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Topic: Tensor field

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	A (Site not responding. Last check: 2007-10-06)
		field is a vector field, the gradient of a vector field is a tensor field, the gradient of
		The divergence of a vector field is a scalar field, the divergence of a tensor field is
		a vector field, the divergence of a tensor field is a vector field, the divergence of a
	www.cchem.berkeley.edu /jsngrp/Che250/HW1.htm (958 words)

	Tensor field - Wikipedia, the free encyclopedia
		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.
		In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus.
		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 - Wikipedia, the free encyclopedia
		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 of the brain.
		Note that the word "tensor" is often used as a shorthand for tensor field, which is a tensor value defined at every point in a manifold.
	en.wikipedia.org /wiki/Tensor (1874 words)

	Tensor - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-06)
		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.
		Examples of physical tensors are the energy-momentum tensor, the inertia tensor and the.
		The "components" of the tensor are the indices of the array.
	www.bexley.us /project/wikipedia/index.php/Tensor_calculus (1625 words)

	What is a tensor?
		Tensors, defined mathematically, are simply arrays of numbers, or functions, that transform according to certain rules under a change of coordinates.
		A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points.
		A tensor may consist of a single number, in which case it is referred to as a tensor of order zero, or simply a scalar.
	www.physlink.com /Education/AskExperts/ae168.cfm (878 words)

	Mirror for Internet Encyclopedia - Wikinfo \| Morphogenic network (Site not responding. Last check: 2007-10-06)
		A morphogenic network is a self-organizing neural network that organizes with respect to an adaptive tensor field : a continuous information space (see information geometry), covariant with physical space.
		An integrator performs temporal bit-voting (Another way to look at this is that the integration field decomposes the output field over the time continuum, into temporal eigenfunctions).
		(or the divergence of the force vector field in state space, + being source, - being sink)
	www.internet-encyclopedia.us /index.php/wiki.php?title=Morphogenic_network (2218 words)

	A Symetric Derivation of the Electromagnetic Energy-Momentum Tensor
		A symmetric derivation of the of the energy-momentum tensor for the electromagnetic field is presented, by treating the interaction between the field and both electric and magnetic charges on equal footing.
		That the specification of the divergence of an antisymmetric tensor and the divergence of it's dual completely determines the tensor (and hence in this case, the fields) is a generalization of Helmholtz's theorem to four dimensional space time [8], the divergence of the dual playing the role of the curl.
		A frequent approach to defining the energy-momentum tensor for the electromagnetic field is to generalize the definition of the Hamiltonian density to a covariant form[9].
	vortex.brynmawr.edu /vortex/eppaper.html (1488 words)

	sciforums.com - covariant derivative
		All tensors are tensor fields in that the tensor is defined throughout the space on which it is defined.
		A tensor is defined at a point P on a manifold according to certain criteria (e.g.
		I have a doubt: you have the stress-energy tensor of General Relativity, and then I have learned that the covariant derivative of the stress-energy tensor is zero.
	www.sciforums.com /showthread.php?t=34327 (547 words)

	UC Davis Geology: AGU Meeting Abstracts
		We have developed a tensor field visualization method tailored specifically to the class of tensor fields exhibiting properties similar to stress and strain tensors, which are commonly encountered in geophysics/geomechanics.
		The main idea it to represent a tensor field as a ``texture-deforming operator,'' which resembles deforming a piece of fabric to express the characteristic properties of a tensor field.
		Here, a white-noise texture is blurred according to the tensor field, resulting in a high correlation of pixels along the principal lines, whereas almost no correlation appears in directions perpendicular to these lines.
	www-geology.ucdavis.edu /pubs/agu04/kellogg1_04.html (546 words)

	Modeling and inversion of 3-D gravity tensor field (Site not responding. Last check: 2007-10-06)
		Modern day instruments are capable of picking up the tensor components of the gravity field, and hence the need arises for applying inversion theory to interpret the tensor data so obtained.
		The developed method is illustrated by the inversion of the synthetic tensor gravity data.
		Jorgensen, G. J., and J. Kisabeth, 2000, Joint 3-D inversion of gravity, magnetic and tensor gravity fields for imaging salt formations in the deepwater Gulf of Mexico: 70th SEG Annual Meeting, Extended Abstracts, Calgary, Canada, 424-426.
	www.mines.utah.edu /cemi/appls/grav/tensgrav/tensor.html (495 words)

