2013:
polar vector. Angular momentum is the cross product of a displacement (a polar vector) and momentum (a polar vector), and is therefore a pseudovector. Torque is angular momentum (a pseudovector) divided by time (a scalar), so is also a pseudovector. Continuing this way, it is straightforward to classify any of the common vectors in physics as either a pseudovector or polar vector. (There are the parity-violating vectors in the theory of weak-interactions, which are neither polar vectors nor pseudovectors. However, these occur very rarely in physics.)
266:
1547:
549:, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.
401:, and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the
1075:
40:
1975:
3537:
3188:
737:
318:(invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged.
1517:
1335:
819:
292:. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but the
1087:
is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any
329:
of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced
3738:
As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product. However, because the cross product does not generalize to other than three
2012:
From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a
3729:
are inverted, then the pseudovector is invariant, but the cross product changes sign. This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.
269:
Each wheel of the car on the left driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car. The fact that the arrows point in the same direction, rather than being mirror images of each other indicates that they are
3302:
1762:
2956:
629:
1678:
3623:
544:
A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the
333:
While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g.
1070:{\displaystyle {\begin{aligned}\mathbf {v_{3}} '=\mathbf {v_{1}} '+\mathbf {v_{2}} '&=(\det R)(R\mathbf {v_{1}} )+(\det R)(R\mathbf {v_{2}} )\\&=(\det R)(R(\mathbf {v_{1}} +\mathbf {v_{2}} ))=(\det R)(R\mathbf {v_{3}} ).\end{aligned}}}
1360:
1152:
2041:. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Nevertheless, there would be no physical consequences, apart from in the
3721:
are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectors
2743:
2571:
2434:
2276:
3532:{\displaystyle \mathbf {a} \wedge \mathbf {b} =\left(a^{2}b^{3}-a^{3}b^{2}\right)\mathbf {e} _{23}+\left(a^{3}b^{1}-a^{1}b^{3}\right)\mathbf {e} _{31}+\left(a^{1}b^{2}-a^{2}b^{1}\right)\mathbf {e} _{12}\ .}
536:
rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the
439:: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of
1970:{\displaystyle \mathbf {v_{3}} '=\mathbf {v_{1}} '\times \mathbf {v_{2}} '=(R\mathbf {v_{1}} )\times (R\mathbf {v_{2}} )=(\det R)(R(\mathbf {v_{1}} \times \mathbf {v_{2}} ))=(\det R)(R\mathbf {v_{3}} ).}
3183:{\displaystyle \mathbf {a} \times \mathbf {b} =\left(a^{2}b^{3}-a^{3}b^{2}\right)\mathbf {e} _{1}+\left(a^{3}b^{1}-a^{1}b^{3}\right)\mathbf {e} _{2}+\left(a^{1}b^{2}-a^{2}b^{1}\right)\mathbf {e} _{3},}
1530:
were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in the
2672:
the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.
824:
634:
2488:
409:
which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.
399:
732:{\displaystyle {\begin{aligned}\mathbf {v} '&=R\mathbf {v} &&{\text{(polar vector)}}\\\mathbf {v} '&=(\det R)(R\mathbf {v} )&&{\text{(pseudovector)}}\end{aligned}}}
296:
angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left, corresponding to the extra sign flip in the reflection of a pseudovector.
427:
The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract
1534:: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. (See
1564:
2369:
330:
by using (left-handed) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.
3708:
2598:
2906:
are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual." To paraphrase Baylis: Given two polar vectors (that is, true vectors)
2756:. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a
3554:
2658:
2629:
2327:
2211:
2176:
2124:
is 2, a sign flip has no effect. Otherwise, the definitions are equivalent, though it should be borne in mind that without additional structure (specifically, either a
1512:{\displaystyle |\mathbf {v_{3}} |=|\mathbf {v_{1}} +\mathbf {v_{2}} |,{\text{ but }}\left|\mathbf {v_{3}} '\right|=\left|\mathbf {v_{1}} '-\mathbf {v_{2}} '\right|}
1330:{\displaystyle \mathbf {v_{3}} '=\mathbf {v_{1}} '+\mathbf {v_{2}} '=(R\mathbf {v_{1}} )+(\det R)(R\mathbf {v_{2}} )=R(\mathbf {v_{1}} +(\det R)\mathbf {v_{2}} ).}
3758:. The idea that "a pseudovector is different from a vector" is only true with a different and more specific definition of the term "vector" as discussed above.
3739:
dimensions, the notion of pseudovector based upon the cross product also cannot be extended to a space of any other number of dimensions. The pseudovector as a
2508:
2296:
3986:
4511:
2104:
This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces. In particular, when
533:
418:
4013:
57:(blue). If the position and current of the wire are reflected across the plane indicated by the dashed line, the magnetic field it generates would
2689:
2513:
4348:
4218:
4153:
4124:
3996:
2374:
3809:
2216:
1698:
are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.)
3904:
560:
in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix
4482:
4442:
4415:
4396:
4329:
4300:
4273:
4245:
4185:
4094:
4065:
4032:
3960:
3914:
3819:
2026:
2022:
440:
1347:
is neither a polar vector nor a pseudovector (although it is still a vector, by the physics definition). For an improper rotation,
4082:
517:
be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)
90:
3657:
2798:
can be expressed as the wedge product of two vectors and is a pseudovector. In four dimensions, however, the pseudovectors are
121:
is a pseudovector because it is often described as a vector, but by just changing the position of reference (and changing the
4384:
158:
will determine which), and is a pseudovector. This has consequences in computer graphics, where it has to be considered when
538:
2439:
4366:
337:
552:
The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider
3754:
Another important note is that pseudovectors, despite their name, are "vectors" in the sense of being elements of a
2179:
2129:
750:; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.
3193:
where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the
1673:{\displaystyle (R\mathbf {v_{1}} )\times (R\mathbf {v_{2}} )=(\det R)(R(\mathbf {v_{1}} \times \mathbf {v_{2}} ))}
3860:
3835:
557:
513:) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box
4506:
2113:
2066:
265:
110:
2332:
3667:
4501:
2826:
basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of the
125:), angular momentum can reverse direction, which is not supposed to happen with true vectors (also known as
2576:
556:, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is
436:
259:
106:
3618:{\displaystyle \mathbf {a} \ \wedge \ \mathbf {b} ={\mathit {i}}\ \mathbf {a} \ \times \ \mathbf {b} \ ,}
2600:
is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on
2144:
521:
2894:
The transformation properties of the pseudovector in three dimensions has been compared to that of the
2634:
2895:
2791:
2109:
2004:
This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.
