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Topic: Tensor notation

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tensor field

tensor algebra

tensor product of fields

Classical treatment of tensors

Tensor intrinsic definition

Tensor product of R algebras

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Component free treatment of tensors

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	Tensor field - Wikipedia, the free encyclopedia
		It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences and engineering.
		The general idea of tensor field combines the requirement of richer geometry — for example an ellipsoid varying from point to point, in the case of a metric tensor — with the idea that we don't want our notion to depend on the particular method of mapping the surface.
		A tensor density is the special case where L is the bundle of densities on a manifold, namely the determinant bundle of the cotangent bundle.
	en.wikipedia.org /wiki/Tensor_field (1198 words)

	Tensor (intrinsic definition) - Wikipedia, the free encyclopedia
		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).
		Here we used the Einstein notation, a convention useful when dealing with coordinate equations: when an index variable appears both raised and lowered on the same side of an equation, we are summing over all its possible values.
		Formally, this means that tensors were introduced as specific representations of the group of all changes of coordinate systems.
	en.wikipedia.org /wiki/Tensor_(intrinsic_definition) (958 words)

	Notation - Wikipedia, the free encyclopedia
		Commonly, "notation" refers to the typographical conventions or rules of symbol usage that are followed, e.g., within a book or article.
		Notation may refer to notation systems, meaning an interpreted system of tokens having a syntax and semantics.
		Dance notation - to document various forms of is used to dance movement such as Labanotation, Benesh Notation, and Eshkol-Wachman notation.
	www.wikipedia.org /wiki/Notation (371 words)

	Glossary of tensor theory
		Einstein notation This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index.
		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.
	www.arikah.com /encyclopedia/Tensor_notation (665 words)

	An Introduction to Tensors
		A tensor of rank 2 (and usually when we use the term tensor we do mean a tensor of rank 2, since rank 0 and 1 tensors have the special names scalars and vectors) in ordinary 3D space will have 9 components.
		Because a tensor of rank 2 can be considered as an nxn array of numbers (where n is the number of dimensions in the vector space), it is usual to represent rank 2 tensors by matrices.
		All of the relationships of general relativity are expressed as tensor relationships, since it is essential (as a fundamental premise of relativity) that the laws themselves are not dependent on the choice of coordinate system (or observer, if you prefer).
	www.mta.ca /faculty/Courses/Physics/4701/EText/TensorIntroduction.html (1727 words)

	7. Description of the Physical Properties in Matrix Notation (Site not responding. Last check: 2007-10-21)
		The introduction of the matrix notation may be encouraged by the usually considerable reduction, due to intrinsic symmetries, of the number of independent components of the tensors representing the physical properties.
		Although the relationship between the suffixes of the tensor components and matrix elements is unambiguous, the relationship between the tensor components and the matrix elements is defined to the extent of a multiplication factor.
		Once the notation of the piezoelectric matrix is accepted as above in (7.5) and a matrix is generated from the components of the stress tensor according to eq.
	www.za.iucr.org /iucr-top/comm/cteach/pamphlets/18/node7.html (774 words)

	Tensor analysis Definition / Tensor analysis Research (Site not responding. Last check: 2007-10-21)
		In mathematicsMathematics is commonly defined as the study of patterns of structure, change, and space; further informally, one might say it is the study of "figures and numbers".
		[click for more], 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.
		Tensors are of importance in physics and engineeringEngineering is the application of science to the needs of humanity.
	www.elresearch.com /Tensor_analysis (181 words)

	Tensor notation (Site not responding. Last check: 2007-10-21)
		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)

	Tensor Project (Site not responding. Last check: 2007-10-21)
		Tensors were first utilized to describe the elastic deformation of solids.
		In the beginning of the 20th century, tensor calculus was refined by the Italian mathematicians Ricci and Levi-Cevita.
		In this project, a tensor notation is advocated from a different point of view.
	user.it.uu.se /~kurt/tensor.html (179 words)

	[No title]
		When a tensor used in physics has the form of the first coordinate being the time coordinate, and the remaining three being spatial coordinates, with zeroes off the diagonal, it’s called a 4-vector.
		Tensor equations with indices in the same relative positions on either side must be generally valid.
		An object formed from a tensor and n powers of the squareroot of -g is called the tensor density of weight n.
	www.geocities.com /jefferywinkler/tensors.html (2918 words)

