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	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 (2237 words)

	PlanetMath: tensor density
		A tensor density is a quantity whose transformation law under change of basis involves the determinant of the transformation matrix (as opposed to a tensor, whose transformation law does not involve the determinant).
		As with tensors, it is possible to define tensor density fields on manifolds.
		This is version 9 of tensor density, born on 2005-01-01, modified 2005-02-18.
	www.planetmath.org /encyclopedia/TensorDensity.html (214 words)

	Tensor Information - tensor lamps
		Tensors may be written down in terms of coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen frame of reference.
		Physicists and engineers are among the first to recognise that vectors and tensors have tensor lamps a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated.
		Examples of physical tensors are the tensor analysis tensor fasciae latae energy-momentum tensor, the inertia tensor and the polarization tensor.
	www.inanot.com /Ina-Electronics_Topics_T-/Tensor.html (1961 words)

	Clifford algebra
		Clifford algebras are associative algebras of importance in mathematics, in particular in the theories of quadratic forms and of orthogonal groups, and in physics.
		The associated graded algebra is canonically isomorphic to the exterior algebra[?] Λ V of the vectorspace.
		Thus by the Artin Wedderburn theorem[?] it is (non canonically) isomorphic to a matrix algebra.
	www.ebroadcast.com.au /lookup/encyclopedia/cl/Clifford_algebra.html (446 words)

	tensor
		In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independent of any chosen frame of reference.
		But tensors are used also within other fields such as continuum mechanics, for example the strain tensor, (see linear elasticity).
		Note that the word "tensor" is often used as a shorthand for tensor field, which is a tensor value defined at every point in a manifold.
	en.mcfly.org /tensor (1356 words)

	PlanetMath: exterior algebra
		The exterior algebra is also known as the Grassmann algebra after its inventor Hermann Grassmann who created it in order to give algebraic treatment of linear geometry.
		Exterior algebra is also an essential prerequisite to understanding de Rham's theory of differential cohomology.
		This is version 20 of exterior algebra, born on 2002-04-07, modified 2006-04-30.
	planetmath.org /encyclopedia/ExteriorAlgebra.html (1003 words)

	Tensor algebra (Site not responding. Last check: 2007-10-31)
		In mathematics, the tensor algebra is an abstract algebra construction of an associative algebra T(V) from a vectorspace V. If we take basis vectors for V, those become non-commutingvariables in T(V), subject to no constraints (beyond associativity, the distributive law and K-linearity, where V is defined over the field K).
		That is, thetensor algebra is representative of algebra with tensors that are formed from V and covariant, of any rank.
		The free algebra point of view is useful for constructions like those of Clifford algebras and universal enveloping algebras, where the existence question can be settled by starting withT(V) and then imposing the required relations.
	www.therfcc.org /tensor-algebra-210862.html (301 words)

	Tensor algebra - InformationBlast
		In mathematics, the tensor algebra is an abstract algebra construction of an associative algebra T(V) from a vector space V. If we take basis vectors for V, those become non-commuting variables in T(V), subject to no constraints (beyond associativity, the distributive law and K-linearity, where V is defined over the field K).
		That is, the tensor algebra is representative of algebra with tensors that are formed from V and covariant, of any rank.
		The free algebra point of view is useful for constructions like those of Clifford algebras and universal enveloping algebras, where the existence question can be settled by starting with T(V) and then imposing the required relations.
	www.informationblast.com /Tensor_algebra.html (313 words)

	Course 423 - Tensors (Site not responding. Last check: 2007-10-31)
		Tensor algebra may be considered as an extension of Linear Algebra.
		Linear algebra is concerned mainly with vectors, which are tensors of type (1,0), and linear maps, which are tensors of type (1,1).
		Tensor calculus is the study of tensor fields on manifolds or varieties.
	www.maths.tcd.ie /pub/official/Courses/423in9596.html (216 words)

