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Topic: Antisymmetric tensor

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	Tools of Tensor calculus - Tensors and Forms
		As we have stressed in the Introduction a tensor in TTC is handled as a single (geometric) object, so, in addition to the components, one has to input also the basis elements.
		The notation for tensors follows a rather straigthforward generalization of the notation for vectors and 1-forms that we have seen in the previous chapter.
		Tensors of type (0, q) which are antisymmetrics on all q indices are called exterior q-forms.
	baldufa.upc.es /xjaen/ttc/tutorial/tens.htm (1223 words)

	Antisymmetric - Wikipedia, the free encyclopedia
		In set theory, the adjective antisymmetric usually refers to an antisymmetric relation.
		The term "antisymmetric function" is sometimes used for odd function, although some meanings of antisymmetric are essentiality f(y,x) = -f (x,y).
		In linear algebra and theoretical physics, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (usually the exchange of two indices, which becomes the transposition of the matrix) is performed.
	en.wikipedia.org /wiki/Antisymmetric (159 words)

	Euclidean Tensors
		The pressure tensor in a liquid is a multiple of this tensor, expressing the property that the pressure is independent of direction.
		The 23 component of the antisymmetric tensor is the same as the 1 component of c, for example.
		If the components of an antisymmetric tensor are the infinitesimal velocities of points of a fluid relative to a certain point, then the associated vector is twice the infinitesimal angle of angular velocity of rotation, the vorticity, and is a pseudovector.
	www.du.edu /~jcalvert/math/eucltens.htm (5403 words)

	NSDL Metadata Record -- Antisymmetric Tensor -- from MathWorld
		An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched.
		For example, a tensor A^{x_1,\dots,x_n} such that A^{x_1,\dots,x_i,\dots,x_j,\dots,x_n}=-A^{x_n,\dots,x_i,\dots,x_j,\dots,x_1} is antisymmetric.
		The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies A^{mn} = -A^{nm}.
	nsdl.org /mr/699003 (90 words)

	Tensors and pseudo-tensors
		Tensors which exhibit tensor behaviour under translations, rotations, special Lorentz transformations, and are invariant under parity inversions, are termed proper tensors, or sometimes polar tensors.
		Tensors such as this, which exhibit tensor behaviour under translations, rotations, and special Lorentz transformations, but are not invariant under parity inversions (in the sense that they correspond to different geometric objects before and after the transformation), are called pseudo-tensors, or sometimes axial tensors.
		The totally antisymmetric tensor is the prototype pseudo-tensor, and is, of course, conventionally defined with respect to a right-handed spatial coordinate system.
	farside.ph.utexas.edu /teaching/em/lectures/node120.html (1018 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.
		Special antisymmetric assignment class: Has specializations for different dimensionalities (for 1, 2, and 3, so far).
	www.tat.physik.uni-tuebingen.de /~rguenth/phd/html/Tensor_8h.html (384 words)

	MathTensor Usage Messages
		The rank of the tensor to be cleared is indicated by the number of indices specified (2 in the above example).
		Detg is the determinant of the metric tensor.
		MaxwellTexpression[la,lb] is the expression for the Maxwell stress tensor in terms of the Maxwell field tensor, MaxwellF.
	smc.vnet.net /usage.html (11090 words)

	Glossary of tensor theory - Wikipedia, the free encyclopedia
		A tensor written in component form is an indexed array.
		A dyadic tensor has rank two, and may be represented as a square matrix.
		Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity.
	en.wikipedia.org /wiki/Glossary_of_tensor_theory (696 words)

	DRAFT: Deducing a Unified Field Theory from Electromagnetism (Site not responding. Last check: 2007-10-18)
		Third, the field strength tensor is antisymmetric, so it must be represented by an odd spin particle, the spin 1 photon.
		The reducible asymmetric tensor can be split into an antisymmetric tensor for the spin 1 photon and a symmetric tensor for the spin 2 graviton.
		By including a symmetric field strength tensor in the GEM action, it becomes possible for the action to account for the changes in metric tensor, and thus remove the metric from the background structure.
	world.std.com /~sweetser/quaternions/gravity/em2gem/em2gem.html (2641 words)

