3866:
2992:
3861:{\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}}}
1446:
1208:
954:
6535:
4209:
4552:
1852:
1441:{\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}}}
607:
3961:
6073:
89:
4359:
578:
6799:
5708:
949:{\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}}.}
4204:{\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}}}
2868:
2520:
4547:{\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}}}
414:
1584:
2638:
5420:
2752:
1089:
315:
4871:
4319:
2376:
2254:
1838:
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
1195:
5042:
2775:
2391:
1462:
573:{\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}}}
6063:
1824:
1721:
2023:
395:
1622:
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.
2531:
5960:
4686:
2653:
5703:{\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} )\,.}
984:
216:
4220:
2283:
2161:
1100:
2863:{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {A} )-({\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }})\mathbf {A} }
4904:
2515:{\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} }
4681:
3953:
2984:
173:
5840:
1929:
321:
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:
1579:{\displaystyle {\frac {\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}{\|{\mathbf {a} }\|\|{\mathbf {b} }\|\|{\mathbf {c} }\|}}=\operatorname {psin} (\mathbf {a} ,\mathbf {b} ,\mathbf {c} )}
5965:
4364:
3966:
2997:
1740:
419:
1640:
1448:
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
1940:
326:
2633:{\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} }
2912:
5871:
5244:
5784:
5180:
5116:
5876:
5074:
2747:{\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} )}
1084:{\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}
5741:
310:{\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} )}
6366:
5142:
1855:
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.)
5412:
5386:
5360:
5334:
5206:
4866:{\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},}
4314:{\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\ \mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\ \mathbf {w} }
5308:
5288:
5268:
4891:
3909:
3889:
2940:
2371:{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )}
2249:{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }
1190:{\displaystyle (\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\,\mathbf {a} =(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}
4593:
1839:
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
5037:{\displaystyle \varepsilon _{ijk}\varepsilon ^{k\ell m}=\delta _{ij}^{\ell m}=\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell }\,,}
6393:
1875:. 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.
6784:
2269:
6248:
3914:
2945:
134:
1896:
5789:
6291:
6221:
6189:
6145:
4349:. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a
6823:
6774:
6736:
6672:
31:
6163:
did not develop the cross product as an algebraic product on vectors, but did use an equivalent form of it in components: see
1595:
Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the
6833:
6058:{\textstyle \mathbf {F} \cdot {\frac {(\mathbf {r} _{u}\times \mathbf {r} _{v})}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}}
2033:
of the scalar triple product. As the exterior product is associative brackets are not needed as it does not matter which of
2874:
1819:{\displaystyle \mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=-\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ).}
1716:{\displaystyle \mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ),}
2018:{\displaystyle |\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} |=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}
6828:
6514:
6386:
390:{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} }
6838:
6619:
6469:
5247:
2385:
Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as:
6524:
6418:
1596:
6764:
6413:
6639:
6756:
6114:
5077:
2879:
2276:"ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is provided
57:
598:
matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as its
2053:
is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector
6802:
6509:
6379:
4557:
The proof follows from the properties of the contraction. The result is the same vector as calculated using
1600:
5845:
5211:
6566:
6499:
6489:
6172:
5955:{\textstyle {\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}}
5746:
6726:
6581:
6576:
6571:
6504:
6449:
6308:
6160:
1604:
5147:
5083:
6591:
6556:
6543:
6434:
6165:
Lagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires".
5047:
1835:
1205:
of two triple products (or the square of a triple product), may be expanded in terms of dot products:
65:
6769:
6649:
6624:
6474:
1603:
of the vectors. This means the product is negated if the orientation is reversed, for example by a
6171:
He may have written a formula similar to the triple product expansion in component form. See also
5724:
6479:
4898:
4894:
4587:
2128:
6209:
1851:
6677:
6634:
6561:
6454:
6287:
6244:
6217:
6185:
6141:
6135:
6109:
4350:
1864:
1731:
406:
6238:
5121:
6682:
6586:
6439:
5718:
2124:
1860:
1449:
402:
61:
4457:
4429:
4378:
6741:
6534:
6494:
6484:
4583:
2766:
2644:
1631:
5391:
5365:
5339:
5313:
5185:
2155:
of one vector with the cross product of the other two. The following relationship holds:
1455:
The ratio of the triple product and the product of the three vector norms is known as a
6746:
6731:
6402:
5293:
5273:
5253:
4876:
3894:
3874:
2925:
1840:
1202:
196:
183:
17:
6325:
6072:
88:
6817:
6779:
6702:
6662:
6629:
6609:
6205:
6177:
2152:
2108:
1615:
117:
6712:
6601:
6551:
6444:
2132:
1619:
1608:
979:
If any two vectors in the scalar triple product are equal, then its value is zero:
73:
38:
6692:
6657:
6614:
6459:
2120:
585:
113:
6240:
Numerical
Modelling of Water Waves: An Introduction to Engineers and Scientists
6721:
6464:
4893:-th component of the resulting vector. This can be simplified by performing a
2030:
1456:
2525:
From
Lagrange's formula it follows that the vector triple product satisfies:
6519:
1872:
976:, since the parallelepiped defined by them would be flat and have no volume.
599:
405:
the triple product. This follows from the circular-shift property and the
6687:
4346:
2762:
2379:
2273:
1868:
973:
45:
30:
This article is about ternary operations on vectors. For other uses, see
2758:
1934:
is a trivector with magnitude equal to the scalar triple product, i.e.
49:
960:
If the scalar triple product is equal to zero, then the three vectors
6697:
2757:
These formulas are very useful in simplifying vector calculations in
179:
4676:{\displaystyle \mathbf {a} \cdot =\varepsilon _{ijk}a^{i}b^{j}c^{k}}
3948:{\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )}
2979:{\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )}
64:. The name "triple product" is used for two different products, the
168:{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}
5835:{\textstyle \iint _{S}\mathbf {F} \cdot {\hat {\mathbf {n} }}\,dS}
2382:"BAC − CAB" is obtained, as in âback of the cabâ.
1924:{\displaystyle \mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} }
1850:
87:
6371:
5250:. We can reason out this identity by recognizing that the index
2873:
This can be also regarded as a special case of the more general
6375:
6367:
Khan
Academy video of the proof of the triple product expansion
6268:. American Elsevier Publishing Company, Inc. pp. 262â263.
6067:
2131:
of vectors with a rank-3 tensor equivalent to the form (or a
6137:
37:"Signed volume" redirects here. For autographed books, see
6309:"Geometric Algebra of One and Many Multivector Variables"
6286:(2nd ed.). Cambridge University Press. p. 46.
2769:
is
Lagrange's formula of vector cross-product identity:
2647:
for the cross product. Another useful formula follows:
584:
The scalar triple product can also be understood as the
6084:
5968:
5879:
5792:
1290:
913:
781:
649:
5848:
5749:
5727:
5423:
5394:
5368:
5342:
5316:
5296:
5276:
5256:
5214:
5188:
5150:
5124:
5086:
5050:
4907:
4879:
4689:
4596:
4362:
4223:
3964:
3917:
3897:
3877:
2995:
2948:
2928:
2882:
2778:
2656:
2534:
2394:
2286:
2272:. Its right hand side can be remembered by using the
2164:
1943:
1899:
1743:
1643:
1465:
1211:
1103:
987:
610:
417:
329:
219:
137:
2123:
of the
Euclidean 3-space applied to the vectors via
6755:
6711:
6648:
6600:
6542:
6427:
4337:of vectors is expressed as their exterior product
1871:, while the exterior product of three vectors is a
6057:
5954:
5865:
5834:
5778:
5735:
5702:
5406:
5380:
5354:
5328:
5302:
5282:
5262:
5238:
5200:
5174:
5136:
5110:
5068:
5036:
4885:
4865:
4675:
4546:
4313:
4203:
3947:
3903:
3883:
3860:
2978:
2934:
2906:
2862:
2746:
2632:
2514:
2370:
2248:
2017:
1923:
1818:
1715:
1578:
1440:
1189:
1083:
948:
572:
389:
309:
167:
6355:. McGraw-Hill Book Company, Inc. pp. 23â25.
1282:
905:
773:
641:
6266:Mathematical Methods in Science and Engineering
4329:If geometric algebra is used the cross product
4214:By combining these three components we obtain:
195:The scalar triple product is unchanged under a
6387:
2065:corresponds to the parallelepiped spanned by
8:
4586:, the triple product is expressed using the
2268:, although the latter name is also used for
1534:
1524:
1521:
1511:
1508:
1498:
6394:
6380:
6372:
6277:
6275:
5743:across the parametrically-defined surface
4462:
4454:
4434:
4426:
4383:
4375:
2135:equivalent to the volume pseudoform); see
1245:
6216:(2nd ed.). MIT Press. p. 1679.
