
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector (spatial) 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). The opposite of a pseudovector is a (true) vector or a ''polar'' vector. A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b: :p = a × b A simple example of an improper rotation is a coordinate inversion: x goes to −x. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector. This concept can be further generalized to pseudoscalars and pseudotensor, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor. 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. However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in R3 every bivector a ∧ b has a unique Hodge dual vector a × b, it is this dual which is more often used. ==Physical examples== Physical examples of pseudovectors include the magnetic field, torque, and the angular momentum. 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. For example, consider the case of an electrical current loop in the ''z''=0 plane: this system is symmetric (invariant) under mirror reflections through the plane (an improper rotation), but the magnetic field is anti-symmetric (flips sign) under that mirror plane—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature. To the extent that physical laws are the same for right-handed and left-handed coordinate systems (i.e. invariant under inversion), the sum of a vector and a pseudovector is not meaningful. However, the weak nuclear force that governs beta decay ''does'' depend on the chirality (physics) of the universe, and in this case pseudovectors and vectors ''are'' added. ==References== * George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'' (Harcourt: San Diego, 2001). (ISBN 0120598159) * John David Jackson, ''Classical Electrodynamics'' (Wiley: New York, 1999). (ISBN 047130932X) == See also == *Grassmann algebra Linear algebra Vector calculus
== Recent work == Recently I've been working on trying to write summaries of Buffy the Vampire Slayer List of Buffy the Vampire Slayer episodes. This is one of the best shows to ever be made for television. Under the guidance of Joss Whedon the writers of the show touch on really weighty subjects such as life, love, and death with great humor and insight. Lately I've been working on List_of_Buffy_the_Vampire_Slayer_episodes#Season_Four_.281999-2000.29. My favorite Buffy characters were Willow_Rosenberg and Tara_Maclay. I loved Tara because she was so gentle and beautiful. By far they were the best couple of all on Buffy. I was so sad when the writers killed off Tara. I really loved the idea that when Willow and Tara were together their magic was made stronger. What a beautiful metaphor about how the right people together in a relationship can make each other stronger. == Favorite Wikis == *Buffy the Vampire Slayer ** Tara Maclay ** Willow Rosenberg ** Wikipedia:WikiProject Buffy ** List of Buffy the Vampire Slayer episodes *Middle-earth **Fëanor **Fëa and hröa ::I love Tolkien's vision of the body and soul, how immortal elves' souls abide in the Halls of Mandos and may return to Middle-earth and no one but Eru_Ilúvatar and Manwë know what happens to the souls of mortal men (called the ''Gift of Men''). *comics **Jean Grey ::Phoenix_(comics) rules. When Willow Rosenberg is after the surviving members of the Evil Trio, one of them refers to her as Dark Phoenix. So cool! == Group Affiliations == I am a member of Wikipedia:WikiProject Buffy.
Vector: Just go to the Wikipedia:WikiProject Buffy and edit your name in under mine if you want to join. I would love your help. - User:CheNuevara 16:27, 28 Apr 2005 (UTC) : Ah, noticed you did that already. Thanks for joining -- talk it up on its subpages, and feel free to invite anyone you know.
See other meanings of words starting from letter:
Words begining with Pseudovector:
Pseudovector
Pseudovector
Pseudovector
Pseudovector_meson
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector (spatial) 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). The opposite of a pseudovector is a (true) vector or a ''polar'' vector. A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b: :p = a × b A simple example of an improper rotation is a coordinate inversion: x goes to −x. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector. This concept can be further generalized to pseudoscalars and pseudotensor, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor. 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. However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in R3 every bivector a ∧ b has a unique Hodge dual vector a × b, it is this dual which is more often used. ==Physical examples== Physical examples of pseudovectors include the magnetic field, torque, and the angular momentum. 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. For example, consider the case of an electrical current loop in the ''z''=0 plane: this system is symmetric (invariant) under mirror reflections through the plane (an improper rotation), but the magnetic field is anti-symmetric (flips sign) under that mirror plane—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature. To the extent that physical laws are the same for right-handed and left-handed coordinate systems (i.e. invariant under inversion), the sum of a vector and a pseudovector is not meaningful. However, the weak nuclear force that governs beta decay ''does'' depend on the chirality (physics) of the universe, and in this case pseudovectors and vectors ''are'' added. ==References== * George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'' (Harcourt: San Diego, 2001). (ISBN 0120598159) * John David Jackson, ''Classical Electrodynamics'' (Wiley: New York, 1999). (ISBN 047130932X) == See also == *Grassmann algebra Linear algebra Vector calculus
Pseudovector
== Recent work == Recently I've been working on trying to write summaries of Buffy the Vampire Slayer List of Buffy the Vampire Slayer episodes. This is one of the best shows to ever be made for television. Under the guidance of Joss Whedon the writers of the show touch on really weighty subjects such as life, love, and death with great humor and insight. Lately I've been working on List_of_Buffy_the_Vampire_Slayer_episodes#Season_Four_.281999-2000.29. My favorite Buffy characters were Willow_Rosenberg and Tara_Maclay. I loved Tara because she was so gentle and beautiful. By far they were the best couple of all on Buffy. I was so sad when the writers killed off Tara. I really loved the idea that when Willow and Tara were together their magic was made stronger. What a beautiful metaphor about how the right people together in a relationship can make each other stronger. == Favorite Wikis == *Buffy the Vampire Slayer ** Tara Maclay ** Willow Rosenberg ** Wikipedia:WikiProject Buffy ** List of Buffy the Vampire Slayer episodes *Middle-earth **Fëanor **Fëa and hröa ::I love Tolkien's vision of the body and soul, how immortal elves' souls abide in the Halls of Mandos and may return to Middle-earth and no one but Eru_Ilúvatar and Manwë know what happens to the souls of mortal men (called the ''Gift of Men''). *comics **Jean Grey ::Phoenix_(comics) rules. When Willow Rosenberg is after the surviving members of the Evil Trio, one of them refers to her as Dark Phoenix. So cool! == Group Affiliations == I am a member of Wikipedia:WikiProject Buffy.
Pseudovector
Vector: Just go to the Wikipedia:WikiProject Buffy and edit your name in under mine if you want to join. I would love your help. - User:CheNuevara 16:27, 28 Apr 2005 (UTC) : Ah, noticed you did that already. Thanks for joining -- talk it up on its subpages, and feel free to invite anyone you know.
See other meanings of words starting from letter:
P
PA | PB | PC | PD | PE | PF | PG | PH | PI | PJ | PK | PL | PM | PN | PO | PR | PS | PT | PU | PW | PX | PY | PZ |Words begining with Pseudovector:
Pseudovector
Pseudovector
Pseudovector
Pseudovector_meson
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