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Topic: Pseudovectors

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Magnetic field

Cartesian coordinates

	The MathematicalLearning Style- Mathematics Magazine
		Vectors are invariant under translation, and they reverse sign upon inversion.
		Objects that resemble vectors but do not reverse sign upon inversion are known as pseudovectors.
		To distinguish vectors from pseudovectors, the former are sometimes called polar vectors.
	www.mathematicsmagazine.com /4-2005/Gr12_4_2005.htm (551 words)

	Pseudovector - Wikipedia, the free encyclopedia
		In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation).
		Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as bivectors, following the calculus of differential forms; the double negation is natural for a bivector.
		Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems.
	en.wikipedia.org /wiki/Pseudovector (381 words)

	Read about Pseudovector at WorldVillage Encyclopedia. Research Pseudovector and learn about Pseudovector here! (Site not responding. Last check: 2007-11-02)
		In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a
		The opposite of a pseudovector is a (true) vector or a polar vector.
		chirality of the universe, and in this case pseudovectors and vectors are added.
	encyclopedia.worldvillage.com /s/b/Pseudovector (338 words)

	Symmetry In Physics Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-02)
		an isometry of position is associated with an isometry of other vectors and pseudovectors, which is the linear part of the isometry of position, combined with an inversion of pseudovectors in the case that the isometry changes orientation.
		In the case of mirror image symmetry in a plane with respect to scalars and vectors, the vectors at the plane are directed in the plane (or zero), and the associated pseudovectors are at the plane directed perpendicular to it (or zero).
		In the case of mirror image symmetry in a plane with respect to scalars and vectors, and an axis of rotation perpendicular to the plane, the vectors at the point of intersection are zero, and the associated pseudovectors are along the axis.
	www.merica.com /encyclopedia/Symmetry_in_physics (1054 words)

	CONK! Encyclopedia: Vector_(spatial) (Site not responding. Last check: 2007-11-02)
		A related concept is that of a pseudovector (or axial vector).
		This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations.
		(This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector.
	www.conk.com /search/encyclopedia.cgi?q=Vector_(spatial) (2808 words)

	Pseudovector -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-02)
		A common way of constructing a pseudovector p is by taking the (A vector that is the product of two other vectors) cross product of two vectors a and b:
		Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as (Click link for more info and facts about bivector) bivectors, following the calculus of (Click link for more info and facts about differential form) differential forms; the double negation is natural for a bivector.
		Physical examples of pseudovectors include the (The lines of force surrounding a permanent magnet or a moving charged particle) magnetic field, (A twisting force) torque, and the (The product of the momentum of a rotating body and its distance from the axis of rotation) angular momentum.
	www.absoluteastronomy.com /encyclopedia/p/ps/pseudovector.htm (325 words)

	Pseudovector (Site not responding. Last check: 2007-11-02)
		In physics and mathematics, apseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip underan improper rotation (a transformation that can be expressed asan inversion followed by a proper rotation).
		Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding andexploiting the effect of symmetry on the solution to physical systems.
		invariant underinversion), the sum of a vector and a pseudovector is not meaningful.
	www.therfcc.org /pseudovector-116046.html (288 words)

	Tensors and tensor algebra (Site not responding. Last check: 2007-11-02)
		They are pseudovectors that do not transform as tensors under a reflection of (an odd number of) coordinate axes.
		Gauss's divergence theorem is made plausible by considering the volume contributions to the integral of the divergence of a vector field cancel, except at the surface.
		and pseudovectors are to be avoided in the construction of equations describing fluid behavior, they can legitimately and usefully be employed in other circumstances.
	astron.berkeley.edu /~jrg/ay202/node185.html (790 words)

	Science and astrology. Zodiac based on a different Plane by Slawomir Stachniewicz
		Scalar product of two vectors or two pseudovectors is a scalar, so its sign does not depend on convention, scalar product of a vector and pseudovector is a pseudoscalar, so its sign depends on convention.
		Its length is equal to product of lengths of both (pseudo)vectors and sine of angle between them, its direction is perpendicular to both (pseudo)vectors and its orientation is determined by the left hand rule (we rotate left hand from the first vector to the other).
		Vector product of two vectors or two pseudovectors is a pseudovector, vector product of a vector and pseudovector is a vector.
	cura.free.fr /xxv/25stach2.html (1541 words)

	Symmetry - Wikipedia, the free encyclopedia
		(v))) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g, see symmetry in physics.
		In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry.
		A corresponding 3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero.
	en.wikipedia.org /wiki/Symmetry (2699 words)