	THE MEANING OF "TENSOR"
		Roughly, according to a classical theorem (Levi's theorem) on the subject, any Lie group is a semidirect product of a solvable algebra by a semisimple algebra, meaning that the structure problem is broken into determining the structure problems for solvable and semisimple algebras.
		A representation of an abstract algebraic structure is a structure preserving homomorphism from the abstract structure into an algebra of matrices.
		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)

	Vectors, Tensors, Spinors
		Algebra of vectors, differentiation of vectors, partial differentiation and associated concepts, integration of vectors, and tensor algebra and analysis.
		Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms; interaction between concept of invariance and calculus of variations.
		Designed to familiarize undergraduates with the methods of vector algebra and vector calculus, this text offers both a clear view of the abstract theory as well as a concise survey of the theory's applications to various branches of pure and applied mathematics.
	store.doverpublications.com /by-subject-science-and-mathematics-mathematics-vectors--tensors--spinors.html (391 words)

	Tensor (Orbital)
		Since every implementation is allowed to choose the tensor's internal data representation freely, you should not rely on such behaviour of mutable arithmetic objects to provoke inner state changes.
		The reason is that one-dimensional indices are meaningless for tensors of rank r>1 and that adding a single component to a tensor is not allowed as it would destroy its rectangular form.
		The semantics of the tensor returned by this method becomes undefined if the backing tensor (i.e., this object) is structurally modified in any way other than via the returned tensor.
	www.functologic.com /orbital/Orbital-doc/api/orbital/math/Tensor.html (820 words)

	Search Results for tensor*
		He discovered a tensor, now called Weyl's conformal curvature tensor, whose vanishing is a necessary condition that the space be conformally flat, that is to say, that the space can be mapped conformally on the Euclidean space.
		The transition from the characteristic tensor to the dynamical variables is conveyed by an analysis of the physical meaning of the constituents.
		Linear algebra proper usually encompasses linear and bilinear forms, and the very beginnings of the theory of multilinear forms as tensor algebras.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=tensor*&CONTEXT=1 (2237 words)

	No Title
		Ricci calculus[1] is a well-known method in general relativity[2, 3] for applying indexed expressions in terms of tensors and symbols.
		The indexed expressions are manipulated according to the rules of the tensor algebra.
		The correct handling of the dummy indices (which means to convert the tensor multiplications in different forms but with the same value to a standard form as well) is the strength of the system.
	www.kfki.hu /cnc/szhkpub/riccir/riccir.html (513 words)

	Numerical Notes: Math Archives (Site not responding. Last check: 2007-10-31)
		Part of what initially confused me about tensor algebra -- and, for that matter, about linear algebra -- is the difference between tensors in an abstract setting and the concrete things with messes of indices which are most often used for computation.
		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)

	Tensor Algebra
		Tensors are probably new to most of us, and they are are a little challenging to get to know.
		The reason tensors are discussed in the course is to empower our understanding of the equations/laws.
		The sum of the diagonal elements of the tensor is
	www.rit.edu /~pnveme/EMEM851n/constitutive/tensors_rect.html (376 words)

	Tensor Analysis and Nonlinear Tensor Functions - Yu I. Dimitrienko, I. Dimitrienko - Kluwer Academic Publishers
		Based on this approach, the author gives a mathematically rigorous definition of a tensor as an individual object in arbitrary linear, Riemannian and other spaces for the first time.
		It is the first book to present a systematized theory of tensor invariants, a theory of nonlinear anisotropic tensor functions and a theory of indifferent tensors describing the physical properties of continua.
		The book will be useful for students and postgraduates of mathematical, mechanical engineering and physical departments of universities and also for investigators and academic scientists working in continuum mechanics, solid physics, general relativity, crystallophysics, quantum chemistry of solids and material science.
	www.libreriauniversitaria.it /BUS/140201015X/Tensor_Analysis_and_Nonlinear_Tensor_Functions.htm (151 words)

	Graduate Algebra II
		Algebra is a subject with extremely concrete origins: it is the language of the everyday manipulation of symbols, and the solution of equations.
		At the same time, it can be quite abstract; modern algebra is also the language of relations between objects, sometimes divorced from the objects themselves.
		Its aim is to introduce the algebraic concepts which are common knowledge for the working mathematician, and to make the link between the abstract definitions and their concrete incarnations.
	www.mast.queensu.ca /~mikeroth/algebra/algebra.html (204 words)

