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Topic: Mixed tensor

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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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	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 /~rfitzp/teaching/jk1/lectures/node10.html (1038 words)

	Tensor -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-26)
		Tensors may be written down in terms of (A system that uses coordinates to establish position) coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen (A system of assumptions and standards that sanction behavior and give it meaning) frame of reference.
		Tensors are of importance in (The science of matter and energy and their interactions) physics and (The discipline dealing with the art or science of applying scientific knowledge to practical problems) engineering.
		Note that the word "tensor" is often used as a shorthand for (additional info and facts about tensor field) tensor field, which is a tensor value defined at every point in a (A pipe that has several lateral outlets to or from other pipes) manifold.
	www.absoluteastronomy.com /encyclopedia/t/te/tensor.htm (2044 words)

	Dr. Ivan Yotov: Papers and Reports (Site not responding. Last check: 2007-10-26)
		The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient).
		For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions.
		Quadrature rules turn the mixed method into a cell-centered finite difference method with a 9 point stencil in 2-D and 19 in 3-D. As shown by theory and experiment, if the modified permeability on the computational domain is smooth, then the convergence rate is optimal and both pressure and velocity are superconvergent at certain points.
	www.ticam.utexas.edu /Groups/SubSurfMod/papers/yotov.html (1661 words)

	Pullback - Wikipedia, the free encyclopedia
		This is precisely because a pullback on mixed tensors would need to be "going in the opposed direction" for the contravariant indeces.
		A general mixed tensor will then transform as a mixture of transposes and inverses, depending on whether the indices are contra- or co-variant.
		In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor; by contrast, the transformation of the contravariant indices is given by a pushforward.
	en.wikipedia.org /wiki/Pullback (888 words)

	Easy Encyclopedia - Online Encyclopedia. Knowledge is Power
		A dyadic tensor has rank two, and may be represented as a square matrix.
		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.easyencyclopedia.com /encyclopedia/g/gl/glossary_of_tensor_theory.html (503 words)

	Mixed Nuts
		Mixed economy 1: In economics and politics, a '''mixed economy''' is an Economicseconomy that combi 17: nited States, have mixed economies.
		Temperate broadleaf and mixed forests 1: '''Temperate broadleaf and mixed forests''' are a temperate and humid biome.
		Mixed martial arts 1: '''Mixed martial arts''' or '''MMA''' is a term for the co 57: The rules for most Mixed Martial Arts competitions have evolved since the 143: [http: www.mma.tv Mixed* Martial Arts TV.
	www.lottery-news.net /dust13451-mixed_nuts.html (487 words)

	Lit-Tensors
		In the case of diffusion tensor visualization, we have made tensors opaque and colored based on their anisotropy, but we lacked a means of shading tensors in the presence of a virtual light source.
		The obvious choice for the ``surface normal'' for a planar anisotropic tensor is the third eigenvector, perpendicular to the plane formed by the span of the first two eigenvectors (associated with the largest two eigenvalues).
		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/tvcg00/node5.html (1496 words)

	Mixed tensor - TheBestLinks.com - Tensor, Contravariant, Covariant, Tensor analysis, ...
		Mixed tensor - TheBestLinks.com - Tensor, Contravariant, Covariant, Tensor analysis,...
		In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant.
		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.thebestlinks.com /Mixed_tensor.html (98 words)

	Maxima Manual - Tensor (Site not responding. Last check: 2007-10-26)
		Tensor Package - will set to zero, in exp, all occurrences of the tensori that have derivative indices.
		Tensor Package - will set to zero, in exp, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.
		Tensor package) This function first computes the covariant components LR[i,j] of the Ricci tensor (LR is a mnemonic for "lower Ricci").
	www.ma.utexas.edu /maxima/maxima_27.html (1080 words)

	ipedia.com: Multilinear algebra Article (Site not responding. Last check: 2007-10-26)
		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.
	www.ipedia.com /multilinear_algebra.html (733 words)

