26937:
1088:
2057:
2439:
1723:
2498:
27201:
55:
43:
12167:
9704:); nevertheless, transversal or not, a product can be defined on this space such that the resulting algebra is isomorphic to the exterior algebra: in the first case the natural choice for the product is just the quotient product (using the available projection), in the second case, this product must be slightly modified as given below (along Arnold setting), but such that the algebra stays isomorphic with the exterior algebra, i.e. the quotient of
6132:; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any exterior product in which the same basis vector appears more than once is zero; any exterior product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis
23561:
23552:
23543:
2052:{\displaystyle {\begin{aligned}\mathbf {v} \wedge \mathbf {w} &=(a\mathbf {e} _{1}+b\mathbf {e} _{2})\wedge (c\mathbf {e} _{1}+d\mathbf {e} _{2})\\&=ac\mathbf {e} _{1}\wedge \mathbf {e} _{1}+ad\mathbf {e} _{1}\wedge \mathbf {e} _{2}+bc\mathbf {e} _{2}\wedge \mathbf {e} _{1}+bd\mathbf {e} _{2}\wedge \mathbf {e} _{2}\\&=\left(ad-bc\right)\mathbf {e} _{1}\wedge \mathbf {e} _{2}\end{aligned}}}
19390:
13199:
3034:
11861:
14370:
3493:
25222:
10123:
7878:
6629:
1664:
21012:
1416:
14078:
22474:
20655:
19141:
12162:{\displaystyle t~{\widehat {\otimes }}~s={\frac {1}{(r+p)!}}\sum _{\sigma \in {\mathfrak {S}}_{r+p}}\operatorname {sgn} (\sigma )t^{i_{\sigma (1)}\cdots i_{\sigma (r)}}s^{i_{\sigma (r+1)}\cdots i_{\sigma (r+p)}}{\mathbf {e} }_{i_{1}}\otimes {\mathbf {e} }_{i_{2}}\otimes \cdots \otimes {\mathbf {e} }_{i_{r+p}}.}
2754:
16414:, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space. Relative to the preferred volume form
10330:
14116:
4613:
23634:
In physics, many quantities are naturally represented by alternating operators. For example, if the motion of a charged particle is described by velocity and acceleration vectors in four-dimensional spacetime, then normalization of the velocity vector requires that the electromagnetic force must be
17363:
3208:
2470:
With the exception of the last property, the exterior product of two vectors satisfies the same properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any chosen parallelogram in a parallel
24972:
21789:
21499:
26187:
for the article. It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See §III.7 and
19814:
the same tensor symbol as the one being used in the definition of the alternating product. Intuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the
18365:
20143:
19805:
18249:
23615:
as well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a
11823:
22603:
22731:
5965:
9959:
7736:
6487:
16954:
10572:
1512:
16545:
16290:
7992:
1249:
20863:
17886:
9438:
are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the
21970:
21657:
1260:
20852:
18722:
15991:
15600:
14777:
13865:
13900:
12672:
22299:
7131:
2255:
The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if
20504:
13706:
of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as
15280:
19385:{\displaystyle \Delta (x_{1}\wedge \cdots \wedge x_{k})=\sum _{p=0}^{k}\;\sum _{\sigma \in Sh(p,k-p)}\;\operatorname {sgn} (\sigma )(x_{\sigma (1)}\wedge \cdots \wedge x_{\sigma (p)})\otimes (x_{\sigma (p+1)}\wedge \cdots \wedge x_{\sigma (k)}).}
16701:
22304:
20515:
23160:
19130:
3029:{\displaystyle \mathbf {u} \wedge \mathbf {v} =(u_{1}v_{2}-u_{2}v_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2})+(u_{3}v_{1}-u_{1}v_{3})(\mathbf {e} _{3}\wedge \mathbf {e} _{1})+(u_{2}v_{3}-u_{3}v_{2})(\mathbf {e} _{2}\wedge \mathbf {e} _{3})}
2154:
24622:
th exterior power of the tangent space. As a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms. Differential forms play a major role in diverse areas of differential geometry.
2234:
24961:
23591:. For instance, it is well known that the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix (with a sign to track orientation). This suggests that the determinant can be
4219:
3197:
2743:
2640:
18454:
16121:
14365:{\displaystyle {\omega {\dot {\wedge }}\eta (x_{1},\ldots ,x_{k+m})}=\sum _{\sigma \in \mathrm {Sh} _{k,m}}\operatorname {sgn} (\sigma )\,\omega (x_{\sigma (1)},\ldots ,x_{\sigma (k)})\,\eta (x_{\sigma (k+1)},\ldots ,x_{\sigma (k+m)}),}
18925:
10186:
5198:
7611:
8733:
7676:
17605:
21853:
4469:
17228:
11066:
3488:{\displaystyle \mathbf {u} \wedge \mathbf {v} \wedge \mathbf {w} =(u_{1}v_{2}w_{3}+u_{2}v_{3}w_{1}+u_{3}v_{1}w_{2}-u_{1}v_{3}w_{2}-u_{2}v_{1}w_{3}-u_{3}v_{2}w_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3})}
22119:
16792:
16358:
25217:{\displaystyle \partial (x_{1}\wedge \cdots \wedge x_{p+1})={\frac {1}{p+1}}\sum _{j<\ell }(-1)^{j+\ell +1}\wedge x_{1}\wedge \cdots \wedge {\hat {x}}_{j}\wedge \cdots \wedge {\hat {x}}_{\ell }\wedge \cdots \wedge x_{p+1}.}
21668:
21566:
21394:
13758:
23030:
19867:
18981:
18257:
24547:
20015:
19685:
18169:
6971:
12347:
11686:
13454:
6326:
3920:
22489:
17194:
13116:
9940:
8268:
22618:
19562:
18053:
17717:
23480:
19492:
18508:
18159:
18106:
17770:
10118:{\displaystyle \operatorname {{\mathcal {A}}^{(r)}} (v_{1}\otimes \cdots \otimes v_{r})=\sum _{\sigma \in {\mathfrak {S}}_{r}}\operatorname {sgn} (\sigma )v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (r)}}
13532:
7873:{\displaystyle {\textstyle \bigwedge }(V)={\textstyle \bigwedge }^{\!0}(V)\oplus {\textstyle \bigwedge }^{\!1}(V)\oplus {\textstyle \bigwedge }^{\!2}(V)\oplus \cdots \oplus {\textstyle \bigwedge }^{\!n}(V)}
6624:{\displaystyle {\textstyle \bigwedge }(V)={\textstyle \bigwedge }^{\!0}(V)\oplus {\textstyle \bigwedge }^{\!1}(V)\oplus {\textstyle \bigwedge }^{\!2}(V)\oplus \cdots \oplus {\textstyle \bigwedge }^{\!n}(V)}
5812:
13344:
is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degree
4952:
4816:
1659:{\displaystyle {\text{Area}}={\Bigl |}\det {\begin{bmatrix}\mathbf {v} &\mathbf {w} \end{bmatrix}}{\Bigr |}={\Biggl |}\det {\begin{bmatrix}a&c\\b&d\end{bmatrix}}{\Biggr |}=\left|ad-bc\right|.}
20340:
20241:
12925:
5676:
16828:
8151:
6854:
5250:
21007:{\displaystyle {\textstyle \bigwedge }^{\!k}(f)={\textstyle \bigwedge }(f)\left|_{{\textstyle \bigwedge }^{\!k}(V)}\right.:{\textstyle \bigwedge }^{\!k}(V)\rightarrow {\textstyle \bigwedge }^{\!k}(W).}
15035:
10479:
8200:
7402:
430:
16444:
16187:
16048:
7893:
1144:
6434:
17810:
14892:
11227:
8910:
11332:
19910:, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on
8101:
1728:
8051:
6760:
6681:
17960:
17459:
15690:
8642:
26639:
24368:
21302:
21261:
21058:
18780:
17651:
17136:
16402:
15382:
9161:
9120:
6244:
6065:
6011:
5489:
5446:
5037:
21864:
10652:
23532:
12423:
1411:{\displaystyle \mathbf {v} ={\begin{bmatrix}a\\b\end{bmatrix}}=a\mathbf {e} _{1}+b\mathbf {e} _{2},\quad \mathbf {w} ={\begin{bmatrix}c\\d\end{bmatrix}}=c\mathbf {e} _{1}+d\mathbf {e} _{2}}
24043:
21582:
15739:
9660:
4884:
4648:
26594:
26551:
5596:
24477:
24118:
23366:
20691:
19607:
19016:
18633:
18551:
14682:
13667:
11485:
10608:
9809:(or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).
9377:
9309:
9196:
9079:
9035:
8987:
8952:
8849:
8724:
8468:
8415:
8332:
5078:
20736:
18650:
14073:{\displaystyle \operatorname {Alt} (\omega )(x_{1},\ldots ,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn} (\sigma )\,\omega (x_{\sigma (1)},\ldots ,x_{\sigma (k)}).}
12856:
19677:
15870:
3808:
25956:
25907:
25308:
15515:
14692:
13769:
4275:
15508:
5780:
1016:
22916:
21382:
20728:
13060:
12529:
10983:
682:
23965:
22196:
16794:
and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an
10925:
10796:
15319:
9848:
7015:
20438:
25975:, and projective modules over a commutative ring. This is thus more general than the result quoted above for direct sums, since not every short exact sequence splits in other
24237:
23771:
12489:
2236:) Note that the coefficient in this last expression is precisely the determinant of the matrix . The fact that this may be positive or negative has the intuitive meaning that
15859:
15830:
15455:
14641:
11414:
5520:
1450:
1125:
22469:{\displaystyle u_{1}\wedge \ldots \wedge u_{k-p}\wedge v_{1}\wedge \ldots \wedge v_{p}\mapsto u_{1}\wedge \ldots \wedge u_{k-p}\otimes g(v_{1})\wedge \ldots \wedge g(v_{p}).}
20650:{\displaystyle {\textstyle \bigwedge }(f)\left|_{{\textstyle \bigwedge }^{\!1}(V)}\right.=f:V={\textstyle \bigwedge }^{\!1}(V)\rightarrow W={\textstyle \bigwedge }^{\!1}(W).}
15126:
13151:
11551:
11171:
10757:
10722:
10466:
9733:
9524:
9342:
11254:
10881:
10403:
10377:
7454:
16623:
7007:
3860:
3623:
23892:
22797:
20189:
13328:
12765:
2317:, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
1713:
1689:
1496:
1474:
22188:
19954:
free for use in the definition of the bialgebra. In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of
11284:
10433:
8372:
4718:
3995:
969:
638:
239:
23034:
22152:
19028:
9781:
8578:
3723:
2073:
25334:
24157:
21126:
2159:
531:
23816:
20426:
19994:
19972:
19952:
10155:
8531:
4086:
24886:
22026:
19932:
16434:
14838:
14806:
10831:
7531:
7508:
7324:
7253:
7229:
7192:
7164:
6876:
5367:
5303:
5272:
4746:
4673:
4298:
4024:
3831:
306:
207:
23926:
18811:
17991:
17800:
16724:
13018:
10325:{\displaystyle \operatorname {Alt} ^{(r)}(v_{1}\otimes \cdots \otimes v_{r})={\frac {1}{r!}}\operatorname {{\mathcal {A}}^{(r)}} (v_{1}\otimes \cdots \otimes v_{r})}
9556:
7488:
6462:
4096:
3749:
3102:
25416:
in 1888. Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably
23306:
22000:
19634:
17093:
16605:
15790:
15116:
14941:
14578:
13601:
13182:
7728:
7280:
6791:
6198:
6169:
6128:
6095:
4435:
4404:
4373:
2651:
2548:
23681:
18397:
16054:
11611:
11580:
11489:, but the product could (or should) be chosen in two ways (or only one). Actually, the product could be chosen in many ways, rescaling it on homogeneous spaces as
11448:
11361:
11118:
10954:
10687:
26157:
Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article.
25253:
22851:
21176:
18864:
15087:
13284:
12519:
5091:
23183:
10178:
7559:
843:
796:
578:
349:
282:
179:
25883:
25863:
25832:
25800:
25637:
25617:
25275:
24876:
24852:
24824:
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24434:
24400:
24311:
24287:
24263:
24203:
24181:
23388:
23328:
23271:
23249:
23227:
23207:
22936:
22819:
22767:
21328:
21218:
21196:
21148:
21100:
21080:
20394:
20374:
20148:
where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely,
18855:
18833:
18573:
18387:
17504:
17218:
17064:
17042:
17010:
16818:
16576:
16177:
16155:
15761:
15644:
15624:
15418:
15339:
14963:
14914:
14604:
14551:
14104:
13888:
13621:
13574:
13554:
13476:
13383:
13363:
12988:
12968:
12948:
12445:
11383:
11138:
11089:
10903:
10855:
9868:
9805:
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9682:
9624:
9600:
9578:
9493:
9473:
9421:
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9274:
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9229:
7616:
7551:
7302:
6709:
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6351:
5804:
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1072:
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874:
820:
769:
749:
709:
598:
555:
479:
459:
371:
326:
259:
152:
4608:{\displaystyle x_{\sigma (1)}\wedge x_{\sigma (2)}\wedge \cdots \wedge x_{\sigma (k)}=\operatorname {sgn}(\sigma )x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k},}
4342:
26339:
17514:
17358:{\displaystyle \left\langle v_{1}\wedge \cdots \wedge v_{k},w_{1}\wedge \cdots \wedge w_{k}\right\rangle =\det {\bigl (}\langle v_{i},w_{j}\rangle {\bigr )},}
21797:
25393:
The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and
Sylvester's theory of multivectors. It was thus a
10991:
25382:. This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a
22031:
16729:
16301:
684:
and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.
21784:{\displaystyle {\textstyle \bigwedge }^{\!k}(V\oplus W)\cong \bigoplus _{p+q=k}{\textstyle \bigwedge }^{\!p}(V)\otimes {\textstyle \bigwedge }^{\!q}(W).}
3548:
and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the
21494:{\displaystyle 0\to {\textstyle \bigwedge }^{\!1}(U)\wedge {\textstyle \bigwedge }(V)\to {\textstyle \bigwedge }(V)\to {\textstyle \bigwedge }(W)\to 0}
23332:-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section.
19018:
by (linear) homomorphism. The correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the
18360:{\displaystyle \mathbf {x} ^{\flat }\in {\textstyle \bigwedge }^{\!l}\left(V^{*}\right)\simeq {\bigl (}{\textstyle \bigwedge }^{\!l}(V){\bigr )}^{*}}
26247:
Chapter XVI sections 6–10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.
21510:
20138:{\displaystyle (\alpha \wedge \beta )(x_{1}\wedge \cdots \wedge x_{k})=(\alpha \otimes \beta )\left(\Delta (x_{1}\wedge \cdots \wedge x_{k})\right)}
19800:{\displaystyle \Delta :{\textstyle \bigwedge }^{k}(V)\to \bigoplus _{p=0}^{k}{\textstyle \bigwedge }^{p}(V)\otimes {\textstyle \bigwedge }^{k-p}(V)}
18244:{\displaystyle \langle \mathbf {x} \wedge \mathbf {v} ,\mathbf {w} \rangle =\langle \mathbf {v} ,\iota _{\mathbf {x} ^{\flat }}\mathbf {w} \rangle }
13713:
22941:
19820:
18934:
24492:
19640:
in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.
14485:. As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of
11818:{\displaystyle t=t^{i_{1}i_{2}\cdots i_{r}}\,{\mathbf {e} }_{i_{1}}\otimes {\mathbf {e} }_{i_{2}}\otimes \cdots \otimes {\mathbf {e} }_{i_{r}},}
26760:
25387:
6893:
22598:{\displaystyle 0\to U\otimes {\textstyle \bigwedge }^{\!k-1}(W)\to {\textstyle \bigwedge }^{\!k}(V)\to {\textstyle \bigwedge }^{\!k}(W)\to 0}
12188:
4821:
Together with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for
13397:
6256:
3868:
24627:
22726:{\displaystyle 0\to {\textstyle \bigwedge }^{k}(U)\to {\textstyle \bigwedge }^{\!k}(V)\to {\textstyle \bigwedge }^{\!k-1}(U)\otimes W\to 0}
20243:
that returns the 0-graded component of its argument. The coproduct and counit, along with the exterior product, define the structure of a
11582:
in the field, as long as the division makes sense (this is such that the redefined product is also associative, i.e. defines an algebra on
6383:
is the number of vectors in the product. The binomial coefficient produces the correct result, even for exceptional cases; in particular,
26461:
17143:
15998:
These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.
13674:
Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose
26757:
A compilation of
English translations of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
13068:
9876:
8210:
26754:
19497:
18003:
17667:
5960:{\displaystyle \{\,e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}~{\big |}~~1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n\,\}}
23395:
10610:
is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace. On the other hand, the image
2244:
may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the
25401:, except focused exclusively on the task of formal reasoning in geometrical terms. In particular, this new development allowed for an
19998:, with the understanding that it works in a different space. Immediately below, an example is given: the alternating product for the
19433:
18464:
18115:
18062:
17726:
26766:
13487:
16949:{\displaystyle \star \circ \star :{\textstyle \bigwedge }^{\!k}(V)\to {\textstyle \bigwedge }^{\!k}(V)=(-1)^{k(n-k)+q}\mathrm {id} }
4892:
4756:
3530:
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product
27594:
26795:
10567:{\displaystyle \operatorname {{\mathcal {A}}^{(r)}} \operatorname {{\mathcal {A}}^{(r)}} =r!\operatorname {{\mathcal {A}}^{(r)}} .}
20259:
20201:
16540:{\displaystyle {\textstyle \bigwedge }^{\!k}(V^{*})\to {\textstyle \bigwedge }^{\!n-k}(V):\alpha \mapsto \iota _{\alpha }\sigma .}
16285:{\displaystyle {\textstyle \bigwedge }^{\!k}(V^{*})\otimes {\textstyle \bigwedge }^{\!n}(V)\to {\textstyle \bigwedge }^{\!n-k}(V)}
12867:
7987:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)\wedge {\textstyle \bigwedge }^{\!p}(V)\subset {\textstyle \bigwedge }^{\!k+p}(V).}
5606:
1244:{\displaystyle {\mathbf {e} }_{1}={\begin{bmatrix}1\\0\end{bmatrix}},\quad {\mathbf {e} }_{2}={\begin{bmatrix}0\\1\end{bmatrix}}.}
8109:
6812:
5208:
3584:. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively oriented
27128:
26667:
Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes.
23229:); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation. In
14981:
8158:
7332:
3089:
of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a
376:
77:(signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of
27186:
16007:
27743:
17881:{\displaystyle \langle x\wedge \mathbf {v} ,\mathbf {w} \rangle =\langle \mathbf {v} ,\iota _{x^{\flat }}\mathbf {w} \rangle }
26740:
26704:
26659:
26502:
26398:
25542:
17662:
With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, for
11615:). Also note, the interior product definition should be changed accordingly, in order to keep its skew derivation property.
6386:
24574:. Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of
23635:
an alternating operator on the velocity. Its six degrees of freedom are identified with the electric and magnetic fields.
14850:
11181:
8857:
26412:
25599:, which in a different context could mean an element of a 4-dimensional vector space. A minority of authors use the term
11289:
26264:
Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.
25740:: The orientations shown here are not correct; the diagram simply gives a sense that an orientation is defined for every
8417:
is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative
8056:
21965:{\displaystyle 0=F^{0}\subseteq F^{1}\subseteq \cdots \subseteq F^{k}\subseteq F^{k+1}={\textstyle \bigwedge }^{\!k}(V)}
8732:
8009:
6718:
6639:
27778:
27457:
23902:
17911:
17420:
15651:
8602:
24332:
21266:
21225:
21022:
18732:
17615:
17100:
16366:
15346:
9125:
9084:
6208:
6021:
5975:
5453:
5410:
5001:
27237:
26599:
26366:
26239:
26175:
26149:
26030:
26002:
25765:
21652:{\displaystyle {\textstyle \bigwedge }(V\oplus W)\cong {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(W).}
13890:
is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the
10613:
26322:
25405:
characterization of dimension, a property that had previously only been examined from the coordinate point of view.
23491:
12359:
27659:
27176:
26732:
24646:
23973:
7730:-vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
24734:: it is just the set of all of the points in the exterior algebra. The topology on this space is essentially the
21576:
In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:
20847:{\displaystyle {\textstyle \bigwedge }(f)(x_{1}\wedge \cdots \wedge x_{k})=f(x_{1})\wedge \cdots \wedge f(x_{k}).}
20002:
can be given in terms of the coproduct. The construction of the bialgebra here parallels the construction in the
18717:{\displaystyle \Delta :{\textstyle \bigwedge }(V)\to {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(V)}
15699:
9633:
4824:
4621:
27138:
27074:
17417:. This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on
15986:{\displaystyle \iota _{\alpha }(a\wedge b)=(\iota _{\alpha }a)\wedge b+(-1)^{\deg a}a\wedge (\iota _{\alpha }b).}
5529:
26564:
26521:
24443:
24048:
23338:
20663:
19579:
18988:
18605:
18523:
15595:{\displaystyle \iota _{\alpha }:{\textstyle \bigwedge }^{\!k}(V)\rightarrow {\textstyle \bigwedge }^{\!k-1}(V).}
14772:{\displaystyle \iota _{\alpha }:{\textstyle \bigwedge }^{\!k}(V)\rightarrow {\textstyle \bigwedge }^{\!k-1}(V).}
14654:
13860:{\displaystyle \omega {\dot {\wedge }}\eta ={\frac {(k+m)!}{k!\,m!}}\operatorname {Alt} (\omega \otimes \eta ),}
13631:
11457:
10580:
9349:
9281:
9168:
9051:
9007:
8959:
8924:
8821:
8696:
8440:
8387:
8304:
5050:
26380:
Calcolo
Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva
26214:
20006:
article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.
