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Topic: Intermediate treatment of tensors

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	Kids.Net.Au - Encyclopedia > Tensor-classical
		The following is a component-based "classical" treatment of tensors.
		See Component-free treatment of tensors for a modern abstract treatment, and Intermediate treatment of tensors for an approach which bridges the two.
		A tensor is a generalization of the concept of vector and matrices.
	www.kids.net.au /encyclopedia-wiki/te/Tensor-classical (223 words)

	Tensor
		In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector and linear operator in a way that is independent of any chosen frame of reference.
		Note that the word "tensor" is often used as a shorthand for tensor field, a concept which defines a tensor value at every point in a manifold.
		This treatment has largely replaced the component-based treatment for advanced study, similar to the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector.
	www.ebroadcast.com.au /lookup/encyclopedia/te/Tensor.html (864 words)

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		Classical treatment of tensors - The following is a component-based "classical" treatment of tensors.
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	Intermediate treatment of tensors - Wikipedia, the free encyclopedia
		Note that the word "tensor" is often used as a shorthand for "tensor field", a concept which defines a tensor value at every point in a manifold.
		A tensor is the mathematical idealization of a geometric or physical quantity whose analytic description, relative to a fixed frame of reference, consists of an array of numbers.
		The number n, the range of the indices, is called the dimension of the tensor; the total number of degrees of freedom required for the specification of a particular tensor is the dimension of the tensor raised to the power of the tensor's rank.
	en.wikipedia.org /wiki/Intermediate_treatment_of_tensors (1108 words)

	Tensor - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-03)
		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.
		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, for example of the brain.
		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.
	www.bexley.us /project/wikipedia/index.php/Tensor_calculus (1625 words)

	Tensor - Open Encyclopedia (Site not responding. Last check: 2007-11-03)
		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.
		While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain the further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components.
		Examples of physical tensors are the energy-momentum tensor, the inertia tensor and the polarization tensor.
	open-encyclopedia.com /Tensor (1361 words)

	Classical treatment of tensors - Wikipedia, the free encyclopedia
		See Component-free treatment of tensors for a modern abstract treatment, and Intermediate treatment of tensors for an approach which bridges the two.
		A tensor is a generalization of the concepts of vectors and matrices.
		A tensor is an invariant multi-dimensional transformation, that takes forms in one coordinate system into another.
	en.wikipedia.org /wiki/Classical_treatment_of_tensors (201 words)

	Intermediate treatment of tensors: Definition and Links by Encyclopedian.com - All about Intermediate treatment of ...
		The valence of a tensor is the pair $(p,q)$ , where $p$ is the number contravariant and $q$ the number of covariant indices, respectively.
		The corresponding array of numbers for a type $(p,q)$ tensor is denoted by the symbol $T^{i-1\ldots i-p}-{j-1\ldots j-q},$ where the superscripts and subscripts are indices that vary from $1$ to $n$ .
		The total degrees of freedom required for the specification of a particular tensor is a power of the dimension; the exponent is the tensor's rank.
	www.encyclopedian.com /in/Intermediate-treatment-of-tensors.html (1134 words)

	Reference.com/Encyclopedia/Tensor
		Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and was made accessible to many mathematicians by the publication of Tullio Levi-Civita's 1900 classic text of the same name (in Italian; translations followed).
		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 of the particular space on which it is defined.
		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; in this technique tensors are in effect made visible.
	www.reference.com /browse/wiki/Tensor (2231 words)

	ipedia.com: Tensor Article (Site not responding. Last check: 2007-11-03)
		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.
		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.
		This treatment has largely replaced the component-based treatment for advanced study, in the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector.
	www.ipedia.com /tensor.html (1234 words)

	TENSOR FACTS AND INFORMATION (Site not responding. Last check: 2007-11-03)
		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''.
		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.
		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.
	www.witwib.com /index.php?s=tensor (1736 words)

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	Tensor - FreeEncyclopedia (Site not responding. Last check: 2007-11-03)
		In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector and linear operator in a way that is independent of any chosen frame of reference.
		Tensors are of importance in differential geometry, physics and engineering.
		This treatment has largely replaced the component-based treatment for advanced study, similar to the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector.
	openproxy.ath.cx /te/Tensor.html (864 words)