	Appendix C
		This appendix describes the general principles for preparing the energy-momentum tensor and it gives the detailed analytical steps that Schwartzschild developed to form the energy-momentum tensor that he used in his solution of the Einstein theory.
		All tensors can be transformed from one coordinate system to another in accordance with this formula (or its equivalent).
		A total of 256 terms are needed to calculate all 16 elements of this tensor.
	www.olduniverse.com /appendix_c.htm (1164 words)

	Glossary of tensor theory - InfoSearchPoint.com (Site not responding. Last check: 2007-10-26)
		If v and w are vectors in vector spaces V and W respectively, then $v \otimes w$ is a tensor in $V \otimes W$ .
		A pure tensor of $V \otimes W$ is one that is of the form $v \otimes w$ .
		In the tensor algebra T(V) of a vector space V, the operation $\otimes$ becomes a normal (internal) binary operation.
	www.infosearchpoint.com /display/Glossary_of_tensor_theory (552 words)

	Tensors
		Equations written in terms of tensors automatically maintain the same pattern amongst the tensor quantities no matter what coordinate system is being referred to.
		This is the prototype equation for covariant tensors.
		If a mixed tensor has contravariant rank p and covariant rank q, then it is said to have type (p, q).
	scholar.uwinnipeg.ca /courses/38/4500.6-001/Cosmology/Tensors.htm (992 words)

	Mixed Finite Elements For Elliptic Problems With Tensor Coefficients As Cell-Centered Finite Differences - Arbogast, ...
		We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coe#cient.
		The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coe#cient times the negative gradient).
		Arbogast, M. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer.
	citeseer.ist.psu.edu /arbogast97mixed.html (721 words)

	General Relativity (Site not responding. Last check: 2007-10-26)
		The metric tensor plays a special role in general relativity in that it determines the geometry of spacetime and is used in the prescription for performing integrals and differentials.
		In order to bring this in the canonical form for the transformation of a tensor, we have to get rid of the second term in the right-hand side of the equation, which arises from the derivative of the transformation coefficients.
		The curvature tensor is applied in the Einstein equation of the gravitational field, and is used to describe the motion of particles in curved spacetime.
	www.nikhef.nl /~henkjan/astro/node13.html (1621 words)

	Maxima Manual - Ctensor
		Indicial Tensor Package) yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (CHR2).
		DIM is the dimension of the manifold with the default 4.
		A function in the CTENSR (Component Tensor Manipulation) package which automatically loads the CTENSR package from within MACSYMA (if it is not already loaded) and then prompts the user to make use of it.
	rene.ma.utexas.edu /maxima/maxima_28.html (1068 words)

	[No title]
		Thus the `natural form' of the Riemann tensor calculated from a metric would be specified as: R(up,dn,dn,dn) The components of the same tensor in a basis would be: R(bup,bdn,bdn,bdn) A full set of default tensors which are defined automatically can be found in the help pages ?grt_objects and ?grt_basis.
		Thus to calculate the covariant derivative of the covariant Einstein tensor, one could use: G(dn,dn,cdn) Similarly, the ordinary partial derivatives can be calculated using the indices `pup' and `pdn', as in G(dn,dn,pdn) - Note that most objects require the calculation of `precursor' objects.
		The program will keep track of which tensors have been calculated for a given metric so that they can always be accessed later and will never be calculated twice.
	grtensor.phy.queensu.ca /Griihelp/grcalc.help (542 words)

	Tensors (Site not responding. Last check: 2007-10-26)
		of the Ricci tensor (LR is a mnemonic for lower Ricci).
		As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor.
		Since all tensors are considered to be symmetric in all indices, the indices are sorted into alphabetical order.
	starship.python.net /crew/mike/maxima/html/macref/node108.html (1789 words)

	Some basic notations from tensor analysis
		The correspondence between vector and tensor notation for a rank one tensor is expressed by
		For a tensor of rank two the correspondence between the two notations is given by, for example in the case of a mixed tensor:
		The divergence theorem in vector and tensor notation
	ta.twi.tudelft.nl /isnas/isnas_mathmanual/node3.html (174 words)

	equivalence Modern Relativity modernrelativity special general black hole mass energy Einstein wormhole time ...
		The covariant metric tensor also acts as a lowering index operator and the contravariant metric tensor acts as a raising index operator.
		From this perspective the "elements of a tensor" are thought of as "projections of the tensor" onto a coordinate dependent template.
		Given that the metric tensor is symmetric, verify that the Affine connections are symmetric in the lower indices.
	www.geocities.com /zcphysicsms/chap4.htm (3433 words)