12800:
17:
12667:{\displaystyle (\iota _{\alpha }t)^{i_{1}\cdots i_{r-1}}=r\sum _{j=0}^{n}\alpha _{j}t^{ji_{1}\cdots i_{r-1}}.}
3757:
27510:
27442:
26684:
26194:
25937:
25888:
22294:{\displaystyle F^{p+1}/F^{p}\cong {\textstyle \bigwedge }^{\!k-p}(U)\otimes {\textstyle \bigwedge }^{\!p}(W)}
19646:
14646:
11677:
4230:
25280:
20009:
In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:
15462:
7126:{\displaystyle \alpha ^{(i)}=\alpha _{1}^{(i)}\wedge \cdots \wedge \alpha _{k}^{(i)},\quad i=1,2,\ldots ,s.}
5733:
1091:
The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.
974:
27886:
27535:
26916:
26788:
25805:
25461:
22856:
21343:
20499:{\displaystyle {\textstyle \bigwedge }(f):{\textstyle \bigwedge }(V)\rightarrow {\textstyle \bigwedge }(W)}
13023:
10959:
643:
23931:
20702:
19609:). The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements
14110:, an equivalent version of the second expression without any factorials or any constants is well-defined:
10910:
10764:
27891:
27773:
27021:
26871:
26679:
24642:
24159:
represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented
15288:
13193:
9815:
15275:{\displaystyle (\iota _{\alpha }w)(u_{1},u_{2},\ldots ,u_{k-1})=w(\alpha ,u_{1},u_{2},\ldots ,u_{k-1}).}
27584:
27404:
26926:
26820:
26716:
introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges.
24698:
24689:
24210:
23693:
13251:
12990:-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on
12464:
5687:
4651:
1018:
which is equivalent in other characteristics). More generally, the exterior algebra can be defined for
106:
26726:
26674:
16696:{\displaystyle \star :{\textstyle \bigwedge }^{\!k}(V)\rightarrow {\textstyle \bigwedge }^{\!n-k}(V).}
15837:
15802:
15427:
14613:
11390:
9604:, hence, a good choice to represent the quotient. But for nonzero characteristic, the vector space of
8277:
studies additional graded structures on exterior algebras, such as those on the exterior algebra of a
5496:
1426:
1101:
27256:
27166:
26815:
14107:
13121:
11492:
11145:
10731:
10696:
10440:
9707:
9498:
9424:
9316:
936:
27738:
25257:, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra
11235:
10862:
10384:
10344:
7410:
3572:
is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns
919:
The definition of the exterior algebra can be extended for spaces built from vector spaces, such as
27840:
27758:
27712:
27419:
27158:
27041:
26467:
26287:
23155:{\displaystyle (-1)^{j}A_{j}A_{0}\wedge A_{j}A_{1}\wedge A_{j}A_{2}\wedge \cdots \wedge A_{j}A_{k}}
21856:
19125:{\displaystyle \Delta (v\wedge w)=1\otimes (v\wedge w)+v\otimes w-w\otimes v+(v\wedge w)\otimes 1.}
17656:
6979:
3836:
3599:
2332:
2149:{\displaystyle \mathbf {e} _{2}\wedge \mathbf {e} _{1}=-(\mathbf {e} _{1}\wedge \mathbf {e} _{2}).}
23821:
22780:
20174:
17198:. The pairing between these two spaces also takes the form of an inner product. On decomposable
13294:
12731:
12172:
The components of this tensor are precisely the skew part of the components of the tensor product
2229:{\displaystyle \mathbf {e} _{1}\wedge \mathbf {e} _{1}=\mathbf {e} _{2}\wedge \mathbf {e} _{2}=0.}
1696:
1672:
1479:
1457:
27810:
27497:
27414:
27384:
27204:
27133:
26911:
26781:
24599:
23617:
22157:
11259:
10408:
8345:
4683:
3968:
2414:
affects neither the base nor the height of the parallelogram and consequently preserves its area.
1132:
942:
611:
498:
212:
25390:
also published similar ideas of exterior calculus for which he claimed priority over
Grassmann.
24956:{\displaystyle \partial :{\textstyle \bigwedge }^{\!p+1}(L)\to {\textstyle \bigwedge }^{\!p}(L)}
22124:
9760:
8536:
3702:
27768:
27624:
27579:
26968:
26901:
26891:
26408:
25436:
25398:
25319:
24650:
24136:
23774:
23629:
23605:
minors of a matrix can be defined by looking at the exterior products of column vectors chosen
21105:
17019:
13386:
9440:
9042:
6137:
5783:
4214:{\displaystyle 0=(x+y)\wedge (x+y)=x\wedge x+x\wedge y+y\wedge x+y\wedge y=x\wedge y+y\wedge x}
3192:{\displaystyle \mathbf {w} =w_{1}\mathbf {e} _{1}+w_{2}\mathbf {e} _{2}+w_{3}\mathbf {e} _{3},}
2454:
parallelogram). The length of the cross product is to the length of the parallel unit vector (
1096:
924:
510:
26761:
C. Burali-Forti, "Introduction to
Differential Geometry, following the method of H. Grassmann"
26692:
25990:
25934:
are projective modules over a commutative ring. Otherwise, it is generally not the case that
23780:
20399:
19979:
19957:
19937:
19643:
Observe that the coproduct preserves the grading of the algebra. Extending to the full space
10131:
9200:. This approach is often used in differential geometry and is described in the next section.
8504:
4059:
2738:{\displaystyle \mathbf {v} =v_{1}\mathbf {e} _{1}+v_{2}\mathbf {e} _{2}+v_{3}\mathbf {e} _{3}}
2635:{\displaystyle \mathbf {u} =u_{1}\mathbf {e} _{1}+u_{2}\mathbf {e} _{2}+u_{3}\mathbf {e} _{3}}
27850:
27805:
27285:
27230:
26983:
26978:
26973:
26906:
26851:
26713:
26510:
Includes applications of the exterior algebra to differential forms, specifically focused on
24677:
24653:
24575:
24567:
23648:
23612:
22005:
19917:
19135:
Expanding this out in detail, one obtains the following expression on decomposable elements:
18449:{\displaystyle \mathbf {x} ^{\flat }(\mathbf {y} )=\langle \mathbf {x} ,\mathbf {y} \rangle }
16419:
16116:{\displaystyle \iota _{\alpha }\circ \iota _{\beta }=-\iota _{\beta }\circ \iota _{\alpha }.}
14823:
14791:
10803:
7516:
7493:
7457:
7309:
7238:
7214:
7177:
7149:
6861:
6479:
5352:
5288:
5257:
4725:
4658:
4283:
4000:
3816:
601:
291:
192:
90:
23911:
18920:{\displaystyle K\simeq {\textstyle \bigwedge }^{\!0}(V)\subseteq {\textstyle \bigwedge }(V)}
18790:
17970:
17779:
16709:
12993:
9535:
7463:
6441:
5193:{\displaystyle x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k},\quad x_{i}\in V,i=1,2,\dots ,k.}
3728:
27825:
27753:
27639:
27505:
27467:
27399:
26993:
26958:
26945:
26836:
26424:
25343:
24693:
24656:. The exterior algebra of differential forms, equipped with the exterior derivative, is a
23688:
23644:
23588:
23279:
21978:
21385:
19612:
17071:
16583:
15768:
15094:
14919:
14556:
13624:
13579:
13160:
11833:
9527:
9252:
8597:
7701:
7606:{\displaystyle {\underset {p}{\underbrace {\alpha \wedge \cdots \wedge \alpha } }}\neq 0\ }
7258:
6769:
6247:
6176:
6147:
6141:
6106:
6073:
4413:
4382:
4351:
1019:
932:
928:
26763:
An
English translation of an early book on the geometric applications of exterior algebras
26347:
24555:
24551:
23657:
20432:. Then, by the universal property, there exists a unique homomorphism of graded algebras
11587:
11556:
11424:
11337:
11094:
10930:
10663:
7671:{\displaystyle \ {\underset {p+1}{\underbrace {\alpha \wedge \cdots \wedge \alpha } }}=0.}
6764:), and therefore its dimension is equal to the sum of the binomial coefficients, which is
8:
27702:
27525:
27515:
27364:
27349:
27305:
27171:
27051:
27026:
26876:
26515:
26202:
25533:
25485:
25456:
25359:
25236:
24638:
23906:
23186:
22830:
21155:
17902:
16132:
15066:
14084:
13263:
12498:
9444:
8777:
6686:
3936:
3677:
3646:
3549:
605:
155:
26769:
An
English translation of one Grassmann's papers on the applications of exterior algebra
26428:
23165:
19974:
by the wedge symbol, with one exception. One can construct an alternating product from
17600:{\displaystyle e_{i_{1}}\wedge \cdots \wedge e_{i_{k}},\quad i_{1}<\cdots <i_{k},}
10160:
825:
778:
560:
331:
264:
161:
27835:
27692:
27545:
27359:
27295:
26881:
26440:
26387:
26307:
25868:
25848:
25817:
25785:
25622:
25602:
25260:
24861:
24837:
24809:
24787:
24757:
24665:
24661:
24631:
24419:
24385:
24296:
24272:
24248:
24188:
24166:
23373:
23313:
23256:
23234:
23212:
23192:
22921:
22804:
22774:
22752:
21848:{\textstyle 0\to U\mathrel {\overset {f}{\to }} V\mathrel {\overset {g}{\to }} W\to 0,}
21313:
21203:
21181:
21133:
21085:
21065:
20379:
20359:
19810:
The tensor symbol ⊗ used in this section should be understood with some caution: it is
18840:
18818:
18558:
18372:
17489:
17203:
17049:
17027:
16995:
16803:
16561:
16162:
16140:
15746:
15629:
15609:
15403:
15324:
14948:
14899:
14589:
14536:
14089:
13873:
13606:
13559:
13539:
13461:
13368:
13348:
13240:
13154:
12973:
12953:
12933:
12787:
12430:
11368:
11123:
11074:
10888:
10840:
9853:
9790:
9738:
9687:
9667:
9609:
9585:
9563:
9478:
9458:
9406:
9382:
9259:
9234:
9214:
8473:
7536:
7287:
6694:
6364:
6336:
6098:
5789:
5713:
5693:
5386:
5330:
5308:
4977:
4444:
4035:
3946:
3682:
3651:
3628:
1057:
1033:
1027:
899:
879:
859:
853:
805:
754:
734:
694:
688:
583:
540:
464:
444:
356:
311:
244:
137:
26463:
Grassmann algebra – Exploring applications of
Extended Vector Algebra with Mathematica
22777:, as to integrate we need a 'differential' object to measure infinitesimal volume. If
4309:
27881:
27830:
27599:
27574:
27389:
27300:
27280:
27079:
27036:
26963:
26856:
26736:
26700:
26655:
26498:
26481:
26394:
26362:
26335:
26235:
26223:
26198:
26171:
26145:
26138:
26133:
26026:
25998:
25761:
25649:
25538:
25490:
25475:
25440:
25429:
25371:
24685:
24613:
24595:
24571:
24482:
23652:
18578:
17995:. This property completely characterizes the inner product on the exterior algebra.
17481:
16795:
15862:
14974:
13390:
11286:
did before, but the product cannot be defined as above. In such a case, isomorphism
11061:{\displaystyle t\wedge s=t~{\widehat {\otimes }}~s=\operatorname {Alt} (t\otimes s).}
9400:
4301:
3585:
2460:) as the size of the exterior product is to the size of the reference parallelogram (
1087:
285:
25711:, Theorem XVI.6.8). More detail on universal properties in general can be found in
25417:
22114:{\displaystyle u_{1}\wedge \ldots \wedge u_{k+1-p}\wedge v_{1}\wedge \ldots v_{p-1}}
16787:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)\to {\textstyle \bigwedge }^{\!k}(V)}
16353:{\displaystyle \iota _{\alpha \wedge \beta }=\iota _{\beta }\circ \iota _{\alpha }.}
8998:
As a consequence of this construction, the operation of assigning to a vector space
27845:
27520:
27487:
27472:
27354:
27223:
26988:
26841:
26722:
26444:
26432:
26328:
26299:
26253:
26227:
26163:
25976:
25809:
25653:
25466:
25444:
24727:
24591:
24486:
16960:
14528:
13341:
9628:-linear antisymmetric tensors could be not transversal to the ideal (actually, for
9209:
8335:
7232:
1023:
485:. The wedge product was introduced originally as an algebraic construction used in
433:
26484:, with a focus on applications. Also includes a history section and bibliography.
25447:
by placing the axiomatic notion of an algebraic system on a firm logical footing.
24641:
gives the exterior algebra of differential forms on a manifold the structure of a
19934:
clearly corresponds to multiplication in the exterior algebra, leaving the symbol
2252:
of the signed area is the ordinary area, and the sign determines its orientation.
876:
in the exterior algebra is also valid in every associative algebra that contains
27815:
27763:
27707:
27687:
27589:
27477:
27344:
27315:
27143:
26936:
26896:
26886:
25509:
25339:
25228:
24657:
24377:
13338:
12723:
10336:
8752:, it is natural to start with the most general associative algebra that contains
8339:
5042:
4027:
2067:
1051:
23579:, the exterior product provides an abstract algebraic manner for describing the
9346:. It will satisfy the analogous universal property. Many of the properties of
27855:
27820:
27717:
27550:
27540:
27530:
27452:
27424:
27409:
27394:
27310:
27148:
27069:
26804:
26647:
26490:
26375:
25496:
25480:
25470:
25425:
25413:
25355:
24240:
23898:
23584:
23576:
20003:
19911:
18590:
18586:
17393:
13703:
11673:
9427:
and projective. Generalizations to the most common situations can be found in
8759:
7884:
3673:
3524:
3085:). The coefficients above are the same as those in the usual definition of the
2514:
2249:
846:
687:
The full exterior algebra contains objects that are not themselves blades, but
27800:
26065:
J Itard, Biography in
Dictionary of Scientific Biography (New York 1970–1990).
25421:
25412:
was lost to mid-19th-century mathematicians, until being thoroughly vetted by
21561:{\displaystyle 0\to {\textstyle \bigwedge }(U)\to {\textstyle \bigwedge }(V).}
16181:. Then the interior product induces a canonical isomorphism of vector spaces
13753:{\displaystyle \omega \wedge \eta =\operatorname {Alt} (\omega \otimes \eta )}
2438:
27875:
27792:
27697:
27609:
27482:
27181:
27104:
27064:
27031:
27011:
25972:
25505:
25311:
24829:
24735:
24719:
24710:
The exterior algebra over the complex numbers is the archetypal example of a
24681:
24603:
24587:
24160:
23025:{\displaystyle A_{0}A_{1}\wedge A_{0}A_{2}\wedge \cdots \wedge A_{0}A_{k}={}}
19396:
17015:
16555:
14402:
10381:. This extends by linearity and homogeneity to an operation, also denoted by
9435:
8773:
8278:
3669:
3086:
2510:
534:
30:"Wedge product" redirects here. For the operation on topological spaces, see
19862:{\displaystyle {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(V)}
18976:{\displaystyle {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(V)}
9455:
For a field of characteristic not 2, the exterior algebra of a vector space
9403:. Where finite dimensionality is used, the properties further require that
4750:, then the following generalization of the alternating property also holds:
27860:
27664:
27649:
27614:
27462:
27447:
27114:
27003:
26953:
26846:
26318:
26018:
25593:
is not equivalent to and should not be confused with similar terms such as
25501:
25383:
24743:
24711:
24542:{\textstyle \mathbf {P} {\bigl (}{\textstyle \bigwedge }^{\!k}(V){\bigr )}}
24405:
22770:
20345:
18582:
2340:
920:
132:
19914:. Here, there is much less of a problem, in that the alternating product
2268:
denotes the signed area of the parallelogram of which the pair of vectors
27748:
27722:
27644:
27333:
27272:
27094:
27059:
27016:
26861:
25409:
23580:
21305:
16406:
16363:
In the geometrical setting, a non-zero element of the top exterior power
14417:
13891:
7197:
6966:{\displaystyle \alpha =\alpha ^{(1)}+\alpha ^{(2)}+\cdots +\alpha ^{(s)}}
5082:
2497:
2245:
1503:
1136:
1128:
728:
723:
712:
102:
18520:
There is a correspondence between the graded dual of the graded algebra
27629:
27123:
26866:
26311:
25354:
The exterior algebra is the main ingredient in the construction of the
24731:
23968:
23595:
in terms of the exterior product of the column vectors. Likewise, the
20429:
14581:
13063:
12342:{\displaystyle (t~{\widehat {\otimes }}~s)^{i_{1}\cdots i_{r+p}}=t^{}.}
8499:
6015:. The reason is the following: given any exterior product of the form
3588:, the exterior product generalizes these notions to higher dimensions.
2276:
form two adjacent sides, then A must satisfy the following properties:
799:
26048:
published a translation of
Grassmann's work in English; he translated
24239:
represents the spanned 3-space weighted by the volume of the oriented
17368:
the determinant of the matrix of inner products. In the special case
13449:{\displaystyle {\bigl (}{\textstyle \bigwedge }^{\!k}(V){\bigr )}^{*}}
10726:, and it inherits the structure of a graded vector space from that on
8273:
In addition to studying the graded structure on the exterior algebra,
6321:{\displaystyle \dim {\textstyle \bigwedge }^{\!k}(V)={\binom {n}{k}},}
3915:{\displaystyle \alpha \wedge \beta =\alpha \otimes \beta {\pmod {I}}.}
2156:(The fact that the exterior product is an alternating map also forces
27604:
27555:
26921:
26436:
26389:
Geometric calculus: According to the Ausdehnungslehre of H. Grassmann
24739:
20244:
19019:
18638:
18597:
18596:
The exterior product of multilinear forms defined above is dual to a
721:, while a more general sum of blades of arbitrary degree is called a
98:
31:
26303:
26213:
This book contains applications of exterior algebras to problems in
26074:
Authors have in the past referred to this calculus variously as the
24714:, which plays a fundamental role in physical theories pertaining to
23611:
at a time. These ideas can be extended not just to matrices but to
22773:. This is also the intimate connection between exterior algebra and
17189:{\displaystyle {\bigl (}{\textstyle \bigwedge }^{\!k}V{\bigr )}^{*}}
13288:, the base field. In this case an alternating multilinear function
54:
42:
27634:
27619:
27089:
26767:"Mechanics, according to the principles of the theory of extension"
26511:
25595:
24583:
24579:
21794:
In greater generality, for a short exact sequence of vector spaces
13111:{\displaystyle \phi :{\textstyle \bigwedge }^{\!k}(V)\rightarrow X}
9935:{\displaystyle v_{1}\otimes \cdots \otimes v_{r},\quad v_{i}\in V.}
8263:{\displaystyle \alpha \wedge \beta =(-1)^{kp}\beta \wedge \alpha .}
6879:
6470:
5276:
3090:
502:
486:
25439:, borrowing from the ideas of Peano and Grassmann, introduced his
23560:
23551:
23542:
19557:{\displaystyle v_{\sigma (p+1)}\wedge \dots \wedge v_{\sigma (k)}}
18048:{\displaystyle \mathbf {v} \in {\textstyle \bigwedge }^{\!k-l}(V)}
17712:{\displaystyle \mathbf {v} \in {\textstyle \bigwedge }^{\!k-1}(V)}
13576:-dimensional, the dimension of the space of alternating maps from
60:
Reversed orientation corresponds to negating the exterior product.
27328:
27290:
26773:
26217:. Rank and related concepts are developed in the early chapters.
25995:
Studies in Econometrics, Time Series, and Multivariate Statistics
24715:
23684:
23620:-independent way to talk about the minors of the transformation.
23475:{\displaystyle (u_{1}+u_{2})\wedge v=u_{1}\wedge v+u_{2}\wedge v}
22824:
11175:, with the above given product, there is a canonical isomorphism
9038:
8772:, and then enforce the alternating property by taking a suitable
1454:, written in components. There is a unique parallelogram having
25443:. This then paved the way for the 20th-century developments of
19871:. Any lingering doubt can be shaken by pondering the equalities
19487:{\displaystyle v_{\sigma (1)}\wedge \dots \wedge v_{\sigma (p)}}
18503:{\displaystyle \mathbf {y} \in {\textstyle \bigwedge }^{\!l}(V)}
18154:{\displaystyle \mathbf {x} \in {\textstyle \bigwedge }^{\!l}(V)}
18101:{\displaystyle \mathbf {w} \in {\textstyle \bigwedge }^{\!k}(V)}
17765:{\displaystyle \mathbf {w} \in {\textstyle \bigwedge }^{\!k}(V)}
16404:(which is a one-dimensional vector space) is sometimes called a
8106:
The exterior product is graded anticommutative, meaning that if
4026:
by the above construction. It follows that the product is also
27654:
27246:
27099:
25993:. In Karlin, Samuel; Amemiya, Takeshi; Goodman, Leo A. (eds.).