6047:
6041:
6036:
6026:
6021:
6015:
6004:
5999:
5989:
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5977:
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5889:
5884:
5880:
5878:
5852:
5850:
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5825:
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5811:
5803:
5797:
5791:
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5748:
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5696:
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5671:
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5522:
5512:
5507:
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5489:
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5474:
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5422:
5393:
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5341:
5315:
5295:
5275:
5255:
5227:
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5213:
5187:
5160:
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5149:
5123:
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5091:
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5060:
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5030:
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5019:
5009:
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4991:
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4608:
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3034:
3022:
3014:
3003:
2996:
2994:
2968:
2960:
2949:
2947:
2927:
2907:{\displaystyle \Delta =d\delta +\delta d}
2881:
2855:
2847:
2839:
2825:
2817:
2809:
2798:
2790:
2779:
2777:
2736:
2728:
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2507:
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2233:
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2214:
2206:
2198:
2184:
2176:
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2163:
2010:
2002:
1994:
1983:
1978:
1970:
1965:
1957:
1949:
1944:
1942:
1916:
1908:
1900:
1898:
1867:the exterior product of two vectors is a
1805:
1797:
1786:
1769:
1758:
1744:
1742:
1702:
1694:
1683:
1669:
1658:
1644:
1642:
1607:, and so is more properly described as a
1568:
1560:
1552:
1528:
1527:
1515:
1514:
1502:
1501:
1488:
1480:
1469:
1466:
1464:
1425:
1417:
1410:
1402:
1395:
1387:
1378:
1370:
1363:
1355:
1348:
1340:
1331:
1323:
1316:
1308:
1301:
1293:
1285:
1271:
1260:
1252:
1237:
1226:
1218:
1210:
1179:
1171:
1157:
1149:
1138:
1137:
1126:
1118:
1107:
1102:
1067:
1059:
1048:
1037:
1029:
1018:
1007:
999:
988:
986:
930:
923:
916:
908:
888:
876:
864:
850:
838:
826:
812:
800:
788:
776:
756:
744:
732:
718:
706:
694:
680:
668:
656:
644:
630:
622:
611:
609:
558:
550:
539:
518:
510:
499:
478:
470:
459:
441:
433:
422:
418:
416:
382:
371:
363:
349:
341:
330:
328:
299:
291:
280:
269:
261:
250:
239:
231:
220:
218:
157:
149:
138:
136:
128:Geometrically, the scalar triple product
6140:. Oxford University Press. p. 215.
6126:
5417:Returning to the triple cross product,
5362:. Likewise, in the second term, we fix
2280:. Some textbooks write the identity as
2119:The triple product is identical to the
401:Swapping any two of the three operands
92:Three vectors defining a parallelepiped
6785:Comparison of linear algebra libraries
6214:Encyclopedic dictionary of mathematics
6182:Encyclopedic Dictionary of Mathematics
5866:{\displaystyle {\hat {\mathbf {n} }}}
5239:{\displaystyle \delta _{ij}^{\ell m}}
7:
5248:generalized Kronecker delta function
4353:can be used, so the formula becomes
2136:
1618:; the cross product transforms as a
186:defined by the three vectors given.
5779:{\displaystyle S=\mathbf {r} (u,v)}
2883:
25:
5175:{\displaystyle \delta _{j}^{i}=1}
5111:{\displaystyle \delta _{j}^{i}=0}
6798:
6797:
6775:Basic Linear Algebra Subprograms
6533:
6071:
6037:
6022:
6000:
5985:
5970:
5934:
5919:
5900:
5885:
5853:
5815:
5804:
5757:
5729:
5689:
5681:
5657:
5649:
5447:
5439:
5428:
5270:will be summed out leaving only
4713:
4705:
4694:
4617:
4609:
4598:
4536:
4528:
4520:
4509:
4501:
4493:
4475:
4464:
4450:
4436:
4422:
4411:
4396:
4388:
4371:
4307:
4296:
4288:
4277:
4266:
4258:
4244:
4236:
4225:
4187:
4178:
4170:
4153:
4144:
4136:
4108:
4100:
4089:
4071:
4062:
4054:
4037:
4028:
4020:
3992:
3984:
3973:
3938:
3930:
3919:
3844:
3835:
3827:
3810:
3801:
3793:
3766:
3754:
3739:
3727:
3712:
3700:
3685:
3667:
3655:
3640:
3628:
3613:
3601:
3586:
3561:
3549:
3537:
3522:
3510:
3498:
3477:
3465:
3450:
3438:
3423:
3405:
3393:
3378:
3366:
3351:
3326:
3314:
3299:
3287:
3272:
3254:
3242:
3227:
3215:
3200:
3175:
3163:
3148:
3136:
3121:
3103:
3091:
3076:
3064:
3049:
3023:
3015:
3004:
2969:
2961:
2950:
2856:
2848:
2840:
2826:
2818:
2810:
2799:
2791:
2780:
2737:
2729:
2718:
2707:
2699:
2688:
2680:
2669:
2661:
2626:
2615:
2607:
2596:
2585:
2577:
2566:
2555:
2547:
2536:
2508:
2500:
2492:
2481:
2473:
2465:
2448:
2440:
2429:
2418:
2407:
2399:
2361:
2353:
2345:
2334:
2326:
2318:
2307:
2299:
2288:
2242:
2234:
2226:
2215:
2207:
2199:
2185:
2177:
2166:
2127:. It also can be expressed as a
2003:
1995:
1984:
1966:
1958:
1950:
1917:
1909:
1901:
1806:
1798:
1787:
1773:
1770:
1762:
1759:
1748:
1745:
1703:
1695:
1684:
1673:
1670:
1662:
1659:
1648:
1645:
1569:
1561:
1553:
1529:
1516:
1503:
1489:
1481:
1470:
1426:
1418:
1411:
1403:
1396:
1388:
1379:
1371:
1364:
1356:
1349:
1341:
1332:
1324:
1317:
1309:
1302:
1294:
1272:
1261:
1253:
1238:
1227:
1219:
1180:
1172:
1158:
1150:
1139:
1127:
1119:
1108:
1068:
1060:
1049:
1038:
1030:
1019:
1008:
1000:
989:
931:
924:
917:
631:
623:
612:
559:
551:
540:
519:
511:
500:
479:
471:
460:
442:
434:
423:
383:
372:
364:
350:
342:
331:
300:
292:
281:
270:
262:
251:
240:
232:
221:
158:
150:
139:
6673:Seven-dimensional cross product
5069:{\displaystyle \delta _{j}^{i}}
2761:. A related identity regarding
1616:handedness of the cross product
1611:if the orientation can change.
116:of one of the vectors with the
32:Triple product (disambiguation)
6048:
6016:
6010:
5980:
5945:
5913:
5857:
5819:
5773:
5761:
5693:
5677:
5661:
5645:
5533:
5467:
5455:
5451:
5435:
5424:
4721:
4717:
4701:
4690:
4621:
4605:
4532:
4516:
4505:
4489:
4468:
4446:
4440:
4418:
4400:
4384:
4300:
4284:
4270:
4254:
4248:
4232:
4182:
4166:
4148:
4132:
4116:
4112:
4096:
4085:
4066:
4050:
4032:
4016:
4000:
3996:
3980:
3969:
3942:
3926:
3839:
3823:
3805:
3789:
3776:
3695:
3677:
3596:
3571:
3493:
3487:
3433:
3415:
3361:
3336:
3282:
3264:
3210:
3185:
3131:
3113:
3059:
3031:
3027:
3011:
3000:
2973:
2957:
2852:
2836:
2830:
2814:
2803:
2787:
2741:
2725:
2711:
2695:
2673:
2657:
2619:
2603:
2589:
2573:
2559:
2543:
2504:
2488:
2477:
2461:
2452:
2436:
2411:
2395:
2365:
2349:
2338:
2322:
2311:
2295:
2238:
2222:
2211:
2195:
2189:
2173:
2011:
2007:
1991:
1979:
1971:
1945:
1810:
1794:
1777:
1755:
1707:
1691:
1677:
1655:
1586:which ranges between â1 and 1.
1573:
1549:
1493:
1477:
1276:
1265:
1249:
1246:
1242:
1231:
1215:
1212:
1184:
1168:
1162:
1146:
1134:
1131:
1115:
1104:
1072:
1056:
1042:
1026:
1012:
996:
635:
619:
563:
547:
523:
507:
483:
467:
446:
430:
376:
360:
354:
338:
304:
288:
274:
258:
244:
228:
162:
146:
1:
6284:Clifford algebras and spinors
2111:faces of the parallelepiped.
6515:Eigenvalues and eigenvectors
6065:is a scalar triple product.
5736:{\displaystyle \mathbf {F} }
5310:. In the first term, we fix
27:Ternary operation on vectors
6184:. MIT Press. p. 1679.
5873:to the surface is given by
6855:
6353:Vector and Tensor Analysis
2378:such that a more familiar
2277:
36:
29:
6793:
6531:
6409:
6243:. Routledge. p. 13.
5842:. The unit normal vector
1614:This also relates to the
1601:parity of the permutation
6282:Pertti Lounesto (2001).
6115:Vector algebra relations
5078:Kronecker delta function
2875:Laplaceâde Rham operator
2262:triple product expansion
1830:Scalar or scalar density
124:Geometric interpretation
56:is a product of three 3-
6824:Mathematical identities
5137:{\displaystyle i\neq j}
4325:Using geometric algebra
2115:As a trilinear function
199:of its three operands (
18:Vector Laplacian/Proofs
6500:Row and column vectors
6134:Wong, Chun Wa (2013).