	Re: CPT theorem
		Pseudovectors have the property than when they are reflected in a mirror they get flipped in their axis whereas vectors don't.
		The classic example is angular momentum: reflect a spinning top in a mirror and the direction of rotation is reversed (in addition to the axis of rotation being reflected) meaning that the reflected angular momentum vector points in the opposite direction.
		This mixing of vectors and pseudovectors is pretty weird if you think about it - but perfectly consistent.
	www.lns.cornell.edu /spr/1998-12/msg0013646.html (255 words)

	pseudovectors
		a vector is not inversion symmetric, where as a pseudovector is. examples of each are the position vector and the angular momentum vector.
		to me, it seems like were defining the cross product differently in right and left handed systems, and then when it doesnt look the same in both, we label it a pseudovector and label the tensor we used to define it a pseudotensor.
		if the right hand rule always applied, there would be no pseudovectors, because you can apply the right hand rule independent of a coordinate system.
	www.physicsforums.com /showthread.php?p=335897 (466 words)

	Hyperspace:
		Once in hyperspace, objects have a pseudovector, that correlates to their direction of movement in normal space.
		The pseudovector is maintained by the object’s hyperdrives as it travels.
		Storms may "blow" ships off their pseudovectors, change the granularity of existing areas (depending on the category of the storm), and generate gravity waves, of awesome power.
	www.aceofangels.com /AoAUniverse/hypermove.html (2462 words)

	[extropy-chat] unidirectional thrust (Site not responding. Last check: 2007-11-02)
		Nor is conservation of parity valid, as Feynman helped prove, and the Co-60 beta decay experiment demonstrated that emission is biased to the spin axis of the nucleus but not in both directions.
		He shows that em energy terms like H^2 and E^2 are scalars by EH are pseudovectors* and conservation doesn't apply in all cases.
		Often complex systems cannot be expressed as pseudovectors owing to the noncommunitivity of finite rotations....
	lists.extropy.org /pipermail/extropy-chat/2005-March/014431.html (1747 words)

	Electromagnetism using Geometric Algebra versus Components
		See reference 4 for an example of how to calculate a magnetic field without invoking the correspondence principle, without using the old-fashioned pseudovector representation of the field.
		You may be familiar with matrix multiplication, which has many of the same axioms as Geometric Algebra, including the associative law, the distributive law, and non-commutative multiplication.
		pseudovector) term on the RHS of equation 4.
	www.av8n.com /physics/maxwell-ga.htm (2806 words)

	Pseudovector (Site not responding. Last check: 2007-11-02)
		En la física y las matemáticas, un pseudovector (o el vector axial) es una cantidad que transforma como un vector bajo rotación apropiada, pero gana un tirón adicional del señal bajo rotación incorrecta (una transformación que se puede expresar como inversión seguida por una rotación apropiada).
		El contrario de un pseudovector es vector (verdadero) de a o un vector polar.
		A menudo, la distinción entre los vectores y los pseudovectors se pasa por alto, pero llega a ser importante en entender y explotar el efecto de la simetría en la solución para los sistemas físicos.
	www.yotor.net /wiki/es/ps/Pseudovector.htm (318 words)

	Vector (spatial) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-02)
		A related concept is that of a (Click link for more info and facts about pseudovector) pseudovector (or axial vector).
		Examples of pseudovectors include (The lines of force surrounding a permanent magnet or a moving charged particle) magnetic field, (A twisting force) torque, and (The product of the momentum of a rotating body and its distance from the axis of rotation) angular momentum.
		Because the cross product depends on the choice of coordinate systems, its result is referred to as a (Click link for more info and facts about pseudovector) pseudovector.
	www.absoluteastronomy.com /encyclopedia/v/ve/vector_(spatial).htm (2114 words)

	Graded algebra
		Given that essentially behave as scalars, they are often referred to as pseudoscalars.
		Similarly, $(n - 1)$ -vectors are also called pseudovectors.
		In order to achieve closure, all these spaces are combined by considering the direct sum of all of them.
	www.ebroadcast.com.au /lookup/encyclopedia/gr/Graded_algebra.html (245 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Measurements of alpha asymmetry in mu /sup -/ absorption in calcium were made using 250 Mev/c pulsed mu /sup -/ mesons stopped in a 12 g/cm/sup 2/ calcium target in a magnetic field.
		The theoretical and measured magnitudes of alpha indicate the presence of pseudoscalar interaction in the mu /sup -/ + A yields A' + n + nu process.
		The sign of the pseudoscalar and pseudovector constant in the ratio g/sub p//g/sub A/ is positive.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4843332&query_id=0 (239 words)

	Pseudovector - Encyclopedia Glossary Meaning Explanation Pseudovector (Site not responding. Last check: 2007-11-02)
		Here you will find more informations about Pseudovector.
		If you find this encyclopedia or its sister projects useful,
		A simple example of an improper rotation is a coordinate inversion: x goes to -x.
	www.encyclopedia-glossary.com /en/Pseudovector.html (377 words)

	Talk:Vector (Site not responding. Last check: 2007-11-02)
		I will confess that I don't know what a pseudovector is. I agree that there's no reason to prefer right handedness over left handedness, but for example Encyclopedia Britannica defines the cross product so that a, b and a×b always form a right handed system; some convention apparently is needed.
		Now the space of pseudovectors and the space of vectors have the same dimension, so it is tempting to associate the two, which would for instance allow one to represent pseudovectors by an arrow.
		In a sense, position is a bound vector, because it depends on the choice of coordinate systems.
	www.termsdefined.net /ta/talk:vector.html (3349 words)