25845:
This part of the statement also holds in greater generality if
24649:
along smooth mappings between manifolds, and it is therefore a
18985:. This definition of the coproduct is lifted to the full space
13527:{\displaystyle {\textstyle \bigwedge }^{\!k}\left(V^{*}\right)}
13212:
6469:
Any element of the exterior algebra can be written as a sum of
494:
24730:. The exterior algebra itself is then just a one-dimensional
9122:
as certain subspaces, one may alternatively define the spaces
4947:{\displaystyle x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k}=0.}
4811:{\displaystyle x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k}=0.}
3540:
can be interpreted as a vector which is perpendicular to both
26341:
Die Lineale Ausdehnungslehre – Ein neuer Zweig der Mathematik
26193:
24554:, and the image of the embedding can be characterized by the
10658:(not yet an algebra, as product is not yet defined), denoted
7208:
2066:, and the last uses the fact that the exterior product is an
26346:(The Linear Extension Theory – A new Branch of Mathematics)
20335:{\displaystyle S(x)=(-1)^{\binom {{\text{deg}}\,x\,+1}{2}}x}
20236:{\displaystyle \varepsilon :{\textstyle \bigwedge }(V)\to K}
19815:
definition of a bialgebra, that is, for creating the object
16959:
where id is the identity mapping, and the inner product has
15389:
12920:{\displaystyle w:V^{k}\to {\textstyle \bigwedge }^{\!k}(V),}
9311:
just as above, as a suitable quotient of the tensor algebra
5671:{\displaystyle \alpha =e_{1}\wedge e_{2}+e_{3}\wedge e_{4}.}
288:, and the names of the product come from the "wedge" symbol
25760:. W.H. Freeman & Co. pp. 58–60, 83, 100–9, 115–9.
24606:
at the point. Equivalently, a differential form of degree
23308:-dimensional oriented areas of the boundary simplexes of a
20944:
20569:
11387:, but then, the product should be modified as given below (
9580:-linear antisymmetric tensors is transversal to the ideal
8146:{\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}
6849:{\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}
5245:{\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}
856:
in the sense that every equation that relates elements of
490:
27215:
24688:. Together, these constructions are used to generate the
19430:}}. It is also convenient to take the pure wedge products
15030:{\displaystyle V^{*}\times V^{*}\times \dots \times V^{*}}
9495:
can be canonically identified with the vector subspace of
8472:. In other words, the exterior algebra has the following
8195:{\displaystyle \beta \in {\textstyle \bigwedge }^{\!p}(V)}
7397:{\displaystyle \alpha =\sum _{i,j}a_{ij}e_{i}\wedge e_{j}}
7203:
Rank is particularly important in the study of 2-vectors (
425:{\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}}
26290:(1878), "Applications of Grassmann's Extensive Algebra",
24664:
of the underlying manifold and plays a vital role in the
16043:{\displaystyle \iota _{\alpha }\circ \iota _{\alpha }=0.}
15394:
The interior product satisfies the following properties:
13198:
12356:
may also be described in index notation as follows. Let
7883:
gives the exterior algebra the additional structure of a
24722:. A single element of the exterior algebra is called a
23273:-dimensional simplex is a scalar multiple of any other.
22190:
The corresponding quotients admit a natural isomorphism
117:-dimensional boundary and on which side the interior is.
26452:
22938:-dimensional volume as the exterior product of vectors
2062:
where the first step uses the distributive law for the
26749:
Chapter 10: The Exterior Product and Exterior Algebras
26605:
26602:
26570:
26567:
26553:
in this text is used to mean the space of alternating
26527:
26524:
25942:
25893:
25677:
This definition is a standard one. See, for instance,
25285:
25283:
24931:
24898:
24684:
on the category of vector spaces, the other being the
24510:
24495:
24338:
24133:-vectors have geometric interpretations: the bivector
23343:
22683:
22656:
22630:
22567:
22540:
22507:
22269:
22236:
21940:
21800:
21756:
21729:
21674:
21631:
21612:
21587:
21540:
21521:
21470:
21451:
21432:
21406:
21272:
21231:
21028:
20979:
20952:
20920:
20895:
20869:
20741:
20705:
20668:
20622:
20589:
20545:
20520:
20481:
20462:
20443:
20212:
19844:
19825:
19770:
19744:
19697:
19651:
19649:
19584:
18993:
18958:
18939:
18902:
18876:
18699:
18680:
18661:
18610:
18528:
18478:
18321:
18278:
18129:
18076:
18017:
17740:
17681:
17621:
17426:
17156:
17106:
16873:
16846:
16762:
16735:
16662:
16635:
16484:
16450:
16372:
16254:
16227:
16193:
15660:
15561:
15534:
15352:
14862:
14738:
14711:
14659:
13635:
13493:
13410:
13080:
12892:
11462:
11309:
11201:
9354:
9286:
9173:
9131:
9090:
9056:
9012:
8964:
8929:
8862:
8826:
8701:
8613:
8445:
8392:
8309:
8170:
8121:
8062:
8015:
7953:
7926:
7899:
7848:
7815:
7788:
7761:
7741:
6824:
6724:
6645:
6599:
6566:
6539:
6512:
6492:
6478:. Hence, as a vector space the exterior algebra is a
6429:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)=\{0\}}
6392:
6268:
6214:
5981:
5459:
5416:
5220:
5055:
5007:
1587:
1539:
1351:
1277:
1217:
1170:
1050:
variables is an exterior algebra over the ring of the
25940:
25891:
25871:
25851:
25820:
25788:
25664:, Theorem IX.2.2). For an elementary treatment, see
25625:
25605:
25322:
25263:
25239:
24975:
24889:
24864:
24840:
24812:
24790:
24760:
24680:, the exterior algebra is one of the two fundamental
24446:
24422:
24388:
24335:
24299:
24275:
24251:
24213:
24191:
24169:
24139:
24051:
23976:
23934:
23914:
23824:
23783:
23696:
23660:
23616:
transformation on the lesser exterior powers gives a
23494:
23398:
23376:
23341:
23316:
23282:
23259:
23237:
23215:
23195:
23168:
23037:
22944:
22924:
22859:
22833:
22807:
22783:
22755:
22621:
22492:
22307:
22199:
22160:
22127:
22034:
22008:
21981:
21867:
21671:
21585:
21513:
21397:
21346:
21316:
21269:
21228:
21206:
21184:
21158:
21136:
21108:
21088:
21068:
21025:
20866:
20739:
20666:
20518:
20441:
20402:
20382:
20362:
20262:
20204:
20177:
20018:
19982:
19960:
19940:
19920:
19823:
19688:
19615:
19582:
19500:
19436:
19144:
19031:
18991:
18937:
18867:
18843:
18821:
18793:
18735:
18653:
18608:
18561:
18526:
18467:
18400:
18375:
18260:
18172:
18118:
18065:
18006:
17973:
17914:
17813:
17782:
17729:
17670:
17618:
17517:
17492:
17423:
17231:
17206:
17146:
17103:
17074:
17052:
17030:
16998:
16831:
16806:
16732:
16712:
16626:
16586:
16564:
16447:
16422:
16369:
16304:
16190:
16165:
16143:
16057:
16010:
15873:
15840:
15805:
15771:
15749:
15702:
15654:
15632:
15612:
15518:
15465:
15430:
15406:
15349:
15327:
15291:
15129:
15097:
15069:
14984:
14951:
14922:
14902:
14887:{\displaystyle w\in {\textstyle \bigwedge }^{\!k}(V)}
14853:
14826:
14794:
14695:
14657:
14616:
14592:
14559:
14539:
14119:
14092:
13903:
13876:
13772:
13716:
13634:
13609:
13582:
13562:
13542:
13490:
13464:
13400:
13371:
13351:
13297:
13266:
13163:
13124:
13071:
13026:
12996:
12976:
12956:
12936:
12870:
12803:
12734:
12532:
12501:
12467:
12433:
12362:
12191:
11864:
11689:
11590:
11559:
11495:
11460:
11427:
11393:
11371:
11340:
11292:
11262:
11238:
11222:{\displaystyle A(V)\cong {\textstyle \bigwedge }(V).}
11184:
11148:
11126:
11097:
11077:
10994:
10962:
10933:
10913:
10891:
10865:
10843:
10806:
10767:
10734:
10699:
10666:
10616:
10583:
10482:
10443:
10411:
10387:
10347:
10189:
10163:
10134:
9962:
9879:
9856:
9818:
9793:
9763:
9741:
9710:
9690:
9670:
9636:
9612:
9588:
9566:
9538:
9501:
9481:
9461:
9443:. More general exterior algebras can be defined for
9409:
9385:
9352:
9319:
9284:
9262:
9237:
9217:
9171:
9128:
9087:
9054:
9010:
8962:
8927:
8905:{\displaystyle {\textstyle \bigwedge }(V)=T(V)\,/\,I}
8860:
8824:
8699:
8605:
8539:
8507:
8443:
8390:
8348:
8307:
8213:
8161:
8112:
8059:
8012:
7896:
7739:
7704:
7619:
7562:
7539:
7519:
7496:
7490:
is therefore even, and is twice the rank of the form
7466:
7413:
7335:
7312:
7290:
7261:
7241:
7217:
7180:
7152:
7018:
6982:
6896:
6864:
6815:
6772:
6721:
6697:
6642:
6490:
6444:
6389:
6367:
6339:
6259:
6211:
6179:
6150:
6109:
6076:
6024:
5978:
5815:
5792:
5736:
5716:
5696:
5609:
5532:
5499:
5456:
5413:
5389:
5355:
5333:
5311:
5291:
5260:
5211:
5094:
5053:
5004:
4980:
4895:
4827:
4759:
4728:
4686:
4661:
4624:
4472:
4447:
4416:
4385:
4354:
4312:
4286:
4233:
4099:
4062:
4038:
4003:
3971:
3949:
3871:
3839:
3819:
3760:
3731:
3705:
3685:
3654:
3631:
3602:
3211:
3105:
2757:
2654:
2551:
2162:
2076:
1726:
1699:
1675:
1515:
1482:
1460:
1429:
1263:
1147:
1104:
1060:
1036:
977:
945:
902:
882:
862:
828:
808:
781:
757:
737:
697:
646:
614:
586:
563:
543:
513:
467:
447:
379:
359:
334:
314:
294:
267:
247:
215:
195:
164:
140:
26413:"A Treatise on Universal Algebra, with Applications"
25715:, Chapter VI), and throughout the works of Bourbaki.
25469:, a generalization of exterior algebra to a nonzero
21504:
is an exact sequence of graded vector spaces, as is
16001:
Further properties of the interior product include:
13157:
characterizes the space of alternating operators on
11327:{\displaystyle A(V)\cong {\textstyle \bigwedge }(V)}
8740:
To construct the most general algebra that contains
8334:
is performed by manipulating symbols and imposing a
3519:
is the basis vector for the one-dimensional space ⋀(
2485:). In other words, the exterior product provides a
25755:
25690:A proof of this can be found in more generality in
25557:
25428:) who applied Grassmann's ideas to the calculus of
24828:, then it is possible to define the structure of a
21304:to itself, and is therefore given by a scalar: the
16550:If, in addition to a volume form, the vector space
8281:(a module that already carries its own gradation).
8096:{\displaystyle {\textstyle \bigwedge }^{\!1}(V)=V.}
26721:
26633:
26588:
26545:
26386:
26298:(4), The Johns Hopkins University Press: 350–358,
26137:
25950:
25922:This statement generalizes only to the case where
25901:
25877:
25857:
25826:
25794:
25631:
25611:
25328:
25302:
25269:
25247:
25216:
24955:
24870:
24846:
24818:
24796:
24766:
24541:
24471:
24428:
24394:
24362:
24305:
24281:
24257:
24231:
24197:
24175:
24151:
24112:
24037:
23959:
23920:
23886:
23810:
23765:
23675:
23526:
23474:
23382:
23360:
23322:
23300:
23265:
23243:
23221:
23201:
23177:
23154:
23024:
22930:
22910:
22845:
22813:
22791:
22761:
22744:
22725:
22597:
22468:
22293:
22182:
22146:
22113:
22020:
21994:
21964:
21847:
21783:
21651:
21560:
21493:
21376:
21322:
21296:
21255:
21212:
21190:
21170:
21142:
21120:
21094:
21074:
21052:
21006:
20846:
20722:
20685:
20649:
20498:
20420:
20388:
20368:
20334:
20235:
20183:
20137:
19988:
19966:
19946:
19926:
19861:
19799:
19671:
19628:
19601:
19556:
19486:
19384:
19124:
19010:
18975:
18919:
18849:
18827:
18805:
18774:
18716:
18627:
18567:
18545:
18502:
18448:
18381:
18359:
18243:
18163:, iteration of the above adjoint properties gives
18153:
18100:
18047:
17985:
17954:
17880:
17794:
17764:
17711:
17645:
17599:
17498:
17453:
17357:
17212:
17188:
17130:
17087:
17058:
17036:
17004:
16948:
16812:
16786:
16718:
16695:
16599:
16570:
16539:
16428:
16396:
16352:
16284:
16171:
16149:
16115:
16042:
15985:
15853:
15824:
15784:
15755:
15733:
15684:
15638:
15618:
15594:
15502:
15449:
15412:
15376:
15333:
15313:
15274:
15110:
15081:
15029:
14957:
14935:
14908:
14886:
14832:
14800:
14771:
14676:
14635:
14598:
14572:
14545:
14364:
14098:
14072:
13882:
13859:
13752:
13661:
13615:
13595:
13568:
13548:
13526:
13470:
13448:
13377:
13357:
13322:
13278:
13258:The above discussion specializes to the case when
13176:
13145:
13110:
13054:
13012:
12982:
12962:
12942:
12919:
12850:
12759:
12666:
12513:
12483:
12439:
12417:
12341:
12161:
11817:
11605:
11574:
11545:
11479:
11442:
11408:
11377:
11355:
11326:
11278:
11248:
11221:
11165:
11132:
11112:
11083:
11060:
10977:
10948:
10919:
10897:
10875:
10849:
10825:
10790:
10751:
10716:
10681:
10646:
10602:
10566:
10460:
10427:
10397:
10371:
10324:
10172:
10149:
10117:
9934:
9862:
9842:
9799:
9775:
9747:
9727:
9696:
9676:
9654:
9618:
9594:
9572:
9550:
9518:
9487:
9467:
9415:
9391:
9371:
9336:
9303:
9268:
9243:
9223:
9190:
9155:
9114:
9073:
9029:
8981:
8946:
8904:
8843:
8718:
8636:
8572:
8525:
8462:
8409:
8366:
8326:
8262:
8194:
8145:
8095:
8046:{\displaystyle {\textstyle \bigwedge }^{\!0}(V)=K}
8045:
7986:
7872:
7722:
7670:
7605:
7545:
7525:
7502:
7482:
7448:
7396:
7318:
7296:
7274:
7247:
7223:
7186:
7158:
7125:
7001:
6965:
6870:
6848:
6785:
6755:{\displaystyle {\textstyle \bigwedge }^{\!1}(V)=V}
6754:
6703:
6676:{\displaystyle {\textstyle \bigwedge }^{\!0}(V)=K}
6675:
6623:
6456:
6428:
6373:
6345:
6320:
6238:
6192:
6163:
6122:
6089:
6059:
6005:
5959:
5798:
5774:
5722:
5702:
5670:
5590:
5514:
5483:
5440:
5395:
5361:
5339:
5317:
5297:
5266:
5244:
5192:
5072:
5031:
4986:
4946:
4886:to be a linearly dependent set of vectors is that
4878:
4810:
4740:
4712:
4667:
4642:
4607:
4453:
4429:
4398:
4367:
4336:
4292:
4269:
4213:
4080:
4044:
4018:
3989:
3955:
3914:
3854:
3825:
3802:
3743:
3717:
3691:
3660:
3637:
3617:
3487:
3191:
3028:
2737:
2634:
2228:
2148:
2051:
1707:
1683:
1658:
1490:
1468:
1444:
1410:
1243:
1119:
1066:
1042:
1010:
963:
908:
888:
868:
837:
814:
790:
763:
743:
703:
676:
632:
592:
572:
549:
525:
473:
453:
424:
365:
343:
320:
300:
276:
253:
233:
201:
173:
146:
26612:
26577:
26534:
25660:, Historical Note). For a modern treatment, see
24938:
24905:
24566:The exterior algebra has notable applications in
24517:
24345:
22690:
22663:
22574:
22547:
22514:
22276:
22243:
21947:
21763:
21736:
21681:
21413:
21279:
21238:
21035:
20986:
20959:
20927:
20876:
20629:
20596:
20552:
18883:
18581:) inherits a bialgebra structure, and, indeed, a
18485:
18328:
18285:
18136:
18083:
18024:
17955:{\displaystyle x^{\flat }(y)=\langle x,y\rangle }
17747:
17688:
17628:
17454:{\displaystyle {\textstyle \bigwedge }^{\!k}(V).}
17433:
17163:
17113:
16880:
16853:
16769:
16742:
16669:
16642:
16491:
16457:
16379:
16261:
16234:
16200:
15685:{\displaystyle ={\textstyle \bigwedge }^{\!1}(V)}
15667:
15568:
15541:
15359:
14869:
14745:
14718:
13500:
13417:
13087:
12970:their exterior product, i.e. their corresponding
12899:
11232:When the characteristic of the field is nonzero,
9138:
9097:
8637:{\displaystyle f:{\textstyle \bigwedge }(V)\to A}
8177:
8128:
8069:
8022:
7960:
7933:
7906:
7855:
7822:
7795:
7768:
6831:
6731:
6652:
6606:
6573:
6546:
6519:
6399:
6309:
6296:
6275:
6221:
6205:By counting the basis elements, the dimension of
5988:
5466:
5423:
5227:
5014:
3081:} is the basis for the three-dimensional space ⋀(
2509:, the exterior algebra is closely related to the
1620:
1574:
1564:
1526:
691:of blades; a sum of blades of homogeneous degree
308:and the fact that the product of two elements of
48:Orientation defined by an ordered set of vectors.
27873:
26634:{\textstyle {\textstyle \bigwedge }^{\!k}V^{*}.}
26222:
25756:Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973).
25712:
25708:
25678:
25661:
24363:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
21297:{\displaystyle {\textstyle \bigwedge }^{\!n}(V)}
21256:{\displaystyle {\textstyle \bigwedge }^{\!n}(f)}
21053:{\displaystyle {\textstyle \bigwedge }^{\!k}(f)}
18775:{\displaystyle \Delta (v)=1\otimes v+v\otimes 1}
17646:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
17306:
17131:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
16397:{\displaystyle {\textstyle \bigwedge }^{\!n}(V)}
15377:{\displaystyle {\textstyle \bigwedge }^{\!0}(V)}
13702:are two anti-symmetric maps. As in the case of
13187:
11839:The exterior product of two alternating tensors
9163:first and then combine them to form the algebra
9156:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
9115:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
6239:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
6060:{\displaystyle v_{1}\wedge \cdots \wedge v_{k},}
6006:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
5484:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
5441:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
5032:{\displaystyle {\textstyle \bigwedge }^{\!k}(V)}
1579:
1531:
26693:"7. Exterior algebra and differential calculus"
26672:
21263:is a mapping of a one-dimensional vector space
16609:, then the resulting isomorphism is called the
13870:where, if the characteristic of the base field
10647:{\displaystyle {\mathcal {A}}(\mathrm {T} (V))}
9081:first and then identifying the exterior powers
1502:of this parallelogram is given by the standard
26132:
25958:converts monomorphisms to monomorphisms. See
23527:{\displaystyle v_{1}\wedge \dots \wedge v_{k}}
17388:, the inner product is the square norm of the
12418:{\displaystyle t=t^{i_{0}i_{1}\cdots i_{r-1}}}
9850:be the space of homogeneous tensors of degree
9684:-linear antisymmetric tensors is contained in
9045:of vector spaces to the category of algebras.
5600:, the following 2-vector is not decomposable:
2373:reverses the orientation of the parallelogram.
497:, and their higher-dimensional analogues: The
27231:
26789:
26755:"The Grassmann method in projective geometry"
26480:An introduction to the exterior algebra, and
26355:Extension Theory (translation of Grassmann's
25408:The import of this new theory of vectors and
25370:The exterior algebra was first introduced by
25277:to be a Lie algebra. Moreover, in that case
24582:over curves, surfaces and higher dimensional
24534:
24503:
24038:{\displaystyle J^{i}=F_{,j}^{ij}=F_{;j}^{ij}}
20322:
20294:
19395:where the second summation is taken over all
18346:
18314:
17347:
17311:
17175:
17149:
13652:
13639:
13435:
13403:
12182:, denoted by square brackets on the indices:
9450:
5888:
2448:vector) in relation to the exterior product (
26699:(2nd ed.), Springer, pp. 275–320,
25909:converts epimorphisms to epimorphisms. See
22769:-dimensional volume and exterior algebra is
18443:
18427:
18238:
18203:
18197:
18173:
17949:
17937:
17875:
17842:
17836:
17814:
17342:
17316:
15734:{\displaystyle \iota _{\alpha }v=\alpha (v)}
15497:
15491:
10366:
10348:
9655:{\displaystyle k\geq \operatorname {char} K}
8995:and satisfies the above universal property.
