Height of curvature in the tooth can be defined as the line encircling a tooth at its greatest bulge to a selected path of insertion. Locations of height of curvature For the outer surfaces of all teeth, the height of curvature is located in the cervical third of the teeth. In the inner surfaces of anterior teeth , both upper & lower teeth, the height of curvature is also located in the cervical third of the tooth. In the posterior teeth , both in upper and lower jaw, the height of contour is found at the middle third of the inner surface of the tooth. The lower second premolar proposes an exception as its height of curvature in inner surface is located in the Occlusion dentistry occlusal third of the inner surface. Functions The height of contours have great functions to mouth oral cavity They allow the food to be deflected allowing proper degree of massage to the gingiva . They prevent the food of being accumlated at the tooth. Holding the gingiva under definite tension. Mal developed height of curvature In case of under developed curvature, gingival recession may result. In case of over developed curvature, food will accumlate and there will be no massage to the gingiva and chronic inflammation may result. Unreferenced date December 2009 Category Dentistry definitions ... more details
to the theory. For example, the Ricci tensor is a non metric contraction of the Riemann curvaturetensor , and the scalar curvature is the unique metric contraction of the Ricci tensor. One can also view ...In multilinear algebra , a tensor contraction is an operation on one or more tensor s that arises from ... vector space dual . In components, it is expressed as a sum of products of scalar components of the tensor ... to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices one a subscript, the other a superscript of the tensor are set equal to each other and summed ... tensor with TensorTensor rank rank or order reduced by 2. Tensor contraction can be seen as a generalization ... from the Component free treatment of tensors Definition Tensor Product of Vector Spaces tensor product of these two spaces to the field k math C V otimes V rightarrow k math corresponding ... the contraction operation on a tensor of type 1,1 , which is an element of math V otimes V math ... obtains a basis free definition of the trace linear algebra trace . In general, a tensor of type m , n ... operation, which is a linear map which yields a tensor of type m &minus 1, n ... dual basis . Since a general mixed dyadic tensor is a linear combination of decomposable tensors ... T T i j mathbf e i e j math be a mixed dyadic tensor. Then its contraction is math T i j mathbf ... being implied by the summation convention . The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type 2,2 on the second and third indices to create a new tensor U of type 1,1 is written as math T ab bc sum b T ab bc T a1 ... dyadic tensor. This tensor does not contract if its base vectors are dotted the result is the contravariant metric mathematics metric tensor , math g ij mathbf e i cdot mathbf e j math , whose rank ... product also known as a Metric tensor metric g , such contractions are possible. One uses the metric ... more details
File Winding Number Around Point.svg thumb 300px This curve has total curvature 6 , and index turning number 3, though it only has winding number 2 about p . In mathematics mathematical study of the differential geometry of curves , the total curvature of an immersion mathematics immersed plane curve is the integral of curvature along a curve taken with respect to arclength math int a b k s ,ds. math The total curvature of a closed curve is always an integer multiple of 2 , called the index of the curve, or turning number it is the winding number of the unit tangent about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces. Comparison to surfaces This relationship between a local invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher dimensional Riemannian geometry such as the Gauss Bonnet theorem . Invariance According to the Whitney Graustein theorem , the total curvature is invariant under a regular homotopy of a curve it is the degree of the Gauss map . However, it is not invariant ... angles have well defined total curvature, interpreting the curvature as point masses at the angles. The total curvature of a curve in a higher dimensional Euclidean space equipped with its arclength ... the total curvature of the resulting curve. That is, the total curvature of a curve in n ... 1 sub is last Frenet curvature the torsion of curves torsion of the curve and sgn is the signum function . The minimum total curvature of any three dimensional curve representing a given Knot mathematics ... Curvature of Knots first John W. last Milnor authorlink John Milnor journal The Annals of Mathematics ... cite arxiv first John M. last Sullivan title Curves of finite total curvature year 2007 eprint math 0606007 class math.GT . Category Curves Category Curvature mathematics ru zh ... more details
