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Topic: Symmetric tensor

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	Springer Online Reference Works
		A tensor which does not change on symmetrization with respect to some group of indices is called a symmetric tensor.
		Symmetrization, with respect to some group, of a tensor which was alternated first (see Alternation) with respect to that group, leads to the zero tensor.
		Symmetrization of tensors, side by side with the alternation operation, is used for the decomposition of a tensor into tensors with a simpler structure.
	eom.springer.de /S/s091750.htm (287 words)

	Springer Online Reference Works
		A tensor which is invariant under transposition of this pair of indices.
		The result of alternation of a symmetric tensor with respect to this pair of indices is zero.
		A tensor is symmetric with respect to a set of indices if it is symmetric with respect to any two indices from this set.
	eom.springer.de /s/s091720.htm (105 words)

	NAMD: Tensor Class Reference
		Copyright (c) 1995, 1996, 1997, 1998, 1999, 2000 by The Board of Trustees of the University of Illinois.
		Referenced by Controller::berendsenPressure(), Molecule::build_gridforce_params(), diagonal(), ComputeFullDirect::doWork(), identity(), Sequencer::langevinPiston(), Controller::langevinPiston1(), Controller::langevinPiston2(), operator+=(), operator-=(), operator<<(), operator=(), outer(), outerAdd(), HomePatch::rattle1(), symmetric(), Tensor(), and triangular().
		Referenced by Molecule::build_gridforce_params(), diagonal(), ComputeFullDirect::doWork(), operator+=(), operator-=(), operator<<(), operator=(), outer(), outerAdd(), symmetric(), Tensor(), and triangular().
	www.ks.uiuc.edu /Research/namd/doxygen/classTensor.html (516 words)

	PlanetMath: second order tensor: symmetric and skew-symmetric parts
		Theorem Every covariant and contravariant tensor of second rank may be expressed univocally as the sum of a symmetric and skew-symmetric tensor.
		"second order tensor: symmetric and skew-symmetric parts" is owned by rspuzio.
		This is version 15 of second order tensor: symmetric and skew-symmetric parts, born on 2006-04-19, modified 2006-10-24.
	planetmath.org /encyclopedia/SecondOrderTensorSymmetricAndSkewSymmetricParts.html (152 words)

	LORENE: Tensor_sym class Reference
		Symmetric tensors (with respect to two of their arguments).
		is intended to store the components of a tensorial field with respect to a specific basis (triad), in the case the tensor has a valence at least 2 and is symmetric with respect to two of its arguments (or in other words, the components are symmetric with respect to two of their indices).
		[input] in the domain of index l0 the tensor is multiplied by the right polynomial (of degree 2n+1), to ensure continuty of the function and its n first derivative at both ends of this domain.
	www.lorene.obspm.fr /Refguide/classTensor__sym.html (1836 words)

	Strain tensor - Wikipedia, the free encyclopedia
		The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation:
		The deformation of an object is defined by a tensor field, i.e., this strain tensor is defined for every point of the object.
		This field is linked to the field of the stress tensor by the generalized Hooke's law.
	en.wikipedia.org /wiki/Strain_tensor (620 words)

	EN175: Mechanics of Solids - Introduction to Tensors and their properties (Site not responding. Last check: )
		The inverse of a tensor may be computed by calculating the inverse of the matrix of its components. Click Here to see how to do this. The result cannot be expressed in a compact form for a general second order tensor, and is best computed by methods such as Gaussian elimination.
		The eigenvalues of a symmetric tensor are always real. The eigenvalues of a skew tensor are always pure imaginary or zero.
		The eigenvalues of a tensor, and the components of the eigenvectors, may be computed by finding the eigenvalues and eigenvectors of the matrix of components. Click here to recall how to do this.
	www.engin.brown.edu /courses/en175/Notes/tensors/tensors.htm (1277 words)

	TLSView Manual
		Given the TLS tensors and their origin, a TLS predicted anisotropic ADP may be calculated for any atom using its position relative to the TLS origin by the set of linear equations given in Table 1.
		The skew tensor is generally non-symmetric, but through a transformation of the TLS model origin a unique origin may be found where it becomes symmetric.
		tensor is drawn at the center of reaction, and the three screw axes drawn with a length proportional to their RMSD rotational magnitude.
	pymmlib.sourceforge.net /doc/tlsview/tlsview.html (4451 words)