8956:). It is then straightforward to show that
6423:
6417:
5954:
5816:
5769:
5737:
5585:
5533:
4879:{\displaystyle \{x_{1},x_{2},\dots ,x_{k}\}}
4873:
4828:
4643:{\displaystyle \operatorname {sgn}(\sigma )}
2542:, the exterior product of a pair of vectors
2492:
896:and in which the square of every element of
26589:{\textstyle {\textstyle \bigwedge }^{\!k}V}
26546:{\textstyle {\textstyle \bigwedge }^{\!k}V}
25526:
25524:
10985:defined by (the same as the wedge product)
9870:. This is spanned by decomposable tensors
8746:and whose multiplication is alternating on
5591:{\displaystyle \{e_{1},e_{2},e_{3},e_{4}\}}
5305:can be expressed as an exterior product of
2434:, since the area of the unit square is one.
27238:
27224:
26796:
26782:
26384:
26352:
26125:
26045:
25885:are modules over a commutative ring: That
24602:is an alternating multilinear form on the
24472:{\displaystyle \operatorname {Gr} _{k}(V)}
24113:{\displaystyle F^{ij}=g^{ik}g^{jl}F_{kl}.}
23361:{\displaystyle {\textstyle \bigwedge }(V)}
20686:{\displaystyle {\textstyle \bigwedge }(f)}
19602:{\displaystyle {\textstyle \bigwedge }(V)}
19248:
19207:
19011:{\displaystyle {\textstyle \bigwedge }(V)}
18628:{\displaystyle {\textstyle \bigwedge }(V)}
18546:{\displaystyle {\textstyle \bigwedge }(V)}
14677:{\displaystyle {\textstyle \bigwedge }(V)}
13662:{\displaystyle \textstyle {\binom {n}{k}}}
13478:is finite-dimensional, then the latter is
11480:{\displaystyle {\textstyle \bigwedge }(V)}
10603:{\displaystyle \operatorname {Alt} ^{(r)}}
9372:{\displaystyle {\textstyle \bigwedge }(M)}
9304:{\displaystyle {\textstyle \bigwedge }(M)}
9191:{\displaystyle {\textstyle \bigwedge }(V)}
9074:{\displaystyle {\textstyle \bigwedge }(V)}
9030:{\displaystyle {\textstyle \bigwedge }(V)}
8982:{\displaystyle {\textstyle \bigwedge }(V)}
8947:{\displaystyle {\textstyle \bigwedge }(V)}
8844:{\displaystyle {\textstyle \bigwedge }(V)}
8736:Universal property of the exterior algebra
8719:{\displaystyle {\textstyle \bigwedge }(V)}
8463:{\displaystyle {\textstyle \bigwedge }(V)}
8410:{\displaystyle {\textstyle \bigwedge }(V)}
8327:{\displaystyle {\textstyle \bigwedge }(V)}
5073:{\displaystyle {\textstyle \bigwedge }(V)}
931:is a vector space. Moreover, the field of
26407:
26334:
26282:, Historical note on chapters II and III)
26278:
26252:
26079:
26025:. Cambridge University Press. p. 1.
25962:, Corollary to Proposition 12, §III.7.9).
25657:
25569:
24645:. The exterior derivative commutes with
22799:is an affine space over the vector space
22785:
20309:
20305:
18835:stands for the unit element of the field
17392:-vector, given by the determinant of the
16438:, the isomorphism is given explicitly by
15390:Axiomatic characterization and properties
14290:
14233:
14010:
13823:
12851:{\displaystyle f(v_{1},\ldots ,v_{k})=0.}
11736:
9953:) of a decomposable tensor is defined by
8898:
8892:
7204:
5953:
5819:
3799:
3202:the exterior product of three vectors is
27595:Covariance and contravariance of vectors
26286:
26279:
26271:
26162:
26111:
26091:
25959:
25910:
25725:
25704:
25691:
25577:
25521:
24671:
24561:
23638:
20344:, the exterior algebra is furthermore a
19672:{\textstyle {\textstyle \bigwedge }(V),}
18577:. The exterior algebra (as well as the
15743:is the dual pairing between elements of
14967:, so it is defined by its values on the
13197:
12693:
9428:
8274:
6878:as a linear combination of decomposable
3935:The exterior product is by construction
3803:{\displaystyle \bigwedge (V):=T(V)/I.\,}
2496:
2437:
1086:
935:may be any field (however for fields of
604:defined by the constituent vectors. The
580:and, more generally, the magnitude of a
70:elements in a real exterior algebra for
26690:
25951:{\displaystyle {\textstyle \bigwedge }}
25902:{\displaystyle {\textstyle \bigwedge }}
25648:This axiomatization of areas is due to
25530:
25303:{\textstyle {\textstyle \bigwedge }(L)}
24779:
24630:defines differential forms in terms of
18929:, so that the above really does lie in
18593:for a detailed treatment of the topic.
11653:is given. Then any alternating tensor
9530:. For characteristic 0 (or higher than
4270:{\displaystyle x\wedge y=-(y\wedge x).}
3925:
640:implies a skew-symmetric property that
14:
27874:
27187:Comparison of linear algebra libraries
26646:
26489:
26459:
26317:
26099:
26083:
26017:
25665:
25349:
24481:, can be naturally identified with an
24318:
23253:-dimensional space, the volume of any
20730:are given on decomposable elements by
20191:is the counit, as defined presently).
18515:
15503:{\displaystyle \Lambda ^{-1}(V)=\{0\}}
10956:carries an associative graded product
10835:, the homogeneous subset of the ideal
8796:generated by all elements of the form
7166:is the minimal number of decomposable
5775:{\displaystyle \{e_{1},\dots ,e_{n}\}}
5681:
3930:
3699:generated by all elements of the form
2063:
1077:
1011:{\displaystyle v\wedge w+w\wedge v=0,}
845:and this makes the exterior algebra a
27219:
26777:
26374:
25988:
25639:-vector, which avoids this confusion.
23928:and thus makes it possible to define
22911:{\displaystyle A_{0},A_{1},...,A_{k}}
21377:{\displaystyle 0\to U\to V\to W\to 0}
21017:The components of the transformation
20723:{\textstyle \bigwedge \left(f\right)}
19576:, respectively (the empty product in
18553:and alternating multilinear forms on
15341:is a pure scalar (i.e., belonging to
13055:{\displaystyle f:V^{k}\rightarrow X,}
13020:given any other alternating operator
12353:
11363:not being a supplement of the ideal
10978:{\displaystyle {\widehat {\otimes }}}
9278:, we can define the exterior algebra
8284:
6798:
5491:is decomposable. For example, given
5381:, by others). Although decomposable
1669:Consider now the exterior product of
1082:
677:{\displaystyle v\wedge w=-w\wedge v,}
26641:Spivak discusses this in Addendum 4.
26453:Other references and further reading
26201:; Gardner, R.B.; Goldschmidt, H.L.;
25997:. Academic Press. pp. 455–464.
24966:defined on decomposable elements by
24776:-fold product of exterior algebras.
24752:-dimensional superspace is just the
23960:{\displaystyle J={\star }d{\star }F}
21662:This is a graded isomorphism; i.e.,
19414:. By convention, one takes that Sh(
19022:article. In this case, one obtains
17610:constitute an orthonormal basis for
10920:{\displaystyle \operatorname {Alt} }
10791:{\displaystyle {\mathcal {A}}^{(r)}}
8919:as the symbol for multiplication in
8435:must contain a homomorphic image of
7196:. This is similar to the notion of
3591:
284:The exterior algebra is named after
23818:or the equivalent Bianchi identity
23368:generalises addition of vectors in
22749:The natural setting for (oriented)
22028:is spanned by elements of the form
20693:preserves homogeneous degree. The
20254:defined on homogeneous elements by
17905:, the linear functional defined by
15314:{\displaystyle \iota _{\alpha }f=0}
14522:
12425:be an antisymmetric tensor of rank
11933:
10040:
9843:{\displaystyle \mathrm {T} ^{r}(V)}
8429:with alternating multiplication on
7681:
3901:
2365:, since interchanging the roles of
1135:consisting of a pair of orthogonal
600:-blade is the (hyper)volume of the
27:Algebra of exterior/ wedge products
24:
27458:Tensors in curvilinear coordinates
26803:
26168:Elements of mathematics, Algebra I
25374:in 1844 under the blanket term of
25342:associated to this complex is the
25323:
24976:
24890:
24122:
20298:
20092:
19689:
19145:
19032:
18736:
18654:
18408:
18268:
18226:
17920:
17863:
16942:
16939:
15467:
14200:
14197:
13643:
11271:
11268:
11265:
11241:
11150:
10868:
10771:
10736:
10701:
10656:alternating tensor graded subspace
10628:
10619:
10552:
10543:
10521:
10512:
10496:
10487:
10445:
10420:
10417:
10414:
10390:
10275:
10266:
10157:(for nonzero characteristic field
9976:
9967:
9821:
9755:generated by elements of the form
9712:
9503:
9321:
9203:
8731:
7513:In characteristic 0, the 2-vector
6300:
66:Geometric interpretation of grade
25:
27903:
26361:, American Mathematical Society,
26258:Lectures on Differential Geometry
25971:Such a filtration also holds for
25558:Wheeler, Misner & Thorne 1973
24232:{\displaystyle u\wedge v\wedge w}
23766:{\displaystyle F_{ij}=A_{}=A_{},}
23645:Einstein's theories of relativity
23570:
13202:Geometric interpretation for the
13184:and can serve as its definition.
12493:is an alternating tensor of rank
12484:{\displaystyle \iota _{\alpha }t}
11618:
8301:. Informally, multiplication in
8295:be a vector space over the field
7172:-vectors in such an expansion of
6858:, then it is possible to express
4957:
3523:). The scalar coefficient is the
2501:Basis Decomposition of a 2-vector
81:vectors can be visualized as any
27200:
27199:
27177:Basic Linear Algebra Subprograms
26935:
26596:is what this article would call
25780:Indeed, the exterior algebra of
25728:, §III.7.5) for generalizations.
24497:
23894:None of this requires a metric.
23559:
23550:
23541:
20396:are a pair of vector spaces and
20351:
18469:
18439:
18431:
18417:
18403:
18263:
18234:
18221:
18207:
18193:
18185:
18177:
18120:
18067:
18008:
17871:
17846:
17832:
17824:
17731:
17672:
17655:, a statement equivalent to the
17097:, and so also an isomorphism of
16985:
16126:
15854:{\displaystyle \iota _{\alpha }}
15825:{\displaystyle \alpha \in V^{*}}
15450:{\displaystyle \alpha \in V^{*}}
14636:{\displaystyle \alpha \in V^{*}}
12132:
12102:
12078:
11794:
11764:
11740:
11409:{\displaystyle {\dot {\wedge }}}
10335:where the sum is taken over the
9785:. Of course, for characteristic
7231:can be identified with half the
6136:-vectors can be computed as the
5515:{\displaystyle \mathbf {R} ^{4}}
5502:
3472:
3457:
3442:
3229:
3221:
3213:
3176:
3151:
3126:
3107:
3013:
2998:
2928:
2913:
2843:
2828:
2767:
2759:
2725:
2700:
2675:
2656:
2622:
2597:
2572:
2553:
2471:plane (here, the one with sides
2210:
2195:
2180:
2165:
2130:
2115:
2094:
2079:
2035:
2020:
1973:
1958:
1937:
1922:
1901:
1886:
1865:
1850:
1819:
1801:
1777:
1759:
1740:
1732:
1701:
1677:
1550:
1543:
1484:
1462:
1445:{\displaystyle \mathbf {R} ^{2}}
1432:
1398:
1380:
1339:
1324:
1306:
1265:
1198:
1151:
1120:{\displaystyle \mathbf {R} ^{2}}
1107:
1026:. In particular, the algebra of
53:
41:
27075:Seven-dimensional cross product
26292:American Journal of Mathematics
26105:
26068:
26059:
26039:
26011:
25982:
25965:
25916:
25839:
25774:
25749:
25731:
24660:whose cohomology is called the
22745:Oriented volume in affine space
22739:
17564:
17508:, then the vectors of the form
17014:a finite-dimensional space, an
16613:, which maps an element to its
13146:{\displaystyle f=\phi \circ w.}
11546:{\displaystyle c(r+p)/c(r)c(p)}
11166:{\displaystyle \mathrm {T} (V)}
10752:{\displaystyle \mathrm {T} (V)}
10717:{\displaystyle \mathrm {T} (V)}
10691:. This is a vector subspace of
10461:{\displaystyle \mathrm {T} (V)}
10339:of permutations on the symbols
9912:
9728:{\displaystyle \mathrm {T} (V)}
9519:{\displaystyle \mathrm {T} (V)}
9337:{\displaystyle \mathrm {T} (M)}
7456:(the matrix of coefficients is
7092:
5140:
3894:
1421:are a pair of given vectors in
1337:
1194:
26697:Functions of Several Variables
26652:Introduction to linear algebra
26215:partial differential equations
25718:
25697:
25684:
25679:Mac Lane & Birkhoff (1999)
25671:
25642:
25583:
25563:
25551:
25508:of the symmetric algebra by a
25297:
25291:
25174:
25146:
25114:
25088:
25067:
25057:
25017:
24979:
24950:
24944:
24926:
24923:
24917:
24804:be a Lie algebra over a field
24586:in a way that generalizes the
24529:
24523:
24466:
24460:
24357:
24351:
23873:
23858:
23845:
23830:
23755:
23743:
23730:
23718:
23425:
23399:
23355:
23349:
23335:The vector space structure on
23295:
23283:
23048:
23038:
22717:
22708:
22702:
22678:
22675:
22669:
22651:
22648:
22642:
22625:
22589:
22586:
22580:
22562:
22559:
22553:
22535:
22532:
22526:
22496:
22460:
22447:
22432:
22419:
22375:
22288:
22282:
22261:
22255:
21959:
21953:
21836:
21825:
21812:
21804:
21775:
21769:
21748:
21742:
21699:
21687:
21643:
21637:
21624:
21618:
21605:
21593:
21571:
21552:
21546:
21536:
21533:
21527:
21517:
21485:
21482:
21476:
21466:
21463:
21457:
21447:
21444:
21438:
21425:
21419:
21401:
21368:
21362:
21356:
21350:
21291:
21285:
21250:
21244:
21047:
21041:
20998:
20992:
20974:
20971:
20965:
20939:
20933:
20907:
20901:
20888:
20882:
20838:
20825:
20810:
20797:
20788:
20756:
20753:
20747:
20680:
20674:
20641:
20635:
20611:
20608:
20602:
20564:
20558:
20532:
20526:
20493:
20487:
20477:
20474:
20468:
20455:
20449:
20412:
20288:
20278:
20272:
20266:
20227:
20224:
20218:
20127:
20095:
20084:
20072:
20066:
20034:
20031:
20019:
19856:
19850:
19837:
19831:
19794:
19788:
19762:
19756:
19718:
19715:
19709:
19663:
19657:
19596:
19590:
19549:
19543:
19521:
19509:
19479:
19473:
19451:
19445:
19376:
19371:
19365:
19343:
19331:
19320:
19314:
19309:
19303:
19281:
19275:
19264:
19261:
19255:
19243:
19225:
19180:
19148:
19113:
19101:
19071:
19059:
19047:
19035:
19005:
18999:
18970:
18964:
18951:
18945:
18914:
18908:
18895:
18889:
18745:
18739:
18711:
18705:
18692:
18686:
18676:
18673:
18667:
18622:
18616:
18540:
18534:
18497:
18491:
18421:
18413:
18340:
18334:
18148:
18142:
18095:
18089:
18042:
18036:
17931:
17925:
17759:
17753:
17706:
17700:
17640:
17634:
17445:
17439:
17125:
17119:
16927:
16915:
16908:
16898:
16892:
16886:
16868:
16865:
16859:
16781:
16775:
16757:
16754:
16748:
16687:
16681:
16657:
16654:
16648:
16518:
16509:
16503:
16479:
16476:
16463:
16391:
16385:
16279:
16273:
16249:
16246:
16240:
16219:
16206:
15977:
15961:
15940:
15930:
15918:
15902:
15896:
15884:
15728:
15722:
15679:
15673:
15586:
15580:
15556:
15553:
15547:
15485:
15479:
15371:
15365:
15266:
15209:
15200:
15149:
15146:
15130:
14881:
14875:
14782:This derivation is called the
14763:
14757:
14733:
14730:
14724:
14671:
14665:
14645:, it is possible to define an
14356:
14351:
14339:
14317:
14305:
14294:
14287:
14282:
14276:
14254:
14248:
14237:
14230:
14224:
14177:
14139:
14064:
14059:
14053:
14031:
14025:
14014:
14007:
14001:
13951:
13919:
13916:
13910:
13851:
13839:
13809:
13797:
13747:
13735:
13429:
13423:
13314:
13102:
13099:
13093:
13043:
12911:
12905:
12887:
12839:
12807:
12751:
12550:
12533:
12331:
12263:
12220:
12192:
12068:
12056:
12040:
12028:
12008:
12002:
11986:
11980:
11964:
11958:
11910:
11898:
11600:
11594:
11569:
11563:
11540:
11534:
11528:
11522:
11511:
11499:
11474:
11468:
11437:
11431:
11350:
11344:
11321:
11315:
11302:
11296:
11249:{\displaystyle {\mathcal {A}}}
11213:
11207:
11194:
11188:
11160:
11154:
11107:
11101:
11052:
11040:
10943:
10937:
10876:{\displaystyle {\mathcal {A}}}
10818:
10812:
10783:
10777:
10746:
10740:
10711:
10705:
10676:
10670:
10641:
10638:
10632:
10624:
10595:
10589:
10555:
10549:
10524:
10518:
10499:
10493:
10455:
10449:
10398:{\displaystyle {\mathcal {A}}}
10372:{\displaystyle \{1,\dots ,r\}}
10319:
10287:
10278:
10272:
10241:
10209:
10201:
10195:
10110:
10104:
10082:
10076:
10065:
10059:
10020:
9988:
9979:
9973:
9837:
9831:
9722:
9716:
9513:
9507:
9366:
9360:
9331:
9325:
9298:
9292:
9185:
9179:
9150:
9144:
9109:
9103:
9068:
9062:
9024:
9018:
8976:
8970:
8941:
8935:
8889:
8883:
8874:
8868:
8838:
8832:
8776:. We thus take the two-sided
8713:
8707:
8628:
8625:
8619:
8561:
8555:
8549:
8543:
8517:
8457:
8451:
8404:
8398:
8321:
8315:
8236:
8226:
8189:
8183:
8140:
8134:
8081:
8075:
8034:
8028:
7978:
7972:
7945:
7939:
7918:
7912:
7867:
7861:
7834:
7828:
7807:
7801:
7780:
7774:
7753:
7747:
7717:
7705:
7449:{\displaystyle a_{ij}=-a_{ji}}
7084:
7078:
7054:
7048:
7030:
7024:
6994:
6988:
6958:
6952:
6933:
6927:
6914:
6908:
6843:
6837:
6743:
6737:
6664:
6658:
6618:
6612:
6585:
6579:
6558:
6552:
6531:
6525:
6504:
6498:
6411:
6405:
6287:
6281:
6233:
6227:
6000:
5994:
5478:
5472:
5435:
5429:
5239:
5233:
5067:
5061:
5026:
5020:
4637:
4631:
4557:
4551:
4537:
4531:
4509:
4503:
4487:
4481:
4331:
4313:
4261:
4249:
4136:
4124:
4118:
4106:
3905:
3895:
3849:
3843:
3785:
3779:
3770:
3764:
3612:
3606:
3482:
3437:
3434:
3236:
3023:
2993:
2990:
2944:
2938:
2908:
2905:
2859:
2853:
2823:
2820:
2774:
2140:
2110:
1829:
1793:
1787:
1751:
13:
1:
27511:Exterior covariant derivative
27443:Tensor (intrinsic definition)
26654:, Wellesley-Cambridge Press,
26207:Exterior differential systems
26120:
25713:Mac Lane & Birkhoff (1999
25709:Mac Lane & Birkhoff (1999
25662:Mac Lane & Birkhoff (1999
24705:
24668:of differentiable manifolds.
24570:, where it is used to define
24207:. Analogously, the 3-vector
22918:, we can define its oriented
13188:Alternating multilinear forms
11678:Einstein summation convention
10435:, on the full tensor algebra
8480:Given any unital associative
7326:can be expressed uniquely as
7002:{\displaystyle \alpha ^{(i)}}
5377:, by some authors; or a
3855:{\displaystyle \bigwedge (V)}
3618:{\displaystyle \bigwedge (V)}
2406:, since adding a multiple of
1131:vector space equipped with a
27536:Raising and lowering indices
26917:Eigenvalues and eigenvectors
26140:Tensor analysis on manifolds
25462:Exterior calculus identities
23887:{\displaystyle F_{}=F_{}=0.}
22792:{\displaystyle \mathbb {A} }
21335:
20184:{\displaystyle \varepsilon }
18637:, giving the structure of a
16295:by the recursive definition
14916:is a multilinear mapping of
13335:alternating multilinear form
13323:{\displaystyle f:V^{k}\to K}
12760:{\displaystyle f:V^{k}\to X}
8685:is the natural inclusion of
4652:signature of the permutation
2335:parallelogram determined by
1708:{\displaystyle \mathbf {w} }
1684:{\displaystyle \mathbf {v} }
1491:{\displaystyle \mathbf {w} }
1469:{\displaystyle \mathbf {v} }
798:The exterior algebra is the
181:which has a product, called
7:
27774:Gluon field strength tensor
27245:
26728:Linear Algebra and Geometry
26680:Encyclopedia of Mathematics
26185:main mathematical reference
25913:, Proposition 3, §III.7.2).
