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Topic: Tensor (intrinsic definition)

Related Topics

curvature tensor

tensor field

tensor algebra

tensor product of fields

Classical treatment of tensors

Tensor intrinsic definition

Tensor product of R algebras

Intermediate treatment of tensors

Component free treatment of tensors

Ricci tensor

rank of a tensor

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	Tensor (intrinsic definition) - Wikipedia, the free encyclopedia
		Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
		In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make references to coordinates at all.
		The tensors of rank zero are just the scalars (elements of the field F), those of contravariant rank 1 the vectors in V, and those of covariant rank 1 the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors).
	en.wikipedia.org /wiki/Tensor_(intrinsic_definition) (972 words)

	Tensor - Wikipedia, the free encyclopedia
		In mathematics, a tensor is (in an informal sense) a generalized linear 'quantity' or 'geometrical entity' that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference.
		In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain.
		A tensor may be expressed as the sequence of values represented by a function with a vector valued domain and a scalar valued range.
	en.wikipedia.org /wiki/Tensor (2230 words)

	PlanetMath: coboundary definition of exterior derivative (Site not responding. Last check: 2007-10-07)
		The nice feature of (1) is that it is equivalent to the definition of the coboundary operator for Lie algebra cohomology.
		"coboundary definition of exterior derivative" is owned by rmilson.
		This is version 11 of coboundary definition of exterior derivative, born on 2006-01-13, modified 2006-01-29.
	planetmath.org /encyclopedia/IntrinsicDefinitionOfExteriorDerivative.html (266 words)

	Pellionisz (1985) Tensor Network Theory of the Metaorganization of Functional Geometries in the Central Nervous System
		Tensor theory simply re-phrases this by stating that the vectorial expression of an object in any system of co-ordinates by means of physical components is of the contravariant type.
		Tensor network theory of the CNS evolved as a mathematical formulation, with the use of the above basic terms of covariants and contravariants, of the geometrical concept of brain function, especially that of the cerebellum.74-78,80-83.
		The covariant metric tensor was established in (3) by calculation as a matrix composed of the cosines among co-ordinate axes.
	usa-siliconvalley.com /inst/pellionisz/85_metaorganization/85_metaorganization.html (9754 words)

	Pellionisz "Dusseldorf-1" 1987
		Given the existence of such intrinsic coordinates, the axiom of generalized coordinates appears inevitable if identification of the internal mathematical language, actually used by neuronal networks, is intended, especially since the mathematical fundamentals of transforming such covariant- contravariant and mixed tensorial expressions in non-orthogonal general frames are well established (30).
		Tensors are mathematical operators connecting general co- ordinates, where one must distinguish between measurement-type orthogonal-projection vectorial representations, physically executable parallelogram-type vectors, and mixed expressions (covariant, contravariant and mixed tensors), 30,53).
		The metric tensor operation was identified as a basic functional characteristics of sensorimotor networks, as elaborated for the cerebellum (53-55, 39- 40).
	www.usa-siliconvalley.com /inst/pellionisz/dussel1dorf/dussel1dorf.html (4571 words)

	Intrinsic conductivity of objects having arbitrary shape and conductivity - Review of conductivity virial expansion ...
		The polarizability α is a second rank tensor [16,17] that generally depends on particle orientation, shape, size, and Δ.
		In some applications it is useful to orient the suspended particles, in which case the effective conductivity of the composite is anisotropic and becomes explicitly dependent on the components of the polarizability tensor [20].
		The intrinsic conductivity is rather insensitive to particle shape when the conductivity of the particles is similar to the embedding medium (Δ ≈ 1), and a formal Taylor expansion about this limit can be made.
	ciks.cbt.nist.gov /~garbocz/paper83/node2.html (1611 words)

	THE MEANING OF "TENSOR"
		That is an antiplatonic viewpoint, which is to say that the body of mathematics is basically a creative aspect of the commonalities of the human nervous system; this body does not exist as the discovery of an 'a priori' existence: Platonic ideals are merely emotionally and perceptually motivated fictions, useful though they may be.
		The structure problem is determining the structure of the category by cataloging the various objects as to their individual properties that are intrinsic and not determined by the category axioms, the relationships of equivalence that exist between the objects by isomorphisms, and the relationships that exist in similarity by homomorphisms.
		Tensor Product The tensor product grows from the idea of the "Cartesian Product" of the one dimensional x-axis as a vector space, and the one dimensional y-axis in elementary algebra to form the two dimensional Cartesian plane.
	graham.main.nc.us /~bhammel/MATH/tensor.html (1952 words)