6059:
5956:
5867:
5836:
5780:
5737:
5704:
5408:
5382:
5356:
5330:
5304:
5284:
5264:
5240:
5202:
5176:
5138:
5112:
5070:
5038:
4887:
4867:
4677:
4548:
4315:
4205:
3949:
3905:
3885:
3862:
2980:
2936:
2908:
2864:
2748:
2634:
2516:
2372:
2270:several other formulas
2250:
2019:
1925:
1856:
1847:As an exterior product
1820:
1717:
1591:Scalar or pseudoscalar
1580:
1442:
1191:
1085:
950:
574:
409:of the cross product:
391:
311:
169:
93:
6834:Operations on vectors
6505:Row and column spaces
6450:Scalar multiplication
6161:Joseph Louis Lagrange
6060:
5957:
5868:
5837:
5781:
5738:
5705:
5409:
5383:
5357:
5331:
5305:
5285:
5265:
5241:
5203:
5177:
5139:
5113:
5071:
5039:
4888:
4868:
4678:
4549:
4316:
4206:
3950:
3906:
3886:
3863:
2981:
2937:
2909:
2865:
2749:
2635:
2517:
2373:
2251:
2149:vector triple product
2143:Vector triple product
2020:
1926:
1854:
1834:Strictly speaking, a
1821:
1718:
1605:parity transformation
1581:
1443:
1192:
1086:
951:
575:
392:
312:
170:
110:triple scalar product
98:scalar triple product
91:
84:Scalar triple product
78:vector triple product
72:and, less often, the
70:scalar triple product
6640:GramâSchmidt process
6592:Gaussian elimination
6351:Lass, Harry (1950).
6326:"Permutation Tensor"
6237:Pengzhi Lin (2008).
6210:"§C: Vector product"
5966:
5877:
5846:
5790:
5747:
5725:
5721:of the vector field
5421:
5392:
5366:
5340:
5314:
5294:
5274:
5254:
5212:
5186:
5148:
5122:
5084:
5048:
4905:
4877:
4687:
4594:
4360:
4221:
3962:
3915:
3895:
3875:
2993:
2946:
2926:
2880:
2776:
2654:
2532:
2392:
2284:
2162:
1941:
1897:
1741:
1641:
1599:of the frame or the
1463:
1209:
1101:
985:
608:
415:
327:
217:
135:
112:) is defined as the
6829:Multilinear algebra
6770:Numerical stability
6650:Multilinear algebra
6625:Inner product space
6475:Linear independence
6264:J. Heading (1970).
6173:Lagrange's identity
5962:, so the integrand
5532:
5517:
5499:
5484:
5407:{\displaystyle l=j}
5381:{\displaystyle i=m}
5355:{\displaystyle j=m}
5329:{\displaystyle i=l}
5235:
5201:{\displaystyle i=j}
5165:
5101:
5065:
5029:
5014:
4996:
4981:
4963:
4899:Levi-Civita symbols
6839:Ternary operations
6480:Linear combination
6083:. You can help by
6055:
5952:
5863:
5832:
5776:
5733:
5700:
5518:
5503:
5485:
5470:
5404:
5378:
5352:
5326:
5300:
5280:
5260:
5236:
5215:
5198:
5172:
5151:
5134:
5108:
5087:
5066:
5051:
5034:
5015:
5000:
4982:
4967:
4943:
4883:
4863:
4673:
4588:Levi-Civita symbol
4544:
4542:
4311:
4201:
4199:
3945:
3901:
3881:
3858:
3856:
2976:
2932:
2904:
2860:
2744:
2630:
2512:
2368:
2266:Lagrange's formula
2246:
2151:is defined as the
2015:
1921:
1857:
1816:
1713:
1576:
1438:
1432:
1187:
1081:
946:
937:
896:
764:
570:
568:
387:
307:
165:
120:of the other two.
94:
6811:
6810:
6678:Geometric algebra
6635:Kronecker product
6470:Linear projection
6455:Vector projection
6250:978-0-415-41578-1
6110:Quadruple product
6101:
6100:
6053:
5950:
5860:
5822:
5303:{\displaystyle j}
5283:{\displaystyle i}
5263:{\displaystyle k}
4886:{\displaystyle i}
4873:referring to the
4305:
4275:
3904:{\displaystyle z}
3884:{\displaystyle y}
2935:{\displaystyle x}
2260:This is known as
2077:, with bivectors
1865:geometric algebra
1732:improper rotation
1538:
407:anticommutativity
100:(also called the
62:Euclidean vectors
60:vectors, usually
16:(Redirected from
6846:
6801:
6800:
6683:Exterior algebra
6620:Hadamard product
6537:
6525:Linear equations
6396:
6389:
6382:
6373:
6356:
6338:
6337:
6335:
6333:
6322:
6316:
6315:
6313:
6304:
6298:
6297:
6279:
6270:
6269:
6261:
6255:
6254:
6234:
6228:
6227:
6202:
6196:
6195:
6170:
6158:
6152:
6151:
6131:
6096:
6093:
6075:
6068:
6064:
6062:
6061:
6056:
6054:
6052:
6051:
6046:
6045:
6040:
6031:
6030:
6025:
6019:
6013:
6009:
6008:
6003:
5994:
5993:
5988:
5978:
5973:
5961:
5959:
5958:
5953:
5951:
5949:
5948:
5943:
5942:
5937:
5928:
5927:
5922:
5916:
5910:
5909:
5908:
5903:
5894:
5893:
5888:
5881:
5872:
5870:
5869:
5864:
5862:
5861:
5856:
5851:
5841:
5839:
5838:
5833:
5824:
5823:
5818:
5813:
5807:
5802:
5801:
5785:
5783:
5782:
5777:
5760:
5742:
5740:
5739:
5734:
5732:
5709:
5707:
5706:
5701:
5692:
5684:
5676:
5675:
5660:
5652:
5644:
5643:
5631:
5630:
5621:
5620:
5611:
5610:
5598:
5597:
5588:
5587:
5578:
5577:
5565:
5564:
5555:
5554:
5545:
5544:
5531:
5526:
5516:
5511:
5498:
5493:
5483:
5478:
5463:
5462:
5450:
5442:
5431:
5413:
5411:
5410:
5405:
5387:
5385:
5384:
5379:
5361:
5359:
5358:
5353:
5335:
5333:
5332:
5327:
5309:
5307:
5306:
5301:
5289:
5287:
5286:
5281:
5269:
5267:
5266:
5261:
5245:
5243:
5242:
5237:
5234:
5226:
5207:
5205:
5204:
5199:
5181:
5179:
5178:
5173:
5164:
5159:
5143:
5141:
5140:
5135:
5117:
5115:
5114:
5109:
5100:
5095:
5075:
5073:
5072:
5067:
5064:
5059:
5043:
5041:
5040:
5035:
5028:
5023:
5013:
5008:
4995:
4990:
4980:
4975:
4962:
4954:
4939:
4938:
4923:
4922:
4892:
4890:
4889:
4884:
4872:
4870:
4869:
4864:
4859:
4858:
4849:
4848:
4839:
4838:
4829:
4828:
4813:
4812:
4794:
4793:
4784:
4783:
4774:
4773:
4758:
4757:
4748:
4747:
4729:
4728:
4716:
4708:
4697:
4682:
4680:
4679:
4674:
4672:
4671:
4662:
4661:
4652:
4651:
4642:
4641:
4620:
4612:
4601:
4553:
4551:
4550:
4545:
4543:
4539:
4531:
4523:
4512:
4504:
4496:
4482:
4478:
4467:
4461:
4460:
4453:
4439:
4433:
4432:
4425:
4414:
4399:
4391:
4382:
4381:
4374:
4351:left contraction
4320:
4318:
4317:
4312:
4310:
4303:
4299:
4291:
4280:
4273:
4269:
4261:
4247:
4239:
4228:
4210:
4208:
4207:
4202:
4200:
4196:
4195:
4190:
4181:
4173:
4162:
4161:
4156:
4147:
4139:
4124:
4123:
4111:
4103:
4092:
4080:
4079:
4074:
4065:
4057:
4046:
4045:
4040:
4031:
4023:
4008:
4007:
3995:
3987:
3976:
3954:
3952:
3951:
3946:
3941:
3933:
3922:
3910:
3908:
3907:
3902:
3890:
3888:
3887:
3882:
3867:
3865:
3864:
3859:
3857:
3853:
3852:
3847:
3838:
3830:
3819:
3818:
3813:
3804:
3796:
3782:
3775:
3774:
3769:
3763:
3762:
3757:
3748:
3747:
3742:
3736:
3735:
3730:
3721:
3720:
3715:
3709:
3708:
3703:
3694:
3693:
3688:
3676:
3675:
3670:
3664:
3663:
3658:
3649:
3648:
3643:
3637:
3636:
3631:
3622:
3621:
3616:
3610:
3609:
3604:
3595:
3594:
3589:
3577:
3570:
3569:
3564:
3558:
3557:
3552:
3546:
3545:
3540:
3531:
3530:
3525:
3519:
3518:
3513:
3507:
3506:
3501:
3486:
3485:
3480:
3474:
3473:
3468:
3459:
3458:
3453:
3447:
3446:
3441:
3432:
3431:
3426:
3414:
3413:
3408:
3402:
3401:
3396:
3387:
3386:
3381:
3375:
3374:
3369:
3360:
3359:
3354:
3342:
3335:
3334:
3329:
3323:
3322:
3317:
3308:
3307:
3302:
3296:
3295:
3290:
3281:
3280:
3275:
3263:
3262:
3257:
3251:
3250:
3245:
3236:
3235:
3230:
3224:
3223:
3218:
3209:
3208:
3203:
3191:
3184:
3183:
3178:
3172:
3171:
3166:
3157:
3156:
3151:
3145:
3144:
3139:
3130:
3129:
3124:
3112:
3111:
3106:
3100:
3099:
3094:
3085:
3084:
3079:
3073:
3072:
3067:
3058:
3057:
3052:
3039:
3038:
3026:
3018:
3007:
2985:
2983:
2982:
2977:
2972:
2964:
2953:
2941:
2939:
2938:
2933:
2913:
2911:
2910:
2905:
2869:
2867:
2866:
2861:
2859:
2851:
2843:
2829:
2821:
2813:
2802:
2794:
2783:
2753:
2751:
2750:
2745:
2740:
2732:
2721:
2710:
2702:
2691:
2683:
2672:
2664:
2639:
2637:
2636:
2631:
2629:
2618:
2610:
2599:
2588:
2580:
2569:
2558:
2550:
2539:
2521:
2519:
2518:
2513:
2511:
2503:
2495:
2484:
2476:
2468:
2451:
2443:
2432:
2421:
2410:
2402:
2377:
2375:
2374:
2369:
2364:
2356:
2348:
2337:
2329:
2321:
2310:
2302:
2291:
2255:
2253:
2252:
2247:
2245:
2237:
2229:
2218:
2210:
2202:
2188:
2180:
2169:
2125:interior product
2106:
2096:
2086:
2052:
2042:
2024:
2022:
2021:
2016:
2014:
2006:
1998:
1987:
1982:
1974:
1969:
1961:
1953:
1948:
1930:
1928:
1927:
1922:
1920:
1912:
1904:
1861:exterior algebra
1825:
1823:
1822:
1817:
1809:
1801:
1790:
1776:
1765:
1751:
1722:
1720:
1719:
1714:
1706:
1698:
1687:
1676:
1665:
1651:
1585:
1583:
1582:
1577:
1572:
1564:
1556:
1539:
1537:
1533:
1532:
1520:
1519:
1507:
1506:
1496:
1492:
1484:
1473:
1467:
1450:Gram determinant
1447:
1445:
1444:
1439:
1437:
1436:
1429:
1421:
1414:
1406:
1399:
1391:
1382:
1374:
1367:
1359:
1352:
1344:
1335:
1327:
1320:
1312:
1305:
1297:
1275:
1264:
1256:
1241:
1230:
1222:
1196:
1194:
1193:
1188:
1183:
1175:
1161:
1153:
1142:
1130:
1122:
1111:
1090:
1088:
1087:
1082:
1071:
1063:
1052:
1041:
1033:
1022:
1011:
1003:
992:
955:
953:
952:
947:
942:
941:
934:
927:
920:
901:
900:
893:
892:
881:
880:
869:
868:
855:
854:
843:
842:
831:
830:
817:
816:
805:
804:
793:
792:
769:
768:
761:
760:
749:
748:
737:
736:
723:
722:
711:
710:
699:
698:
685:
684:
673:
672:
661:
660:
634:
626:
615:
597:
596:
593:
579:
577:
576:
571:
569:
562:
554:
543:
529:
522:
514:
503:
489:
482:
474:
463:
445:
437:
426:
396:
394:
393:
388:
386:
375:
367:
353:
345:
334:
316:
314:
313:
308:
303:
295:
284:
273:
265:
254:
243:
235:
224:
178:is the (signed)
174:
172:
171:
166:
161:
153:
142:
21:
6854:
6853:
6849:
6848:
6847:
6845:
6844:
6843:
6814:
6813:
6812:
6807:
6789:
6751:
6707:
6644:
6596:
6538:
6529:
6495:Change of basis
6485:Multilinear map
6423:
6405:
6400:
6363:
6350:
6347:
6342:
6341:
6331:
6329:
6324:
6323:
6319:
6311:
6307:Janne Pesonen.
6306:
6305:
6301:
6294:
6281:
6280:
6273:
6263:
6262:
6258:
6251:
6236:
6235:
6231:
6224:
6204:
6203:
6199:
6192:
6176:
6164:
6159:
6155:
6148:
6133:
6132:
6128:
6123:
6106:
6097:
6091:
6088:
6081:needs expansion
6035:
6020:
6014:
5998:
5983:
5979:
5964:
5963:
5932:
5917:
5911:
5898:
5883:
5882:
5875:
5874:
5844:
5843:
5793:
5788:
5787:
5745:
5744:
5723:
5722:
5715:
5713:Vector calculus
5667:
5635:
5622:
5612:
5602:
5589:
5579:
5569:
5556:
5546:
5536:
5454:
5419:
5418:
5390:
5389:
5364:
5363:
5338:
5337:
5312:
5311:
5292:
5291:
5272:
5271:
5252:
5251:
5210:
5209:
5184:
5183:
5146:
5145:
5120:
5119:
5082:
5081:
5046:
5045:
4924:
4908:
4903:
4902:
4875:
4874:
4850:
4840:
4830:
4814:
4798:
4785:
4775:
4759:
4749:
4733:
4720:
4685:
4684:
4663:
4653:
4643:
4627:
4592:
4591:
4584:tensor notation
4580:
4578:Tensor calculus
4575:
4573:Interpretations
4541:
4540:
4480:
4479:
4403:
4358:
4357:
4327:
4219:
4218:
4198:
4197:
4185:
4151:
4125:
4115:
4082:
4081:
4069:
4035:
4009:
3999:
3960:
3959:
3913:
3912:
3911:components of
3893:
3892:
3873:
3872:
3871:Similarly, the
3855:
3854:
3842:
3808:
3780:
3779:
3764:
3752:
3737:
3725:
3710:
3698:
3683:
3665:
3653:
3638:
3626:
3611:
3599:
3584:
3575:
3574:
3559:
3547:
3535:
3520:
3508:
3496:
3475:
3463:
3448:
3436:
3421:
3403:
3391:
3376:
3364:
3349:
3340:
3339:
3324:
3312:
3297:
3285:
3270:
3252:
3240:
3225:
3213:
3198:
3189:
3188:
3173:
3161:
3146:
3134:
3119:
3101:
3089:
3074:
3062:
3047:
3040:
3030:
2991:
2990:
2944:
2943:
2924:
2923:
2920:
2878:
2877:
2774:
2773:
2767:vector calculus
2652:
2651:
2645:Jacobi identity
2530:
2529:
2390:
2389:
2282:
2281:
2160:
2159:
2145:
2117:
2098:
2088:
2078:
2044:
2034:
1939:
1938:
1895:
1894:
1849:
1832:
1739:
1738:
1639:
1638:
1632:proper rotation
1593:
1497:
1468:
1461:
1460:
1431:
1430:
1415:
1400:
1384:
1383:
1368:
1353:
1337:
1336:
1321:
1306:
1286:
1207:
1206:
1099:
1098:
983:
982:
936:
935:
928:
921:
909:
895:
894:
884:
882:
872:
870:
860:
857:
856:
846:
844:
834:
832:
822:
819:
818:
808:
806:
796:
794:
784:
777:
763:
762:
752:
750:
740:
738:
728:
725:
724:
714:
712:
702:
700:
690:
687:
686:
676:
674:
664:
662:
652:
645:
606:
605:
594:
591:
589:
567:
566:
527:
526:
487:
486:
449:
413:
412:
325:
324:
215:
214:
192:
133:
132:
126:
86:
42:
35:
28:
23:
22:
15:
12:
11:
5:
6852:
6850:
6842:
6841:
6836:
6831:
6826:
6816:
6815:
6809:
6808:
6806:
6805:
6794:
6791:
6790:
6788:
6787:
6782:
6777:
6772:
6767:
6765:Floating-point
6761:
6759:
6753:
6752:
6750:
6749:
6747:Tensor product
6744:
6739:
6734:
6732:Function space
6729:
6724:
6718:
6716:
6709:
6708:
6706:
6705:
6700:
6695:
6690:
6685:
6680:
6675:
6670:
6668:Triple product
6665:
6660:
6654:
6652:
6646:
6645:
6643:
6642:
6637:
6632:
6627:
6622:
6617:
6612:
6606:
6604:
6598:
6597:
6595:
6594:
6589:
6584:
6582:Transformation
6579:
6574:
6572:Multiplication
6569:
6564:
6559:
6554:
6548:
6546:
6540:
6539:
6532:
6530:
6528:
6527:
6522:
6517:
6512:
6507:
6502:
6497:
6492:
6487:
6482:
6477:
6472:
6467:
6462:
6457:
6452:
6447:
6442:
6437:
6431:
6429:
6428:Basic concepts
6425:
6424:
6422:
6421:
6416:
6410:
6407:
6406:
6403:Linear algebra
6401:
6399:
6398:
6391:
6384:
6376:
6370:
6369:
6362:
6361:External links
6359:
6358:
6357:
6346:
6343:
6340:
6339:
6317:
6299:
6292:
6271:
6256:
6249:
6229:
6222:
6197:
6190:
6169:. Vol. 3.