25450:
24832:on the exterior algebra of
24690:irreducible representations
24643:differential graded algebra
22183:{\displaystyle v_{i}\in V.}
17998:Indeed, more generally for
14553:is finite-dimensional. If
13250:. The "circulations" show
13194:Alternating multilinear map
11279:{\displaystyle {\rm {Alt}}}
10428:{\displaystyle {\rm {Alt}}}
8367:{\displaystyle v\wedge v=0}
8003:is the base field, we have
7460:). The rank of the matrix
7211:). The rank of a 2-vector
6144:that describes the vectors
4713:{\displaystyle x_{i}=x_{j}}
3990:{\displaystyle x\wedge x=0}
3552:of the matrix with columns
3096:Bringing in a third vector
2517:. Using the standard basis
964:{\displaystyle v\wedge v=0}
633:{\displaystyle v\wedge v=0}
234:{\displaystyle v\wedge v=0}
10:
27908:
27585:Cartan formalism (physics)
27405:Penrose graphical notation
26691:Wendell, Fleming (2012) ,
26385:Kannenberg, Lloyd (1999),
26353:Kannenberg, Lloyd (2000),
25365:
25358:, a fundamental object in
24699:Fundamental representation
24414:-dimensional subspaces of
23627:
23623:
23534:is linear in each factor.
22147:{\displaystyle u_{i}\in U}
17046:defines an isomorphism of
16130:
14526:
13191:
12688:
11553:for an arbitrary sequence
11416:product, Arnold setting).
10761:. Moreover, the kernel of
9776:{\displaystyle x\otimes x}
9451:Alternating tensor algebra
8573:{\displaystyle j(v)j(v)=0}
7686:The exterior product of a
3718:{\displaystyle x\otimes x}
2248:of the parallelogram: the
1498:as two of its sides. The
29:
27791:
27731:
27680:
27673:
27565:
27496:
27433:
27377:
27324:
27271:
27264:
27257:Glossary of tensor theory
27253:
27195:
27157:
27113:
27050:
27002:
26944:
26933:
26829:
26811:
26673:Onishchik, A.L. (2001) ,
26324:The Calculus of Extension
26136:; Goldberg, S.I. (1980),
25329:{\displaystyle \partial }
24576:higher-dimensional bodies
24152:{\displaystyle u\wedge v}
23967:or the equivalent tensor
21121:{\displaystyle k\times k}
20247:on the exterior algebra.
11334:still holds, in spite of
11091:has characteristic 0, as
10577:Such that, when defined,
8342:, and using the identity
2493:Cross and triple products
939:two, the above condition
526:{\displaystyle v\wedge w}
85:-dimensional shape (e.g.
27841:Gregorio Ricci-Curbastro
27713:Riemann curvature tensor
27420:Van der Waerden notation
26725:; Remizov, A.O. (2012).
26423:(1504), Cambridge: 385,
25619:-multivector instead of
25515:
23811:{\displaystyle dF=ddA=0}
23689:alternating tensor field
23651:is generally given as a
22827:) collection of ordered
20421:{\displaystyle f:V\to W}
19989:{\displaystyle \otimes }
19967:{\displaystyle \otimes }
19947:{\displaystyle \otimes }
12698:Given two vector spaces
11834:completely antisymmetric
10150:{\displaystyle r!\neq 0}
8526:{\displaystyle j:V\to A}
6355:is the dimension of the
5085:by elements of the form
4081:{\displaystyle x,y\in V}
3560:. The triple product of
2331:, since the area of the
852:The exterior algebra is
27811:Elwin Bruno Christoffel
27744:Angular momentum tensor
27415:Tetrad (index notation)
27385:Abstract index notation
26409:Whitehead, Alfred North
26126:Mathematical references
26096:extended vector algebra
25314:with boundary operator
24600:differentiable manifold
24370:correspond to weighted
22021:{\displaystyle p\geq 1}
21388:of vector spaces, then
21198:is of finite dimension
21062:relative to a basis of
19927:{\displaystyle \wedge }
19422:) equals {id: {1, ...,
16429:{\displaystyle \sigma }
14833:{\displaystyle \alpha }
14801:{\displaystyle \alpha }
11419:Finally, we always get
10826:{\displaystyle I^{(r)}}
9560:), the vector space of
7526:{\displaystyle \alpha }
7503:{\displaystyle \alpha }
7319:{\displaystyle \alpha }
7248:{\displaystyle \alpha }
7224:{\displaystyle \alpha }
7187:{\displaystyle \alpha }
7159:{\displaystyle \alpha }
6871:{\displaystyle \alpha }
6634:(where, by convention,
5450:, not every element of
5362:{\displaystyle \alpha }
5298:{\displaystyle \alpha }
5267:{\displaystyle \alpha }
4741:{\displaystyle i\neq j}
4668:{\displaystyle \sigma }
4293:{\displaystyle \sigma }
4019:{\displaystyle x\in V,}
3826:{\displaystyle \wedge }
822:-th exterior powers of
301:{\displaystyle \wedge }
202:{\displaystyle \wedge }
109:defined by that of its
27625:Levi-Civita connection
26902:Row and column vectors
26635:
26590:
26547:
25991:"On the Wedge Product"
25952:
25903:
25879:
25859:
25828:
25796:
25633:
25613:
25493:, the symmetric analog
25437:Alfred North Whitehead
25399:propositional calculus
25330:
25304:
25271:
25249:
25218:
24957:
24872:
24848:
24820:
24798:
24768:
24550:. This is called the
24543:
24473:
24430:
24404:. In particular, the
24396:
24364:
24307:
24283:
24259:
24233:
24199:
24177:
24153:
24114:
24039:
23961:
23922:
23921:{\displaystyle \star }
23888:
23812:
23775:electromagnetic tensor
23767:
23677:
23630:Electromagnetic tensor
23613:linear transformations
23528:
23476:
23384:
23362:
23324:
23302:
23267:
23245:
23223:
23203:
23179:
23156:
23026:
22932:
22912:
22847:
22815:
22793:
22763:
22727:
22612:is 1-dimensional then
22599:
22483:is 1-dimensional then
22470:
22295:
22184:
22148:
22115:
22022:
21996:
21966:
21849:
21785:
21653:
21562:
21495:
21378:
21324:
21298:
21257:
21214:
21192:
21172:
21144:
21122:
21096:
21076:
21054:
21008:
20848:
20724:
20699:-graded components of
20687:
20651:
20500:
20422:
20390:
20370:
20336:
20237:
20185:
20139:
19990:
19968:
19948:
19928:
19863:
19801:
19741:
19673:
19630:
19603:
19558:
19488:
19386:
19206:
19126:
19012:
18977:
18921:
18851:
18829:
18807:
18806:{\displaystyle v\in V}
18776:
18718:
18629:
18589:. See the article on
18569:
18547:
18504:
18450:
18383:
18361:
18245:
18155:
18102:
18049:
17987:
17986:{\displaystyle y\in V}
17956:
17882:
17796:
17795:{\displaystyle x\in V}
17766:
17713:
17647:
17601:
17500:
17455:
17359:
17214:
17190:
17132:
17089:
17060:
17038:
17006:
16950:
16814:
16788:
16720:
16719:{\displaystyle \star }
16697:
16601:
16572:
16541:
16430:
16398:
16354:
16286:
16173:
16151:
16117:
16044:
15987:
15855:
15826:
15786:
15757:
15735:
15686:
15640:
15620:
15596:
15504:
15451:
15414:
15378:
15335:
15315:
15276:
15112:
15083:
15031:
14959:
14937:
14910:
14888:
14834:
14802:
14773:
14678:
14637:
14600:
14574:
14547:
14366:
14100:
14074:
13884:
13861:
13754:
13663:
13617:
13597:
13570:
13550:
13528:
13472:
13450:
13379:
13359:
13324:
13280:
13255:
13178:
13147:
13112:
13062:there exists a unique
13056:
13014:
13013:{\displaystyle V^{k};}
12984:
12964:
12944:
12921:
12852:
12761:
12668:
12611:
12515:
12485:
12441:
12419:
12343:
12163:
11819:
11607:
11576:
11547:
11481:
11444:
11410:
11379:
11357:
11328:
11280:
11250:
11223:
11167:
11134:
11114:
11085:
11062:
10979:
10950:
10921:
10899:
10877:
10851:
10827:
10792:
10753:
10718:
10683:
10648:
10604:
10568:
10462:
10429:
10399:
10373:
10326:
10174:
10151:
10119:
9936:
9864:
9844:
9801:
9777:
9749:
9729:
9698:
9678:
9664:, the vector space of
9656:
9620:
9596:
9574:
9552:
9551:{\displaystyle \dim V}
9520:
9489:
9469:
9417:
9393:
9373:
9338:
9305:
9270:
9245:
9225:
9192:
9157:
9116:
9075:
9031:
8983:
8948:
8906:
8845:
8737:
8720:
8638:
8574:
8527:
8464:
8411:
8368:
8328:
8264:
8196:
8147:
8097:
8047:
7988:
7874:
7724:
7672:
7607:
7547:
7527:
7504:
7484:
7483:{\displaystyle a_{ij}}
7450:
7398:
7320:
7298:
7276:
7249:
7225:
7188:
7160:
7127:
7003:
6967:
6872:
6850:
6787:
6756:
6705:
6677:
6625:
6458:
6457:{\displaystyle k>n}
6430:
6375:
6347:
6322:
6240:
6194:
6171:in terms of the basis
6165:
6124:
6091:
6061:
6007:
5961:
5800:
5776:
5724:
5704:
5672:
5592:
5516:
5485:
5442:
5397:
5363:
5341:
5319:
5299:
5268:
5246:
5194:
5074:
5033:
4988:
4948:
4880:
4812:
4742:
4714:
4669:
4644:
4609:
4455:
4431:
4400:
4369:
4338:
4294:
4271:
4215:
4082:
4046:
4020:
3991:
3957:
3916:
3856:
3827:
3804:
3745:
3744:{\displaystyle x\in V}
3719:
3693:
3662:
3639:
3619:
3527:of the three vectors.
3489:
3193:
3030:
2739:
2636:
2502:
2467:
2230:
2150:
2053:
1709:
1685:
1660:
1492:
1470:
1446:
1412:
1245:
1121:
1097:Euclidean vector space
1092:
1068:
1044:
1012:
971:must be replaced with
965:
910:
890:
870:
839:
816:
792:
765:
751:-blades is called the
745:
705:
678:
634:
594:
574:
551:
527:
475:
455:
426:
367:
345:
322:
302:
278:
255:
235:
203:
175:
148:
27851:Jan Arnoldus Schouten
27806:Augustin-Louis Cauchy
27286:Differential geometry
26907:Row and column spaces
26852:Scalar multiplication
26714:multivariate calculus
26636:
26591:
26548:
26495:Calculus on manifolds
26460:Browne, J.M. (2007),
26348:alternative reference
26272:Historical references
26076:calculus of extension
26021:(1984). "Chapter 1".
25953:
25904:
25880:
25860:
25829:
25797:
25634:
25614:
25435:A short while later,
25331:
25305:
25272:
25250:
25231:holds if and only if
25219:
24958:
24873:
24849:
24821:
24799:
24769:
24678:representation theory
24672:Representation theory
24654:differential operator
24568:differential geometry
24562:Differential geometry
24544:
24474:
24431:
24397:
24365:
24308:
24284:
24260:
24234:
24200:
24178:
24154:
24115:
24040:
23962:
23923:
23889:
23813:
23768:
23687:or as the equivalent
23678:
23649:electromagnetic field
23639:Electromagnetic field
23529:
23477:
23385:
23363:
23325:
23303:
23301:{\displaystyle (k-1)}
23268:
23246:
23224:
23204:
23180:
23162:(using concatenation
23157:
23027:
22933:
22913:
22848:
22816:
22794:
22764:
22728:
22600:
22471:
22296:
22185:
22149:
22116:
22023:
21997:
21995:{\displaystyle F^{p}}
21967:
21850:
21786:
21654:
21563:
21496:
21379:
21325:
21299:
21258:
21215:
21193:
21173:
21152:. In particular, if
21145:
21123:
21097:
21077:
21055:
21009:
20849:
20725:
20688:
20652:
20501:
20423:
20391:
20371:
20337:
20238:
20186:
20140:
19991:
19969:
19949:
19929:
19864:
19802:
19721:
19674:
19631:
19629:{\displaystyle x_{k}}
19604:
19559:
19489:
19387:
19186:
19127:
19013:
18978:
18922:
18852:
18830:
18808:
18777:
18719:
18645:is a linear function
18630:
18570:
18548:
18505:
18451:
18384:
18362:
18246:
18156:
18103:
18050:
17988:
17957:
17883:
17797:
17767:
17714:
17648:
17602:
17501:
17456:
17360:
17215:
17191:
17133:
17090:
17088:{\displaystyle V^{*}}
17061:
17039:
17007:
16951:
16815:
16789:
16721:
16698:
16602:
16600:{\displaystyle V^{*}}
16573:
16542:
16431:
16399:
16355:
16287:
16174:
16157:has finite dimension
16152:
16118:
16045:
15988:
15856:
15827:
15787:
15785:{\displaystyle V^{*}}
15758:
15736:
15687:
15641:
15621:
15597:
15505:
15459:(where by convention
15452:
15415:
15379:
15336:
15316:
15277:
15113:
15111:{\displaystyle V^{*}}
15084:
15032:
14960:
14938:
14936:{\displaystyle V^{*}}
14911:
14889:
14835:
14803:
14774:
14679:
14638:
14601:
14575:
14573:{\displaystyle V^{*}}
14548:
14367:
14108:finite characteristic
14101:
14075:
13885:
13862:
13755:
13664:
13618:
13598:
13596:{\displaystyle V^{k}}
13571:
13551:
13536:. In particular, if
13529:
13473:
13451:
13380:
13360:
13325:
13281:
13201:
13179:
13177:{\displaystyle V^{k}}
13148:
13113:
13057:
13015:
12985:
12965:
12945:
12922:
12853:
12762:
12706:and a natural number
12694:Alternating operators
12669:
12591:
12516:
12486:
12442:
12420:
12344:
12164:
11820:
11627:has finite dimension
11608:
11577:
11548:
11482:
11445:
11411:
11380:
11358:
11329:
11281:
11251:
11224:
11168:
11135:
11115:
11086:
11063:
10980:
10951:
10922:
10900:
10878:
10852:
10828:
10793:
10754:
10719:
10684:
10649:
10605:
10569:
10463:
10430:
10400:
10374:
10327:
10175:
10152:
10120:
9937:
9865:
9845:
9802:
9778:
9750:
9730:
9699:
9679:
9657:
9621:
9597:
9575:
9553:
9528:antisymmetric tensors
9521:
9490:
9470:
9434:Exterior algebras of
9418:
9394:
9374:
9339:
9306:
9271:
9246:
9226:
9193:
9158:
9117:
9076:
9048:Rather than defining
9032:
9004:its exterior algebra
8984:
8949:
8907:
8846:
8735:
8721:
8639:
8575:
8528:
8465:
8412:
8369:
8329:
8265:
8197:
8148:
8098:
8048:
7989:
7875:
7725:
7723:{\displaystyle (k+p)}
7673:
7608:
7548:
7528:
7505:
7485:
7451:
7399:
7321:
7299:
7277:
7275:{\displaystyle e_{i}}
7255:in a basis. Thus if
7250:
7226:
7189:
7161:
7128:
7009:is decomposable, say
7004:
6968:
6873:
6851:
6788:
6786:{\displaystyle 2^{n}}
6757:
6706:
6678:
6626:
6459:
6431:
6376:
6348:
6323:
6241:
6195:
6193:{\displaystyle e_{i}}
6166:
6164:{\displaystyle v_{j}}
6125:
6123:{\displaystyle e_{i}}
6101:of the basis vectors
6092:
6090:{\displaystyle v_{j}}
6062:
6008:
5962:
5801:
5777:
5725:
5705:
5673:
5593:
5517:
5486:
5443:
5398:
5364:
5342:
5320:
5300:
5269:
5247:
5195:
5075:
5034:
4989:
4949:
4881:
4813:
4743:
4715:
4670:
4645:
4610:
4456:
4432:
4430:{\displaystyle x_{k}}
4401:
4399:{\displaystyle x_{2}}
4370:
4368:{\displaystyle x_{1}}
4339:
4295:
4272:
4216:
4083:
4054:, for supposing that
4047:
4021:
3992:
3958:
3917:
3857:
3828:
3813:The exterior product
3805:
3746:
3720:
3694:
3663:
3640:
3620:
3596:The exterior algebra
3490:
3194:
3031:
2740:
2637:
2500:
2489:formulation of area.
2441:
2309:for any real numbers
2231:
2151:
2054:
1710:
1686:
1661:
1493:
1471:
1447:
1413:
1246:
1122:
1090:
1069:
1045:
1013:
966:
911:
891:
871:
840:
817:
793:
766:
746:
706:
679:
635:
595:
575:
552:
528:
476:
456:
427:
368:
353:The wedge product of
346:
323:
303:
279:
256:
236:
204:
176:
149:
27826:Carl Friedrich Gauss
27759:stress–energy tensor
27754:Cauchy stress tensor
27506:Covariant derivative
27468:Antisymmetric tensor
27400:Multi-index notation
27042:Gram–Schmidt process
26994:Gaussian elimination
26600:
26565:
26522:
25989:James, A.T. (1983).
25938:
25889:
25869:
25849:
25818:
25786:
25623:
25603:
25572:introduced these as
25531:Penrose, R. (2007).