223:
98:
94:
2603:
1546:
3544:
2301:
2185:
2150:
2117:
451:
300:
114:
4290:
2786:
is one of these combinations. This term is attached to a different multivector depending upon the
2038:
326:
163:
4057:
177:
A number of quantities in physics behave as pseudovectors rather than polar vectors, including
65:. The position and current at any point in the wire are "true" vectors, but the magnetic field
4478:
4438:
4426:
4411:
4392:
4344:
4325:
4296:
4269:
4241:
4235:
4214:
4181:
4149:
4120:
4090:
4061:
4052:
Theoretical methods in the physical sciences: an introduction to problem solving using Maple V
4028:
3992:
3956:
3944:
3910:
3815:
2676:
2669:
2046:
553:
196:
4263:
4202:
4173:
4141:
4112:
3772:
3767:
3194:
2773:
2753:
2082:
2042:
1535:
1531:
422:
402:
275:
247:
243:
182:
133:
118:
44:
4375:
2033:" with "left-hand rule" everywhere in math and physics, including in the definition of the
299:
The distinction between polar vectors and pseudovectors becomes important in understanding
4358:
2030:
444:
191:, from which the transformation rules of pseudovectors can be derived. More generally, in
155:
122:
2830:
basis vectors. For instance, in four dimensions where the basis vectors are taken to be {
3945:"Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations"
623:
The transformation rules for polar vectors and pseudovectors can be compactly stated as
4169:
4050:
3789:
2493:
2281:
546:
251:
178:
159:
51:
4495:
3976:
3783:
2034:
443:.) Mathematically, if everything in the universe undergoes a rotation described by a
322:
171:
222:, both of which gain an extra sign-flip under improper rotations compared to a true
3755:
2937:
1550:
Under inversion the two vectors change sign, but their cross product is invariant .
428:
218:
212:
167:
1558:, either proper or improper, the following mathematical equation is always true:
4362:
3980:
3930:
3891:
2799:
2757:
2749:
2125:
747:
78:
33:
4117:
Geometric
Algebra for Computer Science: An Object-Oriented Approach to Geometry
39:
3892:
RP Feynman: §52-5 Polar and axial vectors, Feynman
Lectures in Physics, Vol. 1
3777:
3213:
4477:, Chicago Lectures in Physics, The University of Chicago Press, p. 126,
4083:"Application of conformal geometric algebra in computer vision and graphics"
2787:
315:
255:
17:
3208:
2795:
2738:{\displaystyle \mathbf {ab} =\mathbf {a\cdot b} +\mathbf {a\wedge b} \ ,}
510:
406:
187:
102:
4178:
New foundations for classical mechanics: Fundamental
Theories of Physics
2794:
vectors in the space). In three dimensions, the most general 2-blade or
2566:{\displaystyle {\text{O}}(n)\cong {\text{SO}}(n)\times \mathbb {Z} _{2}}
154:
is a normal to the plane (there are two normals, one on each side – the
4289:
Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000).
3808:
Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000).
185:. In mathematics, in three dimensions, pseudovectors are equivalent to
74:
3909:(Reprint of 1968 Prentice-Hall ed.). Courier Dover. p. 125.
2429:{\displaystyle (\mathbb {R} ^{n},\rho _{\text{pseudo}},{\text{O}}(n))}
2143:
Another way to formalize them is by considering them as elements of a
310:
plane that inside the loop generates a magnetic field oriented in the
529:
239:
227:
2271:{\displaystyle (\mathbb {R} ^{n},\rho _{\text{fund}},{\text{O}}(n))}
2112:, such a pseudovector does not experience a sign flip, and when the
1545:
742:
where the symbols are as described above, and the rotation matrix
432:
264:
38:
321:
In physics, pseudovectors are generally the result of taking the
3903:
Aleksandr
Ivanovich Borisenko; Ivan Evgenʹevich Tarapov (1979).
2950:, the cross product is expressed in terms of its components as:
2371:. Pseudovectors transform in a pseudofundamental representation
1135:. If the universe is transformed by an improper rotation matrix
431:). Under the physics definition, a "vector" is required to have
136:. An oriented plane can be defined by two non-parallel vectors,
199:, pseudovectors are the elements of the algebra with dimension
4237:
Geometric algebra with applications in science and engineering
3674:
3582:
754:
Behavior under addition, subtraction, scalar multiplication
132:
One example of a pseudovector is the normal to an oriented
4431:
Lectures on
Clifford (geometric) algebras and applications
4025:
Lectures on
Clifford (geometric) algebras and applications
3713:
Using the above relations, it is seen that if the vectors
301:
the effect of symmetry on the solution to physical systems
27:
Physical quantity that changes sign with improper rotation
2675:
The basic multiplication in the geometric algebra is the
746:
can be either proper or improper. The symbol det denotes
4087:
Computer algebra and geometric algebra with applications
3780:, a generalization of pseudovector in Clifford algebra
3548:. The cross product and wedge product are related by:
1745:. If the universe is transformed by a rotation matrix
802:. If the universe is transformed by a rotation matrix
4113:"Figure 3.5: Duality of vectors and bivectors in 3-D"
3670:
3557:
3305:
2959:
2914:
in three dimensions, the cross product composed from
2692:
2637:
2606:
2579:
2516:
2496:
2442:
2377:
2335:
2304:
2284:
2219:
2188:
2153:
2057:
One way to formalize pseudovectors is as follows: if
1765:
1567:
1363:
1155:
822:
632:
604:. If it is a pseudovector, it will be transformed to
493:. This important requirement is what distinguishes a
340:
4265:
Multivectors and
Clifford algebra in electrodynamics
210:. The label "pseudo-" can be further generalized to
4292:Linearity and the mathematics of several variables
4146:Geometric Algebra with Applications in Engineering
4049:
3931:Feynman Lectures, 52-7, "Parity is not conserved!"
3861:"Details for IEV number 102-03-34: "polar vector""
3836:"Details for IEV number 102-03-33: "axial vector""
3811:Linearity and the mathematics of several variables
3751:-dimensional space is not restricted in this way.
3702:
3617:
3531:
3182:
2737:
2679:, denoted by simply juxtaposing two vectors as in
2652:
2623:
2592:
2565:
2502:
2482:
2428:
2363:
2321:
2290:
2270:
2205:
2170:
1969:
1672:
1511:
1354:does not in general even keep the same magnitude:
1329:
1069:
731:
393:
4427:"4. Applications of Clifford algebras in physics"
4234:Eduardo Bayro Corrochano; Garret Sobczyk (2001).
4111:Leo Dorst; Daniel Fontijne; Stephen Mann (2007).
