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	Intermediate treatment of tensors - Wikipedia, the free encyclopedia
		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.
		Examples of physical tensors are the energy-momentum tensor and the polarization tensor.
		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)

	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)

	Tensor - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		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.
		The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices.
	www.bexley.us /project/wikipedia/index.php/Tensor_calculus (1625 words)

	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.
		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).
		Formally, this means that tensors were introduced as specific representations of the group of all changes of coordinate systems.
	www.wikipedia.org /wiki/Rank_of_a_tensor (959 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)

	Intermediate treatment of tensors -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Note: The following is a modern component-based treatment of (Any of several muscles that cause an attached structure to become tense or firm) tensors (sometimes called the "classical treatment" of tensors).
		Note that the word "tensor" is often used as a shorthand for " (Click link for more info and facts about tensor field) tensor field", a concept which defines a tensor value at every point in a (A pipe that has several lateral outlets to or from other pipes) manifold.
		tensor valued (A mathematical relation such that each element of one set is associated with at least one element of another set) functions, rather than tensors themselves.
	www.absoluteastronomy.com /encyclopedia/I/In/Intermediate_treatment_of_tensors.htm (1220 words)

	Classical
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		Classical treatment of tensors The following is a component-based "classical" treatment of Intermediate treatment of ten...
	www.brainyencyclopedia.com /topics/classical.html (1331 words)

	Classical treatment of tensors -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		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)

	Mathematics_of_general_relativity (Site not responding. Last check: 2007-10-21)
		Because of the principles of general covariance and local Lorentz invariance, the mathematics of general relativity is done with tensor calculus operating on four-dimensional pseudo-Riemannian manifolds which represent spacetime.
		Tensors are a generalized mechanism for handling arrays of numbers, especially closely related numbers as is the case with tensors in general relativity.
		The Ricci tensor and it's trace are used in the Einstein tensor, which is part of the EFE.
	www.usedaudiparts.com /search.php?title=Mathematics_of_general_relativity (902 words)

	Encyclopedia: Intermediate treatment of tensors
		In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion.
		The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations.
		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 (2120 words)

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	Mathematics of general relativity - Enpsychlopedia (Site not responding. Last check: 2007-10-21)
		These two vector spaces may be used to construct type $(r,s)$ tensors at $p$ , which are real-valued multilinear maps acting on the direct sum of $r$ copies of the cotangent space to $M$ at $p$ with $s$ copies of the tangent space to $M$ at $p$ .
		Tensor fields in GR Tensor fields on a manifold are maps which attach a tensor to each point of the manifold and are discussed more fully in the article tensor field.
		A tensor field is then defined as a map from the manifold to the tensor bundle each point $p$ being associated with a tensor at $p$ .
	www.grohol.com /psypsych/Mathematics_of_general_relativity (3882 words)

	Vectors and Tensors in Engineering and Physics, Second Edition (Site not responding. Last check: 2007-10-21)
		Summary: This book begins as a general treatment of vectors and tensors at an intermediate level, then becomes a treatment of mathematical classical physics with both special and general relativity.
		The book is recommended to readers who have already studied some theory of vectors, tensors, linear algebra, calculus, and both ordinary and partial differential equations, as well as non-quantum physical theory.
		The first six chapters cover vector and tensor theory, including vector and tensor algebra, Cartesian coordinates, general coordinates, and tensor fields of one and many variables.
	www.cap.ca /news/books/Vectors-Maroun.html (626 words)

	Multilinear algebra - FreeEncyclopedia (Site not responding. Last check: 2007-10-21)
		Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concept of a tensor and develops the theory of 'tensor spaces'.
		It developed out of the use of tensors in differential geometry, general relativity, and many branches of applied mathematics.
		The latter deals with tensor fields (tensors varying from point to point on a manifold), but covariance asserts that the language of tensors is essential to the proper formulation of general relativity.
	openproxy.ath.cx /mu/Multilinear_algebra.html (596 words)

	Tensors and Ellipsoids
		The inertia tensor in the dynamics of rigid bodies is an excellent example of a rank-2 tensor where the associated ellipsoid aids in the visualization of the motion.
		The requirement of symmetry comes from several sources, one of which is simply that the tensor should be diagonalizable by an ordinary rotation, which establishes the three orthogonal principal axes of polarization, and the three principal dielectric constants, its eigenvalues.
		From the equation giving the stress tensor in terms of the strain tensor, k and μ, find the relation between the traces of the strain and stress tensors, and from this the equation giving the strain tensor in terms of the stress tensor.
	www.du.edu /~jcalvert/phys/ellipso.htm (5816 words)

	Glossary of tensor theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A dyadic tensor has rank two, and may be represented as a square (A rectangular array of elements (or entries) set out by rows and columns) matrix.
		A pure tensor of is one that is of the form.
		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.
	www.absoluteastronomy.com /encyclopedia/g/gl/glossary_of_tensor_theory.htm (1303 words)

	INTERMEDIATE TREATMENT OF TENSORS
		The valence of a tensor is the pair, where is the number contravariant and the number of covariant indices, respectively.
		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.
		The space of -valent tensors, denoted here by is obtained by taking the tensor product of copies of and copies of the dual vector space.
	www.websters-online-dictionary.org /definition/INTERMEDIATE+TREATMENT+OF+TENSORS (1080 words)