	Electromagnetic Field Gauges
		The anti-symmetric 2-rank electromagnetic field tensor F has 3 properties: its trace is zero, it is antisymmetric, and it contains all the components of the E and B fields.
		Whatever the scalar field is in the first term of the generator gets subtracted away in the second term.
		The field tensor F in the temporal gauge is quite similar to the Coulomb gauge, but some of the signs have changed to target the dphi/dt term.
	world.std.com /~sweetser/quaternions/EandM/gauges/gauges.html (513 words)

	Previous Work
		5] proceeds by multiplying a fixed user-specified vector by the tensor field as sampled on some restricted domain (such as a plane) which acts as a probe to query specific regions of the field.
		When the tensor visualization is not accomplished by showing only some of the information at all locations, it is often done by showing all the tensor information in a restricted subset of locations.
		Streamlines are advected through a vector field of one of the eigenvectors, but instead of simply drawing a line to indicate the path, a surface is formed whose cross-section indicates the orientation of the other two eigenvectors and their associated eigenvalues.
	www.cs.utah.edu /~gk/papers/tvcg00/node2.html (495 words)

	3.4 The New Momentum-Energy Tensor of Field. 6-Formalism
		In Gunnar Nordström's gravitation scalar theory the scalar function, equal to a trace of complemented momentum-energy tensor, thanking to conform multiplier in a linear element is transformed into a source of a scalar gravitation field.
		In this case we obtain finite energy of an electron and degeneration of doublet of vector and scalar fields is eliminated.
		The source of the field may be only the field itself.
	www.geocities.com /elefzaze/indx0304en.html (876 words)

	Field (physics)
		vector fields, such as the magnitude and direction of the force at each point in a magnetic field.
		tensor fields, such as the stress-energy tensor in general relativity.
		Classically, field theory was first used to describe the behavior of particles in electric and magnetic fields.
	www.sciencedaily.com /encyclopedia/field__physics_ (190 words)

	The Stress Tensor of the Electromagnetic Field
		Using the terms as defined in "The classical theory of fields" by Landau and Lifshitz, the antisymmetric 2-rank field tensor F is used to generate the stress tensor T
		I have a practical sense of an E field (the stuff that makes my hair stand on end) and a B field (the invisible hand directing a compass), but have little sense of the field tensor F, a particular combination of the other two.
		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.
	world.std.com /~sweetser/quaternions/EandM/tensor/tensor.html (614 words)

	New Transformation Equations and the Electric Field Four-vector
		In addition, the magnitude of the charge is no longer invariant, but charge density is. The electric field is described by a four-vector and the antisymmetric electromagnetic field tensor is replaced by the electric field tensor.
		Our intention, now, is to present the electric field in its most general form and to express the field and force equations and energy-momentum tensor in terms of the generalized electric field.
		In rectangular coordinates, the derivative of a tensor is identical to the ordinary derivative.
	www.softcom.net /users/der555/node22.html (3756 words)

	Insect Control Using Tensor Fields
		Upon re-instituting the tensor field program being operated 1-2 hours, morning and evening, the miller population again declined to the 3-6 per screen area.
		Results: McCurry was able to harvest at the normal ripening and field drying conditions, whereas other operations in the area were obliged to harvest early and bear the expense of spraying during the growing season and costs of drying the wetter corn in their fields.
		Tensor field modulation by audio in molecular water frequency range appears to very significantly reduce pest insect damage across a range of pest insect species and plant host species.
	www.transformation.net /coils/tensor.html (720 words)

	cs5630: Project #6
		This is the strain tensor field derived from the 50x50x50 challenge vector dataset used in the last project:
		Whereas the vector field contained 3 components (Vx,Vy,Vz) at each point, the strain tensor field consists of 3x3 matrices at each point in the volume.
		This is the scalar 'length' metric of the strain tensor:
	www.cs.utah.edu /classes/cs5630/projects/6 (822 words)

	Tensor Field Tomography for Residual Stress Measurement in Glass Articles. (Site not responding. Last check: 2007-10-06)
		Vector and tensor field tomography makes a number of new and interesting nondestructive methods possible [1]: diagnostics of magnetic field in plasma, of the Kerr-effect field, time of-flight tomography of fluid flow and Doppler tomography of atoms velocities in plasma.
		Reconstruction of the spatial distribution of the stress tensor by interpretation of integrated optical effects of the rays that have passed through the medium may be considered a type of tensor field tomography.
		One of them is connected with the transversal interactions of the two-dimensional vector field and the other with the transversal interactions of the two-dimensional tensor field.
	www.ndt.net /abstract/ecndt98/367.htm (407 words)