3887:
3885:
3883:
3881:
2483:{\displaystyle \rho _{\text{pseudo}}(R)=\det(R)R}
2025:. An alternate approach, more along the lines of
4429:. In Abłamowicz, Rafał; Sobczyk, Garret (eds.).
2465:
1934:
1877:
1619:
1300:
1243:
1030:
973:
930:
894:
694:
394:{\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z})}
3786:— discussion about non-orientable spaces.
2748:where the leading term is the customary vector
2097:form a vector space with the same dimension as
2021:Above, pseudovectors have been discussed using
524:, this requirement is equivalent to defining a
2814:is the dimension of the space and algebra. An
497:(which might be composed of, for example, the
4207:Geometric algebra and applications to physics
4201:Venzo De Sabbata; Bidyut Kumar Datta (2007).
4119:(2nd ed.). Morgan Kaufmann. p. 82.
4081:R Wareham, J Cameron & J Lasenby (2005).
3988:Geometric algebra and applications to physics
3985:Venzo De Sabbata; Bidyut Kumar Datta (2007).
3207:. In this context of geometric algebra, this
2922:is the vector normal to their plane given by
2772:-fold wedge product also is referred to as a
2029:, is to keep the universe fixed, but switch "
588:is a polar vector, it will be transformed to
61:be reflected: Instead, it would be reflected
8:
3906:Vector and tensor analysis with applications
2490:. Another way to view this homomorphism for
1987:is a pseudovector. Similarly, one can show:
3545:Hodge star operator § Three dimensions
2132:), there is no natural identification of ⋀(
303:. Consider an electric current loop in the
238:Physical examples of pseudovectors include
4320:Arfken, George B.; Weber, Hans J. (2001).
3949:Deformations of mathematical structures II
2936:. Given a set of right-handed orthonormal
2000:pseudovector × polar vector = polar vector
1997:polar vector × pseudovector = polar vector
1994:pseudovector × pseudovector = pseudovector
1991:polar vector × polar vector = pseudovector
435:that "transform" in a certain way under a
3865:International Electrotechnical Vocabulary
3840:International Electrotechnical Vocabulary
3679:
3673:
3672:
3669:
3604:
3590:
3581:
3580:
3572:
3558:
3556:
3517:
3512:
3500:
3490:
3477:
3467:
3449:
3444:
3432:
3422:
3409:
3399:
3381:
3376:
3364:
3354:
3341:
3331:
3314:
3306:
3304:
3171:
3166:
3154:
3144:
3131:
3121:
3103:
3098:
3086:
3076:
3063:
3053:
3035:
3030:
3018:
3008:
2995:
2985:
2968:
2960:
2958:
2858:}, the pseudovectors can be written as: {
2718:
2704:
2693:
2691:
2644:
2640:
2639:
2636:
2607:
2605:
2584:
2578:
2557:
2553:
2552:
2534:
2517:
2515:
2495:
2447:
2441:
2409:
2400:
2387:
2383:
2382:
2376:
2340:
2334:
2305:
2303:
2283:
2251:
2242:
2229:
2225:
2224:
2218:
2189:
2187:
2154:
2152:
1954:
1949:
1915:
1910:
1900:
1895:
1861:
1856:
1837:
1832:
1812:
1807:
1792:
1787:
1772:
1767:
1764:
1657:
1652:
1642:
1637:
1603:
1598:
1579:
1574:
1566:
1493:
1488:
1473:
1468:
1444:
1439:
1429:
1421:
1414:
1409:
1399:
1394:
1389:
1381:
1374:
1369:
1364:
1362:
1314:
1309:
1287:
1282:
1263:
1258:
1227:
1222:
1202:
1197:
1182:
1177:
1162:
1157:
1154:
1088:real number yields another pseudovector.
1050:
1045:
1011:
1006:
996:
991:
950:
945:
914:
909:
873:
868:
853:
848:
833:
828:
823:
821:
720:
709:
675:
665:
657:
638:
633:
631:
382:
369:
356:
341:
339:
2364:{\displaystyle \rho _{\text{fund}}(R)=R}
419:Covariance and contravariance of vectors
174:of two polar vectors are pseudovectors.
4339:Doran, Chris; Lasenby, Anthony (2007).
3800:
3703:{\displaystyle {\mathit {i}}^{2}=-1\ .}
4180:(2nd ed.). Springer. p. 60.
1722:is defined to be their cross product,
4203:"The pseudoscalar and imaginary unit"
4014:"§4.2.3 Higher-grade multivectors in
3268:, and so forth. That is, the dual of
3211:is called a pseudovector, and is the
2790:of the space (that is, the number of
2593:{\displaystyle \rho _{\text{pseudo}}}
113:, etc. This can also happen when the
7:
97:does not conform when the object is
89:) is a quantity that behaves like a
4322:Mathematical Methods for Physicists
4101:In three dimensions, a dual may be
2890:Transformations in three dimensions
1105:is known to be a pseudovector, and
405:of the two vectors, which yields a
43:A loop of wire (black), carrying a
2752:and the second term is called the
144:, that span the plane. The vector
25:
4512:Vectors (mathematics and physics)
4295:. World Scientific. p. 340.
3814:. World Scientific. p. 343.
3282:, namely the subspace spanned by
3275:is the subspace perpendicular to
2754:wedge product or exterior product
2631:with the trivial homomorphism on
477:must be similarly transformed to
4449:: The dual of the wedge product
4363:"§52-5: Polar and axial vectors"
4341:Geometric Algebra for Physicists
4268:. World Scientific. p. 11.
3943:William M Pezzaglia Jr. (1992).
3605:
3591:
3573:
3559:
3513:
3445:
3377:
3315:
3307:
3167:
3099:
3031:
2969:
2961:
2764:-fold wedge products of various
2725:
2719:
2711:
2705:
2697:
2694:
2683:. This product is expressed as:
2653:{\displaystyle \mathbb {Z} _{2}}
1955:
1951:
1916:
1912:
1901:
1897:
1862:
1858:
1838:
1834:
1813:
1809:
1793:
1789:
1773:
1769:
1658:
1654:
1643:
1639:
1604:
1600:
1580:
1576:
1494:
1490:
1474:
1470:
1445:
1441:
1415:
1411:
1400:
1396:
1375:
1371:
1315:
1311:
1288:
1284:
1264:
1260:
1228:
1224:
1203:
1199:
1183:
1179:
1163:
1159:
1051:
1047:
1012:
1008:
997:
993:
951:
947:
915:
911:
874:
870:
854:
850:
834:
830:
710:
676:
658:
639:
342:
4379:at Encyclopaedia of Mathematics
3947:. In Julian Ławrynowicz (ed.).