unreferenced date March 2012 In mathematics , the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor . It is therefore a tensor field of TensorTensor rank rank three. It vanishes for the case of Riemannian geometry . Category Differential geometry Geometry stub ... more details
spacetime One more example of a recurrent tensor is the curvaturetensor math mathcal R math on a recurrent ...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 ... Recurrent Tensor Category Riemannian geometry Category Tensors de Rekurrenter Tensor ... more details
Infobox Disease Name PAGENAME Image Spinal column curvature.png Caption Different regions curvatures of the vertebral column DiseasesDB ICD10 ICD10 M 40 m 40 ICD10 M 41 m 40 , ICD10 Q 76 3 q 65 ICD10 Q 76 4 q 65 ICD9 ICD9 737 , ICD9 756.1 ICDO OMIM MedlinePlus eMedicineSubj eMedicineTopic MeshID D013121 Although spinal curvature or curvature of spine can refer to the normal concave and convex curvature of the spine, in clinical contexts, the phrase usually refers to deviations from the expected curvature, even when that difference is a reduction in curvature. Types include kyphosis , lordosis , and scoliosis . The thoracic and sacral pelvic curves develop in the fetus. Around 6 months after birth the cervical curve appears which helps hold the head up. Around one year of age the lumbar curve develops which helps with balance and walking. The cervical and lumbar curves are considered secondary curves whereas the thoracic and sacral curves are primary. medicine stub Dorsopathies Congenital malformations and deformations of musculoskeletal system Category Deforming dorsopathies Category Congenital disorders of musculoskeletal system la Curvatura columnae vertebralis ... more details
orphan date April 2010 Membrane curvature is the geometrical measure or characterization of the curvature ... Geometry of Curvature A biological membrane is commonly described as a two dimensional surface, which ... radii of curvature, and their inverse values are referred to as the two principal curvatures. ref ... ref File Curvature radii.JPG Curvature radii right thumb 300px math c1 1 R1 math math c2 1 R2 math ... shapes, such as cylinder, plane, sphere and saddle. Analysis of the principal curvature is important ... Even though often membrane curvature is thought to be a completely spontaneous process, thermodynamically speaking there must be factors actuating as the driving force for curvature to exist. Currently, there are some postulated mechanisms for accepted theories on curvature nonetheless, undoubtedly .... Driving forces for membrane Curvature Lipid Spontaneous Curvature Perhaps the most simple and intuitive driving force in membrane curvature is the natural spontaneous curvature exhibited by some ... spontaneously negative or positive curvature. Lipids such as DOPC dioleyl phosphatidyl choline , diacyl ... curvature. ref Martens, S., McMahon, H. T. Nature Reviews . 9, 543 556 2008 ref On the other hand ... words they exhibit positive spontaneous curvature. ref Kamal, M et al. Measurement of the membrane curvature preference of phospholipids reveals only weak coupling between lipid shape and leaflet curvature. PNAS 2009 vol. 106 52 pp. 22245 50 ref The table below lists experimentally determined spontaneous ... of the cylinder, J sub B sub is the difference between the spontaneous curvature, J sub s sub ... membrane . Journal of Cell Biology 148, 45 58 2000 . ref So, the spontaneous curvature necessary ... such curvature. The lipids cholesterol, dioleoylphosphatidylethanolamine DOPE and diacylglycerol ... to generate a large membrane curvature. However, even for these lipids, the required J sub B ... Induce Curvature Some biologically occurring lipids do exhibit spontaneous curvature which could explain ... more details
In mathematics , the curvature of a measure defined on the Euclidean plane R sup 2 sup is a quantification of how much the measure s distribution of mass is curved . It is related to notions of curvature in geometry . In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov accordingly, it may be referred to as the Melnikov curvature or Menger Melnikov curvature . Melnikov and Verdera 1995 established a powerful connection between the curvature of measures and the Cauchy integral formula Cauchy kernel . Definition Let be a Borel measure on the Euclidean plane R sup 2 sup . Given three distinct points x , y and z in R sup 2 sup , let R x , y , z be the radius of the Euclidean circle that joins all three of them, or if they are Line geometry collinear . The Menger curvature c x , y , z is defined to be math c x, y, z frac 1 ... 0 if any of the points x , y and z coincide. The Menger Melnikov curvature c sup 2 sup ... to the curvature of at a given point x math c 2 mu x iint mathbb R 2 c x, y, z 2 , mathrm d mu y mathrm ... The trivial measure has zero curvature. A Dirac measure sub a sub supported at any point a has zero curvature. If is any measure whose support measure theory support is contained within a Euclidean line L , then has zero curvature. For example, one dimensional Lebesgue measure on any line or line segment has zero curvature. The Lebesgue measure defined on all of R sup 2 sup has infinite curvature ... has curvature 1 r . Relationship to the Cauchy kernel In this section, R sup 2 sup is thought ... boundedness of the Cauchy kernel to the curvature of measures. They proved that if there is some ... > 0. Here c sub sub denotes a truncated version of the Menger Melnikov curvature in which ... Analytic capacity a discrete approach and the curvature of measure journal Sbornik Mathematics Mat ... 3 Category Curvature mathematics Category Measure theory ... more details