	Comments on M.W. Evans' "Duality and the Antisymmetric Metric"
		Gravitation therefore is a manifestation of curved spacetime with a symmetric metric, and electromagnetism is a manifestation of spinning spacetime with an antisymmetric metric.
		The metric 4-vector in this spacetime is written as an antisymmetric tensor which is used to define a two-form of differential geometry.
		In general curvilinear coordinates both the symmetric and antisymmetric metrics are defined in terms of the same set of scale factors, and this result can be used in principle to measure the effect of one field on the other.
	www.mathematik.tu-darmstadt.de /~bruhn/Comment-Chap2.htm (2608 words)

	Maxwell's equations Modern Relativity modernrelativity special general black hole mass energy Einstein wormhole time ...
		Show that the covariant duel of the contravariant duel of the a covarient antisymmetric tensor of rank two is the original tensor times -1.
		In this case, the local inertial frame observes a gravitational field that is not zero a finite extent away from the charged particle.
		Even so, it is not too difficult to express this relativisticly as a tensor equation in the case of special relativity.
	www.geocities.com /zcphysicsms/chap7.htm (1155 words)

	the coefficient matrix as a tensor sum
		Solving for the antisymmetric part produces a rotation, because the differential equation for the inverse of the solution matrix has the same negative factor as the differential equation for the transpose.
		When the system is two dimensional, there is only one antisymmetric matrix apart from scalar multiples, so that the solution is directly a matrix exponential, similar to what happens when the trace generates an exponential scale factor.
		In any event, once the trace and antisymmetric part have been attended to, the final equation which remains has to deal with a rotating traceless symmetric tensor; this may or may not resemble an actual simplification.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node50.html (285 words)

	tensor, not vector
		tensor field that is not a vector field?
		the "archetype example" of a tensor field the gravity field.
		dervivatives of the"metric" tensor, and the "metric" tensor is often
	www.groupsrv.com /science/post-490.html (689 words)

	Tensors and tensor algebra (Site not responding. Last check: 2007-10-18)
		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)

	Open System for Geniuses - by Chronostalker :: July :: 2005 (Site not responding. Last check: 2007-10-18)
		is not a tensor according to the mathematical definition of a tensor, where by a tensor we understand a geometrical object associated to the bundle of all linear frames.
		Tensor can be represented by a matrix, but this matrix representation changes with the change of basis vectors (in our case, it changes when we rotate basis vectors).
		The elements of the antisymmetric metric tensor and its equivalent metric vector are related in contravariant-covariant tensor notation [7] by
	opensys.blogsome.com /2005/07/01 (2130 words)

	PlanetMath: exterior algebra
		and also known as the wedge product, is an antisymmetric variant of the tensor product.
		For the purposes of the present entry, we define an antisymmetric map to be a
		It is useful to compare the above definition to the categorical definition of the tensor algebra.
	planetmath.org /encyclopedia/ExteriorAlgebra.html (1001 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)

	A Nonsymmetric Metric
		One part is symmetric and one part is antisymmetric, and neither part is c.
		The arrival times at the other clock, as indicated by the receiving clock, can be decomposed into the symmetric and antisymmetric parts.
		The antisymmetric part has the units of seconds/volt.
	www.s-4.com /tensor (536 words)

	The electromagnetic field tensor
		This immediately implies that all of the diagonal components of the tensor are zero.
		In other words, the completely space-like components of the tensor specify the components of the magnetic field, whereas the hybrid space and time-like components specify the components of the electric field.
		, because of the antisymmetry of the electromagnetic field tensor.
	farside.ph.utexas.edu /teaching/jk1/lectures/node22.html (334 words)

	MAX
		As a result of the Machian dynamics in the universe, a body in motion is subject to their potentials and field strengths, which determine its dynamics.
		is antisymmetric in the last two indices, while the torsion T from which it can be built is antisymmetric in the first two.
		supposedly post-Newtonian approximation to the Riemannian connexion, since the contorsion, in contradistinction to the Christofel symbols, is a tensor and it is genuinely antisymmetric.
	www.mypage.bluewin.ch /Bizarre/MAX.htm (1184 words)

	BASIC PRINCIPLES OF CLASSICAL ELECTRODYNAMICS
		In ICED the physical reality is represented uniquely by the mathematical identities—numbers (we relate scalars, vectors, tensors to the category of numbers or identities) which we call physical values.
		The physical values form the exact boundary between physical reality and physical theory and, at the same time, they form the boundary between physics (represented by the physical values) and mathematical apparatus that is used to evaluate the physical values.
		The conservation of the unique energy-momentum tensor is not a consequence – it is a requirement that follows from the minimum action with respect to the variation of a metric tensor in a flat space.
	www.wbabin.net /yuri/keilman10.htm (3987 words)