	From Vectors To Tensors (Site not responding. Last check: 2007-10-26)
		Tensors with symmetries: Tensors with branched symmetry, symmetric tensors
		Extension of the symmetrized tensor associated with a mixed tensor
		Tensors with anti-symmetries: Tensors with branched anti-symmetry; anti-symmetric tensors
	www.booksmatter.com /b354022887X.htm (522 words)

	`shift()` (Site not responding. Last check: 2007-10-26)
		This is accomplished by looking at the altmetric property of the parent to find the tensor metric that existed at the time the parent was created.
		Thus, if the first index of the Riemann tensor is raised, only the anti-symmetry in the third and fourth indices survives, but if all the indices are raised, the offspring has the full Riemann symmetry.
		The tensor metric itself has its indices raised to form the metric inverse via a matrix inverse computed by invert(); this function sets up the shift property on the metric, and also adds the proper delta function to represent a metric with mixed indices.
	www.scar.utoronto.ca /~harper/redten/node32.html (973 words)

	Re: Deriving Schrodinger using Quaternions
		In that case, the rank-1 tensors are 3-vectors.
		The direct tensor sum lets > me staple together things that transform differently, like a tensor of > rank 0 and 1 as is done to define a quaternion initially.
		In matricial form, the direct tensor sum of a polar and an axial vector must be a 2x2 quaternionic matrix, like in the Clifford algebra.
	www.lns.cornell.edu /spr/2000-04/msg0023523.html (664 words)

	Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences
		Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences: SIAM Journal on Numerical Analysis Vol.
		For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions.
		Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation).
	epubs.siam.org /sam-bin/dbq/article/26258 (316 words)

	Big test
		is a mixed tensor of rank equal to the number of components.
		Quotient theorem: If A is an object and the contraction of A with a tensor B with arbitrary components produces a tensor C, then A must be a tensor.
		Hence covariant derivative of a vector along a curve is a tensor.
	newton.ex.ac.uk /teaching/resources/rj/test (555 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Tensors may be represented in either the ambient (F) or coordinate (C) bases with the aid of the procedures C2F and F2C (See below).
		- exterior form or covariant tensor &eta is an involutive automorphism of the exterior algebra or the covariant tensor algebra.
		- exterior form or covariant tensor &xi is an involutive anti-automorphism of the exterior algebra or the covariant tensor algebra.
	www.lancs.ac.uk /depts/spc/staff/chtw/Manifolds8.txt (8273 words)

	Folding Massage Tables , Mixed, bois and aluminium
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		Folding Massage Tables mixed avec Face bung : structure aluminium avec renforcement bois,, tensor-equipped and handles of transport, adjustable height from 57 to 85 cm, weigth 9.5 kg, size close 91/96 x 70 x 17.
		Folding Massage Tables mixed, tensor-equipped, poignées and Face rest, facial cavity, adjustable height from 57 to 85 cm, 12.5 kg, size close 91/96 x 70 x 17.
	www.table-de-massage.ch /massage-tables-folding-mixed-en.php (368 words)

	Natural Strain
		In a Euclidean space, distances are measured by a quadratic form, and an attempt to elude this fact is unlikely to succeed." They advocate using the "topological," quadratic strain fields of Almansi (ref. 5) or Green (ref. 6) instead of the "physical," logarithmic strain field of Hencky (ref. 3).
		A thorough and consistent development of the strain and strain-rate measures affiliated with Hencky was documented (ref. 8), and natural measures for strain and strain-rate were expressed in terms of the fundamental body-metric tensors of Lodge.
		This is a one-to-one mapping (transformation law) between tensor fields defined on a body manifold to tensor fields defined on the spatial manifold.
	www.lerc.nasa.gov /WWW/RT1996/5000/5110f.htm (618 words)

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