	Methods
		Because the anisotropy of a tensor is in general related to the disparity among its three eigenvalues, multiplying a vector by a tensor with high anisotropy will cause a greater relative change among its coordinates, and hence a greater deflection.
		The free parameters in the hue-ball method of tensor visualization are the color assignment on the sphere, and the input vector to use for multiplication with the diffusion matrix.
		The codimension of the diffusion tensor's representative ellipsoid is two in the linear anisotropy case, and one with planar anisotropy.
	www.cs.utah.edu /~gk/papers/vis99/node3.html (2959 words)

	Extrinsic and Intrinsic Curvature
		Intrinsic geometry is a way of describing curvature without appeal to higher dimensions, hence it is appropriate for GTR.
		Note that the one of the principle extrinsic curvatures at any point on the two-dimensional infinite standard cylinder of some radius r would be by definition 1/r, while the intrinsic curvature would be zero (There is no distortion in the parallel transport of a vector on a closed curve embedded in the cylinder).
		The intrinsic curvature is non-zero on a sphere, where parallel-transport of a vector on a closed curve does result in a disagreement.
	www.physicsforums.com /showthread.php?t=65715&goto=nextnewest (1302 words)

	Dyadic tensor: Encyclopedia topic (Site not responding. Last check: 2007-10-07)
		A dyadic tensor in multilinear algebra (multilinear algebra: in mathematics, multilinear algebra extends the methods of linear algebra....
		[follow hyperlink for more...]) is a second rank tensor (tensor: Any of several muscles that cause an attached structure to become tense or firm) written in a special notation, formed by juxtaposing pairs of vectors, i.e.
		A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient.
	www.absoluteastronomy.com /reference/dyadic_tensor (140 words)

	Tensors
		Tensors are of primary importance in connection with coordinate transforms.
		When an entity is described as a tensor it is generally understood that it behaves as a tensor under all non-singular differentiable transformations of the relevant coordinates.
		An entity which only behaves as a tensor under a certain subgroup of non-singular differentiable coordinate transformations is called a qualified tensor, because its name is conventionally qualified by an adjective recalling the subgroup in question.
	farside.ph.utexas.edu /teaching/em/lectures/node111.html (1013 words)

	Tensors, Contravariant and Covariant
		The key attribute of a tensor is that it's representations in different coordinate systems depend only on the relative orientations and scales of the coordinate axes at that point, not on the absolute values of the coordinates.
		We should note that when dealing with a vector (or tensor) field on a manifold each element of the field exists entirely at a single point of the manifold, with a direction and a magnitude, rather than imagining each vector to actually extends from one point in the manifold to another.
		In this way we can also create mixed tensors, i.e., tensors that are contravariant in some of their indices and covariant in others.
	www.mathpages.com /rr/s5-02/5-02.htm (2503 words)

	Mixed tensor: Encyclopedia topic (Site not responding. Last check: 2007-10-07)
		In tensor analysis (tensor analysis: in mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized...
		[follow hyperlink for more...]), a mixed tensor is a tensor (tensor: Any of several muscles that cause an attached structure to become tense or firm) which is neither covariant (covariant: more facts about this subject) nor contravariant (contravariant: contravariant is a mathematical term with a precise definition in tensor analysis....
		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).
	www.absoluteastronomy.com /reference/mixed_tensor (105 words)

	UC Davis Math: Glossary
		An inductive definition of an invariant of knots and links which postulates a linear relation between the invariant of a given link and the invariant of the same link with a crossing switched or otherwise simplified.
		The technical definition is that a topological space is a set X together with distinguished subsets called open sets or open neighborhoods, such that the the empty set and X are open, the intersection of two open sets is open, and the union of an arbitrary collection of open sets is open.
		An algebraic relation arising in statistical mechanics, topological quantum field theory, and quantum groups in which two tensors, one naturally represented by a right-side-up triangle and the other by an upside-down triangle, are equal.
	www.math.ucdavis.edu /glossary.html (9932 words)

	Metric (mathematics) - (Site not responding. Last check: 2007-10-07)
		A metric d on X is called intrinsic if any two points x and y in X can be joined by a curve with length arbitrarily close to d(x, y).
		In differential geometry, one considers metric tensors, which can be thought of as "infinitesimal" metric functions, and are defined as inner products on the tangent space with an appropriate differentiability requirement.
		These are used in the geometric study of the theory of relativity, where the tensor is also called the "invariant distance".
	psychcentral.com /psypsych/Metric_(mathematics) (892 words)

	Riemannian Geometry, Kaluza-Klein Theories...
		Forces acting on objects and influencing their trajectories have also been described in the past in term of various tensor fields defined on a four dimensional manifold modeling the "space of events".
		A general study of the Riemannian geometry of "matter fields", i.e., vector valued functions (or forms) defined on a manifold (in particular the covariant derivative acting on tensors, spinors, p-forms valued in some vector space...) is made in Ch.6 (this chapter could be read independently of the rest of the book).
		The particular case where such matter fields are usual tensors or spinors is studied in Ch.8 (G-spin structures are naturally obtained there as a result of a process of "dimensional reduction").
	quantumfuture.net /quantum_future/kkintro.htm (897 words)