	The Stress Tensor of the Electromagnetic Field
		A tensor is a bookkeeping device designed to keep together elements that transform in a similar way.
		Together, the energy density(W), Poynting's vector (Sa) and the Maxwell stress tensor (m_ab) are all the components of the stress tensor of the electromagnetic field.
		The rest of that tensor is generated by the second line.
	world.std.com /~sweetser/quaternions/EandM/tensor/tensor.html (614 words)

	2.1 Geometry (Site not responding. Last check: )
		and the metric is symmetric and Lorentzian, i.e.
		The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
		The concept of tensors used in Special Relativity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinate-independent (“generally covariant”) form, then the full tensor concept comes into play.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-2/articlesu3.html (2680 words)

	Symmetric tensor - Wikipedia, the free encyclopedia
		A linear operator (or second-rank tensor) A, with components A
		Many physical and engineering properties are symmetrical tensors, e.g., stress and strain.
		Symmetric tensors can always be diagonalized by choosing apropiate Cartesian axes; these are called principal axes.
	en.wikipedia.org /wiki/Symmetric_tensor (121 words)

	Symmetry of the Stress Tensor
		The symmetric part is then sometimes referred to as the "stress tensor"(It is only a part of that), and the anti-symmetric part as the rotation tensor.
		The torque is proportional to the mass, i.e to the density.volume.
		It doesn't introduce the nature of the field but considers a property of the tensor, so we should be careful not to mix the property of the tensor to the nature of a specific field.
	www.physicsforums.com /showthread.php?p=1162269#post1162269 (2463 words)

	Inertia Tensor
		The angular momentum of a rigid body rotating about an axis passing through the origin of the local reference frame is in fact the product of the inertia tensor of the object and the angular velocity.
		This property is explained in detail in Transformation of the Inertia Tensor.
		A 2-dimensional symmetric matrix is not necessarily a tensor of the 2nd rank.
	kwon3d.com /theory/moi/iten.html (237 words)

	pooma: Tensor.h File Reference
		An interface class for an N-dimensional tensor of numeric objects, and engines class for defining a general tensor, using Full and Antisymmetric engine tag classes.
		Tensor is an interface class that takes three template parameters: int D: The number of components in each rank (row or col) of the Tensor.
		The partial specializations are of the Symmetrize functor.
	www.tat.physik.uni-tuebingen.de /~rguenth/phd/html/Tensor_8h.html (384 words)

	9.2.1 The deformation or rate of strain tensor
		In order to attach physical significance to this tensor, it is useful to separately consider its symmetric and anti-symmetric components, which are written
		If the motion is completely rigid, which means that it consists of a translation plus a rotation, then the symmetric part of the velocity derivative tensor vanishes.
		Hence, equation (9.12) says that the rate of change of the infinitesimal distance separating the two parcels, as a fraction of the distance, is related to the relative position of the parcels through the strain tensor.
	www.gfdl.noaa.gov /~smg/MOM/web/guide_parent/s2node93.html (291 words)

	Metric tensor - Wikipedia, the free encyclopedia
		In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space.
		In the later case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.
		In tensor analysis, the metric tensor is often used to provide a canonical isomorphism from the tangent space to the cotangent space.
	en.wikipedia.org /wiki/Metric_tensor (1258 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)

	Mass Tensor
		Since the equation of continuity must be a tensor equation it follows that the complete mathematical representation of mass is a tensor of rank 2.
		to be proportional to the as the mass tensor of dust.
		T is sometimes referred to as the material energy tensor [1], as well as the energy-momentum tensor or the stress-energy tensor.
	www.geocities.com /physics_world/sr/mass_tensor.htm (325 words)

	CanonDefine.html
		CanonDefine is used to define the symmetries of a given tensor.
		The third argument is a set of permutations or minus signed permutations that generate the symmetries of the tensor.
		A minus-signed permutation is represented by a list, where the first element is -1, and the second element is a list of lists describing the permutation, e.g.
	www.cbpf.br /~portugal/Canon/CanonDefine.html (280 words)

	MATTER and SPACE with TORSION
		Solutions of the equations for the spherically symmetric stationary model and the uniform isotropic model are given for a pure gravitational field and a massless fluid with spin
		Therefore, the transitional case for a spherically symmetric field will be described by complex equations, reduced from the condition related to the matter under consideration, and equations (7).
		This is natural, since there is no gravitational field in the center of any spherically symmetric ball: attraction of the external regions leads to their mutual compensation.
	www.acadjournal.com /2003/v9/part4/p1 (4231 words)