25344:Lie algebra homology
25320:
25281:
25261:
25237:
24973:
24887:
24862:
24838:
24810:
24788:
24780:Lie algebra homology
24758:
24694:general linear group
24493:
24483:algebraic subvariety
24444:
24420:
24386:
24333:
24297:
24273:
24249:
24211:
24189:
24167:
24137:
24049:
23974:
23932:
23912:
23822:
23781:
23694:
23676:{\displaystyle F=dA}
23658:
23492:
23396:
23374:
23339:
23314:
23280:
23257:
23235:
23213:
23193:
23166:
23035:
22942:
22922:
22857:
22831:
22805:
22781:
22753:
22619:
22490:
22305:
22197:
22158:
22125:
22032:
22006:
21979:
21865:
21798:
21669:
21583:
21511:
21395:
21386:short exact sequence
21344:
21314:
21267:
21226:
21204:
21182:
21156:
21134:
21106:
21086:
21066:
21023:
20864:
20737:
20703:
20664:
20516:
20439:
20400:
20380:
20360:
20260:
20202:
20198:is the homomorphism
20175:
20016:
19980:
19958:
19938:
19918:
19821:
19686:
19647:
19613:
19580:
19498:
19434:
19142:
19029:
18989:
18935:
18865:
18841:
18819:
18791:
18733:
18726:, which is given by
18651:
18606:
18585:structure, from the
18559:
18524:
18465:
18398:
18373:
18258:
18170:
18116:
18063:
18004:
17971:
17912:
17811:
17780:
17727:
17668:
17657:Cauchy–Binet formula
17616:
17515:
17490:
17421:
17229:
17204:
17144:
17101:
17072:
17050:
17028:
16996:
16829:
16804:
16730:
16710:
16624:
16584:
16562:
16554:is equipped with an
16445:
16420:
16367:
16302:
16188:
16163:
16141:
16055:
16008:
15871:
15838:
15803:
15769:
15747:
15700:
15652:
15630:
15610:
15516:
15463:
15428:
15404:
15347:
15325:
15289:
15127:
15095:
15067:
14982:
14949:
14920:
14900:
14851:
14824:
14792:
14693:
14655:
14614:
14590:
14584:to the vector space
14557:
14537:
14117:
14090:
13901:
13874:
13770:
13714:
13632:
13625:binomial coefficient
13607:
13580:
13560:
13540:
13488:
13480:naturally isomorphic
13462:
13398:
13389:isomorphic with the
13369:
13349:
13295:
13264:
13243:, here planes), for
13161:
13122:
13069:
13024:
12994:
12974:
12954:
12934:
12930:which associates to
12868:
12801:
12732:
12712:alternating operator
12681:is the dimension of
12530:
12499:
12465:
12431:
12360:
12189:
11862:
11687:
11606:{\displaystyle A(V)}
11588:
11575:{\displaystyle c(r)}
11557:
11493:
11458:
11443:{\displaystyle A(V)}
11425:
11391:
11369:
11356:{\displaystyle A(V)}
11338:
11290:
11260:
11236:
11182:
11146:
11124:
11113:{\displaystyle A(V)}
11095:
11075:
10992:
10960:
10949:{\displaystyle A(V)}
10931:
10911:
10889:
10863:
10859:, or the kernel of
10841:
10804:
10765:
10732:
10697:
10682:{\displaystyle A(V)}
10664:
10614:
10581:
10480:
10441:
10409:
10385:
10345:
10187:
10161:
10132:
9960:
9877:
9854:
9816:
9791:
9761:
9739:
9708:
9688:
9668:
9634:
9610:
9586:
9564:
9536:
9499:
9479:
9459:
9407:
9383:
9350:
9317:
9282:
9260:
9235:
9215:
9169:
9126:
9085:
9052:
9008:
8960:
8925:
8858:
8822:
8697:
8603:
8598:algebra homomorphism
8592:, then there exists
8537:
8505:
8441:
8423:-algebra containing
8388:
8346:
8305:
8211:
8159:
8110:
8057:
8010:
7894:
7737:
7702:
7617:
7560:
7537:
7517:
7494:
7464:
7411:
7333:
7310:
7288:
7259:
7239:
7215:
7178:
7150:
7016:
6980:
6894:
6862:
6813:
6770:
6719:
6695:
6640:
6488:
6442:
6387:
6365:
6337:
6257:
6248:binomial coefficient
6209:
6177:
6148:
6107:
6097:can be written as a
6074:
6022:
5976:
5813:
5790:
5734:
5714:
5694:
5607:
5530:
5497:
5454:
5411:
5387:
5353:
5331:
5309:
5289:
5285:. If, furthermore,
5258:
5209:
5092:
5051:
5002:
4978:
4893:
4825:
4757:
4726:
4684:
4659:
4622:
4470:
4445:
4414:
4383:
4352:
4310:
4284:
4231:
4097:
4060:
4036:
4001:
3969:
3947:
3926:Algebraic properties
3869:
3837:
3817:
3758:
3729:
3703:
3683:
3652:
3629:
3600:
3209:
3103:
2755:
2652:
2549:
2402:for any real number
2160:
2074:
2070:, and in particular
1724:
1697:
1673:
1513:
1480:
1458:
1427:
1261:
1145:
1102:
1095:The two-dimensional
1058:
1034:
975:
943:
900:
880:
860:
826:
806:
779:
755:
735:
695:
644:
612:
606:alternating property
584:
561:
541:
511:
465:
445:
377:
357:
332:
312:
292:
265:
245:
213:
193:
162:
138:
123:In mathematics, the
27887:Multilinear algebra
27703:Nonmetricity tensor
27558:(2nd-order tensors)
27526:Hodge star operator
27516:Exterior derivative
27365:Transport phenomena
27350:Continuum mechanics
27306:Multilinear algebra
27172:Numerical stability
27052:Multilinear algebra
27027:Inner product space
26877:Linear independence
26561:; i.e., for Spivak
26429:1898Natur..58..385G
26170:, Springer-Verlag,
26094:), and recently as
25534:The Road to Reality
25506:quantum deformation
25486:Multilinear algebra
25457:Alternating algebra
25380:Theory of Extension
25360:homological algebra
25350:Homological algebra
25248:{\displaystyle {1}}
24639:exterior derivative
24637:In particular, the
24319:Projective geometry
24034:
24010:
23907:Hodge star operator
23653:differential 2-form
23575:In applications to
23187:displacement vector
22846:{\displaystyle k+1}
21855:there is a natural
21171:{\displaystyle V=W}
18516:Bialgebra structure
18391:-vector defined by
17903:musical isomorphism
16706:The composition of
16611:Hodge star operator
16133:Hodge star operator
15082:{\displaystyle k-1}
14810:, or sometimes the
14471:+ 2) < ... <
13279:{\displaystyle X=K}
13241:coordinate surfaces
12770:such that whenever
12514:{\displaystyle r-1}
11631:, and that a basis
11120:is a supplement of
9951:skew-symmetrization
7235:of coefficients of
7088:
7058:
5682:Basis and dimension
4280:More generally, if
3965:, which means that
3931:Alternating product
3833:of two elements of
2442:The cross product (
1078:Motivating examples
689:linear combinations
533:is the area of the
156:associative algebra
101:); with magnitude (
27892:Differential forms
27836:Tullio Levi-Civita
27779:Metric tensor (GR)
27693:Levi-Civita symbol
27546:Tensor contraction
27360:General relativity
27296:Euclidean geometry
26882:Linear combination
26675:"Exterior algebra"
26631:
26609:
26586:
26574:
26543:
26531:
26497:, Addison-Wesley,
26336:Grassmann, Hermann
25977:abelian categories
25948:
25946:
25899:
25897:
25875:
25855:
25824:
25806:enveloping algebra
25792:
25629:
25609:
25430:differential forms
25326:
25300:
25289:
25267:
25245:
25214:
25056:
24953:
24935:
24902:
24868:
24844:
24816:
24794:
24764:
24666:algebraic topology
24662:de Rham cohomology
24632:germs of functions
24628:alternate approach
24594:from calculus. A
24572:differential forms
24539:
24514:
24469:
24426:
24392:
24360:
24342:
24303:
24279:
24255:
24229:
24195:
24173:
24149:
24110:
24035:
24014:
23990:
23957:
23918:
23884:
23808:
23763:
23673:
23524:
23472:
23380:
23358:
23347:
23320:
23298:
23263:
23241:
23219:
23199:
23178:{\displaystyle PQ}
23175:
23152:
23022:
22928:
22908:
22843:
22811:
22789:
22775:differential forms
22759:
22723:
22687:
22660:
22634:
22595:
22571:
22544:
22511:
22479:In particular, if
22466:
22291:
22273:
22240:
22180:
22144:
22111:
22018:
21992:
21962:
21944:
21845:
21781:
21760:
21733:
21726:
21678:
21649:
21635:
21616:
21591:
21558:
21544:
21525:
21491:
21474:
21455:
21436:
21410:
21374:
21320:
21294:
21276:
21253:
21235:
21210:
21188:
21168:
21140:
21118:
21092:
21072:
21050:
21032:
21004:
20983:
20956:
20924:
20899:
20873:
20844:
20745:
20720:
20683:
20672:
20647:
20626:
20593:
20549:
20524:
20496:
20485:
20466:
20447:
20418:
20386:
20366:
20332:
20233:
20216:
20181:
20135:
19986:
19964:
19944:
19924:
19859:
19848:
19829:
19797:
19774:
19748:
19701:
19669:
19655:
19626:
19599:
19588:
19554:
19484:
19382:
19247:
19122:
19008:
18997:
18973:
18962:
18943:
18917:
18906:
18880:
18847:
18825:
18803:
18772:
18714:
18703:
18684:
18665:
18625:
18614:
18565:
18543:
18532:
18500:
18482:
18446:
18379:
18357:
18325:
18282:
18241:
18151:
18133:
18098:
18080:
18045:
18021:
17983:
17952:
17878:
17792:
17762:
17744:
17709:
17685:
17643:
17625:
17597:
17496:
17451:
17430:
17355:
17210:
17186:
17160:
17128:
17110:
17085:
17056:
17034:
17022:inner product) on
17002:
16946:
16877:
16850:
16810:
16784:
16766:
16739:
16716:
16693:
16666:
16639:
16597:
16568:
16537:
16488:
16454:
16426:
16394:
16376:
16350:
16282:
16258:
16231:
16197:
16169:
16147:
16113:
16040:
15983:
15851:
15822:
15782:
15753:
15731:
15682:
15664:
15636:
15616:
15592:
15565:
15538:
15500:
15447:
15410:
15374:
15356:
15331:
15311:
15285:Additionally, let
15272:
15108:
15079:
15027:
14955:
14933:
14906:
14884:
14866:
14830:
14812:insertion operator
14798:
14769:
14742:
14715:
14674:
14663:
14633:
14596:
14570:
14543:
14362:
14217:
14096:
14070:
13994:
13894:of its variables:
13880:
13857:
13750:
13659:
13658:
13613:
13593:
13566:
13546:
13524:
13497:
13468:
13446:
13414:
13375:
13355:
13337:. The set of all
13320:
13276:
13256:
13174:
13155:universal property
13143:
13108:
13084:
13052:
13010:
12980:
12960:
12940:
12917:
12896:
12848:
12788:linearly dependent
12757:
12664:
12511:
12481:
12437:
12415:
12339:
12159:
11951:
11815:
11672:can be written in
11603:
11572:
11543:
11477:
11466:
11440:
11406:
11375:
11353:
11324:
11313:
11276:
11246:
11219:
11205:
11163:
11130:
11110:
11081:
11058:
10975:
10946:
10917:
10895:
10873:
10847:
10823:
10788:
10749:
10714:
10679:
10644:
10600:
10564:
10458:
10425:
10395:
10369:
10322:
10173:{\displaystyle r!}
10170:
10147:
10115:
10052:
9949:(or sometimes the
9947:antisymmetrization
9932:
9860:
9840:
9797:
9773:
9745:
9725:
9694:
9674:
9652:
9616:
9592:
9570:
9548:
9516:
9485:
9465:
9441:Serre–Swan theorem
9425:finitely generated
9413:
9389:
9379:also require that
9369:
9358:
9334:
9301:
9290:
9266:
9241:
9221:
9188:
9177:
9153:
9135:
9112:
9094:
9071:
9060:
9027:
9016:
8979:
8968:
8944:
8933:
8902:
8866:
8841:
8830:
8738:
8716:
8705:
8634:
8617:
8570:
8523:
8474:universal property
8460:
8449:
8407:
8396:
8364:
8324:
8313:
8285:Universal property
8260:
8192:
8174:
8143:
8125:
8093:
8066:
8043:
8019:
7984:
7957:
7930:
7903:
7870:
7852:
7819:
7792:
7765:
7745:
7720:
7668:
7660:
7647:
7603:
7592:
7587:
7543:
7523:
7500:
7480:
7446:
7394:
7357:
7316:
7294:
7272:
7245:
7233:rank of the matrix
7221:
7209:Bryant et al. 1991
7184:
7156:
7123:
7068:
7038:
6999:
6963:
6868:
6846:
6828:
6783:
6752:
6728:
6701:
6673:
6649:
6621:
6603:
6570:
6543:
6516:
6496:
6454:
6426:
6396:
6371:
6343:
6318:
6272:
6236:
6218:
6190:
6161:
6120:
6099:linear combination
6087:
6057:
6003:
5985:
5957:
5796:
5772:
5720:
5700:
5668:
5588:
5512:
5481:
5463:
5438:
5420:
5393:
5359:
5337:
5315:
5295:
5264:
5242:
5224:
5190:
5070:
5059:
5029:
5011:
4984:
4944:
4876:
4808:
4738:
4710:
4680:In particular, if
4665:
4640:
4605:
4463:, it follows that
4451:
4427:
4396:
4365:
4334:
4290:
4267:
4211:
4078:
4042:
4016:
3987:
3953:
3912:
3852:
3823:
3800:
3741:
3715:
3689:
3668:is defined as the
3658:
3635:
3625:of a vector space
3615:
3485:
3189:
3026:
2735:
2632:
2503:
2468:
2226:
2146:
2049:
2047:
1705:
1681:
1656:
1612:
1556:
1488:
1466:
1442:
1408:
1366:
1292:
1241:
1232:
1185:
1117:
1093:
1083:Areas in the plane
1064:
1040:
1028:differential forms
1008:
961:
906:
886:
866:
838:{\displaystyle V,}
835:
812:
791:{\displaystyle V.}
788:
761:
741:
701:
674:
630:
590:
573:{\displaystyle w,}
570:
547:
523:
471:
451:
422:
363:
344:{\displaystyle V.}
341:
318:
298:
277:{\displaystyle V.}
274:
251:
231:
199:
174:{\displaystyle V,}
171:
144:
27869:
27868:
27831:Hermann Grassmann
27787:
27786:
27739:Moment of inertia
27600:Differential form
27575:Affine connection
27390:Einstein notation
27373:
27372:
27301:Exterior calculus
27281:Coordinate system
27213:
27212:
27080:Geometric algebra
27037:Kronecker product
26872:Linear projection
26857:Vector projection
26742:978-3-642-30993-9
26723:Shafarevich, I.R.
26712:This textbook in
26706:978-1-4684-9461-7
26661:978-0-9614088-5-5
26504:978-0-8053-9021-6
26482:geometric algebra
26400:978-0-8176-4126-9
26254:Sternberg, Shlomo
26209:, Springer-Verlag
26164:Bourbaki, Nicolas
26088:extensive algebra
26046:Kannenberg (2000)
25878:{\displaystyle W}
25858:{\displaystyle V}
25827:{\displaystyle V}
25795:{\displaystyle V}
25707:, §III.7.1), and
25650:Leopold Kronecker
25632:{\displaystyle k}
25612:{\displaystyle k}
25544:978-0-679-77631-4
25537:. Vintage books.
25491:Symmetric algebra
25476:Geometric algebra
25441:universal algebra
25372:Hermann Grassmann
25270:{\displaystyle L}
25177:
25149:
25041:
25039:
24871:{\displaystyle K}
24847:{\displaystyle L}
24819:{\displaystyle K}
24797:{\displaystyle L}
24767:{\displaystyle n}
24686:symmetric algebra
24614:linear functional
24596:differential form
24592:surface integrals
24578:, so they can be
24556:Plücker relations
24552:Plücker embedding
24429:{\displaystyle V}
24395:{\displaystyle V}
24306:{\displaystyle w}
24282:{\displaystyle v}
24258:{\displaystyle u}
24198:{\displaystyle v}
24176:{\displaystyle u}
24127:The decomposable
23383:{\displaystyle V}
23323:{\displaystyle k}
23266:{\displaystyle n}
23244:{\displaystyle n}
23222:{\displaystyle Q}
23202:{\displaystyle P}
22931:{\displaystyle k}
22814:{\displaystyle V}
22762:{\displaystyle k}
22608:is exact, and if
21831:
21818:
21705:
21323:{\displaystyle f}
21213:{\displaystyle n}
21191:{\displaystyle V}
21143:{\displaystyle f}
21102:is the matrix of
21095:{\displaystyle W}
21075:{\displaystyle V}
20389:{\displaystyle W}
20369:{\displaystyle V}
20320:
20303:
19208:
18850:{\displaystyle K}
18828:{\displaystyle 1}
18579:symmetric algebra
18568:{\displaystyle V}
18382:{\displaystyle l}
17499:{\displaystyle V}
17482:orthonormal basis
17213:{\displaystyle k}
17059:{\displaystyle V}
17037:{\displaystyle V}
17005:{\displaystyle V}
16822:. In this case,
16813:{\displaystyle V}
16796:orthonormal basis
16726:with itself maps
16571:{\displaystyle V}
16172:{\displaystyle n}
16150:{\displaystyle V}
15863:graded derivation
15756:{\displaystyle V}
15639:{\displaystyle V}
15626:is an element of
15619:{\displaystyle v}
15413:{\displaystyle k}
15334:{\displaystyle f}
14975:Cartesian product
14958:{\displaystyle K}
14909:{\displaystyle w}
14599:{\displaystyle V}
14546:{\displaystyle V}
14401:is the subset of
14184:
14133:
14099:{\displaystyle K}
13972:
13970:
13883:{\displaystyle K}
13831:
13785:
13650:
13616:{\displaystyle K}
13569:{\displaystyle n}
13549:{\displaystyle V}
13471:{\displaystyle V}
13391:dual vector space
13378:{\displaystyle V}
13358:{\displaystyle k}
13342:multilinear forms
13239:-form ("mesh" of
12983:{\displaystyle k}
12963:{\displaystyle V}
12943:{\displaystyle k}
12440:{\displaystyle r}
12215:
12210:
12200:
11919:
11917:
11885:
11880:
11870:
11403:
11378:{\displaystyle I}
11133:{\displaystyle I}
11084:{\displaystyle K}
11027:
11022:
11012:
10972:
10898:{\displaystyle I}
10850:{\displaystyle I}
10260:
10026:
9863:{\displaystyle r}
9800:{\displaystyle 0}
9748:{\displaystyle I}
9697:{\displaystyle I}
9677:{\displaystyle K}
9619:{\displaystyle K}
9595:{\displaystyle I}
9573:{\displaystyle k}
9526:that consists of
9488:{\displaystyle K}
9468:{\displaystyle V}
9416:{\displaystyle M}
9401:projective module
9392:{\displaystyle M}
9269:{\displaystyle M}
9244:{\displaystyle R}
9224:{\displaystyle R}
7626:
7624:
7622:
7602:
7566:
7564:
7546:{\displaystyle p}
7342:
7297:{\displaystyle V}
6704:{\displaystyle V}
6374:{\displaystyle k}
6346:{\displaystyle n}
6307:
5898:
5895:
5885:
5799:{\displaystyle V}
5723:{\displaystyle n}
5703:{\displaystyle V}
5396:{\displaystyle k}
5340:{\displaystyle V}
5318:{\displaystyle k}
4987:{\displaystyle V}
4454:{\displaystyle V}
4045:{\displaystyle V}