	Re: Two definitions of "observable"
		The first definition I remember is that a W-algebra is a -subalgebra of the bounded linear operators on a Hilbert space that is closed in the weak topology.
		You start with a more "intrinsic" definition of C-algebra and then prove the GNS theorem, which says they can all be thought of as norm-closed -subalgebras of the bounded linear operators on a Hilbert space.
		These days, intrinsic definitions are regarded as better than definitions that say something is a "sub-thing" of something else.
	www.lns.cornell.edu /spr/2000-02/msg0022300.html (1325 words)

	[No title]
		In both cases we assumed a metric-compatible connection and used projection tensors defined in terms of the perpendicular or normal to a surface or to a curve.
		Describe a family of submanifolds by giving a projection tensor field whose projected subspaces are the tangent spaces to the submanifolds.
		Shortly we will show that, for a normal projection tensor field, the related rank-$\binom{0}{2}$ tensor is symmetric% \[ g_{H}\left(w,v\right) =w\cdot Hv=g_{H}\left(v,w\right) \]% and that this symmetry is both a necessary and a sufficient condition for a projection-tensor field to be normal.
	www.people.vcu.edu /~rgowdy/phys691/rap/pt.rap (4454 words)

	Numerical Notes: Math Archives
		A definition of a tensor which is based on a particular concrete representation artifically complicates manipulations.
		Tensor products and tensor contractions both are intrinsic operations, which can be defined without reference to any particular coordinate system; however, when one chooses a basis, these operations can be expressed in terms of concrete row and column vectors and matrices.
		I was exposed roughly concurrently to two different perspectives on tensor algebra: in an abstract algebra class we talked about tensor products of groups and modules and vector spaces in a very general setting, and in a Riemannian geometry class we spoke of tensors in terms of multilinear forms.
	www.cs.berkeley.edu /~dbindel/blog/archives/cat_5 (1882 words)

	[No title]
		The Einstein tensor and the energy-momentum tensor are conservative in the covariant sense.
		So the definition of coordinate systems should include all the relevant information which is necessary to describe these results and to relate them in terms of physical laws.
		Although it became definitively clear after Mach's and Einstein's analysis that the concept of an absolute space-time was to be given up and superseded by a space-time depending on its material and energetic content, the present quantum theory of microphysics still assumes space-time to be Minkowskian, i.e.
	www.chez.com /etlefevre/rechell/ukliwo12.htm (9267 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Seventh, and lastly, recall that the Weyl tensor vanishes on a simply connected region iff for some chart on that region, the metric tensor is a scalar multiple of the metric tensor for Minkowski spacetime in the usual Cartesian chart.
		Exercise: show that the extrinsic curvature tensor of the hyperslice T = T0 is K(X) = K_(ab) o^a & o^b, K_(ab) = a(T0) a'(T0)/z^2 diag(1,1,1) Use the Gauss equations to show that the three-dimensional Riemann tensor of this slice is r^2_(323) = r^2_(424) = r^3_(434) = -1/a(T0)^2 IOW, a hyperbolic space H^3 with "radius" a(T0).
		Show that the expansion and vorticity tensors for X vanish (static observers!) and verify that the electrogravitic tensor is ER(X) = E_(ab) o^a & o^b, E_(22) = 4 q^2 r^2/(1 + q^2 r^2)^4 E_(33) = E_(44) = 2 q^2 (1 - q^2 r^2)/(1 + q^2 r^2)^4 while the magnetogravitic tensor BR(X) vanishes.
	math.ucr.edu /home/baez/PUB/electromagneto (5847 words)

	[No title]
		Section 2: Tensors and Tensor Operations A tensor T is defined as an ordered collection of its components: EMBED Equation.DSMT4 where each of the i and each of the j can assume any integer value from 1 to N and all components are real numbers.
		Definition: Let T be the value of a tensor field at a certain point.
		Tensors are usually preferred over “nontensors” because a tensor needs to be defined in only one coordinate system and because change of tensor components under coordinate transformations is predictable and depend only on the coordinates and the nature of the tensor.
	web.mit.edu /dmytro/www/GR_theory.doc (1538 words)

	Discussion
		Large angle polarization of the three components are also constrained by model-independent geometric arguments in its slope and its correlation with the temperature anisotropy.
		If the scalar contributions can be isolated from the vectors, tensors and other foreground sources of polarization from these and other means, these constraints translate into a robust distinction between isocurvature and adiabatic models for structure formation.
		Scalar gravitational redshift effects now dominate over scalar acoustic as well as vector and tensor contributions for the same stress source due the process by which stress perturbations generate metric fluctuations (see Fig.
	background.uchicago.edu /~whu/tamm/webversion/node38.html (363 words)

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