Stress tensor may refer to Stress mechanics , in classical physics Stress energy tensor , in relativistic theories Maxwell stress tensor , in electromagnetism See also Stress disambiguation Tensor disambiguation disambig de Spannungstensor he ... more details
This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection . Special affine curvature , also known as the equi affine curvature or affine curvature , is a particular type of curvature that is defined on a plane curve that remains unchanged ... area . The curves of constant equi affine curvature k are precisely all non singular conic section ... with k 0 are hyperbola s. The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle , the unique circle making second order contact having three point contact with the curve at the point. In the same way, the special affine curvature of a curve at a point P is the special affine curvature of its hyperosculating conic , which is the unique conic ... 1,P 2,P 3,P 4 to P. math In some contexts, the affine curvature refers to a differential invariant of the affine group general affine group , which may readily obtained from the special affine curvature ... group is not used, the special affine curvature k is sometimes also called the affine curvature harv Shirokov 2001b . Formal definition Special affine arclength To define the special affine curvature ... affine curvature Suppose that s is a curve parameterized with its special affine arclength. Then the special affine curvature or equi affine curvature is given by math k s det begin bmatrix beta ... math t mapsto x t , y t , math the special affine curvature is math begin align k t & frac ... 2001a . Affine curvature Suppose as above that s is a curve parameterized by special affine arclength ... is referred to as the affine curvature math kappa frac 1 k 3 2 frac dk ds . math Conics Suppose that s is a curve parameterized by special affine arclength with constant affine curvature k . Let ... . end align math The three cases are now as follows. k 0 If the curvature vanishes ... 2. k 0 If the special affine curvature is positive, then it follows that math beta s left ... more details
In mathematics , the Menger curvature of a triple of points in n dimension al Euclidean space R sup n sup is the Multiplicative inverse reciprocal of the radius of the circle that passes through the three points. It is named after the Austria n United States American mathematician Karl Menger . Definition Let x , y and z be three points in R sup n sup for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let R sup n sup be the Plane mathematics Euclidean plane spanned by x , y and z and let C be the unique circle Euclidean circle in that passes through x , y and z the Circumscribed circle circumcircle of x , y and z . Let R be the radius of C . Then the Menger curvature c x , y , z of x , y and z ... spanned by x , y and z . Another way of computing Menger curvature is the identity math c x,y,z frac ... spanned by x , y , z . Menger curvature may also be defined on a general metric space . If X is a metric ... mathbb R 2 math . Define the Menger curvature of these points to be math c X x,y,z c f x ,f y ... x,y,z is independent of the choice of f . Integral Curvature Rectifiability Menger curvature can be used ... measure restricted to the set math E math . ref cite journal last Leger first J. title Menger curvature ... of Mathematics ref . The basic intuition behind the result is that Menger curvature measures how straight ... der Mosel title Regularizing and self avoidance e ects of integral Menger curvature journal Institut ... journal last Yong Lin and Pertti Mattila title Menger curvature and math C 1 math regularity of fractals ... Capacity, Rectifiability, Menger Curvature and the Cauchy Integral publisher Springer year 2000 ... See also Curvature of a measure Menger Melnikov curvature of a measure External links cite web url ... Curvature last Leymarie first F. accessdate 2007 11 19 year 2003 month September archiveurl http ... 2111&ndash 2119 doi 10.1090 S0002 9939 00 05264 3 issue 7 Category Curvature mathematics Category Multi ... more details
