3855:
2981:
3850:{\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\mathbf {w} _{y}-\mathbf {v} _{y}\mathbf {w} _{x})-\mathbf {u} _{z}(\mathbf {v} _{z}\mathbf {w} _{x}-\mathbf {v} _{x}\mathbf {w} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})+(\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x}-\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x})\\&=\mathbf {v} _{x}(\mathbf {u} _{x}\mathbf {w} _{x}+\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{x}\mathbf {v} _{x}+\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{x}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{x}\end{aligned}}}
1435:
6524:
1197:
943:
1841:
4198:
4541:
78:
6062:
1430:{\displaystyle ((\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} )\;((\mathbf {d} \times \mathbf {e} )\cdot \mathbf {f} )=\det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}}
6788:
596:
3950:
4348:
567:
5697:
938:{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}=\det {\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{bmatrix}}=\det {\begin{bmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{bmatrix}}.}
4193:{\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{y}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{y}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{y}\\(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{z}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{z}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{z}\end{aligned}}}
2857:
2509:
4536:{\displaystyle {\begin{aligned}-\mathbf {a} \;{\big \lrcorner }\;(\mathbf {b} \wedge \mathbf {c} )&=\mathbf {b} \wedge (\mathbf {a} \;{\big \lrcorner }\;\mathbf {c} )-(\mathbf {a} \;{\big \lrcorner }\;\mathbf {b} )\wedge \mathbf {c} \\&=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} \end{aligned}}}
403:
1573:
2627:
5409:
2741:
1078:
304:
4860:
4308:
2365:
2243:
1827:
does not change at all under a coordinate transformation. (For example, the factor of 2 used for doubling a vector does not change if the vector is in spherical vs. rectangular coordinates.) However, if each vector is transformed by a matrix then the triple product ends up being multiplied by the
1184:
5031:
2764:
2380:
1451:
562:{\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )&=-\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\&=-\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\&=-\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}}
6052:
1813:
1710:
2012:
384:
1611:
under parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product (of vectors) must be pseudoscalar-valued.
2520:
5949:
4675:
2642:
5692:{\displaystyle (\mathbf {a} \times )_{i}=(\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell })a^{j}b_{\ell }c_{m}=a^{j}b_{i}c_{j}-a^{j}b_{j}c_{i}=b_{i}(\mathbf {a} \cdot \mathbf {c} )-c_{i}(\mathbf {a} \cdot \mathbf {b} )\,.}
973:
205:
4209:
2272:
2150:
1089:
2852:{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {A} )-({\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }})\mathbf {A} }
4893:
2504:{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} }
4670:
3942:
2973:
162:
5829:
1918:
310:
Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product:
1568:{\displaystyle {\frac {\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}{\|{\mathbf {a} }\|\|{\mathbf {b} }\|\|{\mathbf {c} }\|}}=\operatorname {psin} (\mathbf {a} ,\mathbf {b} ,\mathbf {c} )}
5954:
4353:
3955:
2986:
1729:
408:
1629:
1437:
This restates in vector notation that the product of the determinants of two 3Ă3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is a
1929:
315:
2622:{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=\mathbf {0} }
2901:
5860:
5233:
5773:
5169:
5105:
5865:
5063:
2736:{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )}
1073:{\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )=\mathbf {b} \cdot (\mathbf {a} \times \mathbf {a} )=0}
5730:
299:{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}
6355:
5131:
1844:
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.)
5401:
5375:
5349:
5323:
5195:
4855:{\displaystyle (\mathbf {a} \times )_{i}=\varepsilon _{ijk}a^{j}\varepsilon ^{k\ell m}b_{\ell }c_{m}=\varepsilon _{ijk}\varepsilon ^{k\ell m}a^{j}b_{\ell }c_{m},}
4303:{\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\ \mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\ \mathbf {w} }
5297:
5277:
5257:
4880:
3898:
3878:
2929:
2360:{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )}
2238:{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }
1179:{\displaystyle (\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\,\mathbf {a} =(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}
4582:
1828:
determinant of the transformation matrix, which could be quite arbitrary for a non-rotation. That is, the triple product is more properly described as a
5026:{\displaystyle \varepsilon _{ijk}\varepsilon ^{k\ell m}=\delta _{ij}^{\ell m}=\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell }\,,}
1864:. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element.
3903:
2934:
123:
1885:
5778:
6382:
4339:
4338:. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a
6715:
6773:
2258:
6152:
did not develop the cross product as an algebraic product on vectors, but did use an equivalent form of it in components: see
1584:
Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the
6237:
6047:{\textstyle \mathbf {F} \cdot {\frac {(\mathbf {r} _{u}\times \mathbf {r} _{v})}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}}
2022:
of the scalar triple product. As the exterior product is associative brackets are not needed as it does not matter which of
1808:{\displaystyle \mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=-\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ).}
1705:{\displaystyle \mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ),}
2007:{\displaystyle |\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} |=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}
379:{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} }
5707:
6280:
6210:
6178:
6134:
2374:
Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as:
6812:
6763:
6725:
6661:
20:
6822:
2868:
2265:"ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is provided
587:
matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as its
2863:
2042:
is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector
4546:
The proof follows from the properties of the contraction. The result is the same vector as calculated using
6817:
6503:
6375:
5834:
5200:
4887:
6827:
6608:
6458:
5944:{\textstyle {\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}}
5735:
5236:
6297:
6513:
6407:
5136:
5072:
1585:
6753:
6402:
6154:
Lagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires".
6073:
5036:
1194:
of two triple products (or the square of a triple product), may be expanded in terms of dot products:
6745:
6628:
6103:
5066:
1592:
of the vectors. This means the product is negated if the orientation is reversed, for example by a
46:
6160:
He may have written a formula similar to the triple product expansion in component form. See also
5713:
6791:
6720:
6498:
6368:
1589:
6555:
6488:
6478:
6198:
6161:
6124:
6570:
6565:
6560:
6493:
6438:
6227:
6149:
5110:
1593:
6580:
6545:
6532:
6423:
4446:
4418:
4367:
1824:
54:
8:
6758:
6638:
6613:
6463:
5380:
5354:
5328:
5302:
5174:
2144:
of one vector with the cross product of the other two. The following relationship holds:
1604:
1444:
The ratio of the triple product and the product of the three vector norms is known as a
6468:
5282:
5262:
5242:
4883:
4865:
4576:
3883:
3863:
2914:
2117:
6314:
6666:
6623:
6550:
6443:
6276:
6233:
6206:
6174:
6130:
6098:
1853:
1720:
395:
6671:
6575:
6428:
2113:
1849:
1438:
968:
If any two vectors in the scalar triple product are equal, then its value is zero:
391:
50:
6730:
6523:
6483:
6473:
4572:
2755:
2633:
1620:
6735:
6391:
6229:
Numerical
Modelling of Water Waves: An Introduction to Engineers and Scientists
1840:
1829:
1191:
185:
172:
4882:-th component of the resulting vector. This can be simplified by performing a
6806:
6768:
6691:
6651:
6618:
6598:
2514:
From
Lagrange's formula it follows that the vector triple product satisfies:
2141:
2097:
106:
6194:
6166:
965:, since the parallelepiped defined by them would be flat and have no volume.
6701:
6590:
6540:
6433:
2121:
1608:
1597:
62:
27:
394:
the triple product. This follows from the circular-shift property and the
6681:
6646:
6603:
6448:
2109:
574:
102:
19:
This article is about ternary operations on vectors. For other uses, see
6061:
77:
6710:
6453:
2019:
1923:
is a trivector with magnitude equal to the scalar triple product, i.e.
1445:
949:
If the scalar triple product is equal to zero, then the three vectors
6508:
2746:
These formulas are very useful in simplifying vector calculations in
1861:
588:
4665:{\displaystyle \mathbf {a} \cdot =\varepsilon _{ijk}a^{i}b^{j}c^{k}}
3937:{\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )}
2968:{\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )}
53:. The name "triple product" is used for two different products, the
6676:
4335:
2751:
2368:
2262:
1857:
962:
157:{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}
34:
5824:{\textstyle \iint _{S}\mathbf {F} \cdot {\hat {\mathbf {n} }}\,dS}
2371:"BAC − CAB" is obtained, as in âback of the cabâ.
1913:{\displaystyle \mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} }
6360:
2747:
38:
5239:. We can reason out this identity by recognizing that the index
2862:
This can be also regarded as a special case of the more general
6686:
6356:
Khan
Academy video of the proof of the triple product expansion
168:
6257:. American Elsevier Publishing Company, Inc. pp. 262â263.
2120:
of vectors with a rank-3 tensor equivalent to the form (or a
6126:
26:"Signed volume" redirects here. For autographed books, see
6298:"Geometric Algebra of One and Many Multivector Variables"
6275:(2nd ed.). Cambridge University Press. p. 46.