	Intermediate treatment of tensors (Site not responding. Last check: 2007-11-03)
		It is very grossly a generalization of the concept of vectorss and matrices and allows the writing of equations independently of any given coordinate system.
		The valence of a tensor is the pair
		Finally, it must be mentioned that most physical and geometric applications are concerned with tensor fields, that is to say tensor valued functionss, rather than tensors themselves.
	www.sciencedaily.com /encyclopedia/intermediate_treatment_of_tensors (1042 words)

	physics - Tensor
		In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized 'quantity'.
		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.
		Examples of physical tensors are the energy-momentum tensor, the inertia tensor and the polarization tensor.
	www.physicsdaily.com /physics/Tensor (1437 words)

	Knowledge King - Glossary of tensor theory (Site not responding. Last check: 2007-11-03)
		A tensor written in component form is an indexed array.
		A pure tensor of is one that is of the form.
		In the tensor algebra T(V) of a vector space V, the operation becomes a normal (internal) binary operation.
	www.knowledgeking.net /encyclopedia/g/gl/glossary_of_tensor_theory.html (503 words)

	Classical treatment of tensors -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-03)
		The following is a component-based "classical" treatment of (Any of several muscles that cause an attached structure to become tense or firm) tensors.
		See (Click link for more info and facts about Component-free treatment of tensors) Component-free treatment of tensors for a modern abstract treatment, and (Click link for more info and facts about Intermediate treatment of tensors) Intermediate treatment of tensors for an approach which bridges the two.
		A tensor is an (Click link for more info and facts about invariant) invariant multi-dimensional transformation, that takes forms in one coordinate system into another.
	www.absoluteastronomy.com /encyclopedia/C/Cl/Classical_treatment_of_tensors.htm (368 words)

	Category:Tensors - Wikipedia, the free encyclopedia
		In mathematics, a tensor is a certain kind of geometrical entity and array concept.
		It generalizes the concepts of scalar, vector (spatial) and linear operator, in a way that is independent of any chosen frame of reference.
		Tensors are of importance in pure and applied mathematics, physics and engineering.
	www.wikipedia.org /wiki/Category:Tensors (85 words)

	Encyclopedia: Intermediate treatment of tensors (Site not responding. Last check: 2007-11-03)
		The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations.
		In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range).
		In mathematics, the tensor product, denoted by, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules.
	www.nationmaster.com /encyclopedia/Intermediate-treatment-of-tensors (2128 words)

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	INTERMEDIATE TREATMENT OF TENSORS (Site not responding. Last check: 2007-11-03)
		The valence of a tensor is the pair, where is the number contravariant and the number of covariant indices, respectively.
		Again, it must be emphasized that the tensor and the representing array are not the same thing.
		The formal definition of a tensor quantity begins with a finite-dimensional vector space, which furnishes the uniform "building blocks" for tensors of all valences.
	www.websters-online-dictionary.org /definition/INTERMEDIATE+TREATMENT+OF+TENSORS (1080 words)

	TENSORS (Site not responding. Last check: 2007-11-03)
		Tensors can be represented by arrays of components.
		The point of having a tensor theory is to explain the further implication of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components.
		The tensor calculus achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915, which is formulated completely in the language of tensors.
	www.websters-online-dictionary.org /definition/TENSORS (1124 words)

	Glossary of tensor theory - Encyclopedia, History, Geography and Biography
		A pure tensor of $V \otimes W$ is one that is of the form $v \otimes w$ .
		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.
		These are the derived functors of the tensor product, and feature strongly in homological algebra.
	www.arikah.com /encyclopedia/Tensor_notation (679 words)

	Penn Engineering >> Mechanical Engineering and Applied Mechanics [MEAM] > Undergraduate Program > Course Descriptions
		It continues the treatment of the statics of particles and rigid bodies begun in Physics 150 and leads to the treatment of deformable bodies and their response to loads.
		This is an intermediate course in mechanics of viscous fluid flows.
		Tensors: Einstein summation, tensors of arbitrary order, dyads and polyads, outer and inner products, quotient law, metric tensor, Euclidean and Riemannian spaces, physical components, covariant differentiation, detailed evaluation of Christoffel symbols, Ricci’s theorem, intrinsic differentiation, generalized acceleration, Geodesics.
	www.me.upenn.edu /undergrad/descriptions.html (3338 words)

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