2898:by Baylis. He says: "The terms
1098:is known to be a polar vector,
4343:. Cambridge University Press.
3975:In four dimensions, such as a
2624:{\displaystyle {\text{SO}}(n)}
2618:
2612:
2545:
2539:
2528:
2522:
2474:
2468:
2459:
2453:
2423:
2420:
2414:
2378:
2352:
2346:
2316:
2310:
2265:
2262:
2256:
2220:
2200:
2194:
2165:
2159:
1961:
1943:
1940:
1931:
1925:
1922:
1892:
1886:
1883:
1874:
1868:
1850:
1844:
1826:
1667:
1664:
1634:
1628:
1625:
1616:
1610:
1592:
1586:
1568:
1422:
1390:
1382:
1365:
1321:
1306:
1297:
1279:
1270:
1252:
1249:
1240:
1234:
1216:
1057:
1039:
1036:
1027:
1021:
1018:
988:
982:
979:
970:
957:
939:
936:
927:
921:
903:
900:
891:
714:
703:
700:
691:
388:
349:
1:
3197:or wedge product, denoted by
2322:{\displaystyle {\text{O}}(n)}
2206:{\displaystyle {\text{O}}(n)}
2171:{\displaystyle {\text{O}}(n)}
1715:are known polar vectors, and
1542:Behavior under cross products
772:are known pseudovectors, and
539:Einstein summation convention
117:is changed. For example, the
4371:. Vol. 1. p. 52–6.
1112:is defined to be their sum,
779:is defined to be their sum,
564:, so that a position vector
325:of two polar vectors or the
160:transforming surface normals
93:in many situations, but its
4473:Weinreich, Gabriel (1998),
4368:Feynman Lectures on Physics
4027:. Birkhäuser. p. 100.
3296:. With this understanding,
2782:In the present context the
2178:. Vectors transform in the
1091:On the other hand, suppose
162:. In three dimensions, the
4528:
4433:. Birkhäuser. p. 100
4425:Baylis, William E (2004).
4408:Mathematics for Physicists
4262:Bernard Jancewicz (1988).
4174:"The vector cross product"
4140:Christian Perwass (2009).
3217:of the cross product. The
2180:fundamental representation
2045:phenomena such as certain
416:
314:direction. This system is
274:Consider the pseudovector
31:
4389:Classical Electrodynamics
4240:. Springer. p. 126.
4089:. Springer. p. 330.
4048:William E Baylis (1994).
4012:William E Baylis (2004).
3991:. CRC Press. p. 64.
2510:odd is that in this case
2278:, so that for any matrix
558:inversion through a point
4209:. CRC Press. p. 53
4148:. Springer. p. 17.
4142:"§1.5.2 General vectors"
3979:, the pseudovectors are
3951:. Springer. p. 131
2093:). The pseudovectors of
115:orientation of the space
32:Not to be confused with
3661:. It has the property:
2822:basis vectors and also
2818:-dimensional space has
2027:passive transformations
4406:Lea, Susan M. (2004).
4056:. Birkhäuser. p.
3704:
3619:
3533:
3184:
2802:. In general, it is a
2739:
2654:
2625:
2594:
2567:
2504:
2484:
2430:
2365:
2323:
2292:
2272:
2207:
2172:
2077:is an element of the (
2023:active transformations
1971:
1674:
1554:For a rotation matrix
1551:
1513:
1331:
1071:
733:
441:active transformations
395:
271:
260:magnetic dipole moment
70:
4459:is the cross product
3705:
3620:
3534:
3185:
2740:
2655:
2626:
2595:
2568:
2505:
2485:
2431:
2366:
2324:
2293:
2273:
2208:
2173:
2069:vector space, then a
1972:
1675:
1549:
1514:
1332:
1072:
734:
522:differential geometry
396:
268:
42:
3668:
3555:
3303:
2957:
2896:vector cross product
2792:linearly independent
2690:
2635:
2604:
2577:
2514:
2494:
2440:
2375:
2333:
2302:
2282:
2217:
2186:
2151:
2145:representation space
1763:
1565:
1523:If the magnitude of
1361:
1153:
820:
630:
520:(In the language of
473:, then any "vector"
338:
4475:Geometrical Vectors
2213:with data given by
2081: − 1)-th
2017:The right-hand rule
452:displacement vector
170:at a point and the
99:rigidly transformed
3700:
3615:
3529:
3180:
2760:is a summation of
2735:
2650:
2621:
2590:
2563:
2500:
2480:
2426:
2361:
2319:
2288:
2268:
2203:
2168:
2116:of the underlying
2047:radioactive decays
1967:
1756:is transformed to
1670:
1552:
1509:
1327:
1146:is transformed to
1067:
1065:
813:is transformed to
729:
727:
568:is transformed to
554:improper rotations
457:is transformed to
391:
272:
71:
69:is a pseudovector.
4350:978-0-521-71595-9
4220:978-1-58488-772-0
4155:978-3-540-89067-6
4126:978-0-12-374942-0
3998:978-1-58488-772-0
3696:
3658:unit pseudoscalar
3611:
3603:
3597:
3589:
3571:
3565:
3542:For details, see
3525:
3228:is introduced as
2731:
2677:geometric product
2670:geometric algebra
2664:Geometric algebra
2610:
2587:
2537:
2520:
2503:{\displaystyle n}
2450:
2412:
2403:
2343:
2308:
2291:{\displaystyle R}
2254:
2245:
2192:
2157:
1432:
723:
668:
234:Physical examples
197:geometric algebra
16:(Redirected from
4519:
4487:
4468:
4458:
4448:
4421:
4402:
4372:
4359:Feynman, Richard
4354:
4335:
4307:
4306:
4286:
4280:
4279:
4258:
4252:
4251:
4231:
4225:
4224:
4198:
4192:
4191:
4166:
4160:
4159:
4137:
4131:
4130:
4100:
4078:
4072:
4071:
4060:, see footnote.