	Hyperstreamlines for Unsymmetric Tensors (Site not responding. Last check: )
		The antisymmetric tensor has only three independent components, which are encoded in an additional vector field along the trajectory of the hyperstreamline.
		The tensor is decomposed into a symmetric component representing a stretch and an isometric transformation representing a rotation.
		The symmetric tensor is again visualized as a hyperstreamline, whereas the angle and axis of rotation of the second component define a surface along the trajectory of the hyperstreamline.
	www.cs.auckland.ac.nz /~burkhard/PhD/img3.html (164 words)

	Data Interpolation
		We start by reconstructing a continuous tensor field in the volume through trilinear interpolation.
		Since the coefficients of this linear combination are independent of the tensor indexes, the linear combination of the tensors can be done component-wise.
		We can use trilinear component-wise interpolation because symmetric tensors form a linear subspace in the tensor space: any linear combination of symmetric tensors remains a symmetric tensor, i.e., symmetric tensors are closed under linear combination (the manifold of symmetric tensors is not left).
	www.gg.caltech.edu /~zhukov/research/fiber_tracking/vis02/node5.html (222 words)

	Volume 2 Number 4 Abstracts
		These results can be readily extended to include general tensor fields through linear combination of symmetric tensor fields and vector fields.
		The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other.
		Examples are given on the use of tensor field topology for the interpretation of physical systems.
	www.cs.sunysb.edu /~tvcg/v3n1/abstract.html (1429 words)

	Symmetric versus Antisymmetric Tensors (Site not responding. Last check: )
		Symmetric tensor: no sign or value changes after transposing.
		Antisymmetric tensor: all entries change signs but not value after transposing.
		Any asymmetric tensor can be represented by a symmetric tensor (averaged values of 2 indicies) and an antisymmetric tensor (+ and - diviations from average).
	theworld.com /~sweetser/quaternions/talks/IAP_1/1422.html (73 words)

	Riemann Package
		Maple package for calculating components of user-defined or built-in tensors used in General Relativity.
		The user can add, multiply or contract tensors, and the result of any calculation can be assigned to a new tensor.
		Tensors be symmetric or antisymmetric in all indices, symmetric or antisymmetric in two indices, or they can have the symmetry of a symmetric bivector-tensor (Riemann tensor).
	www.cbpf.br /~portugal/Riemann.html (213 words)

	[No title]
		Section 2: Tensors and Tensor Operations A tensor T is defined as an ordered collection of its components: EMBED Equation.DSMT4 where each of the i and each of the j can assume any integer value from 1 to N and all components are real numbers.
		When a quantity with proper components is defined in multiple coordinate systems or in terms of tensors, the quantity is said to be a tensor if it always obeys the transformation equation.
		Tensors are usually preferred over “nontensors” because a tensor needs to be defined in only one coordinate system and because change of tensor components under coordinate transformations is predictable and depend only on the coordinates and the nature of the tensor.
	web.mit.edu /dmytro/www/GR_theory.doc (1538 words)

	Nonsymmetric gravitational theory - Wikipedia, the free encyclopedia
		In general relativity, the gravitational field is characterized by a symmetric rank-2 tensor, the metric tensor.
		As the electromagnetic field is characterized by an antisymmetric rank-2 tensor, there is an obvious possibility for a unified theory: a nonsymmetric tensor composed of a symmetric part representing gravity, and an antisymmetric part that represents electromagnetism.
		In the weak field approximation where interaction between fields is not taken into account, the resulting theory is characterized by a symmetric rank-2 tensor field (gravity), an antisymmetric tensor field, and a constant characterizing the mass of the antisymmetric tensor field.
	en.wikipedia.org /wiki/Nonsymmetric_Gravitational_Theory (569 words)

	The Electromagnetic tensor
		We have now expressed Maxwell's equations in tensor form as required by Special Relativity.
		This is the energy momentum tensor of the electromagnetic field.
		is symmetric as required and the energy density is [ Assignment 4 ]
	www.mth.uct.ac.za /omei/gr/chap4/node7.html (168 words)

	Lecture # 2
		, are the elements of the diffusivity tensor,
		are reduced to 6, as the maximum number independent tensor elements.
		Although 6 is the maximum number of independent diffusivity elements, it would be rare, indeed, when we would encounter such a low symmetry.
	www.rpi.edu /dept/materials/COURSES/DIFFUSION/LECTURE2/x (379 words)

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