	ABC Online Forum (Site not responding. Last check: 2007-10-21)
		Use of tensor notation makes mathematical manipulations much more convenient, particularly when we are dealing with three dimensional vectors or higher order tensors.
		The notation of a dummy index can be changed while a free index should remain free of the summation rule on repeated indices.
		All this means, is that when using our shorthand notation we cannon replace the free index with any other letter, however, the dummy index is open to change because we are not using the free index as the basis for our summation.
	www2b.abc.net.au /science/k2/stn/archives/archive32/newposts/164/topic164146.shtm (1753 words)

	Nils K. Oeijord's Home Page
		Tensor algebra may be considered as an extension of Linear Algebra.
		Classically, a tensor of type (m,n) is represented by an array with m upper and n lower indices.
		Tensor calculus is the study of tensor fields on manifolds or varieties.
	home.online.no /~n-oeij/tensors.cfm (1643 words)

	Science Fair Projects - Notation
		In its most common usage, notation refers to the typographical conventions or rules of symbol usage that are followed, e.g., within a book or article.
		Under its broader definition, notational systems would include speech (a way of inscribing semantically meaningful vibrations upon the medium of air); money (a combined token-and-medium that attempts to represent value); logical notation; cartography; and of course writing.
		While notational systems have had a great impact on civilization, they are not themselves studied comparatively and longitudinally as a coherent subject, but rather are considered bit layers within the disciplines they support and enable.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Notation (630 words)

	Tensor Operators (Site not responding. Last check: 2007-10-21)
		The problem with this tensor is that it is reducible—combinations of the elements can be arranged in groups such that rotations operate only within these groups.
		The basic idea is that these irreducible subgroups into which Cartesian tensors apparently decompose under rotation (generalizing from our one example) form a more natural basis set of tensors for problems with rotational symmetries.
		Notational note: we have followed Shankar here in having the rank k as a subscript, the “magnetic” quantum number q as a superscript, the same convention used for the spherical harmonics (but not for the D matrices!) Sakurai, Baym and others have the rank above, usually in parentheses, and the magnetic number below.
	landau1.phys.virginia.edu /classes/752.mf1i.spring03/TensorOperators.htm (1443 words)

	Tools of Tensor Calculus - Index notation
		As we have explained Type of a tensor, TTC is very rigurous in the type of tensors.
		The user has to remember that partial differentiation of a tensor does not generally yields another tensor and has to be carefully in using it.
		If you have a tensor with a lower index and want to raise it, just type the tensor in index notation moving the subscript to the upper position.
	baldufa.upc.es /xjaen/ttc/tutorial/inde.htm (1111 words)

	Math Markup Language (Chapter 3)
		Tensor indices are distinct from ordinary subscripts and superscripts in that they must align in vertical columns.
		Because presentation elements should be used to describe the abstract notational structure of expressions, it is important that the base expression in all "scripting" elements (i.e.
		Presubscripts and tensor notations are represented by a single element, .
	www.w3.org /TR/WD-math-980106/chap3_4.html (1901 words)

	Notation article - Notation musical composition music musical notation chemistry chemical - What-Means.com (Site not responding. Last check: 2007-10-21)
		In relativity, for example, Tensor notation is a general way to represent a gravitational field
		In dance various forms of dance notation are used to document movement such as Labanotation and Benesh Notation
		Notation article - Notation definition - what means Notation
	www.what-means.com /encyclopedia/Notation (285 words)

	Dogpile - Web Search: Einstein's tensor notation (Site not responding. Last check: 2007-10-21)
		The equations of motion of a viscous fluid in
		tensor notation, which is probably best known in its application in...
		notation as an example of covariance with respect to...
	dogpile.com /info.dogpl/search/web/Einstein%2527s%2Btensor%2Bnotation (287 words)

	PlanetMath: first order operators in Riemannian geometry
		The gradient operator, which in tensor notation is expressed as
		The divergence operator, which in tensor notation is expressed as
		This is version 2 of first order operators in Riemannian geometry, born on 2005-08-16, modified 2005-08-16.
	planetmath.org /encyclopedia/FirstOrderOperatorsInRiemannianGeometry.html (218 words)

	THEORY MANUAL: 2. FORMULATION OF THE CONTINUUM PROBLEM, 2.1-2.2 (Site not responding. Last check: 2007-10-21)
		Standard tensor notation uses subscripted indices to represent the order and components of a tensor.
		In standard tensor notation, subscripted indices are used to identify the order and components of a tensor.
		The notation for temporal derivatives employs either a derivative operator or a dot (·) to represent the derivative with respect to time.
	www.shef.ac.uk /mecheng/staff/xyl/fidap/help/theory/th02_01-02.htm (2675 words)

	vector matrix tensor (Site not responding. Last check: 2007-10-21)
		What is a tensor and can any examples of their use be given?...
		Research: Nonlinear structure tensor, nonlinear diffusion, image segmentation...
		The evaluation of the elastic and piezoelectric tensors with...
	www.scienceoxygen.com /math/511.html (131 words)

