have released one E.P. titled Break The Silence along with several singles. History Break The Silence 2009 Their debut EP, Break The Silence, was self released in 2009 and generated an instant buzz throughout the Midwest. ref name Biography http restlessmanagement.com ?p 106 , Biography ref The track Nightmare stayed at Number 1 on Alternative Addiction s top ten Unsigned Bands list for several months. The band went on to tour the Midwest extensively and sold over 4,000 copies of BTS. Also, the track from the E.P. Better Off This Way had a clip appear in the hit MTV show Jersey Shore ref name Jersey Shore http www.mtv.com videos jersey shore season 2 ep 18 drunk punch love 1656631 playlist.jhtml series 2211&seriesId 29241&channelId 1 , Jersey Shore ref One Early 2010 The band covered the song One Harry Nilsson song One by Harry Nillson , and more famously covered by Three Dog Night , in 2010. ref name One http www.facebook.com ... 01 the product debuts official theme song for tna final resolution on tnastars com , TNA Wrestling Theme ... to be re released on The Product s follow up release So Alive expected in 2011. The Product was also ... site Daily Unsigned. ref name Daily Unsigned http dailyunsigned.com 2010 12 01 the product , Daily Unsigned ... musicnews article 1911 The Product Ready To Make Their Move , Greg Archilla ref Tour The Product ... more details
Unreferenced date January 2009 In general relativity , the tidal tensor or gravitoelectric tensor is one of the pieces in the Bel decomposition of the Riemann tensor . It is physically interpreted as giving the tidal stresses on small bits of a material object which may also be acted upon by other physical forces , or the tidal accelerations of a small cloud of test particle s in a vacuum solution or electrovacuum solution . Category Tensors in general relativity relativity stub ... more details
Expert subject Physics date December 2008 Unreferenced date December 2008 In general relativity , the magnetogravitic tensor is one of the three pieces appearing in the Bel decomposition of the Riemann tensor . The magnetogravitic tensor can be interpreted physically as a specifying possible spin spin force s on spinning bits of matter, such as spinning test particle s. See also Papapetrou Dixon equations Curvature invariant s References reflist Category Tensors in general relativity relativity stub ... more details
In differential geometry , the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold &mdash see also Curvature of Riemannian manifolds the curvature of an affine connection or covariant derivative on tensors the curvature form of an Ehresmann connection see Ehresmann connection , connection principal bundle or connection vector bundle . See also Tensor disambiguation mathdab zh ... more details
FeatureDetectionCompVisNavbox In mathematics, the structure tensor , also referred to as the second moment ..., and the degree to which those directions are coherent. The structure tensor is often used in image ... 251 year 1989 ref The 2D structure tensor Continuous version For a function math I math of two variables p x , y , the structure tensor is the 2 2 matrix math S w p begin bmatrix int w r I x p r 2 ... as the matrix product math nabla I nabla I math , where math nabla I math denotes the 2 1 single column transpose of the gradient. Note however that the structure tensor math S w p math cannot be factored ... p is a pair of integer indices. The 2D structure tensor at a given pixel is usually taken to be the discrete ... from by math I math by finite difference formulas. The formula of the structure tensor can be written ... The importance of the 2D structure tensor math S w math stems from the fact that its eigenvalue ... tensor they are properly added together. ref cite journal author T. Brox, J. Weickert, B. Burgeth and P ... that is, increasing its variance , one can make the structure tensor more robust in the face of noise ... how the multi scale second moment matrix structure tensor defines a true and uniquely determined multi ... in more detail below, where it is shown that a multi scale formulation of the structure tensor, referred to as the Structure tensor The multi scale structure tensor multi scale structure tensor , constitutes ... of the window function . The 3D structure tensor Definition The structure tensor can be defined also ... 240px Ellipsoidal representation of the 3D structure tensor. In particular, if the ellipsoid is stretched ... cellborder 0px border 0px tr valign top td Image STsurfel.png thumb 180px The structure tensor ellipsoid ... regions of a 3D image. td td Image StepPlane3DST.png thumb 180px The corresponding structure tensor ... 180px The structure tensor of a line like neighborhood curvel , where math lambda 1 approx lambda ... of a 3D image. td td Image curve3DST.png thumb 180px The corresponding structure tensor ellipsoid. td ... more details
In differential geometry and general relativity , the Bach tensor is a tensor of rank 2 which is conformally invariant in dimension n 4. It is 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 . References Reflist Category Tensors Category Tensors in general relativity geometry stub relativity stub ru ... more details