2758:
is
Lagrange's formula of vector cross-product identity:
2636:
for the cross product. Another useful formula follows:
573:
The scalar triple product can also be understood as the
5957:
5868:
5781:
1279:
902:
770:
638:
5837:
5738:
5716:
5412:
5383:
5357:
5331:
5305:
5285:
5265:
5245:
5203:
5177:
5139:
5113:
5075:
5039:
4896:
4868:
4678:
4585:
4351:
4212:
3953:
3906:
3886:
3866:
2984:
2937:
2917:
2871:
2767:
2645:
2523:
2383:
2275:
2261:. Its right hand side can be remembered by using the
2153:
1932:
1888:
1732:
1632:
1454:
1200:
1092:
976:
599:
406:
318:
208:
126:
2112:
of the
Euclidean 3-space applied to the vectors via
4326:of vectors is expressed as their exterior product
1860:, while the exterior product of three vectors is a
6046:
5943:
5854:
5823:
5767:
5724:
5691:
5395:
5369:
5343:
5317:
5291:
5271:
5251:
5227:
5189:
5163:
5125:
5099:
5057:
5025:
4874:
4854:
4664:
4535:
4302:
4192:
3936:
3892:
3872:
3849:
2967:
2923:
2895:
2851:
2735:
2621:
2503:
2359:
2237:
2006:
1912:
1807:
1704:
1567:
1429:
1178:
1072:
937:
561:
378:
298:
156:
6804:
6344:. McGraw-Hill Book Company, Inc. pp. 23â25.
1271:
894:
762:
630:
6270:
6255:Mathematical Methods in Science and Engineering
4318:If geometric algebra is used the cross product
4203:By combining these three components we obtain:
184:The scalar triple product is unchanged under a
6376:
6153:
2054:corresponds to the parallelepiped spanned by
4575:, the triple product is expressed using the
2257:, although the latter name is also used for
1523:
1513:
1510:
1500:
1497:
1487:
6225:
1818:
112:
6383:
6369:
6266:
6264:
6252:
6193:
6165:
5732:across the parametrically-defined surface
4451:
4443:
4423:
4415:
4372:
4364:
4313:
2124:equivalent to the volume pseudoform); see
2103:
1234:
6295:
6205:(2nd ed.). MIT Press. p. 1679.
5814:
5685:
5019:
2896:{\displaystyle \Delta =d\delta +\delta d}
1856:the exterior product of two vectors is a
1835:
1596:, and so is more properly described as a
1579:
1126:
117:Geometrically, the scalar triple product
6129:. Oxford University Press. p. 215.
2131:
1839:
76:
72:
6261:
5406:Returning to the triple cross product,
5351:. Likewise, in the second term, we fix
2269:. Some textbooks write the identity as
2108:The triple product is identical to the
390:Swapping any two of the three operands
81:Three vectors defining a parallelepiped
6805:
6774:Comparison of linear algebra libraries
6289:
6203:Encyclopedic dictionary of mathematics
6171:Encyclopedic Dictionary of Mathematics
6364:
5855:{\displaystyle {\hat {\mathbf {n} }}}
5228:{\displaystyle \delta _{ij}^{\ell m}}
6339:
6122:
6056:
5237:generalized Kronecker delta function
4342:can be used, so the formula becomes
2125:
1607:; the cross product transforms as a
175:defined by the three vectors given.
6116:
5768:{\displaystyle S=\mathbf {r} (u,v)}
13:
6390:
6219:
6187:
5701:
4566:
4561:
2872:
14:
6839:
6349:
5164:{\displaystyle \delta _{j}^{i}=1}
5100:{\displaystyle \delta _{j}^{i}=0}
6787:
6786:
6764:Basic Linear Algebra Subprograms
6522:
6060:
6026:
6011:
5989:
5974:
5959:
5923:
5908:
5889:
5874:
5842:
5804:
5793:
5746:
5718:
5678:
5670:
5646:
5638:
5436:
5428:
5417:
5259:will be summed out leaving only
4702:
4694:
4683:
4606:
4598:
4587:
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4517:
4509:
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4490:
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2707:
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2288:
2277:
2231:
2223:
2215:
2204:
2196:
2188:
2174:
2166:
2155:
2116:. It also can be expressed as a
1992:
1984:
1973:
1955:
1947:
1939:
1906:
1898:
1890:
1795:
1787:
1776:
1762:
1759:
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229:
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147:
139:
128:
6662:Seven-dimensional cross product
5058:{\displaystyle \delta _{j}^{i}}
2750:. A related identity regarding
1605:handedness of the cross product
1600:if the orientation can change.
105:of one of the vectors with the
21:Triple product (disambiguation)
6307:
6246:
6143:
6037:
6005:
5999:
5969:
5934:
5902:
5846:
5808:
5762:
5750:
5682:
5666:
5650:
5634:
5522:
5456:
5444:
5440:
5424:
5413:
4710:
4706:
4690:
4679:
4610:
4594:
4521:
4505:
4494:
4478:
4457:
4435:
4429:
4407:
4389:
4373:
4289:
4273:
4259:
4243:
4237:
4221:
4171:
4155:
4137:
4121:
4105:
4101:
4085:
4074:
4055:
4039:
4021:
4005:
3989:
3985:
3969:
3958:
3931:
3915:
3828:
3812:
3794:
3778:
3765:
3684:
3666:
3585:
3560:
3482:
3476:
3422:
3404:
3350:
3325:
3271:
3253:
3199:
3174:
3120:
3102:
3048:
3020:
3016:
3000:
2989:
2962:
2946:
2841:
2825:
2819:
2803:
2792:
2776:
2730:
2714:
2700:
2684:
2662:
2646:
2608:
2592:
2578:
2562:
2548:
2532:
2493:
2477:
2466:
2450:
2441:
2425:
2400:
2384:
2354:
2338:
2327:
2311:
2300:
2284:
2227:
2211:
2200:
2184:
2178:
2162:
2000:
1996:
1980:
1968:
1960:
1934:
1799:
1783:
1766:
1744:
1696:
1680:
1666:
1644:
1575:which ranges between â1 and 1.
1562:
1538:
1482:
1466:
1265:
1254:
1238:
1235:
1231:
1220:
1204:
1201:
1173:
1157:
1151:
1135:
1123:
1120:
1104:
1093:
1061:
1045:
1031:
1015:
1001:
985:
624:
608:
552:
536:
512:
496:
472:
456:
435:
419:
365:
349:
343:
327:
293:
277:
263:
247:
233:
217:
151:
135:
1:
6333:
6273:Clifford algebras and spinors
2100:faces of the parallelepiped.
178:
6504:Eigenvalues and eigenvectors
6054:is a scalar triple product.
5725:{\displaystyle \mathbf {F} }
5299:. In the first term, we fix
16:Ternary operation on vectors
7:
6173:. MIT Press. p. 1679.
6092:
5862:to the surface is given by
10:
6844:
6342:Vector and Tensor Analysis
2367:such that a more familiar
2266:
25:
18:
6782:
6744:
6700:
6637:
6589:
6531:
6520:
6416:
6398:
6232:. Routledge. p. 13.
5831:. The unit normal vector
1603:This also relates to the
1590:parity of the permutation
6271:Pertti Lounesto (2001).
6109:
6104:Vector algebra relations
5067:Kronecker delta function
2906:
2864:Laplaceâde Rham operator
2251:triple product expansion
1819:Scalar or scalar density
113:Geometric interpretation
45:is a product of three 3-
6813:Mathematical identities
5126:{\displaystyle i\neq j}
4314:Using geometric algebra
2104:As a trilinear function
188:of its three operands (
6489:Row and column vectors
6123:Wong, Chun Wa (2013).
6048:
5945:
5856:
5825:
5769:
5726:
5693:
5397:
5371:
5345:
5319:
5293:
5273:
5253:
5229:
5191:
5165:
5127:
5101:
5059:
5027:
4876:
4856:
4666:
4537:
4304:
4194:
3938:
3894:
3874:
3851:
2969:
2925:
2897:
2853:
2737:
2623:
2505:
2361:
2259:several other formulas
2239:
2008:
1914:
1845:
1836:As an exterior product
1809:
1706:
1580:Scalar or pseudoscalar
1569:
1431:
1180:
1074:
939:
563:
398:of the cross product:
380:
300:
158:
82:
6823:Operations on vectors
6494:Row and column spaces
6439:Scalar multiplication
6150:Joseph Louis Lagrange
6049:
5946:
5857:
5826:
5770:
5727:
5694:
5398:
5372:
5346:
5320:
5294:
5274:
5254:
5230:
5192:
5166:
5128:
5102:
5060:
5028:
4877:
4857:
4667:
4538:
4305:
4195:
3939:
3895:
3875:
3852:
2970:
2926:
2898:
2854:
2738:
2624:
2506:
2362:
2240:
2138:vector triple product
2132:Vector triple product
2009:
1915:
1843:
1823:Strictly speaking, a
1810:
1707:
1594:parity transformation
1570:
1432:
1181:
1075:
940:
564:
381:
301:
159:
99:triple scalar product
87:scalar triple product
80:
73:Scalar triple product
67:vector triple product
61:and, less often, the
59:scalar triple product
6629:GramâSchmidt process
6581:Gaussian elimination
6340:Lass, Harry (1950).
6315:"Permutation Tensor"
6226:Pengzhi Lin (2008).
6199:"§C: Vector product"
5955:
5866:
5835:
5779:
5736:
5714:
5710:of the vector field
5410:
5381:
5355:
5329:
5303:
5283:
5263:
5243:
5201:
5175:
5137:
5111:
5073:
5037:
4894:
4866:
4676:
4583:
4349:
4210:
3951:
3904:
3884:
3864:
2982:
2935:
2915:
2869:
2765:
2643:
2521:
2381:
2273:
2151:
1930:
1886:
1730:
1630:
1588:of the frame or the
1452:
1198:
1090:
974:
597:
404:
316:
206:
124:
101:) is defined as the
6818:Multilinear algebra
6759:Numerical stability
6639:Multilinear algebra
6614:Inner product space
6464:Linear independence
6253:J. Heading (1970).
