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Topic: Solenoidal vector field

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Vector potential

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	Vector potential - Wikipedia, the free encyclopedia
		In vector calculus, a vector potential is a vector field whose curl is a given vector field.
		This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field.
		A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.
	en.wikipedia.org /wiki/Vector_potential (247 words)

	Divergence - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-29)
		In vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point.
		For instance, for a vector field that denotes the velocity of water flowing in a draining bathtub, the divergence would have a negative value over the drain because the water vanishes there (if we only consider two dimensions); away from the drain the divergence would be zero, since there are no other sinks or sources.
		The Laplacian of a scalar field is the divergence of the field's gradient.
	xahlee.org /_p/wiki/Divergence.html (555 words)

	Solenoidal head coil for ultra-low magnetic field MRI
		The solenoidal head coil was studied deeply by Hoult and Richards(1) and by Hoult and Laterbour(2), but is was not used in horizontal magnetic field for imaging porpuses.
		The expression [2] is valid in all the space, the field in the axis of the coil is proportional to:
		They are the approximates expressions of the field in the axis of a rectangular coil; this will be analized and compared with experimental results, from this will be seen that the theoretical appoximation is valid.
	www.geocities.com /CapeCanaveral/2404/mri/head3.htm (1232 words)

	Solenoidal vector field - Wikipedia, the free encyclopedia
		In vector calculus a solenoidal vector field is a vector field v with divergence zero:
		one of Maxwell's equations states that the magnetic field B is solenoidal;
		the velocity field of an incompressible fluid flow is solenoidal.
	en.wikipedia.org /wiki/Solenoidal_vector_field (110 words)

	ESS265: History of Vector Magnetometry:
		The sensor should truly make vector measurements as the angular response should be equal to B cos f where f is the angle between the field vector and the sensor axis.
		The oscillation depends upon the presence of a magnetic field and ceases when the field vector is within "12o of parallel or "7o of normal to the optical axis of the sensor.
		At low frequencies in quiet fields, calculation of the vector field from the sine wave amplitude and phase and the steady component parallel to the spin axis could be provided by either the inboard tilted sensor on the two outboard sensors.
	www-ssc.igpp.ucla.edu /personnel/russell/ESS265/History.html (6585 words)

	[No title]
		4 was initiated with the representation of an irrotational vector field E, this chapter began by focusing on the solenoidal character of the magnetic flux density.
		The boundary conditions obeyed by the vector potential at surfaces of discontinuity (containing surface currents) reflect the discontinuity in tangential H field and the continuity of the normal flux density.
		The homogeneous solution is both irrotational and solenoidal, so it is possible to use either the vector or the scalar potential to represent this part of the field everywhere.
	web.mit.edu /6.013_book/www/chapter8/8.7.html (896 words)

	vector potential
		A vector potential is a vector field which generates a solenoidal vector field.
		This vector potential always points downwards (-z direction) but its magnitude is a radial Gaussian function, centered at the z axis, with inflection points around in a circle of radius 1.
		Imagine the A field on the x-y plane as a group of vertical pillars rising from the ground: their height is the magnitude of A.
	www.fact-library.com /vector_potential.html (1047 words)

	OPTI 501
		Vector integration: Line, surface, and volume integrals, divergence of a vector field, flux of a vector field, the divergence theorem, curl of a vector field, circulation density, Stokes theorem, uniqueness theorem.
		Reflected and refracted fields: Helmholtz equations for the electric and magnetic fields and boundary conditions, TE or s-polarization and TM or p-polarization, reflection and transmission coefficients, normal incidence.
		Scalar and vector potentials: Vector and scalar potentials and their relation to the physical fields, wave equations for the potentials, nonuniqueness of potentials and gauge transformations, Lorentz gauge and Coulomb (or radiation) gauge and associated wave equations for the potentials, seperation into transverse and longitudinal fields in the Coulomb gauge.
	www.optics.arizona.edu /classes/Grad/Opti_501.htm (1056 words)

	PlanetMath: solenoidal field
		A solenoidal vector field is one that satisfies
		Cross-references: Laplace's equation, function, vector, divergence, satisfies, vector field
		This is version 6 of solenoidal field, born on 2002-11-13, modified 2004-10-18.
	planetmath.org /encyclopedia/SolenoidalField.html (58 words)