Multiple issues notability August 2009 refimprove August 2009 Image Tensor logo.png thumb right Tensor Trucks logo Tensor Trucks is a skateboarding truck company founded and designed by professional skateboarder Rodney Mullen in 2000 . Tensor s parent company is Dwindle Distribution . They offer trucks in three different heights lo, mid, hi tailored for differing wheel diameters the hi is designed for 58mm wheels and smaller, the mid for 54mm or smaller, and the lo for 52 and smaller. Mullen holds US patent no. 6,443,471B1 ref http patft.uspto.gov netacgi nph Parser?Sect1 PTO1&Sect2 HITOFF&d PALL&p 1&u 2Fnetahtml 2FPTO 2Fsrchnum.htm&r 1&f G&l 50&s1 6443471.PN.&OS PN 6443471&RS PN 6443471 United States Patent 6,443,471 ref for design features implemented in Tensors. Tensor trucks are manufactured in China. Tensor has also teamed up with Oust Skateboard Bearings to put out a co branded Oust Tensor line of skateboard bearings, as well as working on experimental Tensor trucks with enhanced parts. Tensor released a new truck called the Response in March 2007. It is an all metal design, forgoing the plastic baseplate sliders of the original design, and is touted as the lightest truck, 11 ... Dwindle.com ref ref http www.tensortrucks.com response Tensor Trucks dead link date January 2011 ref In 2008 Tensor released their lightest design yet using magnesium and touted to be 25 lighter than the industry average truck. Key design features Baseplate sliders The most visible feature on a Tensor ... standard aluminum baseplates. Interlocking bushings Tensor bushings or cushions use a patented design .... Baseplate nibs Tensor features four fangs on each baseplate designed to dig into the board to prevent the truck from shifting when mounting hardware loosens. Buttonhead kingpin Tensor implemented ... Song . ref http www.tensortrucks.com Tensor Team ref Notable ex team members Citation needed date ... Oust Bearings Category Skateboarding companies nl Tensor Trucks ... more details
properties of the inner product of Vector geometry vectors in Euclidean space . In the same way as an inner product, a metric tensor is used to define the length of and angle between tangent ... product of two tangent vectors Another interpretation of the metric tensor, also considered by Gauss .... In contemporary terms, the metric tensor allows one to compute the dot product of tangent vectors ... M can support such a structure. Metric as a section of a bundle By the Tensorproduct Universal property universal property of the tensorproduct , any bilinear mapping EquationNote 5 gives rise natural ... math where math tau TM otimes TM stackrel cong to TM otimes TM math is the TensorproductTensor ... with a section fiber bundle section of the tensorproduct bundle math scriptstyle E otimes E math ... dual double dual isomorphism to a section of the tensorproduct math TM otimes TM. math Arclength ...In the mathematics mathematical field of differential geometry , a metric tensor is a type of function ... with a metric tensor is known as a Riemannian manifold . By integral integration , the metric tensor ..., the metric tensor itself is the derivative of the distance function taken in a suitable manner . Thus the metric tensor gives the infinitesimal distance on the manifold. While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci Curbastro and Tullio Levi Civita who first codified the notion of a tensor. The metric tensor is an example of a tensor field , meaning that relative to a local coordinate system on the manifold, a metric tensor takes on the form of a symmetric matrix whose entries transform ... a metric tensor is a covariant symmetric tensor . From the coordinate independent point of view, a metric tensor is defined to be a nondegenerate form nondegenerate symmetric bilinear form on each ... more details
In differential geometry , the Einstein tensor also trace reversed Ricci tensor , named after Albert ... tensor occurs in the Einstein field equations for gravitation describing spacetime curvature in a manner consistent with energy considerations. Definition The Einstein tensor math mathbf G math is a rank 2 tensor defined over Riemannian manifold s. In index free notation it is defined as math mathbf G mathbf R frac 1 2 mathbf g R, math where math mathbf R math is the Ricci tensor , math mathbf g math is the metric tensor and math R math is the scalar curvature . In component form, the previous equation reads as math G mu nu R mu nu 1 over2 g mu nu R. math The Einstein tensor is symmetric math G mu nu G nu mu , math and, like the stress energy tensor , divergenceless math G mu nu nu 0 ,. math Explicit form The Ricci tensor depends only on the metric tensor , so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor ... align math where math delta alpha beta math is the Kronecker tensor and the Christoffel symbol math ... derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably ... denote Antisymmetric tensor antisymmetrization over bracketed indices, i.e. math g alpha beta, gamma ... linear algebra trace of the Einstein tensor can be computed by Tensor contraction contract ing the equation in the Definition definition with the metric tensor math g mu nu math . In math n math dimensions ... 3 space, 1 time gives math G , math , the trace of the Einstein tensor, as the negative of math R , math , the Ricci tensor s trace. Thus another name for the Einstein tensor is the trace reversed Ricci tensor . Use in general relativity The Einstein tensor allows the Einstein field equations without ... explicit form of the Einstein tensor , the Einstein tensor is a nonlinear function of the metric tensor ... more details