4055:
4045:
4039:
4038:
4009:
4003:
4002:
3973:
3967:
3966:
3940:
3934:
3927:
3921:
3920:
3900:
3894:
3889:
3876:
3875:
3873:
3872:
3857:
3851:
3850:
3848:
3847:
3832:
3826:
3825:
3805:
3773:Clifford algebra
3768:Exterior algebra
3746:
3709:
3707:
3706:
3701:
3694:
3684:
3683:
3678:
3677:
3654:
3624:
3622:
3621:
3616:
3609:
3608:
3601:
3595:
3594:
3587:
3586:
3585:
3576:
3569:
3563:
3562:
3538:
3536:
3535:
3530:
3523:
3522:
3521:
3516:
3510:
3506:
3505:
3504:
3495:
3494:
3482:
3481:
3472:
3471:
3454:
3453:
3448:
3442:
3438:
3437:
3436:
3427:
3426:
3414:
3413:
3404:
3403:
3386:
3385:
3380:
3374:
3370:
3369:
3368:
3359:
3358:
3346:
3345:
3336:
3335:
3318:
3310:
3267:
3252:
3237:
3206:
3195:exterior product
3189:
3187:
3186:
3181:
3176:
3175:
3170:
3164:
3160:
3159:
3158:
3149:
3148:
3136:
3135:
3126:
3125:
3108:
3107:
3102:
3096:
3092:
3091:
3090:
3081:
3080:
3068:
3067:
3058:
3057:
3040:
3039:
3034:
3028:
3024:
3023:
3022:
3013:
3012:
3000:
2999:
2990:
2989:
2972:
2964:
2949:
2935:
2809:
2744:
2742:
2741:
2736:
2729:
2728:
2714:
2700:
2659:
2657:
2656:
2651:
2649:
2648:
2643:
2630:
2628:
2627:
2622:
2611:
2608:
2599:
2597:
2596:
2591:
2589:
2588:
2585:
2572:
2570:
2569:
2564:
2562:
2561:
2556:
2538:
2535:
2521:
2518:
2509:
2507:
2506:
2501:
2489:
2487:
2486:
2481:
2452:
2451:
2448:
2435:
2433:
2432:
2427:
2413:
2410:
2405:
2404:
2401:
2392:
2391:
2386:
2370:
2368:
2367:
2362:
2345:
2344:
2341:
2328:
2326:
2325:
2320:
2309:
2306:
2297:
2295:
2294:
2289:
2277:
2275:
2274:
2269:
2255:
2252:
2247:
2246:
2243:
2234:
2233:
2228:
2212:
2210:
2209:
2204:
2193:
2190:
2177:
2175:
2174:
2169:
2158:
2155:
2043:parity-violating
1976:
1974:
1973:
1968:
1960:
1959:
1958:
1921:
1920:
1919:
1906:
1905:
1904:
1867:
1866:
1865:
1843:
1842:
1841:
1822:
1818:
1817:
1816:
1802:
1798:
1797:
1796:
1782:
1778:
1777:
1776:
1744:
1679:
1677:
1676:
1671:
1663:
1662:
1661:
1648:
1647:
1646:
1609:
1608:
1607:
1585:
1584:
1583:
1536:parity violation
1532:weak interaction
1518:
1516:
1515:
1510:
1508:
1504:
1503:
1499:
1498:
1497:
1483:
1479:
1478:
1477:
1458:
1454:
1450:
1449:
1448:
1433:
1430:
1425:
1420:
1419:
1418:
1405:
1404:
1403:
1393:
1385:
1380:
1379:
1378:
1368:
1336:
1334:
1333:
1328:
1320:
1319:
1318:
1293:
1292:
1291:
1269:
1268:
1267:
1233:
1232:
1231:
1212:
1208:
1207:
1206:
1192:
1188:
1187:
1186:
1172:
1168:
1167:
1166:
1134:
1076:
1074:
1073:
1068:
1066:
1056:
1055:
1054:
1017:
1016:
1015:
1002:
1001:
1000:
963:
956:
955:
954:
920:
919:
918:
883:
879:
878:
877:
863:
859:
858:
857:
843:
839:
838:
837:
801:
738:
736:
735:
730:
728:
724:
721:
718:
713:
683:
679:
669:
666:
663:
661:
646:
642:
619:
611:
603:
595:
584:. If the vector
583:
575:
492:
484:
472:
464:
423:Euclidean vector
403:exterior product
400:
398:
397:
392:
387:
386:
374:
373:
361:
360:
345:
309:
291:
276:angular momentum
248:angular momentum
244:angular velocity
205:
183:angular velocity
153:
119:angular momentum
21:
4527:
4526:
4522:
4521:
4520:
4518:
4517:
4516:
4507:Vector calculus
4492:
4491:
4490:
4485:
4472:
4460:
4450:
4445:
4424:
4418:
4405:
4399:
4383:
4357:
4351:
4338:
4332:
4319:
4315:
4310:
4303:
4288:
4287:
4283:
4276:
4261:
4259:
4255:
4248:
4233:
4232:
4228:
4221:
4200:
4199:
4195:
4188:
4168:
4167:
4163:
4156:
4139:
4138:
4134:
4127:
4110:
4097:
4080:
4079:
4075:
4068:
4047:
4046:
4042:
4035:
4020:
4011:
4010:
4006:
3999:
3984:
3974:
3970:
3963:
3942:
3941:
3937:
3928:
3924:
3917:
3902:
3901:
3897:
3890:
3879:
3870:
3868:
3859:
3858:
3854:
3845:
3843:
3834:
3833:
3829:
3822:
3807:
3806:
3802:
3798:
3764:
3740:
3736:
3728:
3671:
3666:
3665:
3653:
3646:
3639:
3629:
3553:
3552:
3511:
3496:
3486:
3473:
3463:
3462:
3458:
3443:
3428:
3418:
3405:
3395:
3394:
3390:
3375:
3360:
3350:
3337:
3327:
3326:
3322:
3301:
3300:
3295:
3288:
3281:
3274:
3266:
3259:
3253:
3250:
3244:
3238:
3235:
3229:
3227:
3198:
3165:
3150:
3140:
3127:
3117:
3116:
3112:
3097:
3082:
3072:
3059:
3049:
3048:
3044:
3029:
3014:
3004:
2991:
2981:
2980:
2976:
2955:
2954:
2947:
2940:
2923:
2892:
2885:
2878:
2871:
2864:
2857:
2850:
2843:
2836:
2803:
2688:
2687:
2666:
2638:
2633:
2632:
2602:
2601:
2580:
2575:
2574:
2551:
2512:
2511:
2492:
2491:
2443:
2438:
2437:
2396:
2381:
2373:
2372:
2336:
2331:
2330:
2300:
2299:
2280:
2279:
2238:
2223:
2215:
2214:
2184:
2183:
2149:
2148:
2055:
2031:right-hand rule
2019:
2010:
1986:
1950:
1911:
1896:
1857:
1833:
1808:
1806:
1788:
1786:
1768:
1766:
1761:
1760:
1755:
1743:
1736:
1729:
1723:
1721:
1714:
1707:
1697:
1690:
1653:
1638:
1599:
1575:
1563:
1562:
1544:
1529:
1489:
1487:
1469:
1467:
1466:
1462:
1440:
1438:
1434:
1431: but
1410:
1395:
1370:
1359:
1358:
1353:
1346:
1310:
1283:
1259:
1223:
1198:
1196:
1178:
1176:
1158:
1156:
1151:
1150:
1145:
1133:
1126:
1119:
1113:
1111:
1104:
1097:
1086:
1064:
1063:
1046:
1007:
992:
961:
960:
946:
910:
884:
869:
867:
849:
847:
829:
827:
818:
817:
812:
800:
793:
786:
780:
778:
771:
764:
756:
726:
725:
717:
684:
674:
671:
670:
662:
647:
637:
628:
627:
609:
605:
593:
589:
573:
569:
509:-components of
482:
478:
462:
458:
445:rotation matrix
437:proper rotation
425:
415:
378:
365:
352:
336:
335:
304:
278:
236:
200:
156:right-hand rule
145:
123:position vector
37:
28:
23:
22:
15:
12:
11:
5:
4525:
4523:
4515:
4514:
4509:
4504:
4502:Linear algebra
4494:
4493:
4489:
4488:
4483:
4470:
4443:
4422:
4416:
4403:
4397:
4385:Jackson, J. D.