6153:
6146:
6125:
6124:
6122:
6119:
6118:
6117:
6112:
6105:
6102:
6099:
6098:
6078:
6076:
6050:
6044:
6039:
6034:
6029:
6024:
6018:
6012:
6007:
6002:
5997:
5992:
5987:
5982:
5976:
5972:
5947:
5941:
5936:
5931:
5926:
5921:
5915:
5907:
5902:
5897:
5892:
5887:
5859:
5855:
5831:
5828:
5821:
5817:
5810:
5806:
5800:
5796:
5775:
5772:
5769:
5766:
5763:
5759:
5755:
5752:
5731:
5714:
5711:
5699:
5695:
5691:
5687:
5683:
5679:
5674:
5670:
5666:
5663:
5659:
5655:
5651:
5647:
5642:
5638:
5634:
5629:
5625:
5619:
5615:
5609:
5605:
5601:
5596:
5592:
5586:
5582:
5576:
5572:
5568:
5563:
5559:
5553:
5549:
5543:
5539:
5535:
5530:
5525:
5521:
5515:
5510:
5506:
5502:
5497:
5492:
5488:
5482:
5477:
5473:
5469:
5466:
5461:
5457:
5453:
5449:
5445:
5441:
5437:
5434:
5430:
5426:
5403:
5400:
5397:
5377:
5374:
5371:
5351:
5348:
5345:
5325:
5322:
5319:
5299:
5279:
5259:
5233:
5230:
5225:
5222:
5218:
5197:
5194:
5191:
5171:
5168:
5163:
5158:
5154:
5133:
5130:
5127:
5107:
5104:
5099:
5094:
5090:
5063:
5058:
5054:
5033:
5027:
5022:
5018:
5012:
5007:
5003:
4999:
4994:
4989:
4985:
4979:
4974:
4970:
4966:
4961:
4958:
4953:
4950:
4946:
4942:
4937:
4934:
4931:
4927:
4921:
4918:
4915:
4911:
4882:
4862:
4857:
4853:
4847:
4843:
4837:
4833:
4827:
4824:
4821:
4817:
4811:
4808:
4805:
4801:
4797:
4792:
4788:
4782:
4778:
4772:
4769:
4766:
4762:
4756:
4752:
4746:
4743:
4740:
4736:
4732:
4727:
4723:
4719:
4715:
4711:
4707:
4703:
4700:
4696:
4692:
4670:
4666:
4660:
4656:
4650:
4646:
4640:
4637:
4634:
4630:
4626:
4623:
4619:
4615:
4611:
4607:
4604:
4600:
4579:
4576:
4574:
4571:
4555:
4554:
4538:
4534:
4530:
4526:
4522:
4518:
4515:
4511:
4507:
4503:
4499:
4495:
4491:
4488:
4485:
4483:
4481:
4477:
4473:
4470:
4466:
4459:
4452:
4448:
4445:
4442:
4438:
4431:
4424:
4420:
4417:
4413:
4409:
4406:
4404:
4402:
4398:
4394:
4390:
4386:
4380:
4373:
4369:
4366:
4365:
4326:
4323:
4322:
4321:
4309:
4302:
4298:
4294:
4290:
4286:
4283:
4279:
4272:
4268:
4264:
4260:
4256:
4253:
4250:
4246:
4242:
4238:
4234:
4231:
4227:
4212:
4211:
4194:
4189:
4184:
4180:
4176:
4172:
4168:
4165:
4160:
4155:
4150:
4146:
4142:
4138:
4134:
4131:
4128:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4095:
4091:
4087:
4084:
4083:
4078:
4073:
4068:
4064:
4060:
4056:
4052:
4049:
4044:
4039:
4034:
4030:
4026:
4022:
4018:
4015:
4012:
4010:
4006:
4002:
3998:
3994:
3990:
3986:
3982:
3979:
3975:
3971:
3968:
3967:
3955:are given by:
3944:
3940:
3936:
3932:
3928:
3925:
3921:
3900:
3880:
3869:
3868:
3851:
3846:
3841:
3837:
3833:
3829:
3825:
3822:
3817:
3812:
3807:
3803:
3799:
3795:
3791:
3788:
3785:
3783:
3781:
3778:
3773:
3768:
3761:
3756:
3751:
3746:
3741:
3734:
3729:
3724:
3719:
3714:
3707:
3702:
3697:
3692:
3687:
3682:
3679:
3674:
3669:
3662:
3657:
3652:
3647:
3642:
3635:
3630:
3625:
3620:
3615:
3608:
3603:
3598:
3593:
3588:
3583:
3580:
3578:
3576:
3573:
3568:
3563:
3556:
3551:
3544:
3539:
3534:
3529:
3524:
3517:
3512:
3505:
3500:
3495:
3492:
3489:
3484:
3479:
3472:
3467:
3462:
3457:
3452:
3445:
3440:
3435:
3430:
3425:
3420:
3417:
3412:
3407:
3400:
3395:
3390:
3385:
3380:
3373:
3368:
3363:
3358:
3353:
3348:
3345:
3343:
3341:
3338:
3333:
3328:
3321:
3316:
3311:
3306:
3301:
3294:
3289:
3284:
3279:
3274:
3269:
3266:
3261:
3256:
3249:
3244:
3239:
3234:
3229:
3222:
3217:
3212:
3207:
3202:
3197:
3194:
3192:
3190:
3187:
3182:
3177:
3170:
3165:
3160:
3155:
3150:
3143:
3138:
3133:
3128:
3123:
3118:
3115:
3110:
3105:
3098:
3093:
3088:
3083:
3078:
3071:
3066:
3061:
3056:
3051:
3046:
3043:
3041:
3037:
3033:
3029:
3025:
3021:
3017:
3013:
3010:
3006:
3002:
2999:
2998:
2975:
2971:
2967:
2963:
2959:
2956:
2952:
2942:component of
2931:
2919:
2916:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2871:
2870:
2858:
2854:
2850:
2846:
2842:
2838:
2835:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2805:
2801:
2797:
2793:
2789:
2786:
2782:
2765:and useful in
2755:
2754:
2743:
2739:
2735:
2731:
2727:
2724:
2720:
2716:
2713:
2709:
2705:
2701:
2697:
2694:
2690:
2686:
2682:
2678:
2675:
2671:
2667:
2663:
2659:
2641:
2640:
2628:
2624:
2621:
2617:
2613:
2609:
2605:
2602:
2598:
2594:
2591:
2587:
2583:
2579:
2575:
2572:
2568:
2564:
2561:
2557:
2553:
2549:
2545:
2542:
2538:
2523:
2522:
2510:
2506:
2502:
2498:
2494:
2490:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2460:
2457:
2454:
2450:
2446:
2442:
2438:
2435:
2431:
2427:
2424:
2420:
2416:
2413:
2409:
2405:
2401:
2397:
2367:
2363:
2359:
2355:
2351:
2347:
2343:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2313:
2309:
2305:
2301:
2297:
2294:
2290:
2258:
2257:
2244:
2240:
2236:
2232:
2228:
2224:
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2194:
2191:
2187:
2183:
2179:
2175:
2172:
2168:
2144:
2141:
2116:
2113:
2027:
2026:
2013:
2009:
2005:
2001:
1997:
1993:
1990:
1986:
1981:
1977:
1973:
1968:
1964:
1960:
1956:
1952:
1947:
1932:
1931:
1919:
1915:
1911:
1907:
1903:
1890:, the product
1878:Given vectors
1848:
1845:
1841:scalar density
1831:
1828:
1827:
1826:
1815:
1812:
1808:
1804:
1800:
1796:
1793:
1789:
1785:
1782:
1779:
1775:
1772:
1768:
1764:
1761:
1757:
1754:
1750:
1747:
1724:
1723:
1712:
1709:
1705:
1701:
1697:
1693:
1690:
1686:
1682:
1679:
1675:
1672:
1668:
1664:
1661:
1657:
1654:
1650:
1647:
1592:
1589:
1588:
1587:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1548:
1545:
1542:
1536:
1531:
1526:
1523:
1518:
1513:
1510:
1505:
1500:
1495:
1491:
1487:
1483:
1479:
1476:
1472:
1453:
1435:
1428:
1424:
1420:
1416:
1413:
1409:
1405:
1401:
1398:
1394:
1390:
1386:
1385:
1381:
1377:
1373:
1369:
1366:
1362:
1358:
1354:
1351:
1347:
1343:
1339:
1338:
1334:
1330:
1326:
1322:
1319:
1315:
1311:
1307:
1304:
1300:
1296:
1292:
1291:
1289:
1284:
1281:
1278:
1274:
1270:
1267:
1263:
1259:
1255:
1251:
1248:
1244:
1240:
1236:
1233:
1229:
1225:
1221:
1217:
1214:
1203:simple product
1199:
1198:
1197:
1186:
1182:
1178:
1174:
1170:
1167:
1164:
1160:
1156:
1152:
1148:
1145:
1141:
1136:
1133:
1129:
1125:
1121:
1117:
1114:
1110:
1106:
1093:
1092:
1091:
1080:
1077:
1074:
1070:
1066:
1062:
1058:
1055:
1051:
1047:
1044:
1040:
1036:
1032:
1028:
1025:
1021:
1017:
1014:
1010:
1006:
1002:
998:
995:
991:
977:
958:
957:
956:
945:
940:
933:
929:
926:
922:
919:
915:
914:
912:
907:
904:
899:
891:
887:
883:
879:
875:
871:
867:
863:
859:
858:
853:
849:
845:
841:
837:
833:
829:
825:
821:
820:
815:
811:
807:
803:
799:
795:
791:
787:
783:
782:
780:
775:
772:
767:
759:
755:
751:
747:
743:
739:
735:
731:
727:
726:
721:
717:
713:
709:
705:
701:
697:
693:
689:
688:
683:
679:
675:
671:
667:
663:
659:
655:
651:
650:
648:
643:
640:
637:
633:
629:
625:
621:
618:
614:
582:
581:
580:
565:
561:
557:
553:
549:
546:
542:
538:
535:
532:
530:
528:
525:
521:
517:
513:
509:
506:
502:
498:
495:
492:
490:
488:
485:
481:
477:
473:
469:
466:
462:
458:
455:
452:
450:
448:
444:
440:
436:
432:
429:
425:
421:
420:
399:
398:
397:
385:
381:
378:
374:
370:
366:
362:
359:
356:
352:
348:
344:
340:
337:
333:
319:
318:
317:
306:
302:
298:
294:
290:
287:
283:
279:
276:
272:
268:
264:
260:
257:
253:
249:
246:
242:
238:
234:
230:
227:
223:
197:circular shift
191:
188:
184:parallelepiped
176:
175:
164:
160:
156:
152:
148:
145:
141:
125:
122:
85:
82:
54:triple product
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6851:
6840:
6837:
6835:
6832:
6830:
6827:
6825:
6822:
6821:
6819:
6804:
6796:
6795:
6792:
6786:
6783:
6781:
6780:Sparse matrix
6778:
6776:
6773:
6771:
6768:
6766:
6763:
6762:
6760:
6758:
6754:
6748:
6745:
6743:
6740:
6738:
6735:
6733:
6730:
6728:
6725:
6723:
6720:
6719:
6717:
6715:constructions
6714:
6710:
6704:
6703:Outermorphism
6701:
6699:
6696:
6694:
6691:
6689:
6686:
6684:
6681:
6679:
6676:
6674:
6671:
6669:
6666:
6664:
6663:Cross product
6661:
6659:
6656:
6655:
6653:
6651:
6647:
6641:
6638:
6636:
6633:
6631:
6630:Outer product
6628:
6626:
6623:
6621:
6618:
6616:
6613:
6611:
6610:Orthogonality
6608:
6607:
6605:
6603:
6599:
6593:
6590:
6588:
6587:Cramer's rule
6585:
6583:
6580:
6578:
6575:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6557:Decomposition
6555:
6553:
6550:
6549:
6547:
6545:
6541:
6536:
6526:
6523:
6521:
6518:
6516:
6513:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6471:
6468:
6466:
6463:
6461:
6458:
6456:
6453:
6451:
6448:
6446:
6443:
6441:
6438:
6436:
6433:
6432:
6430:
6426:
6420:
6417:
6415:
6412:
6411:
6408:
6404:
6397:
6392:
6390:
6385:
6383:
6378:
6377:
6374:
6368:
6365:
6364:
6360:
6354:
6349:
6348:
6344:
6327:
6321:
6318:
6314:. p. 37.
6310:
6303:
6300:
6295:
6293:0-521-00551-5
6289:
6285:
6278:
6276:
6272:
6267:
6260:
6257:
6252:
6246:
6242:
6241:
6233:
6230:
6225:
6223:0-262-59020-4
6219:
6215:
6211:
6207:
6201:
6198:
6193:
6191:0-262-59020-4
6187:
6183:
6179:
6174:
6168:
6162:
6157:
6154:
6149:
6147:9780199641390
6143:
6139:
6138:
6130:
6127:
6120:
6116:
6113:
6111:
6108:
6107:
6103:
6095:
6086:
6082:
6079:This section
6077:
6074:
6070:
6069:
6066:
6042:
6032:
6027:
6005:
5995:
5990:
5974:
5939:
5929:
5924:
5905:
5895:
5890:
5829:
5826:
5808:
5798:
5794:
5770:
5767:
5764:
5753:
5750:
5720:
5719:flux integral
5717:Consider the
5712:
5710:
5697:
5685:
5672:
5668:
5664:
5653:
5640:
5636:
5632:
5627:
5623:
5617:
5613:
5607:
5603:
5599:
5594:
5590:
5584:
5580:
5574:
5570:
5566:
5561:
5557:
5551:
5547:
5541:
5537:
5528:
5523:
5519:
5513:
5508:
5504:
5500:
5495:
5490:
5486:
5480:
5475:
5471:
5464:
5459:
5443:
5432:
5415:
5401:
5398:
5395:
5375:
5372:
5369:
5349:
5346:
5343:
5323:
5320:
5317:
5297:
5277:
5257:
5249:
5231:
5228:
5223:
5220:
5216:
5195:
5192:
5189:
5169:
5166:
5161:
5156:
5152:
5131:
5128:
5125:
5105:
5102:
5097:
5092:
5088:
5079:
5061:
5056:
5052:
5031:
5025:
5020:
5016:
5010:
5005:
5001:
4997:
4992:
4987:
4983:
4977:
4972:
4968:
4964:
4959:
4956:
4951:
4948:
4944:
4940:
4935:
4932:
4929:
4925:
4919:
4916:
4913:
4909:
4900:
4896:
4880:
4860:
4855:
4851:
4845:
4841:
4835:
4831:
4825:
4822:
4819:
4815:
4809:
4806:
4803:
4799:
4795:
4790:
4786:
4780:
4776:
4770:
4767:
4764:
4760:
4754:
4750:
4744:
4741:
4738:
4734:
4730:
4725:
4709:
4698:
4668:
4664:
4658:
4654:
4648:
4644:
4638:
4635:
4632:
4628:
4624:
4613:
4602:
4589:
4585:
4577:
4572:
4570:
4568:
4564:
4560:
4524:
4513:
4497:
4486:
4484:
4471:
4443:
4415:
4407:
4405:
4392:
4367:
4356:
4355:
4354:
4352:
4348:
4344:
4340:
4336:
4332:
4324:
4292:
4281:
4262:
4251:
4240:
4229:
4217:
4216:
4215:
4192:
4174:
4163:
4158:
4140:
4129:
4127:
4120:
4104:
4093:
4076:
4058:
4047:
4042:
4024:
4013:
4011:
4004:
3988:
3977:
3958:
3957:
3956:
3934:
3923:
3898:
3878:
3849:
3831:
3820:
3815:
3797:
3786:
3784:
3771:
3759:
3749:
3744:
3732:
3722:
3717:
3705:
3690:
3680:
3672:
3660:
3650:
3645:
3633:
3623:
3618:
3606:
3591:
3581:
3579:
3566:
3554:
3542:
3532:
3527:
3515:
3503:
3490:
3482:
3470:
3460:
3455:
3443:
3428:
3418:
3410:
3398:
3388:
3383:
3371:
3356:
3346:
3344:
3331:
3319:
3309:
3304:
3292:
3277:
3267:
3259:
3247:
3237:
3232:
3220:
3205:
3195:
3193:
3180:
3168:
3158:
3153:
3141:
3126:
3116:
3108:
3096:
3086:
3081:
3069:
3054:
3044:
3042:
3035:
3019:
3008:
2989:
2988:
2987:
2986:is given by:
2965:
2954:
2929:
2917:
2915:
2901:
2898:
2895:
2892:
2889:
2886:
2876:
2844:
2833:
2822:
2806:
2795:
2784:
2772:
2771:
2770:
2768:
2764:
2760:
2733:
2722:
2714:
2703:
2692:
2684:
2676:
2665:
2650:
2649:
2648:
2646:
2643:which is the
2622:
2611:
2600:
2592:
2581:
2570:
2562:
2551:
2540:
2528:
2527:
2526:
2496:
2485:
2469:
2458:
2455:
2444:
2433:
2425:
2422:
2414:
2403:
2388:
2387:
2386:
2383:
2381:
2357:
2341:
2330:
2314:
2303:
2292:
2279:
2275:
2271:
2267:
2263:
2230:
2219:
2203:
2192:
2181:
2170:
2158:
2157:
2156:
2154:
2153:cross product
2150:
2142:
2140:
2138:
2134:
2130:
2126:
2122:
2114:
2112:
2110:
2109:parallelogram
2107:matching the
2105:
2101:
2095:
2091:
2085:
2081:
2076:
2072:
2068:
2064:
2060:
2056:
2051:
2047:
2041:
2037:
2032:
1999:
1988:
1975:
1962:
1954:
1937:
1936:
1935:
1913:
1905:
1893:
1892:
1891:
1889:
1885:
1881:
1876:
1874:
1870:
1866:
1862:
1853:
1846:
1844:
1842:
1837:
1829:
1813:
1802:
1791:
1783:
1780:
1766:
1752:
1737:
1736:
1735:
1733:
1729:
1710:
1699:
1688:
1680:
1666:
1652:
1637:
1636:
1635:
1633:
1629:
1624:
1621:
1617:
1612:
1610:
1606:
1602:
1598:
1590:
1565:
1557:
1546:
1543:
1540:
1485:
1474:
1458:
1454:
1451:
1433:
1422:
1407:
1392:
1375:
1360:
1345:
1328:
1313:
1298:
1287:
1279:
1268:
1257:
1234:
1223:
1204:
1200:
1176:
1165:
1154:
1143:
1123:
1112:
1097:
1096:
1094:
1078:
1075:
1064:
1053:
1045:
1034:
1023:
1015:
1004:
993:
981:
980:
978:
975:
971:
967:
963:
959:
943:
938:
910:
902:
897:
889:
885:
877:
873:
865:
861:
851:
847:
839:
835:
827:
823:
813:
809:
801:
797:
789:
785:
778:
770:
765:
757:
753:
745:
741:
733:
729:
719:
715:
707:
703:
695:
691:
681:
677:
669:
665:
657:
653:
646:
638:
627:
616:
604:
603:
601:
587:
583:
555:
544:
536:
533:
531:
515:
504:
496:
493:
491:
475:
464:
456:
453:
451:
438:
427:
411:
410:
408:
404:
400:
379:
368:
357:
346:
335:
323:
322:
320:
296:
285:
277:
266:
255:
247:
236:
225:
213:
212:
210:
206:
202:
198:
194:
193:
189:
187:
185:
181:
154:
143:
131:
130:
129:
123:
121:
119:
118:cross product
115:
111:
107:
103:
102:mixed product
99:
90:
83:
81:
79:
75:
71:
67:
63:
59:
55:
51:
47:
40:
33:
19:
6713:Vector space
6667:
6445:Vector space
6352:
6330:. Retrieved
6320:
6302:
6283:
6265:
6259:
6239:
6232:
6213:
6200:
6181:
6166:
6156:
6136:
6129:
6092:January 2014
6089:
6085:adding to it
6080:
5716:
5416:
4581:
4566:
4562:
4558:
4556:
4342:
4338:
4334:
4330:
4328:
4213:
3870:
2921:
2872:
2756:
2642:
2524:
2384:
2265:
2261:
2259:
2148:
2146:
2133:pseudotensor
2118:
2103:
2099:
2093:
2089:
2083:
2079:
2074:
2070:
2066:
2062:
2058:
2054:
2049:
2045:
2039:
2035:
2028:
1933:
1887:
1883:
1879:
1877:
1858:
1833:
1727:
1725:
1627:
1625:
1620:pseudovector
1613:
1609:pseudoscalar
1594:
969:
965:
961:
208:
204:
200:
177:
127:
109:
105:
101:
97:
95:
77:
69:
53:
43:
39:Bibliophilia
6693:Multivector
6658:Determinant
6615:Dot product
6460:Linear span
4895:contraction
2129:contraction
2121:volume form
2029:and is the
1597:orientation
586:determinant
114:dot product
106:box product
58:dimensional
6818:Categories
6727:Direct sum
6562:Invertible
6465:Linear map
6345:References
6206:Kiyosi ItĂ´
6178:Kiyosi ItĂ´
2031:Hodge dual
1457:polar sine
190:Properties
6757:Numerical
6520:Transpose
6328:. Wolfram
6033:×
5996:×
5975:⋅
5930:×
5896:×
5858:^
5820:^
5809:⋅
5795:∬
5686:⋅
5665:−
5654:⋅
5600:−
5552:ℓ
5529:ℓ
5520:δ
5505:δ
5501:−
5487:δ
5481:ℓ
5472:δ
5444:×
5433:×
5388:and thus
5336:and thus
5229:ℓ
5217:δ
5153:δ
5129:≠
5089:δ
5053:δ
5026:ℓ
5017:δ
5002:δ
4998:−
4984:δ
4978:ℓ
4969:δ
4957:ℓ
4945:δ
4933:ℓ
4926:ε
4910:ε
4846:ℓ
4823:ℓ
4816:ε
4800:ε
4781:ℓ
4768:ℓ
4761:ε
4735:ε
4710:×
4699:×
4629:ε
4614:×
4603:⋅
4525:⋅
4514:−
4498:⋅
4472:∧
4444:−
4416:∧
4393:∧
4368:−
4293:⋅
4282:−
4263:⋅
4241:×
4230:×
4175:⋅
4164:−
4141:⋅
4105:×
4094:×
4059:⋅
4048:−
4025:⋅
3989:×
3978:×
3935:×
3924:×
3832:⋅
3821:−
3798:⋅
3681:−
3533:−
3419:−
3268:−
3159:−
3117:−
3087:−
3020:×
3009:×
2966:×
2955:×
2899:δ
2893:δ
2884:Δ
2849:∇
2845:⋅
2841:∇
2834:−
2823:⋅
2819:∇
2811:∇
2796:×
2792:∇
2785:×
2781:∇
2763:gradients
2734:×
2723:×
2715:−
2704:×
2693:×
2677:×
2666:×
2612:×
2601:×
2582:×
2571:×
2552:×
2541:×
2497:⋅
2470:⋅
2459:−
2445:×
2434:×
2426:−
2415:×
2404:×
2358:⋅
2342:−
2331:⋅
2304:×
2293:×
2231:⋅
2220:−
2204:⋅
2182:×
2171:×
2000:×
1989:⋅
1963:∧
1955:∧
1914:∧
1906:∧
1873:trivector
1803:×
1792:⋅
1784:−
1767:×
1753:⋅
1700:×
1689:⋅
1667:×
1653:⋅
1547:
1535:‖
1525:‖
1522:‖
1512:‖
1509:‖
1499:‖
1486:×
1475:⋅
1423:⋅
1408:⋅
1393:⋅
1376:⋅
1361:⋅
1346:⋅
1329:⋅
1314:⋅
1299:⋅
1269:⋅
1258:×
1235:⋅
1224:×
1177:×
1166:×
1155:×
1124:×
1113:⋅
1065:×
1054:⋅
1035:×
1024:⋅
1005:×
994:⋅
628:×
617:⋅
600:transpose
556:×
545:⋅
537:−
516:×
505:⋅
497:−
476:×
465:⋅
457:−
439:×
428:⋅
380:⋅
369:×
347:×
336:⋅
297:×
286:⋅
267:×
256:⋅
237:×
226:⋅
155:×
144:⋅
6803:Category
6742:Subspace
6737:Quotient
6688:Bivector
6602:Bilinear
6544:Matrices
6419:Glossary
6208:(1993).
6180:(1987).
6104:See also
4458:⌟
4430:⌟
4379:⌟
4347:bivector
2380:mnemonic
2274:mnemonic
1869:bivector
974:coplanar
76:-valued
68:-valued
46:geometry
6414:Outline
6167:Oeuvres
5246:is the
5076:is the
4897:on the
2759:physics
1726:but if
588:of the
403:negates
182:of the
50:algebra
6698:Tensor
6510:Kernel
6440:Vector
6435:Scalar
6332:21 May
6290:
6247:
6220:
6188:
6144:
5208:) and
5044:where
4304:
4274:
2073:, and
1836:scalar
1730:is an
1095:Also:
968:, and
180:volume
74:vector
66:scalar
52:, the
6567:Minor
6552:Block
6490:Basis
6312:(PDF)
6121:Notes
5182:when
5118:when
2918:Proof
2278:below
2264:, or
2137:below
1734:then
1634:then
1630:is a
108:, or
6722:Dual
6577:Rank
6334:2014
6288:ISBN
6245:ISBN
6218:ISBN
6186:ISBN
6175:and
6142:ISBN
5290:and
5144:and
4683:and
4345:, a
3891:and
2922:The
2147:The
2097:and
2043:or
1886:and
1863:and
1544:psin
1201:The
972:are
96:The
48:and
6087:.
4582:In
4569:).
4561:Ă (
1859:In
1626:If
1283:det
906:det
774:det
642:det
602:):
211:):
44:In
6820::
6274:^
6212:.
5786::
5414:.
4901:,
4590::
4565:Ă
4333:Ă
2914:.
2139:.
2102:â§
2092:â§
2087:,
2082:â§
2069:,
2061:â§
2057:â§
2048:â§
2038:â§
1882:,
1843:.
964:,
207:,
203:,
104:,
80:.
6395:e
6388:t
6381:v
6336:.
6296:.
6253:.
6226:.
6194:.
6150:.