6162:Lagrange's identity
5951:, so the integrand
5521:
5506:
5488:
5473:
5396:{\displaystyle l=j}
5370:{\displaystyle i=m}
5344:{\displaystyle j=m}
5318:{\displaystyle i=l}
5224:
5190:{\displaystyle i=j}
5154:
5090:
5054:
5018:
5003:
4985:
4970:
4952:
4888:Levi-Civita symbols
6828:Ternary operations
6469:Linear combination
6072:. You can help by
6044:
5941:
5852:
5821:
5765:
5722:
5689:
5507:
5492:
5474:
5459:
5393:
5367:
5341:
5315:
5289:
5269:
5249:
5225:
5204:
5187:
5161:
5140:
5123:
5097:
5076:
5055:
5040:
5023:
5004:
4989:
4971:
4956:
4932:
4872:
4852:
4662:
4577:Levi-Civita symbol
4533:
4531:
4300:
4190:
4188:
3934:
3890:
3870:
3847:
3845:
2965:
2921:
2893:
2849:
2733:
2619:
2501:
2357:
2255:Lagrange's formula
2235:
2140:is defined as the
2004:
1910:
1846:
1805:
1702:
1565:
1427:
1421:
1176:
1070:
935:
926:
885:
753:
559:
557:
376:
296:
154:
109:of the other two.
83:
6800:
6799:
6667:Geometric algebra
6624:Kronecker product
6459:Linear projection
6444:Vector projection
6239:978-0-415-41578-1
6099:Quadruple product
6090:
6089:
6042:
5939:
5849:
5811:
5292:{\displaystyle j}
5272:{\displaystyle i}
5252:{\displaystyle k}
4875:{\displaystyle i}
4862:referring to the
4294:
4264:
3893:{\displaystyle z}
3873:{\displaystyle y}
2924:{\displaystyle x}
2249:This is known as
2066:, with bivectors
1854:geometric algebra
1721:improper rotation
1527:
396:anticommutativity
89:(also called the
51:Euclidean vectors
49:vectors, usually
6835:
6790:
6789:
6672:Exterior algebra
6609:Hadamard product
6526:
6514:Linear equations
6385:
6378:
6371:
6362:
6361:
6345:
6327:
6326:
6324:
6322:
6311:
6305:
6304:
6302:
6293:
6287:
6286:
6268:
6259:
6258:
6250:
6244:
6243:
6223:
6217:
6216:
6191:
6185:
6184:
6159:
6147:
6141:
6140:
6120:
6085:
6082:
6064:
6057:
6053:
6051:
6050:
6045:
6043:
6041:
6040:
6035:
6034:
6029:
6020:
6019:
6014:
6008:
6002:
5998:
5997:
5992:
5983:
5982:
5977:
5967:
5962:
5950:
5948:
5947:
5942:
5940:
5938:
5937:
5932:
5931:
5926:
5917:
5916:
5911:
5905:
5899:
5898:
5897:
5892:
5883:
5882:
5877:
5870:
5861:
5859:
5858:
5853:
5851:
5850:
5845:
5840:
5830:
5828:
5827:
5822:
5813:
5812:
5807:
5802:
5796:
5791:
5790:
5774:
5772:
5771:
5766:
5749:
5731:
5729:
5728:
5723:
5721:
5698:
5696:
5695:
5690:
5681:
5673:
5665:
5664:
5649:
5641:
5633:
5632:
5620:
5619:
5610:
5609:
5600:
5599:
5587:
5586:
5577:
5576:
5567:
5566:
5554:
5553:
5544:
5543:
5534:
5533:
5520:
5515:
5505:
5500:
5487:
5482:
5472:
5467:
5452:
5451:
5439:
5431:
5420:
5402:
5400:
5399:
5394:
5376:
5374:
5373:
5368:
5350:
5348:
5347:
5342:
5324:
5322:
5321:
5316:
5298:
5296:
5295:
5290:
5278:
5276:
5275:
5270:
5258:
5256:
5255:
5250:
5234:
5232:
5231:
5226:
5223:
5215:
5196:
5194:
5193:
5188:
5170:
5168:
5167:
5162:
5153:
5148:
5132:
5130:
5129:
5124:
5106:
5104:
5103:
5098:
5089:
5084:
5064:
5062:
5061:
5056:
5053:
5048:
5032:
5030:
5029:
5024:
5017:
5012:
5002:
4997:
4984:
4979:
4969:
4964:
4951:
4943:
4928:
4927:
4912:
4911:
4881:
4879:
4878:
4873:
4861:
4859:
4858:
4853:
4848:
4847:
4838:
4837:
4828:
4827:
4818:
4817:
4802:
4801:
4783:
4782:
4773:
4772:
4763:
4762:
4747:
4746:
4737:
4736:
4718:
4717:
4705:
4697:
4686:
4671:
4669:
4668:
4663:
4661:
4660:
4651:
4650:
4641:
4640:
4631:
4630:
4609:
4601:
4590:
4542:
4540:
4539:
4534:
4532:
4528:
4520:
4512:
4501:
4493:
4485:
4471:
4467:
4456:
4450:
4449:
4442:
4428:
4422:
4421:
4414:
4403:
4388:
4380:
4371:
4370:
4363:
4340:left contraction
4309:
4307:
4306:
4301:
4299:
4292:
4288:
4280:
4269:
4262:
4258:
4250:
4236:
4228:
4217:
4199:
4197:
4196:
4191:
4189:
4185:
4184:
4179:
4170:
4162:
4151:
4150:
4145:
4136:
4128:
4113:
4112:
4100:
4092:
4081:
4069:
4068:
4063:
4054:
4046:
4035:
4034:
4029:
4020:
4012:
3997:
3996:
3984:
3976:
3965:
3943:
3941:
3940:
3935:
3930:
3922:
3911:
3899:
3897:
3896:
3891:
3879:
3877:
3876:
3871:
3856:
3854:
3853:
3848:
3846:
3842:
3841:
3836:
3827:
3819:
3808:
3807:
3802:
3793:
3785:
3771:
3764:
3763:
3758:
3752:
3751:
3746:
3737:
3736:
3731:
3725:
3724:
3719:
3710:
3709:
3704:
3698:
3697:
3692:
3683:
3682:
3677:
3665:
3664:
3659:
3653:
3652:
3647:
3638:
3637:
3632:
3626:
3625:
3620:
3611:
3610:
3605:
3599:
3598:
3593:
3584:
3583:
3578:
3566:
3559:
3558:
3553:
3547:
3546:
3541:
3535:
3534:
3529:
3520:
3519:
3514:
3508:
3507:
3502:
3496:
3495:
3490:
3475:
3474:
3469:
3463:
3462:
3457:
3448:
3447:
3442:
3436:
3435:
3430:
3421:
3420:
3415:
3403:
3402:
3397:
3391:
3390:
3385:
3376:
3375:
3370:
3364:
3363:
3358:
3349:
3348:
3343:
3331:
3324:
3323:
3318:
3312:
3311:
3306:
3297:
3296:
3291:
3285:
3284:
3279:
3270:
3269:
3264:
3252:
3251:
3246:
3240:
3239:
3234:
3225:
3224:
3219:
3213:
3212:
3207:
3198:
3197:
3192:
3180:
3173:
3172:
3167:
3161:
3160:
3155:
3146:
3145:
3140:
3134:
3133:
3128:
3119:
3118:
3113:
3101:
3100:
3095:
3089:
3088:
3083:
3074:
3073:
3068:
3062:
3061:
3056:
3047:
3046:
3041:
3028:
3027:
3015:
3007:
2996:
2974:
2972:
2971:
2966:
2961:
2953:
2942:
2930:
2928:
2927:
2922:
2902:
2900:
2899:
2894:
2858:
2856:
2855:
2850:
2848:
2840:
2832:
2818:
2810:
2802:
2791:
2783:
2772:
2742:
2740:
2739:
2734:
2729:
2721:
2710:
2699:
2691:
2680:
2672:
2661:
2653:
2628:
2626:
2625:
2620:
2618:
2607:
2599:
2588:
2577:
2569:
2558:
2547:
2539:
2528:
2510:
2508:
2507:
2502:
2500:
2492:
2484:
2473:
2465:
2457:
2440:
2432:
2421:
2410:
2399:
2391:
2366:
2364:
2363:
2358:
2353:
2345:
2337:
2326:
2318:
2310:
2299:
2291:
2280:
2244:
2242:
2241:
2236:
2234:
2226:
2218:
2207:
2199:
2191:
2177:
2169:
2158:
2114:interior product
2095:
2085:
2075:
2041:
2031:
2013:
2011:
2010:
2005:
2003:
1995:
1987:
1976:
1971:
1963:
1958:
1950:
1942:
1937:
1919:
1917:
1916:
1911:
1909:
1901:
1893:
1850:exterior algebra
1814:
1812:
1811:
1806:
1798:
1790:
1779:
1765:
1754:
1740:
1711:
1709:
1708:
1703:
1695:
1687:
1676:
1665:
1654:
1640:
1574:
1572:
1571:
1566:
1561:
1553:
1545:
1528:
1526:
1522:
1521:
1509:
1508:
1496:
1495:
1485:
1481:
1473:
1462:
1456:
1439:Gram determinant
1436:
1434:
1433:
1428:
1426:
1425:
1418:
1410:
1403:
1395:
1388:
1380:
1371:
1363:
1356:
1348:
1341:
1333:
1324:
1316:
1309:
1301:
1294:
1286:
1264:
1253:
1245:
1230:
1219:
1211:
1185:
1183:
1182:
1177:
1172:
1164:
1150:
1142:
1131:
1119:
1111:
1100:
1079:
1077:
1076:
1071:
1060:
1052:
1041:
1030:
1022:
1011:
1000:
992:
981:
944:
942:
941:
936:
931:
930:
923:
916:
909:
890:
889:
882:
881:
870:
869:
858:
857:
844:
843:
832:
831:
820:
819:
806:
805:
794:
793:
782:
781:
758:
757:
750:
749:
738:
737:
726:
725:
712:
711:
700:
699:
688:
687:
674:
673:
662:
661:
650:
649:
623:
615:
604:
586:
585:
582:
568:
566:
565:
560:
558:
551:
543:
532:
518:
511:
503:
492:
478:
471:
463:
452:
434:
426:
415:
385:
383:
382:
377:
375:
364:
356:
342:
334:
323:
305:
303:
302:
297:
292:
284:
273:
262:
254:
243:
232:
224:
213:
167:is the (signed)
163:
161:
160:
155:
150:
142:
131:
6843:
6842:
6838:
6837:
6836:
6834:
6833:
6832:
6803:
6802:
6801:
6796:
6778:
6740:
6696:
6633:
6585:
6527:
6518:
6484:Change of basis
6474:Multilinear map
6412:
6394:
6389:
6352:
6336:
6331:
6330:
6320:
6318:
6313:
6312:
6308:
6300:
6296:Janne Pesonen.