	PlanetMath: tensor algebra
		From the point of view of category theory, one can describe the tensor algebra construction as a functor
		Cross-references: homomorphism, algebras, forgetful functor, category, functor, category theory, point, right, bimodule, module, non-commutative, ground ring, cover, tensor product, power, tensor, component, commutative ring
		This is version 7 of tensor algebra, born on 2002-12-18, modified 2005-09-30.
	www.planetmath.org /encyclopedia/TensorAlgebra.html (95 words)

	Tensors and tensor algebra (Site not responding. Last check: 2007-10-31)
		transforming it according to the rules for a second rank tensor it is the same in all rotated frames of reference.
		Clearly the sum or difference of two tensors of the same rank is also a tensor, and similarly if one multiplies all elements of a tensor by a scalar it is still a tensor.
		Setting two indices of a tensor equal and summing reduces the the rank of the tensor by two.
	astron.berkeley.edu /~jrg/ay202/node185.html (790 words)

	Tensor Product of Algebras
		We're going to prove that the tensor product of two r algebras is another r algebra, but first we need a lemma about associative homomorphisms.
		This satisfies the criteria of an r algebra, as described in the introduction.
		Tensor the domains of these homomorphisms to get u, and tensor the ranges to get t.
	www.mathreference.com /ring-alg,prod.html (848 words)

	Glossary of tensor theory
		A tensor written in component form is an indexed array.
		A dyadic tensor has rank two, and may be represented as a square matrix.
		These are the derived functors of the tensor product, and feature strongly in homological algebra.
	en.mcfly.org /Glossary_of_tensor_theory (548 words)

	Tensor notation (Site not responding. Last check: 2007-10-31)
		The divergence theorem in vector and tensor notation...
		The equations of motion of a viscous fluid in tensor notation...
		Computer formulation of the equations of motion using tensor notation...
	www.scienceoxygen.com /math/517.html (118 words)

	Predmety - Predmety
		We define the notions of tensor and exterior algebras, differential forms in R^n and their integrals over k-dimensional surfaces in R^n.
		We define also smooth manifolds with border, tangent vectors, vector and tensor fields, integral of a differential form on a manifold and the highlight is the proof of the general Stokes theorem.
		A summary of properties of tensor algebra of a vector space; outer algebra of a vector space, basic properties of outer multiplication; symmetric algerba of a vector space, orientation of a vector space, volume of a paralleliped using outer product and the Gramm matrix.
	www.mff.cuni.cz /vnitro/is/sis/predmety/kod.php?kod=GEM002 (255 words)

	Glossary of tensor theory - InformationBlast
		Such pure tensors are not generic: if both V and W have dimension > 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure.
		The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
		The quotient space of T(V) on which it becomes an internal operation is the exterior algebra of V; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V. Symmetric power, symmetric algebra
	www.informationblast.com /Tensor_notation.html (517 words)

	Tensor notation (Site not responding. Last check: 2007-10-31)
		Such pure tensors are not generic: if both V and W have dimension > 1, there will be tensors thatare not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure.
		The free algebra on a set X is for practical purposes the same as thetensor algebra on the vector space with X as basis.
		The quotient space of T(V) on which it becomes aninternal operation is the exterior algebra of V; it is a gradedalgebra, with the graded piece of weight k being called the k-th exterior power of V. Symmetric power, symmetric algebra
	www.therfcc.org /tensor-notation-133576.html (484 words)

	Scalars vectors and tensors (Site not responding. Last check: 2007-10-31)
		are used as subscripts to denote the components of vectors and tensors.
		To obey the laws of algebra the components of a vector should be written as a column.
		It is not necessary to distinguish between covariant and contravariant tensors in Cartesian geometry; subscripts are therefore always denoted by subscripts.
	astron.berkeley.edu /~jrg/ay202/node184.html (652 words)

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