6094:)
6090:(
6049:|
6043:v
6038:r
6028:u
6023:r
6017:|
6011:)
6006:v
6001:r
5991:u
5986:r
5981:(
5971:F
5946:|
5940:v
5935:r
5925:u
5920:r
5914:|
5906:v
5901:r
5891:u
5886:r
5854:n
5830:S
5827:d
5816:n
5805:F
5799:S
5774:)
5771:v
5768:,
5765:u
5762:(
5758:r
5754:=
5751:S
5730:F
5698:.
5694:)
5690:b
5682:a
5678:(
5673:i
5669:c
5662:)
5658:c
5650:a
5646:(
5641:i
5637:b
5633:=
5628:i
5624:c
5618:j
5614:b
5608:j
5604:a
5595:j
5591:c
5585:i
5581:b
5575:j
5571:a
5567:=
5562:m
5558:c
5548:b
5542:j
5538:a
5534:)
5524:j
5514:m
5509:i
5496:m
5491:j
5476:i
5468:(
5465:=
5460:i
5456:)
5452:]
5448:c
5440:b
5436:[
5429:a
5425:(
5402:j
5399:=
5396:l
5376:m
5373:=
5370:i
5350:m
5347:=
5344:j
5324:l
5321:=
5318:i
5298:j
5278:i
5258:k
5232:m
5224:j
5221:i
5196:j
5193:=
5190:i
5170:1
5167:=
5162:i
5157:j
5132:j
5126:i
5106:0
5103:=
5098:i
5093:j
5080:(
5062:i
5057:j
5032:,
5021:j
5011:m
5006:i
4993:m
4988:j
4973:i
4965:=
4960:m
4952:j
4949:i
4941:=
4936:m
4930:k
4920:k
4917:j
4914:i
4881:i
4861:,
4856:m
4852:c
4842:b
4836:j
4832:a
4826:m
4820:k
4810:k
4807:j
4804:i
4796:=
4791:m
4787:c
4777:b
4771:m
4765:k
4755:j
4751:a
4745:k
4742:j
4739:i
4731:=
4726:i
4722:)
4718:]
4714:c
4706:b
4702:[
4695:a
4691:(
4669:k
4665:c
4659:j
4655:b
4649:i
4645:a
4639:k
4636:j
4633:i
4625:=
4622:]
4618:c
4610:b
4606:[
4599:a
4567:c
4563:b
4559:a
4537:c
4533:)
4529:b
4521:a
4517:(
4510:b
4506:)
4502:c
4494:a
4490:(
4487:=
4476:c
4469:)
4465:b
4451:a
4447:(
4441:)
4437:c
4423:a
4419:(
4412:b
4408:=
4401:)
4397:c
4389:b
4385:(
4372:a
4343:c
4341:â§
4339:b
4335:c
4331:b
4308:w
4301:)
4297:v
4289:u
4285:(
4278:v
4271:)
4267:w
4259:u
4255:(
4252:=
4249:)
4245:w
4237:v
4233:(
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4121:z
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4113:)
4109:w
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4097:(
4090:u
4086:(
4077:y
4072:w
4067:)
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4055:u
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4029:w
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4017:(
4014:=
4005:y
4001:)
3997:)
3993:w
3985:v
3981:(
3974:u
3970:(
3943:)
3939:w
3931:v
3927:(
3920:u
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3828:u
3824:(
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3811:v
3806:)
3802:w
3794:u
3790:(
3787:=
3777:)
3772:z
3767:v
3760:z
3755:u
3750:+
3745:y
3740:v
3733:y
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3723:+
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3673:z
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3656:u
3651:+
3646:y
3641:w
3634:y
3629:u
3624:+
3619:x
3614:w
3607:x
3602:u
3597:(
3592:x
3587:v
3582:=
3572:)
3567:x
3562:w
3555:x
3550:v
3543:x
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3528:x
3523:w
3516:x
3511:v
3504:x
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3494:(
3491:+
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3424:w
3416:)
3411:z
3406:w
3399:z
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3389:+
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3362:(
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3352:v
3347:=
3337:)
3332:z
3327:v
3320:z
3315:u
3310:+
3305:y
3300:v
3293:y
3288:u
3283:(
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3273:w
3265:)
3260:z
3255:w
3248:z
3243:u
3238:+
3233:y
3228:w
3221:y
3216:u
3211:(
3206:x
3201:v
3196:=
3186:)
3181:z
3176:w
3169:x
3164:v
3154:x
3149:w
3142:z
3137:v
3132:(
3127:z
3122:u
3114:)
3109:x
3104:w
3097:y
3092:v
3082:y
3077:w
3070:x
3065:v
3060:(
3055:y
3050:u
3045:=
3036:x
3032:)
3028:)
3024:w
3016:v
3012:(
3005:u
3001:(
2974:)
2970:w
2962:v
2958:(
2951:u
2930:x
2902:d
2896:+
2890:d
2887:=
2857:A
2853:)
2837:(
2831:)
2827:A
2815:(
2807:=
2804:)
2800:A
2788:(
2742:)
2738:c
2730:a
2726:(
2719:b
2712:)
2708:c
2700:b
2696:(
2689:a
2685:=
2681:c
2674:)
2670:b
2662:a
2658:(
2627:0
2623:=
2620:)
2616:b
2608:a
2604:(
2597:c
2593:+
2590:)
2586:a
2578:c
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2563:+
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2489:(
2486:+
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2478:)
2474:b
2466:c
2462:(
2456:=
2453:)
2449:b
2441:a
2437:(
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2423:=
2419:c
2412:)
2408:b
2400:a
2396:(
2366:)
2362:b
2354:a
2350:(
2346:c
2339:)
2335:c
2327:a
2323:(
2319:b
2315:=
2312:)
2308:c
2300:b
2296:(
2289:a
2256:.
2243:c
2239:)
2235:b
2227:a
2223:(
2216:b
2212:)
2208:c
2200:a
2196:(
2193:=
2190:)
2186:c
2178:b
2174:(
2167:a
2104:c
2100:a
2094:c
2090:b
2084:b
2080:a
2075:c
2071:b
2067:a
2063:c
2059:b
2055:a
2050:c
2046:b
2040:b
2036:a
2025:,
2012:|
2008:)
2004:c
1996:b
1992:(
1985:a
1980:|
1976:=
1972:|
1967:c
1959:b
1951:a
1946:|
1918:c
1910:b
1902:a
1888:c
1884:b
1880:a
1814:.
1811:)
1807:c
1799:b
1795:(
1788:a
1781:=
1778:)
1774:c
1771:T
1763:b
1760:T
1756:(
1749:a
1746:T
1728:T
1711:,
1708:)
1704:c
1696:b
1692:(
1685:a
1681:=
1678:)
1674:c
1671:T
1663:b
1660:T
1656:(
1649:a
1646:T
1628:T
1574:)
1570:c
1566:,
1562:b
1558:,
1554:a
1550:(
1541:=
1530:c
1517:b
1504:a
1494:)
1490:c
1482:b
1478:(
1471:a
1459::
1452:.
1434:]
1427:f
1419:c
1412:e
1404:c
1397:d
1389:c
1380:f
1372:b
1365:e
1357:b
1350:d
1342:b
1333:f
1325:a
1318:e
1310:a
1303:d
1295:a
1288:[
1280:=
1277:)
1273:f
1266:)
1262:e
1254:d
1250:(
1247:(
1243:)
1239:c
1232:)
1228:b
1220:a
1216:(
1213:(
1185:)
1181:c
1173:a
1169:(
1163:)
1159:b
1151:a
1147:(
1144:=
1140:a
1135:)
1132:)
1128:c
1120:b
1116:(
1109:a
1105:(
1079:0
1076:=
1073:)
1069:a
1061:a
1057:(
1050:b
1046:=
1043:)
1039:a
1031:b
1027:(
1020:a
1016:=
1013:)
1009:b
1001:a
997:(
990:a
970:c
966:b
962:a
944:.
939:]
932:c
925:b
918:a
911:[
903:=
898:]
890:3
886:c
878:3
874:b
866:3
862:a
852:2
848:c
840:2
836:b
828:2
824:a
814:1
810:c
802:1
798:b
790:1
786:a
779:[
771:=
766:]
758:3
754:c
746:2
742:c
734:1
730:c
720:3
716:b
708:2
704:b
696:1
692:b
682:3
678:a
670:2
666:a
658:1
654:a
647:[
639:=
636:)
632:c
624:b
620:(
613:a
595:3
592:Ă
590:3
564:)
560:a
552:b
548:(
541:c
534:=
524:)
520:c
512:a
508:(
501:b
494:=
484:)
480:b
472:c
468:(
461:a
454:=
447:)
443:c
435:b
431:(
424:a
384:c
377:)
373:b
365:a
361:(
358:=
355:)
351:c
343:b
339:(
332:a
305:)
301:b
293:a
289:(
282:c
278:=
275:)
271:a
263:c
259:(
252:b
248:=
245:)
241:c
233:b
229:(
222:a
209:c
205:b
201:a
163:)
159:c
151:b
147:(
140:a
41:.
34:.
20:)
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