	Amazon.ca: Books: Classical Mechanics: A Modern Perspective (Site not responding. Last check: 2007-10-21)
		The details behind the theory of classical mechanics are presented very quickly in the book, and this might make the book difficult to read for students first exposed to mechanics at this level.
		Their approach to tensors though is kind of antiquated, for it motivates them via the outer product, which is reminiscent of the dyadic approach that is currently "out of fashion".
		Chapter 8 is an overview of gravitational physics, and the authors show the effects of a body moving in a non-uniform gravitational field, with an example dealing with the tides.
	www.amazon.ca /exec/obidos/ASIN/0070037345 (1595 words)

	Books : Tensors, Differential Forms, and Variational Principles (Site not responding. Last check: 2007-10-21)
		Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations.
		They connect the classical and modern views of physics on curved geometries, (as the back cover says, by successive abstraction), from parallel displacement to affine connections to curvature tensors to integration theorems to invariant field theories.
		Coordinate-free treatment is restricted to a brief appendix.
	www.cellphonegamesdownload.com /0486658406/Tensors_Differential_Forms_and_Variational_Principles.shtml (378 words)

	Tensor (intrinsic definition) (Site not responding. Last check: 2007-10-21)
		If you are baffled by this article, try reading the main tensor article and the classical treatment first.
		The modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept.
		(The term "tensor" is sometimes used as a shorthand for "tensor field".) For instance, the curvature tensor is discussed in differential geometry and the stress-energy tensor is important in physics and engineering.
	www.free-download-soft.com /info/add-in.html (750 words)

	Classical treatment of tensors: Definition and Links by Encyclopedian.com - All about Classical treatment of tensors
		Classical treatment of tensors: Definition and Links by Encyclopedian.com - All about Classical treatment of tensors
		A covariant tensor of order 1( $T-i$ ) is defined as:
		Definition / meaning of Classical treatment of tensors:
	www.encyclopedian.com /te/Tensor-classical.html (235 words)

	Killing equations in classical mechanics (Site not responding. Last check: 2007-10-21)
		The relation between the Killing vectors and tensors admitted by the four-dimensional manifold and the constants of the motion of the geodesics is largely known to people working in the General Relativity theory.
		In general, the property also holds that contracting a Killing tensor of rank p by p tangent vectors one obtains a quantity which results to be constant on the geodesics [3], that is a first integral of the geodesic equations.
		However, when the invariant is non-homogeneous and its quadratic part is associated to the tensor M, we have the non-trivial case of the well-known Stäckel potentials (to see this we refer to [11]).
	www.sif.it /cimento/tocb/112.02-03/05/05.html (6640 words)

	Physics Help and Math Help - Physics Forums - intro tensors book
		While this is operationally useful, it tends to obscure the deeper meaning of tensors as geometric entities with a life independent of any chosen coordinate system." On page 15 he describes the scalar or dot product as a familiar example of a tensor of type (0,2).
		Anyone who knows conceptually what a tensor is would immediately realize that a homogeneous polynomial of degree d in the entries of a tangent vector, is a (symmetric) tensor of type (0,d), and that the components of the tensor are merely the coefficients of the polynomial (written as a non commutative polynomial, i.e.
		In chapter 5 Spivak presents the classical Ricci calculus, subtitled "the debauch of indices", and proves computationally the "test case" that a manifold with zero curvature tensor is locally isometric to (flat) euclidean space.
	www.physicsforums.com /printthread.php?t=39268&pp=40 (2860 words)

	Classical Dynamics Books for the Lincoln Automotive Enthusiast (Site not responding. Last check: 2007-10-21)
		Chronologically, "Classical Dynamics" was Donald Greewood's second major publication on Analytical Dynamics, covering more advanced topics than the ones in "Principles of Dynamics," whose first edition preceded "Classical Dynamics" by some 12 years.
		"Classical Dynamics" is a somewhat more readable text, but just like its companion book, it fails to address issues like how one can use Lagrange's equations (or Hamilton's, for that matter) to correctly account for the effects of nonlinear dissipative forces.
		Also, its treatment of velocity-dependent potentials could be substantially extended, as could the chapter on Relativity.
	www.lincolnsofdistinction.com /books/book.php?isbn=0486696901.html (393 words)

	Books : Mathematics of Classical and Quantum Physics (Site not responding. Last check: 2007-10-21)
		The first chapter on vector analysis from the component view is the best treatment of the subject I've seen anywhere.
		The treatment of each subject is framed in a way that makes immediate application to physics natural and intuitive.
		The treatment given here is concise and complete.
	www.ajeno.com /ItemId/048667164X (458 words)

	Amazon.com: Books: Tensors, Differential Forms, and Variational Principles (Site not responding. Last check: 2007-10-21)
		Tensors are generalizations of vectors to any number of dimensions (vectors are type (1,0) tensors, diff.
		One of the key principles of General Relativity is that if physical laws are expressed in tensor form, then they are independent of local coordinate systems, and valid everywhere.
		Many years ago, this became the first book I had ever read about tensor calculus, differential geometry, or classical field theories, and I still have not found a much better treatment of any of these subjects anywhere else.
	www.amazon.com /exec/obidos/tg/detail/-/0486658406?v=glance (1327 words)

	Amazon.com: Books: Lectures on Classical Differential Geometry (Site not responding. Last check: 2007-10-21)
		A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises.
		This is the language of modern geometry.It leads on naturally to tensors, fibre bundles, de Rham cohomology and so on and so forth.The emphasis in modern geometry is on global phenomena, the interaction between local and global (e.g.
		Morse theory or De Rham cohomology), and the attempt to do everything in an algebraic setting (projective modules, spectral sequences, categories etc.) For this purpose, Struik is useless, though he does have some coverage of forms (he calls them by their earlier name of 'pfaffians').
	www.amazon.com /exec/obidos/tg/detail/-/0486656098?v=glance (793 words)

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