6294:
6290:
6283:
6269:
6262:
6251:
6247:
6240:
6224:
6220:
6213:
6192:
6188:
6181:
6148:
6144:
6137:
6121:
6117:
6112:
6095:
6086:
6080:
6077:
6070:needs expansion
6036:
6030:
6025:
6024:
6015:
6010:
6009:
6004:
6003:
5993:
5988:
5987:
5978:
5973:
5972:
5968:
5966:
5958:
5956:
5953:
5952:
5933:
5927:
5922:
5921:
5912:
5907:
5906:
5901:
5900:
5893:
5888:
5887:
5878:
5873:
5872:
5871:
5869:
5867:
5864:
5863:
5841:
5839:
5838:
5836:
5833:
5832:
5803:
5801:
5800:
5792:
5786:
5782:
5780:
5777:
5776:
5745:
5737:
5734:
5733:
5717:
5715:
5712:
5711:
5704:
5702:Vector calculus
5677:
5669:
5660:
5656:
5645:
5637:
5628:
5624:
5615:
5611:
5605:
5601:
5595:
5591:
5582:
5578:
5572:
5568:
5562:
5558:
5549:
5545:
5539:
5535:
5529:
5525:
5516:
5511:
5501:
5496:
5483:
5478:
5468:
5463:
5447:
5443:
5435:
5427:
5416:
5411:
5408:
5407:
5382:
5379:
5378:
5356:
5353:
5352:
5330:
5327:
5326:
5304:
5301:
5300:
5284:
5281:
5280:
5264:
5261:
5260:
5244:
5241:
5240:
5216:
5208:
5202:
5199:
5198:
5176:
5173:
5172:
5149:
5144:
5138:
5135:
5134:
5112:
5109:
5108:
5085:
5080:
5074:
5071:
5070:
5049:
5044:
5038:
5035:
5034:
5013:
5008:
4998:
4993:
4980:
4975:
4965:
4960:
4944:
4936:
4917:
4913:
4901:
4897:
4895:
4892:
4891:
4867:
4864:
4863:
4843:
4839:
4833:
4829:
4823:
4819:
4807:
4803:
4791:
4787:
4778:
4774:
4768:
4764:
4752:
4748:
4742:
4738:
4726:
4722:
4713:
4709:
4701:
4693:
4682:
4677:
4674:
4673:
4656:
4652:
4646:
4642:
4636:
4632:
4620:
4616:
4605:
4597:
4586:
4584:
4581:
4580:
4573:tensor notation
4569:
4567:Tensor calculus
4564:
4562:Interpretations
4530:
4529:
4524:
4516:
4508:
4497:
4489:
4481:
4469:
4468:
4463:
4452:
4445:
4444:
4438:
4424:
4417:
4416:
4410:
4399:
4392:
4384:
4376:
4366:
4365:
4359:
4352:
4350:
4347:
4346:
4316:
4295:
4284:
4276:
4265:
4254:
4246:
4232:
4224:
4213:
4211:
4208:
4207:
4187:
4186:
4180:
4175:
4174:
4166:
4158:
4146:
4141:
4140:
4132:
4124:
4114:
4108:
4104:
4096:
4088:
4077:
4071:
4070:
4064:
4059:
4058:
4050:
4042:
4030:
4025:
4024:
4016:
4008:
3998:
3992:
3988:
3980:
3972:
3961:
3954:
3952:
3949:
3948:
3926:
3918:
3907:
3905:
3902:
3901:
3900:components of
3885:
3882:
3881:
3865:
3862:
3861:
3860:Similarly, the
3844:
3843:
3837:
3832:
3831:
3823:
3815:
3803:
3798:
3797:
3789:
3781:
3769:
3768:
3759:
3754:
3753:
3747:
3742:
3741:
3732:
3727:
3726:
3720:
3715:
3714:
3705:
3700:
3699:
3693:
3688:
3687:
3678:
3673:
3672:
3660:
3655:
3654:
3648:
3643:
3642:
3633:
3628:
3627:
3621:
3616:
3615:
3606:
3601:
3600:
3594:
3589:
3588:
3579:
3574:
3573:
3564:
3563:
3554:
3549:
3548:
3542:
3537:
3536:
3530:
3525:
3524:
3515:
3510:
3509:
3503:
3498:
3497:
3491:
3486:
3485:
3470:
3465:
3464:
3458:
3453:
3452:
3443:
3438:
3437:
3431:
3426:
3425:
3416:
3411:
3410:
3398:
3393:
3392:
3386:
3381:
3380:
3371:
3366:
3365:
3359:
3354:
3353:
3344:
3339:
3338:
3329:
3328:
3319:
3314:
3313:
3307:
3302:
3301:
3292:
3287:
3286:
3280:
3275:
3274:
3265:
3260:
3259:
3247:
3242:
3241:
3235:
3230:
3229:
3220:
3215:
3214:
3208:
3203:
3202:
3193:
3188:
3187:
3178:
3177:
3168:
3163:
3162:
3156:
3151:
3150:
3141:
3136:
3135:
3129:
3124:
3123:
3114:
3109:
3108:
3096:
3091:
3090:
3084:
3079:
3078:
3069:
3064:
3063:
3057:
3052:
3051:
3042:
3037:
3036:
3029:
3023:
3019:
3011:
3003:
2992:
2985:
2983:
2980:
2979:
2957:
2949:
2938:
2936:
2933:
2932:
2916:
2913:
2912:
2909:
2870:
2867:
2866:
2844:
2836:
2828:
2814:
2806:
2798:
2787:
2779:
2768:
2766:
2763:
2762:
2756:vector calculus
2725:
2717:
2706:
2695:
2687:
2676:
2668:
2657:
2649:
2644:
2641:
2640:
2634:Jacobi identity
2614:
2603:
2595:
2584:
2573:
2565:
2554:
2543:
2535:
2524:
2522:
2519:
2518:
2496:
2488:
2480:
2469:
2461:
2453:
2436:
2428:
2417:
2406:
2395:
2387:
2382:
2379:
2378:
2349:
2341:
2333:
2322:
2314:
2306:
2295:
2287:
2276:
2274:
2271:
2270:
2230:
2222:
2214:
2203:
2195:
2187:
2173:
2165:
2154:
2152:
2149:
2148:
2134:
2106:
2087:
2077:
2067:
2033:
2023:
1999:
1991:
1983:
1972:
1967:
1959:
1954:
1946:
1938:
1933:
1931:
1928:
1927:
1905:
1897:
1889:
1887:
1884:
1883:
1838:
1821:
1794:
1786:
1775:
1758:
1747:
1733:
1731:
1728:
1727:
1691:
1683:
1672:
1658:
1647:
1633:
1631:
1628:
1627:
1621:proper rotation
1582:
1557:
1549:
1541:
1517:
1516:
1504:
1503:
1491:
1490:
1486:
1477:
1469:
1458:
1457:
1455:
1453:
1450:
1449:
1420:
1419:
1414:
1406:
1404:
1399:
1391:
1389:
1384:
1376:
1373:
1372:
1367:
1359:
1357:
1352:
1344:
1342:
1337:
1329:
1326:
1325:
1320:
1312:
1310:
1305:
1297:
1295:
1290:
1282:
1275:
1274:
1260:
1249:
1241:
1226:
1215:
1207:
1199:
1196:
1195:
1168:
1160:
1146:
1138:
1127:
1115:
1107:
1096:
1091:
1088:
1087:
1056:
1048:
1037:
1026:
1018:
1007:
996:
988:
977:
975:
972:
971:
925:
924:
919:
917:
912:
910:
905:
898:
897:
884:
883:
877:
873:
871:
865:
861:
859:
853:
849:
846:
845:
839:
835:
833:
827:
823:
821:
815:
811:
808:
807:
801:
797:
795:
789:
785:
783:
777:
773:
766:
765:
752:
751:
745:
741:
739:
733:
729:
727:
721:
717:
714:
713:
707:
703:
701:
695:
691:
689:
683:
679:
676:
675:
669:
665:
663:
657:
653:
651:
645:
641:
634:
633:
619:
611:
600:
598:
595:
594:
583:
580:
578:
556:
555:
547:
539:
528:
516:
515:
507:
499:
488:
476:
475:
467:
459:
448:
438:
430:
422:
411:
407:
405:
402:
401:
371:
360:
352:
338:
330:
319:
317:
314:
313:
288:
280:
269:
258:
250:
239:
228:
220:
209:
207:
204:
203:
181:
146:
138:
127:
125:
122:
121:
115:
75:
31:
24:
17:
12:
11:
5:
6841:
6831:
6830:
6825:
6820:
6815:
6798:
6797:
6795:
6794:
6783:
6780:
6779:
6777:
6776:
6771:
6766:
6761:
6756:
6754:Floating-point
6750:
6748:
6742:
6741:
6739:
6738:
6736:Tensor product
6733:
6728:
6723:
6721:Function space
6718:
6713:
6707:
6705:
6698:
6697:
6695:
6694:
6689:
6684:
6679:
6674:
6669:
6664:
6659:
6657:Triple product
6654:
6649:
6643:
6641:
6635:
6634:
6632:
6631:
6626:
6621:
6616:
6611:
6606:
6601:
6595:
6593:
6587:
6586:
6584:
6583:
6578:
6573:
6571:Transformation
6568:
6563:
6561:Multiplication
6558:
6553:
6548:
6543:
6537:
6535:
6529:
6528:
6521:
6519:
6517:
6516:
6511:
6506:
6501:
6496:
6491:
6486:
6481:
6476:
6471:
6466:
6461:
6456:
6451:
6446:
6441:
6436:
6431:
6426:
6420:
6418:
6417:Basic concepts
6414:
6413:
6411:
6410:
6405:
6399:
6396:
6395:
6392:Linear algebra
6388:
6387:
6380:
6373:
6365:
6359:
6358:
6351:
6350:External links
6348:
6347:
6346:
6335:
6332:
6329:
6328:
6306:
6288:
6281:
6260:
6245:
6238:
6218:
6211:
6186:
6179:
6158:. Vol. 3.
6142:
6135:
6114:
6113:
6111:
6108:
6107:
6106:
6101:
6094:
6091:
6088:
6087:
6067:
6065:
6039:
6033:
6028:
6023:
6018:
6013:
6007:
6001:
5996:
5991:
5986:
5981:
5976:
5971:
5965:
5961:
5936:
5930:
5925:
5920:
5915:
5910:
5904:
5896:
5891:
5886:
5881:
5876:
5848:
5844:
5820:
5817:
5810:
5806:
5799:
5795:
5789:
5785:
5764:
5761:
5758:
5755:
5752:
5748:
5744:
5741:
5720:
5703:
5700:
5688:
5684:
5680:
5676:
5672:
5668:
5663:
5659:
5655:
5652:
5648:
5644:
5640:
5636:
5631:
5627:
5623:
5618:
5614:
5608:
5604:
5598:
5594:
5590:
5585:
5581:
5575:
5571:
5565:
5561:
5557:
5552:
5548:
5542:
5538:
5532:
5528:
5524:
5519:
5514:
5510:
5504:
5499:
5495:
5491:
5486:
5481:
5477:
5471:
5466:
5462:
5458:
5455:
5450:
5446:
5442:
5438:
5434:
5430:
5426:
5423:
5419:
5415:
5392:
5389:
5386:
5366:
5363:
5360:
5340:
5337:
5334:
5314:
5311:
5308:
5288:
5268:
5248:
5222:
5219:
5214:
5211:
5207:
5186:
5183:
5180:
5160:
5157:
5152:
5147:
5143:
5122:
5119:
5116:
5096:
5093:
5088:
5083:
5079:
5052:
5047:
5043:
5022:
5016:
5011:
5007:
5001:
4996:
4992:
4988:
4983:
4978:
4974:
4968:
4963:
4959:
4955:
4950:
4947:
4942:
4939:
4935:
4931:
4926:
4923:
4920:
4916:
4910:
4907:
4904:
4900:
4871:
4851:
4846:
4842:
4836:
4832:
4826:
4822:
4816:
4813:
4810:
4806:
4800:
4797:
4794:
4790:
4786:
4781:
4777:
4771:
4767:
4761:
4758:
4755:
4751:
4745:
4741:
4735:
4732:
4729:
4725:
4721:
4716:
4712:
4708:
4704:
4700:
4696:
4692:
4689:
4685:
4681:
4659:
4655:
4649:
4645:
4639:
4635:
4629:
4626:
4623:
4619:
4615:
4612:
4608:
4604:
4600:
4596:
4593:
4589:
4568:
4565:
4563:
4560:
4544:
4543:
4527:
4523:
4519:
4515:
4511:
4507:
4504:
4500:
4496:
4492:
4488:
4484:
4480:
4477:
4474:
4472:
4470:
4466:
4462:
4459:
4455:
4448:
4441:
4437:
4434:
4431:
4427:
4420:
4413:
4409:
4406:
4402:
4398:
4395:
4393:
4391:
4387:
4383:
4379:
4375:
4369:
4362:
4358:
4355:
4354:
4315:
4312:
4311:
4310:
4298:
4291:
4287:
4283:
4279:
4275:
4272:
4268:
4261:
4257:
4253:
4249:
4245:
4242:
4239:
4235:
4231:
4227:
4223:
4220:
4216:
4201:
4200:
4183:
4178:
4173:
4169:
4165:
4161:
4157:
4154:
4149:
4144:
4139:
4135:
4131:
4127:
4123:
4120:
4117:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4084:
4080:
4076:
4073:
4072:
4067:
4062:
4057:
4053:
4049:
4045:
4041:
4038:
4033:
4028:
4023:
4019:
4015:
4011:
4007:
4004:
4001:
3999:
3995:
3991:
3987:
3983:
3979:
3975:
3971:
3968:
3964:
3960:
3957:
3956:
3944:are given by:
3933:
3929:
3925:
3921:
3917:
3914:
3910:
3889:
3869:
3858:
3857:
3840:
3835:
3830:
3826:
3822:
3818:
3814:
3811:
3806:
3801:
3796:
3792:
3788:
3784:
3780:
3777:
3774:
3772:
3770:
3767:
3762:
3757:
3750:
3745:
3740:
3735:
3730:
3723:
3718:
3713:
3708:
3703:
3696:
3691:
3686:
3681:
3676:
3671:
3668:
3663:
3658:
3651:
3646:
3641:
3636:
3631:
3624:
3619:
3614:
3609:
3604:
3597:
3592:
3587:
3582:
3577:
3572:
3569:
3567:
3565:
3562:
3557:
3552:
3545:
3540:
3533:
3528:
3523:
3518:
3513:
3506:
3501:
3494:
3489:
3484:
3481:
3478:
3473:
3468:
3461:
3456:
3451:
3446:
3441:
3434:
3429:
3424:
3419:
3414:
3409:
3406:
3401:
3396:
3389:
3384:
3379:
3374:
3369:
3362:
3357:
3352:
3347:
3342:
3337:
3334:
3332:
3330:
3327:
3322:
3317:
3310:
3305:
3300:
3295:
3290:
3283:
3278:
3273:
3268:
3263:
3258:
3255:
3250:
3245:
3238:
3233:
3228:
3223:
3218:
3211:
3206:
3201:
3196:
3191:
3186:
3183:
3181:
3179:
3176:
3171:
3166:
3159:
3154:
3149:
3144:
3139:
3132:
3127:
3122:
3117:
3112:
3107:
3104:
3099:
3094:
3087:
3082:
3077:
3072:
3067:
3060:
3055:
3050:
3045:
3040:
3035:
3032:
3030:
3026:
3022:
3018:
3014:
3010:
3006:
3002:
2999:
2995:
2991:
2988:
2987:
2964:
2960:
2956:
2952:
2948:
2945:
2941:
2931:component of
2920:
2908:
2905:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2860:
2859:
2847:
2843:
2839:
2835:
2831:
2827:
2824:
2821:
2817:
2813:
2809:
2805:
2801:
2797:
2794:
2790:
2786:
2782:
2778:
2775:
2771:
2754:and useful in
2744:
2743:
2732:
2728:
2724:
2720:
2716:
2713:
2709:
2705:
2702:
2698:
2694:
2690:
2686:
2683:
2679:
2675:
2671:
2667:
2664:
2660:
2656:
2652:
2648:
2630:
2629:
2617:
2613:
2610:
2606:
2602:
2598:
2594:
2591:
2587:
2583:
2580:
2576:
2572:
2568:
2564:
2561:
2557:
2553:
2550:
2546:
2542:
2538:
2534:
2531:
2527:
2512:
2511:
2499:
2495:
2491:
2487:
2483:
2479:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2449:
2446:
2443:
2439:
2435:
2431:
2427:
2424:
2420:
2416:
2413:
2409:
2405:
2402:
2398:
2394:
2390:
2386:
2356:
2352:
2348:
2344:
2340:
2336:
2332:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2302:
2298:
2294:
2290:
2286:
2283:
2279:
2247:
2246:
2233:
2229:
2225:
2221:
2217:
2213:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2183:
2180:
2176:
2172:
2168:
2164:
2161:
2157:
2133:
2130:
2105:
2102:
2016:
2015:
2002:
1998:
1994:
1990:
1986:
1982:
1979:
1975:
1970:
1966:
1962:
1957:
1953:
1949:
1945:
1941:
1936:
1921:
1920:
1908:
1904:
1900:
1896:
1892:
1879:, the product
1867:Given vectors
1837:
1834:
1830:scalar density
1820:
1817:
1816:
1815:
1804:
1801:
1797:
1793:
1789:
1785:
1782:
1778:
1774:
1771:
1768:
1764:
1761:
1757:
1753:
1750:
1746:
1743:
1739:
1736:
1713:
1712:
1701:
1698:
1694:
1690:
1686:
1682:
1679:
1675:
1671:
1668:
1664:
1661:
1657:
1653:
1650:
1646:
1643:
1639:
1636:
1581:
1578:
1577:
1576:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1537:
1534:
1531:
1525:
1520:
1515:
1512:
1507:
1502:
1499:
1494:
1489:
1484:
1480:
1476:
1472:
1468:
1465:
1461:
1442:
1424:
1417:
1413:
1409:
1405:
1402:
1398:
1394:
1390:
1387:
1383:
1379:
1375:
1374:
1370:
1366:
1362:
1358:
1355:
1351:
1347:
1343:
1340:
1336:
1332:
1328:
1327:
1323:
1319:
1315:
1311:
1308:
1304:
1300:
1296:
1293:
1289:
1285:
1281:
1280:
1278:
1273:
1270:
1267:
1263:
1259:
1256:
1252:
1248:
1244:
1240:
1237:
1233:
1229:
1225:
1222:
1218:
1214:
1210:
1206:
1203:
1192:simple product
1188:
1187:
1186:
1175:
1171:
1167:
1163:
1159:
1156:
1153:
1149:
1145:
1141:
1137:
1134:
1130:
1125:
1122:
1118:
1114:
1110:
1106:
1103:
1099:
1095:
1082:
1081:
1080:
1069:
1066:
1063:
1059:
1055:
1051:
1047:
1044:
1040:
1036:
1033:
1029:
1025:
1021:
1017:
1014:
1010:
1006:
1003:
999:
995:
991:
987:
984:
980:
966:
947:
946:
945:
934:
929:
922:
918:
915:
911:
908:
904:
903:
901:
896:
893:
888:
880:
876:
872:
868:
864:
860:
856:
852:
848:
847:
842:
838:
834:
830:
826:
822:
818:
814:
810:
809:
804:
800:
796:
792:
788:
784:
780:
776:
772:
771:
769:
764:
761:
756:
748:
744:
740:
736:
732:
728:
724:
720:
716:
715:
710:
706:
702:
698:
694:
690:
686:
682:
678:
677:
672:
668:
664:
660:
656:
652:
648:
644:
640:
639:
637:
632:
629:
626:
622:
618:
614:
610:
607:
603:
571:
570:
569:
554:
550:
546:
542:
538:
535:
531:
527:
524:
521:
519:
517:
514:
510:
506:
502:
498:
495:
491:
487:
484:
481:
479:
477:
474:
470:
466:
462:
458:
455:
451:
447:
444:
441:
439:
437:
433:
429:
425:
421:
418:
414:
410:
409:
388:
387:
386:
374:
370:
367:
363:
359:
355:
351:
348:
345:
341:
337:
333:
329:
326:
322:
308:
307:
306:
295:
291:
287:
283:
279:
276:
272:
268:
265:
261:
257:
253:
249:
246:
242:
238:
235:
231:
227:
223:
219:
216:
212:
186:circular shift
180:
177:
173:parallelepiped
165:
164:
153:
149:
145:
141:
137:
134:
130:
114:
111:
74:
71:
43:triple product
15:
9:
6:
4:
3:
2:
6840:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6810:
6808:
6793:
6785:
6784:
6781:
6775:
6772:
6770:
6769:Sparse matrix
6767:
6765:
6762:
6760:
6757:
6755:
6752:
6751:
6749:
6747:
6743:
6737:
6734:
6732:
6729:
6727:
6724:
6722:
6719:
6717:
6714:
6712:
6709:
6708:
6706:
6704:constructions
6703:
6699:
6693:
6692:Outermorphism
6690:
6688:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6658:
6655:
6653:
6652:Cross product
6650:
6648:
6645:
6644:
6642:
6640:
6636:
6630:
6627:
6625:
6622:
6620:
6619:Outer product
6617:
6615:
6612:
6610:
6607:
6605:
6602:
6600:
6599:Orthogonality
6597:
6596:
6594:
6592:
6588:
6582:
6579:
6577:
6576:Cramer's rule
6574:
6572:
6569:
6567:
6564:
6562:
6559:
6557:
6554:
6552:
6549:
6547:
6546:Decomposition
6544:
6542:
6539:
6538:
6536:
6534:
6530:
6525:
6515:
6512:
6510:
6507:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6475:
6472:
6470:
6467:
6465:
6462:
6460:
6457:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6432:
6430:
6427:
6425:
6422:
6421:
6419:
6415:
6409:
6406:
6404:
6401:
6400:
6397:
6393:
6386:
6381:
6379:
6374:
6372:
6367:
6366:
6363:
6357:
6354:
6353:
6343:
6338:
6337:
6316:
6310:
6303:. p. 37.
6299:
6292:
6284:
6282:0-521-00551-5
6278:
6274:
6267:
6265:
6256:
6249:
6241:
6235:
6231:
6230:
6222:
6214:
6212:0-262-59020-4
6208:
6204:
6200:
6196:
6190:
6182:
6180:0-262-59020-4
6176:
6172:
6168:
6163:
6157:
6151:
6146:
6138:
6136:9780199641390
6132:
6128:
6127:
6119:
6115:
6105:
6102:
6100:
6097:
6096:
6084:
6075:
6071:
6068:This section
6066:
6063:
6059:
6058:
6055:
6031:
6021:
6016:
5994:
5984:
5979:
5963:
5928:
5918:
5913:
5894:
5884:
5879:
5818:
5815:
5797:
5787:
5783:
5759:
5756:
5753:
5742:
5739:
5709:
5708:flux integral
5706:Consider the
5699:
5686:
5674:
5661:
5657:
5653:
5642:
5629:
5625:
5621:
5616:
5612:
5606:
5602:
5596:
5592:
5588:
5583:
5579:
5573:
5569:
5563:
5559:
5555:
5550:
5546:
5540:
5536:
5530:
5526:
5517:
5512:
5508:
5502:
5497:
5493:
5489:
5484:
5479:
5475:
5469:
5464:
5460:
5453:
5448:
5432:
5421:
5404:
5390:
5387:
5384:
5364:
5361:
5358:
5338:
5335:
5332:
5312:
5309:
5306:
5286:
5266:
5246:
5238:
5220:
5217:
5212:
5209:
5205:
5184:
5181:
5178:
5158:
5155:
5150:
5145:
5141:
5120:
5117:
5114:
5094:
5091:
5086:
5081:
5077:
5068:
5050:
5045:
5041:
5020:
5014:
5009:
5005:
4999:
4994:
4990:
4986:
4981:
4976:
4972:
4966:
4961:
4957:
4953:
4948:
4945:
4940:
4937:
4933:
4929:
4924:
4921:
4918:
4914:
4908:
4905:
4902:
4898:
4889:
4885:
4869:
4849:
4844:
4840:
4834:
4830:
4824:
4820:
4814:
4811:
4808:
4804:
4798:
4795:
4792:
4788:
4784:
4779:
4775:
4769:
4765:
4759:
4756:
4753:
4749:
4743:
4739:
4733:
4730:
4727:
4723:
4719:
4714:
4698:
4687:
4657:
4653:
4647:
4643:
4637:
4633:
4627:
4624:
4621:
4617:
4613:
4602:
4591:
4578:
4574:
4559:
4557:
4553:
4549:
4513:
4502:
4486:
4475:
4473:
4460:
4432:
4404:
4396:
4394:
4381:
4356:
4345:
4344:
4343:
4341:
4337:
4333:
4329:
4325:
4321:
4281:
4270:
4251:
4240:
4229:
4218:
4206:
4205:
4204:
4181:
4163:
4152:
4147:
4129:
4118:
4116:
4109:
4093:
4082:
4065:
4047:
4036:
4031:
4013:
4002:
4000:
3993:
3977:
3966:
3947:
3946:
3945:
3923:
3912:
3887:
3867:
3838:
3820:
3809:
3804:
3786:
3775:
3773:
3760:
3748:
3738:
3733:
3721:
3711:
3706:
3694:
3679:
3669:
3661:
3649:
3639:
3634:
3622:
3612:
3607:
3595:
3580:
3570:
3568:
3555:
3543:
3531:
3521:
3516:
3504:
3492:
3479:
3471:
3459:
3449:
3444:
3432:
3417:
3407:
3399:
3387:
3377:
3372:
3360:
3345:
3335:
3333:
3320:
3308:
3298:
3293:
3281:
3266:
3256:
3248:
3236:
3226:
3221:
3209:
3194:
3184:
3182:
3169:
3157:
3147:
3142:
3130:
3115:
3105:
3097:
3085:
3075:
3070:
3058:
3043:
3033:
3031:
3024:
3008:
2997:
2978:
2977:
2976:
2975:is given by:
2954:
2943:
2918:
2904:
2890:
2887:
2884:
2881:
2878:
2875:
2865:
2833:
2822:
2811:
2795:
2784:
2773:
2761:
2760:
2759:
2757:
2753:
2749:
2722:
2711:
2703:
2692:
2681:
2673:
2665:
2654:
2639:
2638:
2637:
2635:
2632:which is the
2611:
2600:
2589:
2581:
2570:
2559:
2551:
2540:
2529:
2517:
2516:
2515:
2485:
2474:
2458:
2447:
2444:
2433:
2422:
2414:
2411:
2403:
2392:
2377:
2376:
2375:
2372:
2370:
2346:
2330:
2319:
2303:
2292:
2281:
2268:
2264:
2260:
2256:
2252:
2219:
2208:
2192:
2181:
2170:
2159:
2147:
2146:
2145:
2143:
2142:cross product
2139:
2129:
2127:
2123:
2119:
2115:
2111:
2101:
2099:
2098:parallelogram
2096:matching the
2094:
2090:
2084:
2080:
2074:
2070:
2065:
2061:
2057:
2053:
2049:
2045:
2040:
2036:
2030:
2026:
2021:
1988:
1977:
1964:
1951:
1943:
1926:
1925:
1924:
1902:
1894:
1882:
1881:
1880:
1878:
1874:
1870:
1865:
1863:
1859:
1855:
1851:
1842:
1833:
1831:
1826:
1802:
1791:
1780:
1772:
1769:
1755:
1741:
1726:
1725:
1724:
1722:
1718:
1699:
1688:
1677:
1669:
1655:
1641:
1626:
1625:
1624:
1622:
1618:
1613:
1610:
1606:
1601:
1599:
1595:
1591:
1587:
1554:
1546:
1535:
1532:
1529:
1474:
1463:
1447:
1443:
1440:
1422:
1411:
1396:
1381:
1364:
1349:
1334:
1317:
1302:
1287:
1276:
1268:
1257:
1246:
1223:
1212:
1193:
1189:
1165:
1154:
1143:
1132:
1112:
1101:
1086:
1085:
1083:
1067:
1064:
1053:
1042:
1034:
1023:
1012:
1004:
993:
982:
970:
969:
967:
964:
960:
956:
952:
948:
932:
927:
899:
891:
886:
878:
874:
866:
862:
854:
850:
840:
836:
828:
824:
816:
812:
802:
798:
790:
786:
778:
774:
767:
759:
754:
746:
742:
734:
730:
722:
718:
708:
704:
696:
692:
684:
680:
670:
666:
658:
654:
646:
642:
635:
627:
616:
605:
593:
592:
590:
576:
572:
544:
533:
525:
522:
520:
504:
493:
485:
482:
480:
464:
453:
445:
442:
440:
427:
416:
400:
399:
397:
393:
389:
368:
357:
346:
335:
324:
312:
311:
309:
285:
274:
266:
255:
244:
236:
225:
214:
202:
201:
199:
195:
191:
187:
183:
182:
176:
174:
170:
143:
132:
120:
119:
118:
110:
108:
107:cross product
104:
100:
96:
92:
91:mixed product
88:
79:
70:
68:
64:
60:
56:
52:
48:
44:
40:
36:
29:
22:
6702:Vector space
6656:
6434:Vector space
6341:
6319:. Retrieved
6309:
6291:
6272:
6254:
6248:
6228:
6221:
6202:
6189:
6170:
6155:
6145:
6125:
6118:
6081:January 2014
6078:
6074:adding to it
6069:
5705:
5405:
4570:
4555:
4551:
4547:
4545:
4331:
4327:
4323:
4319:
4317:
4202:
3859:
2910:
2861:
2745:
2631:
2513:
2373:
2254:
2250:
2248:
2137:
2135:
2122:pseudotensor
2107:
2092:
2088:
2082:
2078:
2072:
2068:
2063:
2059:
2055:
2051:
2047:
2043:
2038:
2034:
2028:
2024:
2017:
1922:
1876:
1872:
1868:
1866:
1847:
1822:
1716:
1714:
1616:
1614:
1609:pseudovector
1602:
1598:pseudoscalar
1583:
958:
954:
950:
197:
193:
189:
166:
116:
98:
94:
90:
86:
84:
66:
58:
42:
32:
28:Bibliophilia
6682:Multivector
6647:Determinant
6604:Dot product
6449:Linear span
4884:contraction
2118:contraction
2110:volume form
2018:and is the
1586:orientation
575:determinant
103:dot product
95:box product
47:dimensional
6807:Categories
6716:Direct sum
6551:Invertible
6454:Linear map
6334:References
6195:Kiyosi ItĂ´
6167:Kiyosi ItĂ´
2020:Hodge dual
1446:polar sine
179:Properties
6746:Numerical
6509:Transpose
6317:. Wolfram
6022:×
5985:×
5964:⋅
5919:×
5885:×
5847:^
5809:^
5798:⋅
5784:∬
5675:⋅
5654:−
5643:⋅
5589:−
5541:ℓ
5518:ℓ
5509:δ
5494:δ
5490:−
5476:δ
5470:ℓ
5461:δ
5433:×
5422:×
5377:and thus
5325:and thus
5218:ℓ
5206:δ
5142:δ
5118:≠
5078:δ
5042:δ
5015:ℓ
5006:δ
4991:δ
4987:−
4973:δ
4967:ℓ
4958:δ
4946:ℓ
4934:δ
4922:ℓ
4915:ε
4899:ε
4835:ℓ
4812:ℓ
4805:ε
4789:ε
4770:ℓ
4757:ℓ
4750:ε
4724:ε
4699:×
4688:×
4618:ε
4603:×
4592:⋅
4514:⋅
4503:−
4487:⋅
4461:∧
4433:−
4405:∧
4382:∧
4357:−
4282:⋅
4271:−
4252:⋅
4230:×
4219:×
4164:⋅
4153:−
4130:⋅
4094:×
4083:×
4048:⋅
4037:−
4014:⋅
3978:×
3967:×
3924:×
3913:×
3821:⋅
3810:−
3787:⋅
3670:−
3522:−
3408:−
3257:−
3148:−
3106:−
3076:−
3009:×
2998:×
2955:×
2944:×
2888:δ
2882:δ
2873:Δ
2838:∇
2834:⋅
2830:∇
2823:−
2812:⋅
2808:∇
2800:∇
2785:×
2781:∇
2774:×
2770:∇
2752:gradients
2723:×
2712:×
2704:−
2693:×
2682:×
2666:×
2655:×
2601:×
2590:×
2571:×
2560:×
2541:×
2530:×
2486:⋅
2459:⋅
2448:−
2434:×
2423:×
2415:−
2404:×
2393:×
2347:⋅
2331:−
2320:⋅
2293:×
2282:×
2220:⋅
2209:−
2193:⋅
2171:×
2160:×
1989:×
1978:⋅
1952:∧
1944:∧
1903:∧
1895:∧
1862:trivector
1792:×
1781:⋅
1773:−
1756:×
1742:⋅
1689:×
1678:⋅
1656:×
1642:⋅
1536:
1524:‖
1514:‖
1511:‖
1501:‖
1498:‖
1488:‖
1475:×
1464:⋅
1412:⋅
1397:⋅
1382:⋅
1365:⋅
1350:⋅
1335:⋅
1318:⋅
1303:⋅
1288:⋅
1258:⋅
1247:×
1224:⋅
1213:×
1166:×
1155:×
1144:×
1113:×
1102:⋅
1054:×
1043:⋅
1024:×
1013:⋅
994:×
983:⋅
617:×
606:⋅
589:transpose
545:×
534:⋅
526:−
505:×
494:⋅
486:−
465:×
454:⋅
446:−
428:×
417:⋅
369:⋅
358:×
336:×
325:⋅
286:×
275:⋅
256:×
245:⋅
226:×
215:⋅
144:×
133:⋅
6792:Category
6731:Subspace
6726:Quotient
6677:Bivector
6591:Bilinear
6533:Matrices
6408:Glossary
6197:(1993).
6169:(1987).