In mathematics, the term symmetric product can refer to The Symmetric tensor Symmetric part of a tensor symmetric part of a tensor The symmetric product of an algebraic curve The infinite symmetric product SP X of a space X in algebraic topology disambig ... more details
concretely as a tensor , or as a vector valued form vector valued two form on the manifold. If is an affine connection on a differential manifold , then the torsion tensor is defined, in terms ... of Einstein Cartan theory . The torsion tensor Let M be a manifold with a connection on the tangent bundle. The torsion tensor sometimes called the Cartan torsion tensor is a vector valued form ... rule generalized product rule Leibniz rule , T fX , Y T X , fY fT X , Y for any smooth function f . So ... and the Bianchi identities The Riemann curvature tensor curvature tensor of is a mapping T M ... are extended to vector fields away from the point thus it defines a tensor, much like the torsion ... nabla XR Y,Z R T X,Y ,Z right 0 math Components of the torsion tensor The components of the torsion tensor math T c ab math in terms of a local basis of a vector space basis of section fiber bundle ... geodesic equations determine the symmetric part of the connection, the torsion tensor determines ... e i mathbf e i, mathbf e j math are the frame components of the torsion tensor, as given in the previous ..., is a tensor of type 1,2 carrying one contravariant and two covariant indices . Alternatively, the solder ... form for further details. Irreducible decomposition The torsion tensor can be decomposed into two ... of T is then math T 0 T frac 1 n 1 iota operatorname tr ,T math where denotes the interior product ... away from p . By the Leibniz product rule, one sees that does not actually depend on how X and Y are extended so it defines a tensor on M . Let S and A be the symmmetric and alternating parts of ... method . See also Curvature tensor Contortion tensor Levi Civita connection Torsion of curves Notes reflist References citation last1 Bishop first1 R.L. last2 Goldberg first2 S.I. title Tensor ... year 1999 isbn 0914098713 curvature DEFAULTSORT Torsion Tensor Category Differential geometry ... more details
In differential geometry , the Cotton tensor on a pseudo Riemannian manifold of dimension n is a third order tensor field tensor concomitant of the metric tensor metric , like the Weyl curvature Weyl tensor . The concept is named after mile Cotton . Just as the vanishing of the Weyl tensor for n ... flat , the same is true for the Cotton tensor for n 3, while for n 3 it is identically zero. In coordinates, and denoting the Ricci tensor by R sub ij sub and the scalar curvature by R , the components of the Cotton tensor are math C ijk nabla k R ij nabla j R ik frac 1 2 n 1 left nabla j Rg ik nabla k Rg ij right . math The Cotton tensor can be regarded as a vector valued ... order trace free tensor density math C i j nabla k left R li frac 1 4 Rg li right epsilon klj , math sometimes called the Cotton York tensor . The proof of the classical result that for n 3 the vanishing of the Cotton tensor is equivalent the metric being conformally flat is given .... This tensor density is uniquely characterized by its conformal properties coupled with the demand ... math is the tensor math S alpha beta gamma delta lambda mu partial beta omega delta lambda beta partial mu omega g beta mu partial lambda omega math The Riemann curvature tensor transforms as math widetilde ... manifolds, we obtain the Ricci tensor by contracting the transformed Riemann tensor to see ... math Combining all these facts together permits us to conclude the Cotton York tensor transforms as math ... W, math where the gradient is plugged into the symmetric part of the Weyl tensor W . Symmetries The Cotton tensor has the following symmetries math C ijk C ikj , math and therefore math C ijk 0. , math In addition the Bianchi formula for the Weyl tensor for can be rewritten as math delta W ... reflist Cite journal first S. J. last Aldersley title Comments on certain divergence free tensor ... origyear 1925 year 1977 isbn 0691080267 A. Garcia, F.W. Hehl, C. Heinicke, A. Macias, The Cotton tensor ... more details