	Description (Site not responding. Last check: 2007-10-21)
		Based on the rank 1 irreducible tensor components given in equation (10-12) - the reason is shown in the section on tensor products which follows in the Chapter - we shall set the constant C to be 1/2.
		The rank one tensors are given in equation (10-18) and the three Wigner 3-j coefficients are obtainable from equations sosi, sosi, and sosi.
		In that instance the Hamiltonian is rotated by keeping the space/spin tensor aligned along the z-axis in the laboratory frame (the axis of the static spectrometer magnetic field) and re-orienting the pure spatial tensor.
	gamma.magnet.fsu.edu /html/modules/level1/spint7.htm (2605 words)

	A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) (James G. Simmonds)
		The mathematics of tensor analysis is introduced in well-separated stages: the concept of a tensor as an operator; the representation of a tensor in terms of its Cartesian components; the components of a tensor relative to a general basis, tensor notation, and finally, tensor calculus.
		The physical interpretation and application of vectors and tensors are stressed throughout.
		This book is great for anyone who was to now the basics on tensors for use in general relativity or continuum mechanics, and is a great introduction book for someone who wants to study Differential geometry.
	johnkeyes.com /a/038794088X-a-brief-on-tensor-analysis-undergraduate-texts-in-mathematics.html (540 words)

	Tensors and tensor algebra (Site not responding. Last check: 2007-10-21)
		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)

	proof that grad V is a (1,1) tensor - Information Technology Services (Site not responding. Last check: 2007-10-21)
		Since apparently tensors arose in riemann's study of differential geometry, and in particular to measure curvature, perhaps that is the place to begin.
		You can consider the grad to be a (1,1) tensor in the sense that for any fixed v, the tensor takes in a contravariant del f and covariant df such that the definition is true.
		Such a thing is a candidate for a tensor "of type (1,1)" since it associates to every point a linear map from the tangent space to itself, or equivalently an element of the tensor product of the tangent space with its dual.
	www.physicsforums.com /archive/t-48019_proof_that_grad_V_is_a_(1,1)_tensor.html (3403 words)

	Arrays in Blitz++
		Users are free to declare their own tensor indices with different names if they prefer.
		Tensor indices specify how arrays are oriented in the domain of the array receiving the expression (Fig.
		Figure 1: Illustration of Blitz++ tensor notation: the indices specify how arrays are oriented in the domain of the array receiving the result.
	www.osl.iu.edu /~tveldhui/papers/iscope98 (1928 words)

	Tensor Mobility Model of Shockley Equations
		As is known from the consideration on crystallographic symmetry, the mobility is a second-rank tensor.
		Moreover, the mobility should be expressed as a fourth-rank tensor when strain or magnetic field is applied.
		Because of this formula, tensor expressions for the mobility are obtained.
	www.nsti.org /Nanotech2004/showabstract.html?absno=543 (277 words)

	Introduction to Vector and Tensor Analysis - Interactive Reviews
		For example, he never even defines what a tensor IS. This is problematic for a number of mathematical terms.
		Much of tensor analysis is understanding notation, then realizing that much of calculus is applicable to a tensor, with tweaks here and there.
		No matter how elegant you get with differential forms or manifold notation; when it comes time to use a tensor you have to break it down into components; and no other book is as great as this one.
	www.interactivereviews.com /product/048661879X (776 words)

	Math 251 Lecture 04 (Site not responding. Last check: 2007-10-21)
		Tensor Notation was introduced, because I think this makes the proofs of vector identities less tedious.
		Tensor notation was then used to prove that the direction of a cross-product is perpendicular to both original vectors.
		These can be proved easily using tensor notation, and in fact, a couple of these proofs will be shown next lecture.
	www.math.sfu.ca /~hebron/archive/1999-3/math251/lec_notes/lec04.html (221 words)

	C. Calculating tensor components (ResearchIndex)
		Calculating tensor components C2 The goal of the GRTensorII program is the calculation of components of indexed objects, in particular tensors.
		1 Algebraic invariants of the Riemann tensor in a fourdimensio..
		1 Scalar polynomial invariants of the Riemann tensor for spher..
	citeseer.ist.psu.edu /452420.html (234 words)

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