	How many theories?
		The antisymmetric tensor field carries a force that is difficult to describe in this short space.
		Scalar tachyon, massless vector boson, antisymmetric tensor, graviton and dilaton
		It's just as well that bosonic string theory is unstable, because it's not a realistic theory to begin with.
	superstringtheory.com /basics/basic5a.html (1102 words)

	Worldline approach to vector and antisymmetric tensor fields (Site not responding. Last check: 2007-10-18)
		The N = 2 spinning particle action describes the propagation of antisymmetric tensor fields, including vector fields as a special case.
		Its quantization on the torus produces the one-loop effective action for a single antisymmetric tensor.
		We use this worldline representation to calculate the first few Seeley-DeWitt coefficients for antisymmetric tensor fields of arbitrary rank in arbitrary dimensions.
	stacks.iop.org /1126-6708/2005/i=04/a=010 (442 words)

	[No title]
		Notice that the above 5 x 5 matrix is an antisymmetric tensor.
		For a general SU(n) grand unification, in order to ensure the vector nature of the color gauge interactions, to have only three colors, and three anticolors, the simplest choice is to assign fermions only to antisymmetric representations.
		The dimensionality of an mth rank antisymmetric tensor under SU(n) is
	www.geocities.com /jefferywinkler/beyondstandardmodel.html (3275 words)

	Heisenberg Durr Schroedinger Bohm
		In the theories of scalar, vector,and tensor fields (another way of denoting spins 0, 1, and 2) the fields are described by scalar, vector, or tensor potential functions:
		From the point of view that 4x4 real antisymmetric matrices correspond to Spin(1,3) Lorentz transformations of 4-dim Minkowski space, which are the 6 bivectors of the 1 4 6 4 1 graded Clifford algebra structure
		But it was soon realized that the antisymmetric part g[uv] in the decomposition guv = g(uv) + g[uv] could not describe physically the electromagnetic field.
	www.valdostamuseum.org /hamsmith/HeisHist.html (3256 words)

	Einstein's quest for unification (January 2005) - Physics World - PhysicsWeb
		Another recurring theme in Einstein's quest for unification was to generalize the "metric" of relativity - the symmetric tensor that describes the curvature of space-time - so that it could also describe the electromagnetic field.
		For example, in his first unification paper in 1925 the antisymmetric part of his tensor field was not suitable for describing all the components of the electric and magnetic fields.
		However, in his final years following 1945 he returned to a theory with a fundamental tensor that was not symmetric and would include both the metric and the electromagnetic tensor, which avoided some of these problems.
	physicsweb.org /articles/world/18/1/15/1 (1414 words)

	PlanetMath: Hodge star operator
		The Hodge star operation occurs most frequently in differential geometry in the case where
		Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise.
		Cross-references: pointwise, fields, antisymmetric, cotangent, tensor, manifold, geometry, operation, metric tensor, Cartesian coordinates, dimensions, identity operator, metric, volume, unit, coordinates, operator, Levi-Civita permutation symbol, components, dual basis, basis, terms, mapping, linear operator, inner product, vector space, finite
	planetmath.org /encyclopedia/HodgeStarOperator.html (169 words)

	POOMA Tutorial 4: Further Topics
		POOMA includes two "tiny" classes that are optimized to represent small vectors and tensors.
		This is a data-parallel statement that works in a way analogous to the loop at lines 22-25, except that the POOMA evaluator will calculate patches in parallel.
		Tensor, equal to the transpose of the tensor (element (i,j) of the transpose is equal to element (j,i) of the input tensor
	www.nongnu.org /freepooma/tutorial/tut-04.html (3740 words)

	The Tom Bearden Website (Site not responding. Last check: 2007-10-18)
		A differential p form is a (0, p) tensor which is completely antisymmetric.
		Thus scalars are 0 forms, dual vectors are one forms, the antisymmetric tensor is a two form, the current density is a three form and the Levi Civita tensor in 4-D is a four form.
		The tetrad is a one form, the spin connection is a one form but not a tensor, the torsion tensor is a vector valued two form, the Riemann form is a (1, 1) tensor valued two form.
	www.cheniere.org /correspondence/062503.htm (2367 words)

	antisymmetric - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "antisymmetric" is defined.
		antisymmetric, antisymmetric, antisymmetric : PlanetMath Encyclopedia [home, info]
		Phrases that include antisymmetric: antisymmetric matrix, antisymmetric tensor, antisymmetric part
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=antisymmetric (135 words)

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