6093:See also
4447:⌟
4419:⌟
4368:⌟
4336:bivector
2369:mnemonic
2263:mnemonic
1858:bivector
963:coplanar
65:-valued
57:-valued
35:geometry
6403:Outline
6156:Oeuvres
5235:is the
5065:is the
4886:on the
2748:physics
1715:but if
577:of the
392:negates
171:of the
39:algebra
6687:Tensor
6499:Kernel
6429:Vector
6424:Scalar
6321:21 May
6279:
6236:
6209:
6177:
6133:
5197:) and
5033:where
4293:
4263:
2062:, and
1825:scalar
1719:is an
1084:Also:
957:, and
169:volume
63:vector
55:scalar
41:, the
6556:Minor
6541:Block
6479:Basis
6301:(PDF)
6110:Notes
5171:when
5107:when
2907:Proof
2267:below
2253:, or
2126:below
1723:then
1623:then
1619:is a
97:, or
6711:Dual
6566:Rank
6323:2014
6277:ISBN
6234:ISBN
6207:ISBN
6175:ISBN
6164:and
6131:ISBN
5279:and
5133:and
4672:and
4334:, a
3880:and
2911:The
2136:The
2086:and
2032:or
1875:and
1852:and
1533:psin
1190:The
961:are
85:The
37:and
6076:.
4571:In
4558:).
4550:Ă (
1848:In
1615:If
1272:det
895:det
763:det
631:det
591:):
200:):
33:In
6809::
6263:^
6201:.
5775::
5403:.
4890:,
4579::
4554:Ă
4322:Ă
2903:.
2128:.
2091:â§
2081:â§
2076:,
2071:â§
2058:,
2050:â§
2046:â§
2037:â§
2027:â§
1871:,
1832:.
953:,
196:,
192:,
93:,
69:.
6384:e
6377:t
6370:v
6325:.
6285:.
6242:.
6215:.
6183:.
6139:.
6083:)
6079:(
6038:|
6032:v
6027:r
6017:u
6012:r
6006:|
6000:)
5995:v
5990:r
5980:u
5975:r
5970:(
5960:F
5935:|
5929:v
5924:r
5914:u
5909:r
5903:|
5895:v
5890:r
5880:u
5875:r
5843:n
5819:S
5816:d
5805:n
5794:F
5788:S
5763:)
5760:v
5757:,
5754:u
5751:(
5747:r
5743:=
5740:S
5719:F
5687:.
5683:)
5679:b
5671:a
5667:(
5662:i
5658:c
5651:)
5647:c
5639:a
5635:(
5630:i
5626:b
5622:=
5617:i
5613:c
5607:j
5603:b
5597:j
5593:a
5584:j
5580:c
5574:i
5570:b
5564:j
5560:a
5556:=
5551:m
5547:c
5537:b
5531:j
5527:a
5523:)
5513:j
5503:m
5498:i
5485:m
5480:j
5465:i
5457:(
5454:=
5449:i
5445:)
5441:]
5437:c
5429:b
5425:[
5418:a
5414:(
5391:j
5388:=
5385:l
5365:m
5362:=
5359:i
5339:m
5336:=
5333:j
5313:l
5310:=
5307:i
5287:j
5267:i
5247:k
5221:m
5213:j
5210:i
5185:j
5182:=
5179:i
5159:1
5156:=
5151:i
5146:j
5121:j
5115:i
5095:0
5092:=
5087:i
5082:j
5069:(
5051:i
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5021:,
5010:j
5000:m
4995:i
4982:m
4977:j
4962:i
4954:=
4949:m
4941:j
4938:i
4930:=
4925:m
4919:k
4909:k
4906:j
4903:i
4870:i
4850:,
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4831:b
4825:j
4821:a
4815:m
4809:k
4799:k
4796:j
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4785:=
4780:m
4776:c
4766:b
4760:m
4754:k
4744:j
4740:a
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4720:=
4715:i
4711:)
4707:]
4703:c
4695:b
4691:[
4684:a
4680:(
4658:k
4654:c
4648:j
4644:b
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4628:k
4625:j
4622:i
4614:=
4611:]
4607:c
4599:b
4595:[
4588:a
4556:c
4552:b
4548:a
4526:c
4522:)
4518:b
4510:a
4506:(
4499:b
4495:)
4491:c
4483:a
4479:(
4476:=
4465:c
4458:)
4454:b
4440:a
4436:(
4430:)
4426:c
4412:a
4408:(
4401:b
4397:=
4390:)
4386:c
4378:b
4374:(
4361:a
4332:c
4330:â§
4328:b
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4320:b
4297:w
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4286:v
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4274:(
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4260:)
4256:w
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4241:=
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4234:w
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3994:y
3990:)
3986:)
3982:w
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3963:u
3959:(
3932:)
3928:w
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3805:x
3800:v
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3791:w
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3779:(
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3761:z
3756:v
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3744:u
3739:+
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3645:u
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3635:y
3630:w
3623:y
3618:u
3613:+
3608:x
3603:w
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3591:u
3586:(
3581:x
3576:v
3571:=
3561:)
3556:x
3551:w
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3500:v
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3477:)
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3405:)
3400:z
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3388:z
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3351:(
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3336:=
3326:)
3321:z
3316:v
3309:z
3304:u
3299:+
3294:y
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3282:y
3277:u
3272:(
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3249:z
3244:w
3237:z
3232:u
3227:+
3222:y
3217:w
3210:y
3205:u
3200:(
3195:x
3190:v
3185:=
3175:)
3170:z
3165:w
3158:x
3153:v
3143:x
3138:w
3131:z
3126:v
3121:(
3116:z
3111:u
3103:)
3098:x
3093:w
3086:y
3081:v
3071:y
3066:w
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3049:(
3044:y
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3034:=
3025:x
3021:)
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3013:w
3005:v
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2994:u
2990:(
2963:)
2959:w
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2947:(
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2891:d
2885:+
2879:d
2876:=
2846:A
2842:)
2826:(
2820:)
2816:A
2804:(
2796:=
2793:)
2789:A
2777:(
2731:)
2727:c
2719:a
2715:(
2708:b
2701:)
2697:c
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2685:(
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2674:=
2670:c
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2659:b
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2647:(
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2609:)
2605:b
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2593:(
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2451:(
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2430:a
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2397:b
2389:a
2385:(
2355:)
2351:b
2343:a
2339:(
2335:c
2328:)
2324:c
2316:a
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2308:b
2304:=
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2297:c
2289:b
2285:(
2278:a
2245:.
2232:c
2228:)
2224:b
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2205:b
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2197:c
2189:a
2185:(
2182:=
2179:)
2175:c
2167:b
2163:(
2156:a
2093:c
2089:a
2083:c
2079:b
2073:b
2069:a
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2060:b
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2048:b
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2039:c
2035:b
2029:b
2025:a
2014:,
2001:|
1997:)
1993:c
1985:b
1981:(
1974:a
1969:|
1965:=
1961:|
1956:c
1948:b
1940:a
1935:|
1907:c
1899:b
1891:a
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1873:b
1869:a
1803:.
1800:)
1796:c
1788:b
1784:(
1777:a
1770:=
1767:)
1763:c
1760:T
1752:b
1749:T
1745:(
1738:a
1735:T
1717:T
1700:,
1697:)
1693:c
1685:b
1681:(
1674:a
1670:=
1667:)
1663:c
1660:T
1652:b
1649:T
1645:(
1638:a
1635:T
1617:T
1563:)
1559:c
1555:,
1551:b
1547:,
1543:a
1539:(
1530:=
1519:c
1506:b
1493:a
1483:)
1479:c
1471:b
1467:(
1460:a
1448::
1441:.
1423:]
1416:f
1408:c
1401:e
1393:c
1386:d
1378:c
1369:f
1361:b
1354:e
1346:b
1339:d
1331:b
1322:f
1314:a
1307:e
1299:a
1292:d
1284:a
1277:[
1269:=
1266:)
1262:f
1255:)
1251:e
1243:d
1239:(
1236:(
1232:)
1228:c
1221:)
1217:b
1209:a
1205:(
1202:(
1174:)
1170:c
1162:a
1158:(
1152:)
1148:b
1140:a
1136:(
1133:=
1129:a
1124:)
1121:)
1117:c
1109:b
1105:(
1098:a
1094:(
1068:0
1065:=
1062:)
1058:a
1050:a
1046:(
1039:b
1035:=
1032:)
1028:a
1020:b
1016:(
1009:a
1005:=
1002:)
998:b
990:a
986:(
979:a
959:c
955:b
951:a
933:.
928:]
921:c
914:b
907:a
900:[
892:=
887:]
879:3
875:c
867:3
863:b
855:3
851:a
841:2
837:c
829:2
825:b
817:2
813:a
803:1
799:c
791:1
787:b
779:1
775:a
768:[
760:=
755:]
747:3
743:c
735:2
731:c
723:1
719:c
709:3
705:b
697:2
693:b
685:1
681:b
671:3
667:a
659:2
655:a
647:1
643:a
636:[
628:=
625:)
621:c
613:b
609:(
602:a
584:3
581:Ă
579:3
553:)
549:a
541:b
537:(
530:c
523:=
513:)
509:c
501:a
497:(
490:b
483:=
473:)
469:b
461:c
457:(
450:a
443:=
436:)
432:c
424:b
420:(
413:a
373:c
366:)
362:b
354:a
350:(
347:=
344:)
340:c
332:b
328:(
321:a
294:)
290:b
282:a
278:(
271:c
267:=
264:)
260:a
252:c
248:(
241:b
237:=
234:)
230:c
222:b
218:(
211:a
198:c
194:b
190:a
152:)
148:c
140:b
136:(
129:a
30:.
23:.
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