	Vortical -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		In (Click link for more info and facts about fluid dynamics) fluid dynamics, the movement of a fluid can be said to be vortical if the fluid moves around in a circle, or in a helix, or if it tends to spin around some axis.
		where v is the velocity vector field of the fluid.
		Vector: (Click link for more info and facts about solenoidal vector field) solenoidal vector field, (Click link for more info and facts about vector potential) vector potential
	www.absoluteastronomy.com /encyclopedia/v/vo/vortical.htm (265 words)

	Abstracts for Complexity in Geophysical Systems, October 8-12, 2001
		These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (i) the suggestion that a small p-value may be a precursor of a large earthquake, (ii) the relative seismic quiescence sometimes observed before large aftershocks, and (iii) the increase of seismic activity preceding large earthquakes.
		The determination of the main geomagnetic field at the core-mantle boundary (CMB) from observations made at nearly twice the distance from the geocentre, i.e.
		The more complex CMB field has several pairs of dip poles and several null flux lines, not all of which are nested and upon some of which there are pairs of touch points.
	www.ima.umn.edu /geoscience/abstracts/10-8abs.html (2556 words)

	CTAC 2001 Abstract: A Vector Spherical Harmonic Spectral Code for Linearised Magnetohydrodynamics - David Ivers
		The magnetic field, the velocity, the pressure and the temperature in the sphere are governed by the magnetic induction equation, the heat equation and the Navier-Stokes equation in the Boussinesq approximation with uniform diffusivities.
		The linearised equations are discretised in angle using vector spherical harmonic expansions of all vector fields and spherical harmonic expansions of all scalar fields, for both the basic and perturbation states.
		Thus there are five independent perturbation scalar fields: the temperature, and the toroidal and poloidal potentials of the magnetic field and the velocity.
	conference.maths.uq.edu.au /ctac2001/abstracts/DavidIvers2.html (311 words)

	Nonsolenoidal fields (Site not responding. Last check: 2007-10-29)
		A general vector field (of sufficient smoothness) can be decomposed as follows.
		It is advantageous here to apply the present procedure to all vector fields, whether or not they are solenoidal.
		This is because if a field is both irrotational and solenoidal, so that it has both scaloidal and poloidal representations, the scaloidal representation is usually easier to work with.
	www.ae.su.oz.au /~mcbain/papers/thesis/node144.html (253 words)

	[No title]
		] The theorem that the moment of a force is the algebraic sum of the moments of its vector components acting at a common point on the line of action of the force.
		The field of vectors arising from considering a system of differential equations on a differentiable manifold.
		A function whose range is in a vector space.
	www.accessscience.com /Dictionary/V/V3/DictV3.html (2291 words)

	Symbolic Analysis of Magnetic Field with
		This elementary field is zero on the direction of the current and decreases inversely proportional to the square of the distance to the source.
		The vector potential given by (4) is solenoidal (has div A = 0), if the divergence of the current density is zero in the whole conducting domain (respectively, if the normal component of the current density, Jn, is zero on the surface of the conductor).
		In this case, the vector potential has only one component, normal to the problem's plane (A is represented by a scalar rather than a vector), and the magnetic field lines coincide with the equipotential lines.
	www.ewh.ieee.org /soc/es/May2001/03/bsl1.html (1266 words)

	Divergence (Site not responding. Last check: 2007-10-29)
		The divergence of a vector field is defined
		One of the most important results in vector field theory is the so-called divergence theorem or Gauss' theorem.
		Such a field is called a solenoidal vector field.
	farside.ph.utexas.edu /teaching/em1/lectures/node19.html (422 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Provide an introduction to vector calculus and vector fields, with particular interest in conservative and irrotational vector fields.
		state what a conservative vector field is, what a potential function is, and be able to calculate a potential function for a conservative vector field.
		state what an irrotational vector field is and the relationship between conservative and irrotational vector fields.
	www.aero.gla.ac.uk /UGrad/courses2004-5/Year2/AIMathemAE2X.doc (249 words)

	MAGNETIC PETROLOGY
		When establishing the form of the basis functions used to represent ionospheric fields, it was assumed that the source currents flowed in a region that was entirely below the Magsat and POGO satellite sampling shells.
		In addition, a relationship was assumed between the external and internal fields from these currents based upon the concept of an equivalent sheet current flowing on a sphere at r = a + 110 km.
		If displacement currents are neglected, then the source currents are solenoidal, and these assumptions may be used for current loops or circuits which do not pierce the sampling shell, which is true for the E-region.
	denali.gsfc.nasa.gov /cm/toroidal.html (255 words)