	Physics Help and Math Help - Physics Forums - Christoffel symbol as tensor
		defines a map of dual vectors at p (as opposed to dual vector fields defined in a neighborhood of p) to tensors of type (0, 2) at p.
		So maybe the confusion is that when he changes coordinates, in order to have a tensor, he would have to transform both derivative operators by the tensor law, whereas he is merely taking one of the coordinate derivative operators as his delta a, rather than taking whatever he gets when he transforms the primed one.
		For example, a Lorentz tensor is not a tensor in the normal sense of the term since the Lorentz tensor is only a tensor under a Lorentz transformation.
	www.physicsforums.com /printthread.php?t=40177 (2872 words)

	Articles - Exact solutions in general relativity (Site not responding. Last check: 2007-10-06)
		In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field.
		Here, the tensor field standing on the left hand side, the Einstein tensor, is computed uniquely from the metric tensor which is part of the definition of a Lorentzian manifold.
		These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor.
	lastring.com /articles/Exact_solutions_in_general_relativity?... (2304 words)

	Vector/Tensor Visualization (Site not responding. Last check: 2007-10-06)
		The Tk interface offers two choices for tensor glyph visualization: Either generate the glyphs inside a sphere with radius that can be changed by the user or generate the glyphs at the vertices of a isosurface with isovalue in the range 0-10 (that can be changed).
		Thus tensor glyphs are always inside a sphere centered at the critical point (if not on the isosurface vertices) and vector glyphs on the surface of the same sphere.
		With tensor fields we visualize eigenvalues, eigenvectors and these dont have a direct physical meaning in some cases (although they do in other cases).
	www.physics.utah.edu /~zona/cs6630/Project5.html (4051 words)

	Fundamental theorem of calculus - Open Encyclopedia (Site not responding. Last check: 2007-10-06)
		This means that if a continuous function is first integrated and then differentiated, the original function is retrieved.
		This theorem is of such central importance in calculus that it deserves to be called the fundamental theorem for the entire field of study.
		An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.
	open-encyclopedia.com /Fundamental_theorem_of_calculus (932 words)

	A Modern Formulation of Riemann's Theory: A Supplement to Nineteenth Century Geometry
		A section of any of these bundles is a tensor field on M (of rank r, if the bundled objects are r-linear functions).
		A Riemannian metric g on the n-manifold is a tensor field of rank 2 on M.
		Nowadays this object is conceived as a tensor field of rank 4, which assigns to each point P in a Riemannian n-manifold a 4-linear function on the tangent space T
	plato.stanford.edu /entries/geometry-19th/supplement.html (1124 words)

	Tensor Field Visualization (Site not responding. Last check: 2007-10-06)
		This image superimposes the hyperstreamline method and the deformation based method for visualizing tensor fields.
		Two planes perpendicular to the applied force (red arrow) are selected and deformed by tensor field on those planes.
		Additionally, they cover a large region of the field and hence provide more direct information over a greater portion of the field.
	www.cse.ucsc.edu /~pang/tensor.html (352 words)

	Description (Site not responding. Last check: 2007-10-06)
		Evidently, the general effect of the applied static field (Zeeman) is to cause some rotation about the z-axis (the axis of the static field), and at common field strengths this rotation will be in hundreds of Megahertz.
		Because only the rank 2 dipolar spatial tensors will be non-zero, it is sufficient to utilize only the irreducible spherical rank 2 components of both the spatial and spin dipolar tensors in construction of the dipolar Hamiltonian.
		The scalar coupling tensor values could in principle be experimentally determined in the tensor principal axes, and then less values are necessary to specify the tensor because many more tensor elements are zero.
	gamma.magnet.fsu.edu /html/modules/hslib/hshama2.htm (9175 words)

	The Electromagnetic tensor
		is the electric field parallel and perpendicular to
		in an electromagnetic field is [ Assignment 4 ]:
		This is the energy momentum tensor of the electromagnetic field.
	www.mth.uct.ac.za /omei/gr/chap4/node7.html (168 words)

	Tensor Field Visualization
		The physical interpretation of mathematical features of tensor fields is highly application-specific.
		Our technique is a global method that represents the physical meaning of these tensor fields with their central features: regions of compression and expansion.
		The eigenvector fields are represented using a texture-based approach resembling LIC methods.
	graphics.cs.ucdavis.edu /~lfeng/research/tensor (194 words)

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