	Tuning-Math Archive Section 9: 8500 - 8524
		A cross product takes vectors to vectors (or > pseudovectors, if you are a physicist) and in fact a three > dimensional real vector space with cross product is the real Lie > algebra o(3).
		Why pull the rug out from under me? I wish you had commented > when I posted this: > > Yahoo groups: /tuning-math/message/7798 * [with cont.] > > I took this as an important step in my learning about bra and ket > vectors.
		Maybe I'm prejudiced; as a physics major it's up to you to make use of the distinction for our purposes, perhaps, but to a mathematician there isn't a lot of difference between pseudovector and bivector.
	www.robertinventor.com /tuning-math/s___9/msg_8500-8524.html (3349 words)

	Ebla Forum: View topic - Physics Descriptions (Site not responding. Last check: 2007-11-02)
		Yes, they're dangerous to be "around," but they're also dangerous to calculate because they are associated with pseudovectors, of which torque is one example.
		A pseudovector acts like a real vector in the sense that it has magnitude and a direction.
		Other pseudovectors include angular velocity (ω) and angular momentum, which we will save for later.
	www.eblaforum.org /main/viewtopic.php?t=790 (4021 words)

	Dirac for Dunces
		When you say 'reverses sign when one of the co-ordinate axes is reversed you mean something along the lines of a spin 1/2 thingy?
		Are they flows of pseudovectors that have similar properties?
		If so I can see there might be some fun maths describing these.
	www.lns.cornell.edu /spr/2002-01/msg0038039.html (958 words)

	CONTEXTS FOR SIMPLE SPINOR ALGEBRA
		But, the secondary geometrical invariants, bivector = pseudovector (cross product) and volume = pseudoscalar change sign only under a reflection of an odd number of coordinate axes, so all possible flips fall into two (chiral) equivalence classes [(+++), (+--)], [(++-), (---)] that arbitrarily called right handed and left handed.
		Rank: 0 1 2 3 scalars vectors bivectors pseudoscalars pseudovectors Dimension: 1 3 3 1 A bivector is equivalent to a pseudovector.
		Rank: 0 1 2 3 4 scalars vectors bivectors pseudovectors pseudoscalars Dimension: 1 4 6 4 1 The linear space of bivectors is Hodge-* selfdual.
	graham.main.nc.us /~bhammel/PHYS/spinor.html (5134 words)

	Homework Solutions (Site not responding. Last check: 2007-11-02)
		The cross product of two pseudovectors is a pseudovector.
		The cross product of a vector and a pseudovector is a vector.
		The triple cross product produces a vector, Two examples include torque, and angular momentum, Finally, the scalar triple product transforms in the following way: A' * B' = - A * D. Therfore, it does not transform like a scalar, and is called a pseudoscalar.
	www.towson.edu /users/schaefer/354solutions.htm (265 words)

	What is a 3/2 spin particle called?
		== Starblade13(at)Yahoo.com (Starblade Darksquall) writes: == === Also, there are pseudoscalars, pseudovectors, and pseudotensors...
		It is called a "spinor" when the context is clear; it is sometimes called a "vector-spinor" when it is not, and also a "Rarita-Schwinger particle" after the people who first wrote down the wave equation it would obey.
		= Also, there are pseudoscalars, pseudovectors, and pseudotensors...
	www.pych-one.com /new-5403645-4394.html (1110 words)

	PHYS-L archives -- December 2001 (#7)
		When you introduce pseudovectors you have to explain the direction in terms of (I presume, in your case) the right hand rule.
		You need to explain that the choice is arbitrary - Martians, for example, use the left tentacle rule (and get all tangled up explaining it).
		I sometimes see pseudovectors defined when the cross product is introduced, but then I don't think I've ever explicitly seen their transformation properties used later on.
	lists.nau.edu /cgi-bin/wa?A2=ind0112&L=phys-l&D=0&P=610 (269 words)

	Axial Vectors in Rotating Coordinate Systems
		Ordinary vectors are shown in red and axial vectors in violet.
		Sometimes axial vectors are called pseudovectors and ordinary position vectors are called polar vectors.
		The significant difference between axial vectors and polar vectors is the effect on their coordinates of an inversion of the coordinate system; i.e.,
	www.applet-magic.com /axial.htm (252 words)

	Summary_Chapter_4_JJ
		It turns out that quantities that behaved in a vectorial or scalar way under rotations, now can have an additional property: namely they can be odd, or they can be even under parity.
		This is what makes the difference between scalars, and pseudoscalars, and vectors and pseudovectors.
		The wave function is nothing else but the representation of kets in the position basis.
	perso.wanadoo.fr /patrick.vanesch/nrqmJJ/Summary_Chapter_4_JJ.html (3226 words)

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