In mathematics and theoretical physics , a tensor is antisymmetric on two indices i and j if it alternates Sign mathematics sign when the two indices are interchanged ref cite book author K.F. Riley, M.P. Hobson, S.J. Bence title Mathematical methods for physics and engineering publisher Cambridge University Press year 2010 isbn 978 0 521 86153 3 ref math T ijk dots T jik dots math An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric. If a tensor changes sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form . Antisymmetric and symmetric tensors A tensor A which is antisymmetric on indices i and j has the property that the Tensor contraction contraction with a tensor B , which is symmetric on indices i and j , is identically 0. For a general tensor U with components math U ijk dots math and a pair of indices i and j , U has symmetric and antisymmetric parts defined as math U ij k dots frac 1 2 U ijk dots U jik dots math symmetric part math U ij k dots frac 1 2 U ijk dots U jik dots math antisymmetric part Similar definitions can be given for other pairs of indices. As the term part suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in math U ijk dots U ij k dots U ij k dots math Notation A shorthand notation for anti symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for a rank 2 covariant tensor M, math M ab frac 1 2 M ab M ba ,, math and for a rank 3 covariant tensor T, math T abc frac 1 3 T abc T acb T bca T bac T cab T cba ,. math ... c 1 dots c i b 1 dots b n i S c 1 dots c i ,. math Example An important antisymmetric tensor in physics is the electromagnetic tensor F in electromagnetism . See also Symmetric tensor Antisymmetric matrix ... ko he nl Anti symmetrische tensor pt Tensor antissim trico ru ... more details
Expert subject Physics date November 2008 There are two different tensor s sometimes referred to as the Lanczos tensor both named after Cornelius Lanczos A tensor in the theory of quadratic Lagrangian s, which vanishes in Four dimensional space four dimensions . The potential tensor H for the Weyl tensor C , this can be expressed as math C abcd H abc d H abd c H cda b H cdb a , math math g ac H bd H db g ad H bc H cb g bd H ac H ca g bc H ad H da 2 , math math 2H ef e f g ac g bd g ad g bc 3, , math where the Lanczos tensor has the symmetries math H abc H bac 0, , math math H abc H bca H cab 0, , math and where math H bd math is defined by math H bd stackrel mathrm def H e b d e H e b e d . math Thus, the Lanczos potential tensor is a gravitational field analog of the vector potential A for the electromagnetic field . See also Introduction to 2 spinors in general relativity World Scientific, 2003 by Peter O Donnell for a more detailed discussion of the Lanczos tensor and spinor. External links http www.worldscibooks.com physics 5222.html Introduction to 2 spinors in general relativity http www.arXiv.org abs gr qc 9904006 gr qc 9904006 Category Tensors Category Differential geometry Category Tensors in general relativity relativity stub ... more details
multiple issues context September 2011 no footnotes September 2011 one source September 2011 orphan September 2011 technical September 2011 In Scientific Visualization a Tensor Glyph is an object that can visualize all or most of the nine degrees of freedom, such as acceleration , twist or shear of a math 3 times 3 math matrix. It is used for tensor field visualization, where a data matrix is available at every point in the grid. There are certain types of glyphs that can be used ellipsoid cuboid cylindrical superquadrics References reflist http www.cs.utah.edu gk papers vissym04 Superquadric Tensor Glyphs Images and Examples Category Computer graphics Category Scientific modeling Category Visualization graphic ... more details
The contorsion tensor in differential geometry expresses the difference between a metric compatible affine connection with Christoffel symbol math Gamma ij k math and the unique torsion free Levi Civita connection for the same metric. The contortion tensor math K ab c math is defined in terms of the torsion tensor math T ij k Gamma ij k Gamma ji k math as math K ijk frac 1 2 T ijk T jki T kij , math where the indices are being raised and lowered with respect to the metric math T ijk equiv g kl T ij l math . The reason for the non obvious sum in the definition is that the contortion tensor, being the difference between two metric compatible Christoffel symbols, must be antisymmetric in the last two indices, whilst the torsion tensor itself is antisymmetric in its first two indices. The connection can now be written as math Gamma kj i bar Gamma kj i K kj i, math where math bar Gamma kj i math is the torsion free Levi Civita connection. Category Tensors ... more details