about the measure of curvature degree angle Degree of curve or degree of curvature is a measure of curvature of a circular arc used in civil engineering for its easy use in layout surveying . A n degree curve turns the forward direction by n degree angle degrees over some agreed upon distance. The usual distance in US road work is 100 ft 30.48 m of Arc geometry arc ref Davis, Foote, and Kelly. Surveying Theory and Practice, 1966 ref . US railroad work traditionally used 100 ft of Chord geometry chord and this may be used in other places for road work. Other lengths may be used&mdash such as 100 metres where SI is favoured, or a shorter length for sharper curves. Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential and this made work easier before electronic calculators. The convert 100 ft 2 abbr on is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1 00, 2 00 etc. Metric work may use similar notation, such as kilometers plus meters 1 000. See also multicol Curve Highway engineering Lateral motion device Minimum railway curve radius multicol break Railway systems engineering Track transition curve Transition curve multicol end References Reflist External links Note the variation in usage among these samples. http www.du.edu jcalvert railway degcurv.htm http www.tpub.com content engineering 14071 css 14071 242.htm http www.steamlocomotive.com model curve.html http www.cee.mtu.edu balkire ce3401tc ce3401Lec20.doc http www.trainweb.org freemoslo Modules Tips and Techniques degrees of curve.htm http www.fairview industries.com standardmodule circurve.htm http ceprofs.tamu.edu rbruner curves circularcrvs.htm http www.memun.org SchoolsProject html Resources Roads Fundamentals.htm http www.cityoffrederick.com departments Planning APPENDIX C SUBDIVISION REGULATIONS Sec 5 Final subdivision plat.html Rail tracks Railway track layouts Category Surveying Category ... more details
Unreferenced stub auto yes date December 2009 In general relativity , the topogravitic tensor is one of the three pieces of the Bel decomposition of the Riemann tensor . The topogravitic tensor can be interpreted as representing the sectional curvatures for the spatial part of a frame fields in general relativity frame field . DEFAULTSORT Topogravitic Tensor Category Tensors in general relativity Relativity stub ... more details
a 2 form on tangent vectors, while the covariant derivative is only defined for vector fields. Curvature and the Bianchi identities The Riemann curvaturetensorcurvaturetensor of is a mapping T M ... method . See also Curvaturetensor Contortion tensor Levi Civita connection Torsion of curves ... year 1999 isbn 0914098713 curvature DEFAULTSORT Torsion Tensor Category Differential geometry ... twists about a curve on the surface. The companion notion of curvature measures how moving ... and curvature form the two fundamental invariants of the connection. In this context, torsion gives ... ed whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described 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 ... are extended to vector fields away from the point thus it defines a tensor, much like the torsion . The Bianchi identities relate the curvature and torsion as follows. ref See Kobayashi Nomizu 1996 ... 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 ... representation on R sup n sup . The curvature form and Bianchi identities The curvature form is the gl ... covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi ... math D Omega 0. , math Moreover, one can recover the curvature and torsion tensors from the curvature ... the curvature and torsion forms are horizontal they vanish on the ambiguous vertical vectors . Torsion ... 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
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
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
General relativity In differential geometry , a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system .... A distinction is made among authentic tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle . Definition Unreferenced section date June 2011 Some authors classify tensor densities into the two types called authentic tensor densities and pseudotensor ... tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence ... elucidate the different ways that tensor densities transform under the somewhat ... into these types, there is only one way that tensor densities transform under orientation preserving ... than 2 to the determinant of the metric tensor expressed with Covariance and contravariance of vectors covariant indices. cn date June 2011 Tensor and pseudotensor densities For example, a mixed rank two authentic tensor density of weight W transforms as math mathfrak T alpha beta left det left frac ... authentic tensor density of integer weight W where math bar mathfrak T math is the rank two tensor density in the math bar x math coordinate system, math mathfrak T math is the transformed tensor density ... is applicable only when W is an integer. However, see even and odd tensor densities below. We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation ... 1 when its argument is positive or 1 when its argument is negative. Even and odd tensor densities The transformations for even and odd tensor densities have the benefit of being well defined even when W is not an integer. Thus one can speak of, say, an odd tensor density of weight 2 or an even tensor density of weight 1 2. When W is an even integer the above formula for an authentic 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
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