4381:
4373:
4355:
4349:
4336:
4330:
4316:
4314:
4311:
4309:
4308:
4301:
4281:
4274:
4253:
4246:
4226:
4219:
4193:
4186:
4170:David Hestenes
4161:
4154:
4132:
4125:
4095:
4073:
4066:
4040:
4033:
4018:
4004:
3997:
3968:
3961:
3935:
3922:
3915:
3895:
3877:
3852:
3827:
3820:
3799:
3797:
3794:
3793:
3792:
3790:Tensor density
3787:
3781:
3775:
3770:
3763:
3760:
3735:
3732:
3726:
3711:
3710:
3699:
3693:
3690:
3687:
3682:
3676:
3655:is called the
3651:
3644:
3637:
3626:
3625:
3614:
3607:
3600:
3593:
3584:
3579:
3575:
3568:
3561:
3540:
3539:
3528:
3520:
3515:
3509:
3503:
3499:
3493:
3489:
3485:
3480:
3476:
3470:
3466:
3461:
3457:
3452:
3447:
3441:
3435:
3431:
3425:
3421:
3417:
3412:
3408:
3402:
3398:
3393:
3389:
3384:
3379:
3373:
3367:
3363:
3357:
3353:
3349:
3344:
3340:
3334:
3330:
3325:
3321:
3317:
3313:
3309:
3293:
3286:
3279:
3272:
3264:
3257:
3248:
3242:
3233:
3225:
3191:
3190:
3179:
3174:
3169:
3163:
3157:
3153:
3147:
3143:
3139:
3134:
3130:
3124:
3120:
3115:
3111:
3106:
3101:
3095:
3089:
3085:
3079:
3075:
3071:
3066:
3062:
3056:
3052:
3047:
3043:
3038:
3033:
3027:
3021:
3017:
3011:
3007:
3003:
2998:
2994:
2988:
2984:
2979:
2975:
2971:
2967:
2963:
2945:
2891:
2888:
2883:
2876:
2869:
2862:
2855:
2848:
2841:
2834:
2810:-blade, where
2746:
2745:
2734:
2727:
2724:
2721:
2717:
2713:
2710:
2707:
2703:
2699:
2696:
2665:
2662:
2647:
2642:
2620:
2617:
2614:
2583:
2560:
2555:
2550:
2547:
2544:
2541:
2533:
2530:
2527:
2524:
2499:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2446:
2425:
2422:
2419:
2416:
2408:
2399:
2395:
2390:
2385:
2380:
2360:
2357:
2354:
2351:
2348:
2339:
2318:
2315:
2312:
2287:
2267:
2264:
2261:
2258:
2250:
2241:
2237:
2232:
2227:
2222:
2202:
2199:
2196:
2167:
2164:
2161:
2114:characteristic
2083:exterior power
2054:
2051:
2018:
2015:
2009:
2006:
2002:
2001:
1998:
1995:
1992:
1984:
1978:
1977:
1966:
1963:
1957:
1953:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1918:
1914:
1909:
1903:
1899:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1864:
1860:
1855:
1852:
1849:
1846:
1840:
1836:
1831:
1828:
1825:
1821:
1815:
1811:
1805:
1801:
1795:
1791:
1785:
1781:
1775:
1771:
1753:
1741:
1734:
1727:
1719:
1712:
1705:
1695:
1688:
1682:
1681:
1669:
1666:
1660:
1656:
1651:
1645:
1641:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1606:
1602:
1597:
1594:
1591:
1588:
1582:
1578:
1573:
1570:
1543:
1540:
1527:
1521:
1520:
1507:
1502:
1496:
1492:
1486:
1482:
1476:
1472:
1465:
1461:
1457:
1453:
1447:
1443:
1437:
1428:
1424:
1417:
1413:
1408:
1402:
1398:
1392:
1388:
1384:
1377:
1373:
1367:
1351:
1344:
1338:
1337:
1326:
1323:
1317:
1313:
1308:
1305:
1302:
1299:
1296:
1290:
1286:
1281:
1278:
1275:
1272:
1266:
1262:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1230:
1226:
1221:
1218:
1215:
1211:
1205:
1201:
1195:
1191:
1185:
1181:
1175:
1171:
1165:
1161:
1143:
1131:
1124:
1117:
1109:
1102:
1095:
1084:
1078:
1077:
1062:
1059:
1053:
1049:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1014:
1010:
1005:
999:
995:
990:
987:
984:
981:
978:
975:
972:
969:
966:
964:
962:
959:
953:
949:
944:
941:
938:
935:
932:
929:
926:
923:
917:
913:
908:
905:
902:
899:
896:
893:
890:
887:
885:
882:
876:
872:
866:
862:
856:
852:
846:
842:
836:
832:
826:
825:
810:
798:
791:
784:
776:
769:
762:
755:
752:
740:
739:
722:(pseudovector)
719:
716:
712:
708:
705:
702:
699:
696:
693:
690:
687:
685:
682:
678:
673:
672:
667:(polar vector)
664:
660:
656:
653:
650:
648:
645:
641:
636:
635:
547:dyadic product
414:
411:
390:
385:
381:
377:
372:
368:
364:
359:
355:
351:
348:
344:
270:pseudovectors.