3956:{\displaystyle V}
3692:{\displaystyle I}
3676:by the two-sided
3661:{\displaystyle K}
3638:{\displaystyle V}
3592:Formal definition
3586:orthonormal basis
2487:basis-independent
1519:
1067:{\displaystyle k}
1043:{\displaystyle k}
909:{\displaystyle V}
889:{\displaystyle V}
869:{\displaystyle V}
815:{\displaystyle k}
773:th exterior power
764:{\displaystyle k}
744:{\displaystyle k}
704:{\displaystyle k}
593:{\displaystyle k}
550:{\displaystyle v}
474:{\displaystyle k}
454:{\displaystyle k}
366:{\displaystyle k}
321:{\displaystyle V}
286:Hermann Grassmann
254:{\displaystyle v}
241:for every vector
189:and denoted with
147:{\displaystyle V}
129:Grassmann algebra
16:(Redirected from
27899:
27846:Bernhard Riemann
27678:
27677:
27521:Exterior product
27488:Two-point tensor
27473:Symmetric tensor
27355:Electromagnetism
27269:
27268:
27240:
27233:
27226:
27217:
27216:
27203:
27202:
27085:Exterior algebra
27022:Hadamard product
26939:
26927:Linear equations
26798:
26791:
26784:
26775:
26774:
26746:
26709:
26687:
26664:
26640:
26638:
26637:
26632:
26627:
26626:
26617:
26616:
26610:
26595:
26593:
26592:
26587:
26582:
26581:
26575:
26552:
26550:
26549:
26544:
26539:
26538:
26532:
26518:. The notation
26516:Stokes's theorem
26507:
26477:
26476:
26475:
26466:, archived from
26447:
26437:10.1038/058385a0
26403:
26392:
26382:
26371:
26357:Ausdehnungslehre
26345:
26331:
26329:Internet Archive
26314:
26283:
26261:
26244:
26210:
26180:
26154:
26143:
26115:
26109:
26103:
26072:
26066:
26063:
26057:
26054:Extension Theory
26050:Ausdehnungslehre
26043:
26037:
26036:
26015:
26009:
26008:
25986:
25980:
25969:
25963:
25957:
25955:
25954:
25949:
25947:
25933:
25927:
25920:
25914:
25908:
25906:
25905:
25900:
25898:
25884:
25882:
25881:
25876:
25864:
25862:
25861:
25856:
25843:
25837:
25835:
25833:
25831:
25830:
25825:
25810:Lie superalgebra
25803:
25801:
25799:
25798:
25793:
25778:
25772:
25771:
25753:
25747:
25745:
25735:
25729:
25722:
25716:
25701:
25695:
25688:
25682:
25675:
25669:
25654:Karl Weierstrass
25646:
25640:
25638:
25636:
25635:
25630:
25618:
25616:
25615:
25610:
25587:
25581:
25570:Grassmann (1844)
25567:
25561:
25555:
25549:
25548:
25528:
25467:Clifford algebra
25445:abstract algebra
25397:, much like the
25376:Ausdehnungslehre
25337:
25335:
25333:
25332:
25327:
25309:
25307:
25306:
25301:
25290:
25276:
25274:
25273:
25268:
25256:
25254:
25252:
25251:
25246:
25244:
25223:
25221:
25220:
25215:
25210:
25209:
25185:
25184:
25179:
25178:
25170:
25157:
25156:
25151:
25150:
25142:
25129:
25128:
25113:
25112:
25100:
25099:
25087:
25086:
25055:
25040:
25038:
25024:
25016:
25015:
24991:
24990:
24962:
24960:
24959:
24954:
24943:
24942:
24936:
24916:
24915:
24903:
24880:-linear mapping
24879:
24877:
24875:
24874:
24869:
24855:
24853:
24851:
24850:
24845:
24827:
24825:
24823:
24822:
24817:
24803:
24801:
24800:
24795:
24775:
24773:
24771:
24770:
24765:
24751:
24728:Grassmann number
24621:
24611:
24598:at a point of a
24549:
24548:
24546:
24545:
24540:
24538:
24537:
24522:
24521:
24515:
24507:
24506:
24500:
24487:projective space
24480:
24478:
24476:
24475:
24470:
24456:
24455:
24437:
24435:
24433:
24432:
24427:
24413:
24403:
24401:
24399:
24398:
24393:
24378:linear subspaces
24375:
24369:
24367:
24366:
24361:
24350:
24349:
24343:
24328:
24314:
24312:
24310:
24309:
24304:
24290:
24288:
24286:
24285:
24280:
24266:
24264:
24262:
24261:
24256:
24238:
24236:
24235:
24230:
24206:
24204:
24202:
24201:
24196:
24182:
24180:
24179:
24174:
24158:
24156:
24155:
24150:
24132:
24119:
24117:
24116:
24111:
24106:
24105:
24093:
24092:
24080:
24079:
24064:
24063:
24044:
24042:
24041:
24036:
24033:
24025:
24009:
24001:
23986:
23985:
23966:
23964:
23963:
23958:
23953:
23945:
23927:
23925:
23924:
23919:
23893:
23891:
23890:
23885:
23877:
23876:
23849:
23848:
23817:
23815:
23814:
23809:
23772:
23770:
23769:
23764:
23759:
23758:
23734:
23733:
23709:
23708:
23682:
23680:
23679:
23674:
23610:
23604:
23563:
23554:
23545:
23533:
23531:
23530:
23525:
23523:
23522:
23504:
23503:
23487:
23482:and similarly a
23481:
23479:
23478:
23473:
23465:
23464:
23446:
23445:
23424:
23423:
23411:
23410:
23391:
23389:
23387:
23386:
23381:
23367:
23365:
23364:
23359:
23348:
23331:
23329:
23327:
23326:
23321:
23307:
23305:
23304:
23299:
23272:
23270:
23269:
23264:
23252:
23250:
23248:
23247:
23242:
23228:
23226:
23225:
23220:
23208:
23206:
23205:
23200:
23184:
23182:
23181:
23176:
23161:
23159:
23158:
23153:
23151:
23150:
23141:
23140:
23122:
23121:
23112:
23111:
23099:
23098:
23089:
23088:
23076:
23075:
23066:
23065:
23056:
23055:
23031:
23029:
23028:
23023:
23021:
23016:
23015:
23006:
23005:
22987:
22986:
22977:
22976:
22964:
22963:
22954:
22953:
22937:
22935:
22934:
22929:
22917:
22915:
22914:
22909:
22907:
22906:
22882:
22881:
22869:
22868:
22852:
22850:
22849:
22844:
22822:
22820:
22818:
22817:
22812:
22798:
22796:
22795:
22790:
22788:
22768:
22766:
22765:
22760:
22732:
22730:
22729:
22724:
22701:
22700:
22688:
22668:
22667:
22661:
22641:
22640:
22635:
22604:
22602:
22601:
22596:
22579:
22578:
22572:
22552:
22551:
22545:
22525:
22524:
22512:
22475:
22473:
22472:
22467:
22459:
22458:
22431:
22430:
22412:
22411:
22387:
22386:
22374:
22373:
22355:
22354:
22342:
22341:
22317:
22316:
22300:
22298:
22297:
22292:
22281:
22280:
22274:
22254:
22253:
22241:
22230:
22229:
22220:
22215:
22214:
22189:
22187:
22186:
22181:
22170:
22169:
22153:
22151:
22150:
22145:
22137:
22136:
22120:
22118:
22117:
22112:
22110:
22109:
22088:
22087:
22075:
22074:
22044:
22043:
22027:
22025:
22024:
22019:
22001:
21999:
21998:
21993:
21991:
21990:
21971:
21969:
21968:
21963:
21952:
21951:
21945:
21934:
21933:
21915:
21914:
21896:
21895:
21883:
21882:
21854:
21852:
21851:
21846:
21832:
21824:
21819:
21811:
21790:
21788:
21787:
21782:
21768:
21767:
21761:
21741:
21740:
21734:
21725:
21686:
21685:
21679:
21658:
21656:
21655:
21650:
21636:
21617:
21592:
21567:
21565:
21564:
21559:
21545:
21526:
21500:
21498:
21497:
21492:
21475:
21456:
21437:
21418:
21417:
21411:
21383:
21381:
21380:
21375:
21331:
21329:
21327:
21326:
21321:
21303:
21301:
21300:
21295:
21284:
21283:
21277:
21262:
21260:
21259:
21254:
21243:
21242:
21236:
21221:
21219:
21217:
21216:
21211:
21197:
21195:
21194:
21189:
21177:
21175:
21174:
21169:
21151:
21149:
21147:
21146:
21141:
21127:
21125:
21124:
21119:
21101:
21099:
21098:
21093:
21081:
21079:
21078:
21073:
21061:
21059:
21057:
21056:
21051:
21040:
21039:
21033:
21013:
21011:
21010:
21005:
20991:
20990:
20984:
20964:
20963:
20957:
20946:
20943:
20942:
20932:
20931:
20925:
20900:
20881:
20880:
20874:
20853:
20851:
20850:
20845:
20837:
20836:
20809:
20808:
20787:
20786:
20768:
20767:
20746:
20729:
20727:
20726:
20721:
20719:
20698:
20692:
20690:
20689:
20684:
20673:
20656:
20654:
20653:
20648:
20634:
20633:
20627:
20601:
20600:
20594:
20571:
20568:
20567:
20557:
20556:
20550:
20525:
20505:
20503:
20502:
20497:
20486:
20467:
20448:
20427:
20425:
20424:
20419:
20395:
20393:
20392:
20387:
20375:
20373:
20372:
20367:
20343:
20341:
20339:
20338:
20333:
20328:
20327:
20326:
20325:
20316:
20304:
20301:
20297:
20242:
20240:
20239:
20234:
20217:
20190:
20188:
20187:
20182:
20170:
20144:
20142:
20141:
20136:
20134:
20130:
20126:
20125:
20107:
20106:
20065:
20064:
20046:
20045:
19997:
19995:
19993:
19992:
19987:
19973:
19971:
19970:
19965:
19953:
19951:
19950:
19945:
19933:
19931:
19930:
19925:
19909:
19890:
19870:
19868:
19866:
19865:
19860:
19849:
19830:
19806:
19804:
19803:
19798:
19787:
19786:
19775:
19755:
19754:
19749:
19740:
19735:
19708:
19707:
19702:
19678:
19676:
19675:
19670:
19656:
19635:
19633:
19632:
19627:
19625:
19624:
19608:
19606:
19605:
19600:
19589:
19563:
19561:
19560:
19555:
19553:
19552:
19525:
19524:
19493:
19491:
19490:
19485:
19483:
19482:
19455:
19454:
19411:
19391:
19389:
19388:
19383:
19375:
19374:
19347:
19346:
19313:
19312:
19285:
19284:
19246:
19205:
19200:
19179:
19178:
19160:
19159:
19131:
19129:
19128:
19123:
19017:
19015:
19014:
19009:
18998:
18984:
18982:
18980:
18979:
18974:
18963:
18944:
18928:
18926:
18924:
18923:
18918:
18907:
18888:
18887:
18881:
18858:
18856:
18854:
18853:
18848:
18834:
18832:
18831:
18826:
18814:
18812:
18810:
18809:
18804:
18781:
18779:
18778:
18773:
18725:
18723:
18721:
18720:
18715:
18704:
18685:
18666:
18636:
18634:
18632:
18631:
18626:
18615:
18576:
18574:
18572:
18571:
18566:
18552:
18550:
18549:
18544:
18533:
18511:
18509:
18507:
18506:
18501:
18490:
18489:
18483:
18472:
18455:
18453:
18452:
18447:
18442:
18434:
18420:
18412:
18411:
18406:
18390:
18388:
18386:
18385:
18380:
18366:
18364:
18363:
18358:
18356:
18355:
18350:
18349:
18333:
18332:
18326:
18318:
18317:
18308:
18304:
18303:
18290:
18289:
18283:
18272:
18271:
18266:
18250:
18248:
18247:
18242:
18237:
18232:
18231:
18230:
18229:
18224:
18210:
18196:
18188:
18180:
18162:
18160:
18158:
18157:
18152:
18141:
18140:
18134:
18123:
18109:
18107:
18105:
18104:
18099:
18088:
18087:
18081:
18070:
18056:
18054:
18052:
18051:
18046:
18035:
18034:
18022:
18011:
17994:
17992:
17990:
17989:
17984:
17961:
17959:
17958:
17953:
17924:
17923:
17900:
17887:
17885:
17884:
17879:
17874:
17869:
17868:
17867:
17866:
17849:
17835:
17827:
17803:
17801:
17799:
17798:
17793:
17773:
17771:
17769:
17768:
17763:
17752:
17751:
17745:
17734:
17720:
17718:
17716:
17715:
17710:
17699:
17698:
17686:
17675:
17654:
17652:
17650:
17649:
17644:
17633:
17632:
17626:
17606:
17604:
17603:
17598:
17593:
17592:
17574:
17573:
17560:
17559:
17558:
17557:
17534:
17533:
17532:
17531:
17507:
17505:
17503:
17502:
17497:
17479:
17460:
17458:
17457:
17452:
17438:
17437:
17431:
17416:
17387:
17364:
17362:
17361:
17356:
17351:
17350:
17341:
17340:
17328:
17327:
17315:
17314:
17302:
17298:
17297:
17296:
17278:
17277:
17265:
17264:
17246:
17245:
17221:
17219:
17217:
17216:
17211:
17197:
17195:
17193:
17192:
17187:
17185:
17184:
17179:
17178:
17168:
17167:
17161:
17153:
17152:
17137:
17135:
17134:
17129:
17118:
17117:
17111:
17096:
17094:
17092:
17091:
17086:
17084:
17083:
17065:
17063:
17062:
17057:
17045:
17043:
17041:
17040:
17035:
17020:pseudo-Euclidean
17013:
17011:
17009:
17008:
17003:
16973:
16961:metric signature
16955:
16953:
16952:
16947:
16945:
16937:
16936:
16885:
16884:
16878:
16858:
16857:
16851:
16821:
16819:
16817:
16816:
16811:
16793:
16791:
16790:
16785:
16774:
16773:
16767:
16747:
16746:
16740:
16725:
16723:
16722:
16717:
16702:
16700:
16699:
16694:
16680:
16679:
16667:
16647:
16646:
16640:
16608:
16606:
16604:
16603:
16598:
16596:
16595:
16577:
16575:
16574:
16569:
16546:
16544:
16543:
16538:
16530:
16529:
16502:
16501:
16489:
16475:
16474:
16462:
16461:
16455:
16437:
16435:
16433:
16432:
16427:
16412:orientation form
16403:
16401:
16400:
16395:
16384:
16383:
16377:
16359:
16357:
16356:
16351:
16346:
16345:
16333:
16332:
16320:
16319:
16291:
16289:
16288:
16283:
16272:
16271:
16259:
16239:
16238:
16232:
16218:
16217:
16205:
16204:
16198:
16180:
16178:
16176:
16175:
16170:
16156:
16154:
16153:
16148:
16122:
16120:
16119:
16114:
16109:
16108:
16096:
16095:
16080:
16079:
16067:
16066:
16049:
16047:
16046:
16041:
16033:
16032:
16020:
16019:
15992:
15990:
15989:
15984:
15973:
15972:
15954:
15953:
15914:
15913:
15883:
15882:
15860:
15858:
15857:
15852:
15850:
15849:
15833:
15831:
15829:
15828:
15823:
15821:
15820:
15793:
15791:
15789:
15788:
15783:
15781:
15780:
15763:and elements of
15762:
15760:
15759:
15754:
15742:
15740:
15738:
15737:
15732:
15712:
15711:
15693:
15691:
15689:
15688:
15683:
15672:
15671:
15665:
15645:
15643:
15642:
15637:
15625:
15623:
15622:
15617:
15601:
15599:
15598:
15593:
15579:
15578:
15566:
15546:
15545:
15539:
15528:
15527:
15509:
15507:
15506:
15501:
15478:
15477:
15458:
15456:
15454:
15453:
15448:
15446:
15445:
15421:
15419:
15417:
15416:
15411:
15385:
15383:
15381:
15380:
15375:
15364:
15363:
15357:
15340:
15338:
15337:
15332:
15320:
15318:
15317:
15312:
15301:
15300:
15281:
15279:
15278:
15273:
15265:
15264:
15240:
15239:
15227:
15226:
15199:
15198:
15174:
15173:
15161:
15160:
15142:
15141:
15119:
15117:
15115:
15114:
15109:
15107:
15106:
15088:
15086:
15085:
15080:
15038:
15036:
15034:
15033:
15028:
15026:
15025:
15007:
15006:
14994:
14993:
14972:
14966:
14964:
14962:
14961:
14956:
14942:
14940:
14939:
14934:
14932:
14931:
14915:
14913:
14912:
14907:
14895:
14893:
14891:
14890:
14885:
14874:
14873:
14867:
14841:
14839:
14837:
14836:
14831:
14809:
14807:
14805:
14804:
14799:
14784:interior product
14778:
14776:
14775:
14770:
14756:
14755:
14743:
14723:
14722:
14716:
14705:
14704:
14685:
14683:
14681:
14680:
14675:
14664:
14644:
14642:
14640:
14639:
14634:
14632:
14631:
14608:, then for each
14607:
14605:
14603:
14602:
14597:
14579:
14577:
14576:
14571:
14569:
14568:
14552:
14550:
14549:
14544:
14529:Interior product
14523:Interior product
14518:
14484:
14453:
14444:(2) < ⋯ <
14434:
14413:
14400:
14371:
14369:
14368:
14363:
14355:
14354:
14321:
14320:
14286:
14285:
14258:
14257:
14216:
14215:
14214:
14203:
14180:
14176:
14175:
14151:
14150:
14135:
14134:
14126:
14105:
14103:
14102:
14097:
14079:
14077:
14076:
14071:
14063:
14062:
14035:
14034:
13993:
13992:
13991:
13971:
13969:
13958:
13950:
13949:
13931:
13930:
13889:
13887:
13886:
13881:
13866:
13864:
13863:
13858:
13832:
13830:
13815:
13795:
13787:
13786:
13778:
13759:
13757:
13756:
13751:
13701:
13687:
13670:
13668:
13666:
13665:
13660:
13657:
13656:
13655:
13642:
13622:
13620:
13619:
13614:
13602:
13600:
13599:
13594:
13592:
13591:
13575:
13573:
13572:
13567:
13555:
13553:
13552:
13547:
13535:
13533:
13531:
13530:
13525:
13523:
13519:
13518:
13505:
13504:
13498:
13481:
13477:
13475:
13474:
13469:
13457:
13455:
13453:
13452:
13447:
13445:
13444:
13439:
13438:
13422:
13421:
13415:
13407:
13406:
13384:
13382:
13381:
13376:
13364:
13362:
13361:
13356:
13329:
13327:
13326:
13321:
13313:
13312:
13287:
13285:
13283:
13282:
13277:
13249:
13238:
13232:
13226:
13220:
13211:
13204:exterior product
13183:
13181:
13180:
13175:
13173:
13172:
13152:
13150:
13149:
13144:
13117:
13115:
13114:
13109:
13092:
13091:
13085:
13061:
13059:
13058:
13053:
13042:
13041:
13019:
13017:
13016:
13011:
13006:
13005:
12989:
12987:
12986:
12981:
12969:
12967:
12966:
12961:
12949:
12947:
12946:
12941:
12926:
12924:
12923:
12918:
12904:
12903:
12897:
12886:
12885:
12857:
12855:
12854:
12849:
12838:
12837:
12819:
12818:
12766:
12764:
12763:
12758:
12750:
12749:
12673:
12671:
12670:
12665:
12660:
12659:
12658:
12657:
12639:
12638:
12621:
12620:
12610:
12605:
12584:
12583:
12582:
12581:
12563:
12562:
12545:
12544:
12522:
12520:
12518:
12517:
12512:
12492:
12490:
12488:
12487:
12482:
12477:
12476:
12458:
12448:
12446:
12444:
12443:
12438:
12424:
12422:
12421:
12416:
12414:
12413:
12412:
12411:
12393:
12392:
12383:
12382:
12354:interior product
12348:
12346:
12345:
12340:
12335:
12334:
12330:
12329:
12311:
12310:
12290:
12289:
12288:
12287:
12275:
12274:
12254:
12253:
12252:
12251:
12233:
12232:
12213:
12212:
12211:
12203:
12198:
12181:
12168:
12166:
12165:
12160:
12155:
12154:
12153:
12152:
12136:
12135:
12119:
12118:
12117:
12116:
12106:
12105:
12095:
12094:
12093:
12092:
12082:
12081:
12074:
12073:
12072:
12071:
12044:
12043:
12014:
12013:
12012:
12011:
11990:
11989:
11950:
11949:
11948:
11937:
11936:
11918:
11916:
11893:
11883:
11882:
11881:
11873:
11868:
11836:in its indices.