Polder tensor is a tensor used in description of magnetic Permeability electromagnetism permeability of ferrite s. ref http www.nature.com nature journal v182 n4642 abs 1821080a0.html G. G. Robbrecht, J. L. Verhaeghe, Measurements of the Permeability Tensor for Ferroxcube , Letters to Nature, Nature 182, 1080 18 October 1958 , doi 10.1038 1821080a0 ref The tensor notation needs to be used because ferrimagnetic materials become anisotropy anisotropic in the presence of magnetizing field. The tensor is described mathematically as ref http books.google.co.uk books?id lqHsnZoa7wAC&pg PA93&dq polder tensor&cd 5 v onepage&q polder 20tensor&f false Ricardo Marqu s, Ferran Mart n, Mario Sorolla, Metamaterials with negative parameter theory, design, and microwave applications, Willey, New Jersey, USA, 2009, ISBN 978 0 471 7458 2, page 93 ref math B begin bmatrix mu & j kappa & 0 j kappa & mu & 0 0 & 0 & mu 0 end bmatrix H math where math mu mu 0 left 1 frac omega 0 omega m omega 0 2 omega 2 right math math kappa mu 0 frac omega omega m omega 0 2 omega 2 math math omega 0 gamma mu 0 H 0 math math omega m gamma mu 0 M math and math gamma 17.6 cdot g math kHz A m is a gyromagnetic ratio and g is a factor between 1.9 2.4 depending on ferrite material. Magnetizing frequency f is expressed as math omega 2 pi f math , H sub 0 sub is a bias field, M is magnetization and math mu 0 math is magnetic permeability of free space . References Reflist Use dmy dates date September 2010 DEFAULTSORT Polder Tensor Category Ferrites ... more details
The gyration tensor is a tensor that describes the second moment mathematics moment s of position of a collection of Elementary particle particle s math S mn stackrel mathrm def frac 1 N sum i 1 N r m i r n i math where math r m i math is the math mathrm m th math Cartesian coordinate system Cartesian coordinate of the position vector geometric vector math mathbf r i math of the math mathrm i th math particle. The origin mathematics origin of the coordinate system has been chosen such that math sum i 1 N mathbf r i 0 math i.e. in the system of the center of mass math r CM math . Where math r CM frac 1 N sum i 1 N mathbf r i math In the continuum limit, math S mn stackrel mathrm def int d mathbf r rho mathbf r r m r n math where math rho mathbf r math represents the number density of particles at position math mathbf r math . Although they have different units, the gyration tensor is related to the moments of inertia moment of inertia tensor . The key difference is that the particle positions are weighted by mass in the inertia tensor, whereas the gyration tensor depends only on the particle positions mass plays no role in defining the gyration tensor. Thus, the gyration tensor would be proportional to the inertial tensor if all the particle masses were identical. Diagonalization Since the gyration tensor is a symmetric 3x3 matrix mathematics matrix , a Cartesian coordinate system can be found in which it is diagonal math mathbf S begin bmatrix lambda x 2 & 0 & 0 0 & lambda y 2 & 0 0 & 0 & lambda z 2 end bmatrix math where the axes are chosen such that the diagonal elements are ordered math lambda x 2 leq lambda y 2 leq lambda z 2 math . These diagonal elements are called the principal moments of the gyration tensor. Shape descriptors The principal moments can be combined to give several parameters that describe the distribution of particles. The squared radius of gyration ... Article Feedback 5 pl Tensor yracji ... more details
context date January 2010 In mathematics, a recurrent tensor , with respect to a connection mathematics connection math nabla math on a manifold M , is a Tensor field tensor T for which there is a differential form one form on M such that math nabla T omega otimes T. , math Examples Parallel Tensors An example for recurrent Tensor field tensor s are parallel tensors which are defined by math nabla A 0 math with respect to some connection math nabla math . If we take a pseudo Riemannian manifold math M,g math then the metric g is a parallel and therefore recurrent tensor with respect to its Levi Civita connection , which is defined via math nabla LC g 0 math and its property to be torsion free. Parallel vector fields math nabla X 0 math are examples of recurrent tensors that find importance in mathematical research. For example, if math X math is a recurrent non null vector field on a pseudo Riemannian manifold satisfying math nabla X omega otimes X math for some closed one form math omega math , then X can be rescaled to a parallel vector field ref Alekseevsky, Baum 2008 ref . In particular, non parallel recurrent vector fields are null vector fields. Metric space Another example appears in connection with Weyl structure s. Historically, Weyl structures emerged from the considerations of Hermann Weyl with regards to properties of parallel transport of vectors and their length ref Weyl 1918 ref . By demanding that a manifold have an affine parallel transport in such a way that the manifold is locally an affine space , it was shown that the induced connection had a vanishing torsion tensor math T nabla X,Y nabla XY nabla YX X,Y 0 math . Additionally, he claimed that the manifold ... tensor with respect to math nabla math . As a result, Weyl called the resulting manifold math M,g math ... spacetime One more example of a recurrent tensor is the curvature tensor math mathcal R math on a recurrent ... Recurrent Tensor Category Riemannian geometry Category Tensors de Rekurrenter Tensor ... more details
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