252:magnetic field
235:
232:
179:magnetic field
52:magnetic field
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4524:
4513:
4510:
4508:
4505:
4503:
4500:
4499:
4497:
4486:
4484:9780226890487
4480:
4476:
4471:
4467:
4463:
4457:
4453:
4446:
4444:0-8176-3257-3
4440:
4436:
4432:
4428:
4423:
4419:
4417:0-534-37997-4
4413:
4409:
4404:
4400:
4398:0-471-30932-X
4394:
4390:
4386:
4382:
4380:
4378:
4374:
4370:
4369:
4364:
4360:
4356:
4352:
4346:
4342:
4337:
4333:
4331:0-12-059815-9
4327:
4323:
4318:
4317:
4312:
4304:
4302:981-02-4196-8
4298:
4294:
4293:
4285:
4282:
4277:
4275:9971-5-0290-9
4271:
4267:
4266:
4260:For example,
4257:
4254:
4249:
4247:0-8176-4199-8
4243:
4239:
4238:
4230:
4227:
4222:
4216:
4212:
4208:
4204:
4197:
4194:
4189:
4187:0-7923-5302-1
4183:
4179:
4175:
4171:
4165:
4162:
4157:
4151:
4147:
4143:
4136:
4133:
4128:
4122:
4118:
4114:
4108:
4104:
4098:
4096:3-540-26296-2
4092:
4088:
4084:
4077:
4074:
4069:
4067:0-8176-3715-X
4063:
4059:
4054:
4053:
4044:
4041:
4036:
4034:0-8176-3257-3
4030:
4026:
4022:
4017:
4008:
4005:
4000:
3994:
3990:
3989:
3982:
3978:
3977:Dirac algebra
3972:
3969:
3964:
3962:0-7923-2576-1
3958:
3954:
3950:
3946:
3939:
3936:
3932:
3926:
3923:
3918:
3916:0-486-63833-2
3912:
3908:
3907:
3899:
3896:
3893:
3888:
3886:
3884:
3882:
3878:
3867:(in Japanese)
3866:
3862:
3856:
3853:
3842:(in Japanese)
3841:
3837:
3831:
3828:
3823:
3821:981-02-4196-8
3817:
3813:
3812:
3804:
3801:
3795:
3791:
3788:
3785:
3784:Orientability
3782:
3779:
3776:
3774:
3771:
3769:
3766:
3765:
3761:
3759:
3757:
3752:
3750:
3747:-blade in an
3744:
3734:Note on usage
3733:
3731:
3725:
3720:
3716:
3697:
3691:
3688:
3685:
3680:
3664:
3663:
3662:
3660:
3659:
3650:
3643:
3636:
3632:
3612:
3598:
3577:
3566:
3551:
3550:
3549:
3547:
3546:
3526:
3518:
3507:
3501:
3497:
3491:
3487:
3483:
3478:
3474:
3468:
3464:
3459:
3455:
3450:
3439:
3433:
3429:
3423:
3419:
3415:
3410:
3406:
3400:
3396:
3391:
3387:
3382:
3371:
3365:
3361:
3355:
3351:
3347:
3342:
3338:
3332:
3328:
3323:
3319:
3311:
3299:
3298:
3297:
3292:
3285:
3278:
3271:
3263:
3256:
3247:
3241:
3232:
3224:
3220:
3216:
3215:
3210:
3205:
3201:
3196:
3177:
3172:
3161:
3155:
3151:
3145:
3141:
3137:
3132:
3128:
3122:
3118:
3113:
3109:
3104:
3093:
3087:
3083:
3077:
3073:
3069:
3064:
3060:
3054:
3050:
3045:
3041:
3036:
3025:
3019:
3015:
3009:
3005:
3001:
2996:
2992:
2986:
2982:
2977:
2973:
2965:
2953:
2952:
2951:
2944:
2939:
2938:basis vectors
2934:
2930:
2926:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2889:
2887:
2882:
2875:
2868:
2861:
2854:
2847:
2840:
2833:
2829:
2825:
2821:
2817:
2813:
2807:
2801:
2797:
2793:
2789:
2785:
2780:
2778:
2776:
2771:
2767:
2763:
2759:
2755:
2751:
2732:
2722:
2715:
2708:
2701:
2686:
2685:
2684:
2682:
2678:
2673:
2671:
2663:
2661:
2645:
2615:
2581:
2558:
2548:
2542:
2531:
2525:
2497:
2477:
2471:
2462:
2456:
2444:
2417:
2406:
2397:
2393:
2388:
2358:
2355:
2349:
2337:
2313:
2285:
2259:
2248:
2239:
2235:
2230:
2197:
2181:
2162:
2146:
2141:
2139:
2135:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2102:
2100:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2053:Formalization
2052:
2050:
2048:
2044:
2040:
2036:
2035:cross product
2032:
2028:
2024:
2016:
2014:
2007:
2005:
1999:
1996:
1993:
1990:
1989:
1988:
1983:
1964:
1946:
1937:
1928:
1907:
1889:
1880:
1871:
1853:
1847:
1829:
1823:
1819:
1803:
1799:
1783:
1779:
1759:
1758:
1757:
1752:
1748:
1740:
1733:
1726:
1718:
1711:
1704:
1699:
1694:
1687:
1649:
1631:
1622:
1613:
1595:
1589:
1571:
1561:
1560:
1559:
1557:
1548:
1541:
1539:
1537:
1533:
1526:
1505:
1500:
1484:
1480:
1463:
1459:
1455:
1451:
1435:
1426:
1406:
1386:
1357:
1356:
1355:
1350:
1343:
1324:
1303:
1294:
1276:
1273:
1255:
1246:
1237:
1219:
1213:
1209:
1193:
1189:
1173:
1169:
1149:
1148:
1147:
1142:
1138:
1130:
1123:
1116:
1108:
1101:
1094:
1089:
1083:
1060:
1042:
1033:
1024:
1003:
985:
976:
967:
965:
942:
933:
924:
906:
897:
888:
886:
880:
864:
860:
844:
840:
816:
815:
814:
809:
805:
797:
790:
783:
775:
768:
761:
753:
751:
749:
745:
706:
697:
688:
686:
680:
654:
651:
649:
643:
626:
625:
624:
621:
618:
615:
608:
602:
599:
592:
587:
582:
579:
572:
567:
563:
559:
555:
550:
548:
542:
540:
535:
534:contravariant
531:
527:
523:
518:
516:
512:
508:
504:
500:
496:
491:
488:
481:
476:
471:
468:
461:
456:
453:
449:
446:
442:
438:
434:
430:
424:
420:
412:
410:
408:
404:
383:
379:
375:
370:
366:
362:
357:
353:
346:
331:
328:
324:
323:cross product
319:
317:
313:
307:
302:
297:
295:
289:
285:
281:
277:
267:
263:
261:
257:
253:
249:
245:
241:
233:
231:
229:
225:
221:
220:
219:pseudotensors
215:
214:
213:pseudoscalars
209:
203:
198:
195:-dimensional
194:
190:
189:
184:
180:
175:
173:
172:cross product
169:
165:
161:
157:
152:
148:
143:
139:
135:
130:
128:
127:polar vectors
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
68:
64:
60:
56:
53:
49:
46:
41:
35:
30:
19:
4474:
4465:
4461:
4455:
4451:
4434:
4430:
4410:. Thompson.