	Reconstructed Potential Functions In Bounded Domain Vector Tomography - Osman, Prince (ResearchIndex)
		Abstract: The tomographic imaging of vector fields on bounded domains is considered.
		It is then shown that potential functions reconstructed from these formulas include a component that arises from the harmonic component of the vector field, which is due to boundary conditions alone.
		Closed form expressions for the harmonic field contribution to the reconstructed scalar and vector potential functions are...
	citeseer.ist.psu.edu /osman97reconstructed.html (478 words)

	Topics: Vector Fields on Manifolds
		vector field v on a manifold X, the flow of v on X is the mapping
		Vertical vector field: A vector field in a fiber bundle is vertical if it is tangent to the fiber.
		Vertical covector field: Given a preferred horizontal subspace on a fb, a covector field is vertical if its contraction with any horizontal vector vanishes.
	www.phy.olemiss.edu /~luca/Topics/v/vector_field.html (299 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The application of external magnetic field causes both an alignment of the magnetic moments of the spinning electrons and an induced magnetic moment due to a change in the orbital motion of electrons.
		The magnetic field of a small bar magnet is the same as that of a magnetic dipole.
		For a bar magnet the fictitious magnetic charges +qm and —qm are assumed to be separated by a distance d and to form an equivalent magnetic dipole of moment.
	www.ece.sc.edu /classes/Summer01/elct361/EM-CH06.doc (1064 words)

	The magnetic vector potential (Site not responding. Last check: 2007-10-29)
		In other words, the vector potential is undetermined to the gradient of a scalar field.
		Note that the vector potential is parallel to the direction of the current.
		This would seem to suggest that there is a more direct relationship between the vector potential and the current than there is between the magnetic field and the current.
	farside.ph.utexas.edu /teaching/em1/lectures/node35.html (641 words)

	Sections20-01.html
		The constant and positive value of the divergence signals the uniform spread of the field, evidenced by the flow pointing radially outwards from the origin.
		Connections between conservative fields and solenoidal fields are detailed in Sections21.5.
		is the field of velocity vectors for a steady fluid flow with constant density, then its divergence must be zero, as we discuss in Sections23.3.
	www.adeptscience.co.uk /products/mathsim/maple/powertools/engineeringmath/html/Section20-01.html (670 words)

	Solenoidal vector field -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		In (Click link for more info and facts about vector calculus) vector calculus a solenoidal vector field is a (Click link for more info and facts about vector field) vector field v with (An infinite series that has no limit) divergence zero:
		This condition is clearly satisfied whenever v has a (Click link for more info and facts about vector potential) vector potential,
		the velocity field of an (Click link for more info and facts about incompressible fluid flow) incompressible fluid flow is solenoidal.
	www.absoluteastronomy.com /encyclopedia/s/so/solenoidal_vector_field.htm (151 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-10-29)
		Solenoids can be constructed to use electricity, compressed air (pneumatic solenoids), or pressurized fluids (hydraulic solenoids).
		An electrical solenoid is a form of electromagnet.
		In vector calculus a solenoidal vector field is a vector field v with divergence zero: [ \nabla \cdot \mathbf{v} = 0 ].
	www.hostingciamca.com /browse.php?title=S/SO/SOL (11109 words)

	Section20-01.html
		, the ratio of the flux of the field
		Connections between conservative fields and solenoidal fields are detailed in Section 21.5.
		is the field of velocity vectors for a steady fluid flow with constant density, then its divergence must be zero, as we discuss in Section 23.3.
	adept.maplesoft.com /powertools/engineeringmath/html/Section20-01.html (690 words)

	Mathematical formulation
		The purpose of PHI-3D is to provide an efficient solver for the equations of incompressible MHD in a three-dimensional, Cartesian, periodic domain.
		In dimensionless form where the magnetic field intensity is measured in units of velocity the equations can be written as follows:
		are inverse measures of the diffusivity of vorticity and magnetic field respectively and consist of ratios of diffusive to advective timescales.
	astro.uchicago.edu /Computing/HPCC/Codes/mps_50gf/html/notes/node3.html (147 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		When the fields created by these time sources are confined to propagate as a wave along a transmission line or inside a waveguide, the wave is usually referred to as a guided wave.
		The expression for the electric field intensity as given in (5.39) is the same as that produced by a static electric dipole.
		Thus, the fields are zero along the axis of the dipole and are maximum in a plane perpendicular to its axis (broadside to the dipole).
	www.ece.sc.edu /classes/Spring01/elct362/Ch5_362.doc (5850 words)

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