4407:
4388:
4377:Axial vector
4376:
4367:
4340:
4324:. Harcourt.
4321:
4291:
4284:
4264:
4256:
4236:
4229:
4210:
4206:
4196:
4177:
4164:
4145:
4135:
4116:
4106:
4103:right-handed
4102:
4086:
4076:
4051:
4043:
4024:
4015:
4007:
3987:
3971:
3952:
3948:
3938:
3925:
3905:
3898:
3869:. Retrieved
3864:
3855:
3844:. Retrieved
3839:
3830:
3810:
3803:
3756:vector space
3753:
3748:
3742:
3737:
3723:
3718:
3714:
3712:
3656:
3648:
3641:
3634:
3630:
3627:
3543:
3541:
3290:
3283:
3276:
3269:
3261:
3254:
3245:
3239:
3230:
3222:
3218:
3212:
3203:
3199:
3192:
2942:
2932:
2928:
2924:
2919:
2915:
2911:
2907:
2904:pseudovector
2903:
2900:axial vector
2899:
2893:
2880:
2873:
2866:
2859:
2852:
2845:
2838:
2831:
2827:
2823:
2819:
2815:
2811:
2805:
2784:pseudovector
2783:
2781:
2774:
2769:
2765:
2761:
2747:
2680:
2674:
2667:
2142:
2137:
2133:
2121:
2105:
2103:
2098:
2094:
2090:
2086:
2078:
2074:
2071:pseudovector
2070:
2062:
2058:
2056:
2020:
2011:
2003:
1981:
1979:
1750:
1746:
1738:
1731:
1724:
1716:
1709:
1702:
1700:
1692:
1685:
1683:
1555:
1553:
1524:
1522:
1348:
1341:
1339:
1140:
1136:
1128:
1121:
1114:
1106:
1099:
1092:
1090:
1081:
1079:
807:
803:
795:
788:
781:
773:
766:
759:
757:
743:
741:
622:
616:
613:
606:
600:
597:
590:
585:
580:
577:
570:
565:
561:
551:
543:
525:
519:
514:
506:
502:
498:
494:
489:
486:
479:
474:
469:
466:
459:
454:
450:, so that a
447:
429:vector space
426:
332:
320:
311:
305:
298:
293:
287:
283:
279:
273:
237:
217:
211:
207:
201:
192:
186:
176:
168:vector field
150:
146:
141:
137:
131:
126:
87:axial vector
86:
83:pseudovector
82:
72:
66:
63:and reversed
62:
58:
54:
50:, creates a
47:
29:
18:Axial vector
4107:left-handed
2768:-values. A
2758:multivector
2750:dot product
2130:orientation
2126:volume form
2067:dimensional
1340:Therefore,
748:determinant
206:, written ⋀
166:of a polar
107:translation
79:mathematics
34:Free vector
4496:Categories
4313:References
3981:trivectors
3871:2023-11-07
3846:2023-11-07
3778:Antivector
3214:Hodge dual
2800:trivectors
2788:dimensions
2329:, one has
433:components
417:See also:
111:reflection
4391:. Wiley.
3689:−
3599:×
3567:∧
3484:−
3416:−
3348:−
3312:∧
3138:−
3070:−
3002:−
2966:×
2723:∧
2709:⋅
2582:ρ
2549:×
2532:≅
2445:ρ
2398:ρ
2338:ρ
2240:ρ
1908:×
1848:×
1804:×
1650:×
1590:×
1485:−
316:symmetric
256:vorticity
188:bivectors
95:direction
4387:(1999).
4172:(1999).
4021:: Duals"
3762:See also
3209:bivector
2796:bivector
2037:and the
2008:Examples
1820:′
1800:′
1780:′
1701:Suppose
1501:′
1481:′
1452:′
1210:′
1190:′
1170:′
881:′
861:′
841:′
758:Suppose
681:′
644:′
528:to be a
511:velocity
407:bivector
103:rotation
2573:. Then
2436:, with
2136:) with
1749:, then
1139:, then
806:, then
505:-, and
413:Details
75:physics
45:current
4481:
4441:
4414:
4395:
4347:
4328:
4299:
4272:
4244:
4217:
4184:
4152:
4123:
4109:; see
4093:
4064:
4031:
3995:
3959:
3913:
3818:
3695:
3628:where
3610:
3602:
3596:
3588:
3570:
3564:
3524:
2777:-blade
2730:
2586:pseudo
2449:pseudo
2402:pseudo
2128:or an
2061:is an
1684:where
530:tensor
526:vector
515:cannot
495:vector
294:actual
240:torque
228:tensor
224:scalar
91:vector
3796:Notes
2118:field
134:plane
4479:ISBN
4439:ISBN
4412:ISBN
4393:ISBN
4345:ISBN
4326:ISBN
4297:ISBN
4270:ISBN
4242:ISBN
4215:ISBN
4182:ISBN
4150:ISBN
4121:ISBN
4091:ISBN
4062:ISBN
4029:ISBN
3993:ISBN
3957:ISBN
3929:See
3911:ISBN
3816:ISBN
3745:– 1)
3717:and
3289:and
3219:dual
2918:and
2910:and
2902:and
2808:− 1)
2342:fund
2244:fund
2147:for
2110:even
2089:: ⋀(
2039:curl
1708:and
1691:and
765:and
421:and
327:curl
282:= Σ(
258:and
216:and
181:and
164:curl
140:and
85:(or
81:, a
77:and
4105:or
4058:234
3221:of
2886:}.
2884:123
2877:124
2870:134
2863:234
2668:In
2466:det
2298:in
2182:of
2120:of
2108:is
2085:of
2073:of
1980:So
1935:det
1878:det
1620:det
1538:.)
1301:det
1244:det
1080:So
1031:det
974:det
931:det
895:det
695:det
612:= −
541:.)
532:of
501:-,
308:= 0
226:or
204:− 1
129:).
101:by
73:In
59:not
4498::
4464:×
4454:∧
4437:.
4435:ff
4365:.
4361:.
4213:.
4211:ff
4205:.
4176:.
4144:.
4115:.
4085:.
4023:.
4016:Cℓ
3983:.
3955:.
3953:ff
3880:^
3863:.
3838:.
3647:∧
3640:∧
3633:=
3519:12
3451:31
3383:23
3260:∧
3234:23
3202:∧
2941:{
2931:×
2927:=
2879:,
2872:,
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