11824:
11822:
11821:
11816:
11811:
11810:
11809:
11808:
11798:
11797:
11781:
11780:
11779:
11778:
11768:
11767:
11757:
11756:
11755:
11754:
11744:
11743:
11735:
11734:
11733:
11732:
11720:
11719:
11710:
11709:
11671:
11648:
11614:
11612:
11610:
11609:
11604:
11581:
11579:
11578:
11573:
11552:
11550:
11549:
11544:
11518:
11488:
11486:
11484:
11483:
11478:
11467:
11452:isomorphic with
11451:
11449:
11447:
11446:
11441:
11415:
11413:
11412:
11407:
11405:
11404:
11396:
11386:
11384:
11382:
11381:
11376:
11362:
11360:
11359:
11354:
11333:
11331:
11330:
11325:
11314:
11285:
11283:
11282:
11277:
11275:
11274:
11255:
11253:
11252:
11247:
11245:
11244:
11228:
11226:
11225:
11220:
11206:
11174:
11172:
11170:
11169:
11164:
11153:
11139:
11137:
11136:
11131:
11119:
11117:
11116:
11111:
11090:
11088:
11087:
11082:
11067:
11065:
11064:
11059:
11025:
11024:
11023:
11015:
11010:
10984:
10982:
10981:
10976:
10974:
10973:
10965:
10955:
10953:
10952:
10947:
10926:
10924:
10923:
10918:
10906:
10904:
10902:
10901:
10896:
10882:
10880:
10879:
10874:
10872:
10871:
10858:
10856:
10854:
10853:
10848:
10834:
10832:
10830:
10829:
10824:
10822:
10821:
10797:
10795:
10794:
10789:
10787:
10786:
10775:
10774:
10760:
10758:
10756:
10755:
10750:
10739:
10725:
10723:
10721:
10720:
10715:
10704:
10690:
10688:
10686:
10685:
10680:
10653:
10651:
10650:
10645:
10631:
10623:
10622:
10609:
10607:
10606:
10601:
10599:
10598:
10573:
10571:
10570:
10565:
10560:
10559:
10558:
10547:
10546:
10529:
10528:
10527:
10516:
10515:
10504:
10503:
10502:
10491:
10490:
10469:
10467:
10465:
10464:
10459:
10448:
10434:
10432:
10431:
10426:
10424:
10423:
10404:
10402:
10401:
10396:
10394:
10393:
10380:
10378:
10376:
10375:
10370:
10331:
10329:
10328:
10323:
10318:
10317:
10299:
10298:
10283:
10282:
10281:
10270:
10269:
10261:
10259:
10248:
10240:
10239:
10221:
10220:
10205:
10204:
10179:
10177:
10176:
10171:
10156:
10154:
10153:
10148:
10124:
10122:
10121:
10116:
10114:
10113:
10086:
10085:
10051:
10050:
10049:
10044:
10043:
10019:
10018:
10000:
9999:
9984:
9983:
9982:
9971:
9970:
9941:
9939:
9938:
9933:
9922:
9921:
9908:
9907:
9889:
9888:
9869:
9867:
9866:
9861:
9849:
9847:
9846:
9841:
9830:
9829:
9824:
9808:
9806:
9804:
9803:
9798:
9784:
9782:
9780:
9779:
9774:
9754:
9752:
9751:
9746:
9734:
9732:
9731:
9726:
9715:
9703:
9701:
9700:
9695:
9683:
9681:
9680:
9675:
9663:
9661:
9659:
9658:
9653:
9627:
9625:
9623:
9622:
9617:
9603:
9601:
9599:
9598:
9593:
9579:
9577:
9576:
9571:
9559:
9557:
9555:
9554:
9549:
9525:
9523:
9522:
9517:
9506:
9494:
9492:
9491:
9486:
9474:
9472:
9471:
9466:
9422:
9420:
9419:
9414:
9398:
9396:
9395:
9390:
9378:
9376:
9375:
9370:
9359:
9345:
9343:
9341:
9340:
9335:
9324:
9310:
9308:
9307:
9302:
9291:
9277:
9275:
9273:
9272:
9267:
9250:
9248:
9247:
9242:
9230:
9228:
9227:
9222:
9210:commutative ring
9199:
9197:
9195:
9194:
9189:
9178:
9162:
9160:
9159:
9154:
9143:
9142:
9136:
9121:
9119:
9118:
9113:
9102:
9101:
9095:
9080:
9078:
9077:
9072:
9061:
9036:
9034:
9033:
9028:
9017:
9003:
8994:
8988:
8986:
8985:
8980:
8969:
8955:
8953:
8951:
8950:
8945:
8934:
8918:
8911:
8909:
8908:
8903:
8897:
8867:
8851:as the quotient
8850:
8848:
8847:
8842:
8831:
8817:
8811:
8805:
8795:
8784:
8771:
8757:
8751:
8745:
8727:
8725:
8723:
8722:
8717:
8706:
8690:
8684:
8678:
8672:
8666:
8643:
8641:
8640:
8635:
8618:
8591:
8585:
8579:
8577:
8576:
8571:
8532:
8530:
8529:
8524:
8497:
8491:
8485:
8471:
8469:
8467:
8466:
8461:
8450:
8434:
8428:
8422:
8416:
8414:
8413:
8408:
8397:
8383:
8373:
8371:
8370:
8365:
8336:distributive law
8333:
8331:
8330:
8325:
8314:
8300:
8294:
8269:
8267:
8266:
8261:
8247:
8246:
8203:
8201:
8199:
8198:
8193:
8182:
8181:
8175:
8152:
8150:
8149:
8144:
8133:
8132:
8126:
8102:
8100:
8099:
8094:
8074:
8073:
8067:
8052:
8050:
8049:
8044:
8027:
8026:
8020:
8002:
7993:
7991:
7990:
7985:
7971:
7970:
7958:
7938:
7937:
7931:
7911:
7910:
7904:
7879:
7877:
7876:
7871:
7860:
7859:
7853:
7827:
7826:
7820:
7800:
7799:
7793:
7773:
7772:
7766:
7746:
7729:
7727:
7726:
7721:
7697:
7691:
7682:Graded structure
7677:
7675:
7674:
7669:
7661:
7659:
7648:
7643:
7620:
7612:
7610:
7609:
7604:
7600:
7593:
7588:
7583:
7552:
7550:
7549:
7544:
7532:
7530:
7529:
7524:
7509:
7507:
7506:
7501:
7489:
7487:
7486:
7481:
7479:
7478:
7455:
7453:
7452:
7447:
7445:
7444:
7426:
7425:
7403:
7401:
7400:
7395:
7393:
7392:
7380:
7379:
7370:
7369:
7356:
7325:
7323:
7322:
7317:
7305:
7303:
7301:
7300:
7295:
7281:
7279:
7278:
7273:
7271:
7270:
7254:
7252:
7251:
7246:
7230:
7228:
7227:
7222:
7195:
7193:
7191:
7190:
7185:
7171:
7165:
7163:
7162:
7157:
7145:
7132:
7130:
7129:
7124:
7087:
7076:
7057:
7046:
7034:
7033:
7008:
7006:
7005:
7000:
6998:
6997:
6972:
6970:
6969:
6964:
6962:
6961:
6937:
6936:
6918:
6917:
6884:
6877:
6875:
6874:
6869:
6857:
6855:
6853:
6852:
6847:
6836:
6835:
6829:
6794:
6792:
6790:
6789:
6784:
6782:
6781:
6763:
6761:
6759:
6758:
6753:
6736:
6735:
6729:
6712:
6710:
6708:
6707:
6702:
6684:
6682:
6680:
6679:
6674:
6657:
6656:
6650:
6630:
6628:
6627:
6622:
6611:
6610:
6604:
6578:
6577:
6571:
6551:
6550:
6544:
6524:
6523:
6517:
6497:
6475:
6465:
6463:
6461:
6460:
6455:
6435:
6433:
6432:
6427:
6404:
6403:
6397:
6382:
6380:
6378:
6377:
6372:
6354:
6352:
6350:
6349:
6344:
6327:
6325:
6324:
6319:
6314:
6313:
6312:
6299:
6280:
6279:
6273:
6245:
6243:
6242:
6237:
6226:
6225:
6219:
6201:
6199:
6197:
6196:
6191:
6189:
6188:
6170:
6168:
6167:
6162:
6160:
6159:
6135:
6131:
6129:
6127:
6126:
6121:
6119:
6118:
6096:
6094:
6093:
6088:
6086:
6085:
6066:
6064:
6063:
6058:
6053:
6052:
6034:
6033:
6014:
6012:
6010:
6009:
6004:
5993:
5992:
5986:
5966:
5964:
5963:
5958:
5946:
5945:
5927:
5926:
5914:
5913:
5896:
5893:
5892:
5891:
5883:
5882:
5881:
5880:
5879:
5856:
5855:
5854:
5853:
5836:
5835:
5834:
5833:
5805:
5803:
5802:
5797:
5781:
5779:
5778:
5773:
5768:
5767:
5749:
5748:
5729:
5727:
5726:
5721:
5709:
5707:
5706:
5701:
5677:
5675:
5674:
5669:
5664:
5663:
5651:
5650:
5638:
5637:
5625:
5624:
5599:
5597:
5595:
5594:
5589:
5584:
5583:
5571:
5570:
5558:
5557:
5545:
5544:
5523:
5521:
5519:
5518:
5513:
5511:
5510:
5505:
5490:
5488:
5487:
5482:
5471:
5470:
5464:
5449:
5447:
5445:
5444:
5439:
5428:
5427:
5421:
5404:
5402:
5400:
5399:
5394:
5368:
5366:
5365:
5360:
5348:
5346:
5344:
5343:
5338:
5324:
5322:
5321:
5316:
5304:
5302:
5301:
5296:
5281:
5274:is said to be a
5273:
5271:
5270:
5265:
5253:
5251:
5249:
5248:
5243:
5232:
5231:
5225:
5199:
5197:
5196:
5191:
5150:
5149:
5136:
5135:
5117:
5116:
5104:
5103:
5081:
5079:
5077:
5076:
5071:
5060:
5040:
5038:
5036:
5035:
5030:
5019:
5018:
5012:
4995:
4993:
4991:
4990:
4985:
4967:
4953:
4951:
4950:
4945:
4937:
4936:
4918:
4917:
4905:
4904:
4885:
4883:
4882:
4877:
4872:
4871:
4853:
4852:
4840:
4839:
4817:
4815:
4814:
4809:
4801:
4800:
4782:
4781:
4769:
4768:
4749:
4747:
4745:
4744:
4739:
4719:
4717:
4716:
4711:
4709:
4708:
4696:
4695:
4676:
4674:
4672:
4671:
4666:
4649:
4647:
4646:
4641:
4614:
4612:
4611:
4606:
4601:
4600:
4582:
4581:
4569:
4568:
4541:
4540:
4513:
4512:
4491:
4490:
4462:
4460:
4458:
4457:
4452:
4439:are elements of
4438:
4436:
4434:
4433:
4428:
4426:
4425:
4407:
4405:
4403:
4402:
4397:
4395:
4394:
4376:
4374:
4372:
4371:
4366:
4364:
4363:
4345:
4343:
4341:
4340:
4337:{\displaystyle }
4335:
4304:of the integers
4299:
4297:
4296:
4291:
4276:
4274:
4273:
4268:
4220:
4218:
4217:
4212:
4089:
4087:
4085:
4084:
4079:
4053:
4051:
4049:
4048:
4043:
4025:
4023:
4022:
4017:
3996:
3994:
3993:
3988:
3964:
3962:
3960:
3959:
3954:
3921:
3919:
3918:
3913:
3908:
3861:
3859:
3858:
3853:
3832:
3830:
3829:
3824:
3809:
3807:
3806:
3801:
3792:
3751:. Symbolically,
3750:
3748:
3747:
3742:
3724:
3722:
3721:
3716:
3698:
3696:
3695:
3690:
3670:quotient algebra
3667:
3665:
3664:
3659:
3644:
3642:
3641:
3636:
3624:
3622:
3621:
3616:
3539:
3494:
3492:
3491:
3486:
3481:
3480:
3475:
3466:
3465:
3460:
3451:
3450:
3445:
3433:
3432:
3423:
3422:
3413:
3412:
3400:
3399:
3390:
3389:
3380:
3379:
3367:
3366:
3357:
3356:
3347:
3346:
3334:
3333:
3324:
3323:
3314:
3313:
3301:
3300:
3291:
3290:
3281:
3280:
3268:
3267:
3258:
3257:
3248:
3247:
3232:
3224:
3216:
3198:
3196:
3195:
3190:
3185:
3184:
3179:
3173:
3172:
3160:
3159:
3154:
3148:
3147:
3135:
3134:
3129:
3123:
3122:
3110:
3035:
3033:
3032:
3027:
3022:
3021:
3016:
3007:
3006:
3001:
2989:
2988:
2979:
2978:
2966:
2965:
2956:
2955:
2937:
2936:
2931:
2922:
2921:
2916:
2904:
2903:
2894:
2893:
2881:
2880:
2871:
2870:
2852:
2851:
2846:
2837:
2836:
2831:
2819:
2818:
2809:
2808:
2796:
2795:
2786:
2785:
2770:
2762:
2744:
2742:
2741:
2736:
2734:
2733:
2728:
2722:
2721:
2709:
2708:
2703:
2697:
2696:
2684:
2683:
2678:
2672:
2671:
2659:
2641:
2639:
2638:
2633:
2631:
2630:
2625:
2619:
2618:
2606:
2605:
2600:
2594:
2593:
2581:
2580:
2575:
2569:
2568:
2556:
2541:
2464:
2458:
2452:
2446:
2433:
2401:
2364:
2330:
2308:
2267:
2235:
2233:
2232:
2227:
2219:
2218:
2213:
2204:
2203:
2198:
2189:
2188:
2183:
2174:
2173:
2168:
2155:
2153:
2152:
2147:
2139:
2138:
2133:
2124:
2123:
2118:
2103:
2102:
2097:
2088:
2087:
2082:
2064:exterior product
2058:
2056:
2055:
2050:
2048:
2044:
2043:
2038:
2029:
2028:
2023:
2017:
2013:
1986:
1982:
1981:
1976:
1967:
1966:
1961:
1946:
1945:
1940:
1931:
1930:
1925:
1910:
1909:
1904:
1895:
1894:
1889:
1874:
1873:
1868:
1859:
1858:
1853:
1835:
1828:
1827:
1822:
1810:
1809:
1804:
1786:
1785:
1780:
1768:
1767:
1762:
1743:
1735:
1716:
1714:
1712:
1711:
1706:
1704:
1690:
1688:
1687:
1682:
1680:
1665:
1663:
1662:
1657:
1652:
1648:
1624:
1623:
1617:
1616:
1578:
1577:
1568:
1567:
1561:
1560:
1553:
1546:
1530:
1529:
1520:
1517:
1497:
1495:
1494:
1489:
1487:
1475:
1473:
1472:
1467:
1465:
1453:
1451:
1449:
1448:
1443:
1441:
1440:
1435:
1417:
1415:
1414:
1409:
1407:
1406:
1401:
1389:
1388:
1383:
1371:
1370:
1342:
1333:
1332:
1327:
1315:
1314:
1309:
1297:
1296:
1268:
1250:
1248:
1247:
1242:
1237:
1236:
1208:
1207:
1202:
1201:
1190:
1189:
1161:
1160:
1155:
1154:
1126:
1124:
1123:
1118:
1116:
1115:
1110:
1073:
1071:
1070:
1065:
1052:smooth functions
1049:
1047:
1046:
1041:
1024:commutative ring
1017:
1015:
1014:
1009:
970:
968:
967:
962:
915:
913:
912:
907:
895:
893:
892:
887:
875:
873:
872:
867:
844:
842:
841:
836:
821:
819:
818:
813:
797:
795:
794:
789:
770:
768:
767:
762:
750:
748:
747:
742:
715:
710:
708:
707:
702:
683:
681:
680:
675:
639:
637:
636:
631:
599:
597:
596:
591:
579:
577:
576:
571:
556:
554:
553:
548:
532:
530:
529:
524:
505:
480:
478:
477:
472:
460:
458:
457:
452:
431:
429:
428:
423:
421:
420:
402:
401:
389:
388:
372:
370:
369:
364:
350:
348:
347:
342:
327:
325:
324:
319:
307:
305:
304:
299:
283:
281:
280:
275:
260:
258:
257:
252:
240:
238:
237:
232:
208:
206:
205:
200:
183:exterior product
180:
178:
177:
172:
153:
151:
150:
145:
125:exterior algebra
116:
76:
57:
45:
21:
27907:
27906:
27902:
27901:
27900:
27898:
27897:
27896:
27872:
27871:
27870:
27865:
27816:Albert Einstein
27783:
27764:Einstein tensor
27727:
27708:Ricci curvature
27688:Kronecker delta
27674:Notable tensors
27669:
27590:Connection form
27567:
27561:
27492:
27478:Tensor operator
27435:
27429:
27369:
27345:Computer vision
27338:
27320:
27316:Tensor calculus
27260:
27249:
27244:
27214:
27209:
27191:
27153:
27109:
27046:
26998:
26940:
26931:
26897:Change of basis
26887:Multilinear map
26825:
26807:
26802:
26772:
26743:
26707:
26662:
26622:
26618:
26611:
26604:
26603:
26601:
26598:
26597:
26576:
26569:
26568:
26566:
26563:
26562:
26533:
26526:
26525:
26523:
26520:
26519:
26505:
26491:Spivak, Michael
26473:
26471:
26455:
26450:
26401:
26376:Peano, Giuseppe
26369:
26304:10.2307/2369379
26274:
26269:
26260:, Prentice Hall
26242:
26234:, AMS Chelsea,
26203:Griffiths, P.A.
26178:
26152:
26128:
26123:
26118:
26110:
26106:
26073:
26069:
26064:
26060:
26044:
26040:
26033:
26016:
26012:
26005:
25987:
25983:
25970:
25966:
25941:
25939:
25936:
25935:
25929:
25923:
25921:
25917:
25892:
25890:
25887:
25886:
25870:
25867:
25866:
25850:
25847:
25846:
25844:
25840:
25819:
25816:
25815:
25813:
25808:of the abelian
25787:
25784:
25783:
25781:
25779:
25775:
25768:
25754:
25750:
25741:
25736:
25732:
25723:
25719:
25702:
25698:
25692:Bourbaki (1989)
25689:
25685:
25676:
25672:
25658:Bourbaki (1989b
25647:
25643:
25624:
25621:
25620:
25604:
25601:
25600:
25588:
25584:
25568:
25564:
25556:
25552:
25545:
25529:
25522:
25518:
25510:symplectic form
25453:
25368:
25352:
25321:
25318:
25317:
25315:
25284:
25282:
25279:
25278:
25262:
25259:
25258:
25240:
25238:
25235:
25234:
25232:
25229:Jacobi identity
25199:
25195:
25180:
25169:
25168:
25167:
25152:
25141:
25140:
25139:
25124:
25120:
25108:
25104:
25095:
25091:
25070:
25066:
25045:
25028:
25023:
25005:
25001:
24986:
24982:
24974:
24971:
24970:
24937:
24930:
24929:
24904:
24897:
24896:
24888:
24885:
24884:
24863:
24860:
24859:
24857:
24839:
24836:
24835:
24833:
24811:
24808:
24807:
24805:
24789:
24786:
24785:
24782:
24759:
24756:
24755:
24753:
24747:
24708:
24674:
24658:cochain complex
24617:
24607:
24564:
24533:
24532:
24516:
24509:
24508:
24502:
24501:
24496:
24494:
24491:
24490:
24489:
24451:
24447:
24445:
24442:
24441:
24439:
24421:
24418:
24417:
24415:
24409:
24387:
24384:
24383:
24381:
24371:
24344:
24337:
24336:
24334:
24331:
24330:
24324:
24321:
24298:
24295:
24294:
24292:
24274:
24271:
24270:
24268:
24250:
24247:
24246:
24244:
24212:
24209:
24208:
24190:
24187:
24186:
24184:
24168:
24165:
24164:
24138:
24135:
24134:
24128:
24125:
24123:Linear geometry
24098:
24094:
24085:
24081:
24072:
24068:
24056:
24052:
24050:
24047:
24046:
24026:
24018:
24002:
23994:
23981:
23977:
23975:
23972:
23971:
23949:
23941:
23933:
23930:
23929:
23913:
23910:
23909:
23857:
23853:
23829:
23825:
23823:
23820:
23819:
23782:
23779:
23778:
23742:
23738:
23717:
23713:
23701:
23697:
23695:
23692:
23691:
23659:
23656:
23655:
23641:
23632:
23626:
23606:
23596:
23573:
23568:
23567:
23566:
23565:
23564:
23556:
23555:
23547:
23546:
23518:
23514:
23499:
23495:
23493:
23490:
23489:
23483:
23460:
23456:
23441:
23437:
23419:
23415:
23406:
23402:
23397:
23394:
23393:
23375:
23372:
23371:
23369:
23342:
23340:
23337:
23336:
23315:
23312:
23311:
23309:
23281:
23278:
23277:
23276:The sum of the
23258:
23255:
23254:
23236:
23233:
23232:
23230:
23214:
23211:
23210:
23194:
23191:
23190:
23167:
23164:
23163:
23146:
23142:
23136:
23132:
23117:
23113:
23107:
23103:
23094:
23090:
23084:
23080:
23071:
23067:
23061:
23057:
23051:
23047:
23036:
23033:
23032:
23020:
23011:
23007:
23001:
22997:
22982:
22978:
22972:
22968:
22959:
22955:
22949:
22945:
22943:
22940:
22939:
22923:
22920:
22919:
22902:
22898:
22877:
22873:
22864:
22860:
22858:
22855:
22854:
22832:
22829:
22828:
22806:
22803:
22802:
22800:
22784:
22782:
22779:
22778:
22754:
22751:
22750:
22747:
22742:
22689:
22682:
22681:
22662:
22655:
22654:
22636:
22629:
22628:
22620:
22617:
22616:
22573:
22566:
22565:
22546:
22539:
22538:
22513:
22506:
22505:
22491:
22488:
22487:
22454:
22450:
22426:
22422:
22401:
22397:
22382:
22378:
22369:
22365:
22350:
22346:
22331:
22327:
22312:
22308:
22306:
22303:
22302:
22275:
22268:
22267:
22242:
22235:
22234:
22225:
22221:
22216:
22204:
22200:
22198:
22195:
22194:
22165:
22161:
22159:
22156:
22155:
22132:
22128:
22126:
22123:
22122:
22099:
22095:
22083:
22079:
22058:
22054:
22039:
22035:
22033:
22030:
22029:
22007:
22004:
22003:
21986:
21982:
21980:
21977:
21976:
21946:
21939:
21938:
21923:
21919:
21910:
21906:
21891:
21887:
21878:
21874:
21866:
21863:
21862:
21823:
21810:
21799:
21796:
21795:
21762:
21755:
21754:
21735:
21728:
21727:
21709:
21680:
21673:
21672:
21670:
21667:
21666:
21630:
21611:
21586:
21584:
21581:
21580:
21574:
21539:
21520:
21512:
21509:
21508:
21469:
21450:
21431:
21412:
21405:
21404:
21396:
21393:
21392:
21345:
21342:
21341:
21338:
21315:
21312:
21311:
21309:
21278:
21271:
21270:
21268:
21265:
21264:
21237:
21230:
21229:
21227:
21224:
21223:
21205:
21202:
21201:
21199:
21183:
21180:
21179:
21157:
21154:
21153:
21135:
21132:
21131:
21129:
21107:
21104:
21103:
21087:
21084:
21083:
21067:
21064:
21063:
21034:
21027:
21026:
21024:
21021:
21020:
21018:
20985:
20978:
20977:
20958:
20951:
20950:
20926:
20919:
20918:
20917:
20914:
20910:
20894:
20875:
20868:
20867:
20865:
20862:
20861:
20832:
20828:
20804:
20800:
20782:
20778:
20763:
20759:
20740:
20738:
20735:
20734:
20709:
20704:
20701:
20700:
20694:
20667:
20665:
20662:
20661:
20660:In particular,
20628:
20621:
20620:
20595:
20588:
20587:
20551:
20544:
20543:
20542:
20539:
20535:
20519:
20517:
20514:
20513:
20480:
20461:
20442:
20440:
20437:
20436:
20401:
20398:
20397:
20381:
20378:
20377:
20361:
20358:
20357:
20354:
20321:
20300:
20299:
20293:
20292:
20291:
20287:
20261:
20258:
20257:
20255:
20211:
20203:
20200:
20199:
20176:
20173:
20172:
20149:
20121:
20117:
20102:
20098:
20091:
20087:
20060:
20056:
20041:
20037:
20017:
20014:
20013:
19981:
19978:
19977:
19975:
19959:
19956:
19955:
19939:
19936:
19935:
19919:
19916:
19915:
19912:tensor algebras
19892:
19872:
19843:
19824:
19822:
19819:
19818:
19816:
19776:
19769:
19768:
19750:
19743:
19742:
19736:
19725:
19703:
19696:
19695:
19687:
19684:
19683:
19650:
19648:
19645:
19644:
19620:
19616:
19614:
19611:
19610:
19583:
19581:
19578:
19577:
19564:to equal 1 for
19539:
19535:
19505:
19501:
19499:
19496:
19495:
19469:
19465:
19441:
19437:
19435:
19432:
19431:
19397:
19361:
19357:
19327:
19323:
19299:
19295:
19271:
19267:
19212:
19201:
19190:
19174:
19170:
19155:
19151:
19143:
19140:
19139:
19030:
19027:
19026:
18992:
18990:
18987:
18986:
18957:
18938:
18936:
18933:
18932:
18930:
18901:
18882:
18875:
18874:
18866:
18863:
18862:
18860:
18859:. Recall that
18842:
18839:
18838:
18836:
18820:
18817:
18816:
18792:
18789:
18788:
18786:
18734:
18731:
18730:
18698:
18679:
18660:
18652:
18649:
18648:
18646:
18609:
18607:
18604:
18603:
18601:
18591:tensor algebras
18560:
18557:
18556:
18554:
18527:
18525:
18522:
18521:
18518:
18484:
18477:
18476:
18468:
18466:
18463:
18462:
18460:
18438:
18430:
18416:
18407:
18402:
18401:
18399:
18396:
18395:
18374:
18371:
18370:
18368:
18351:
18345:
18344:
18343:
18327:
18320:
18319:
18313:
18312:
18299:
18295:
18291:
18284:
18277:
18276:
18267:
18262:
18261:
18259:
18256:
18255:
18233:
18225:
18220:
18219:
18218:
18214:
18206:
18192:
18184:
18176:
18171:
18168:
18167:
18135:
18128:
18127:
18119:
18117:
18114:
18113:
18111:
18082:
18075:
18074:
18066:
18064:
18061:
18060:
18058:
18023:
18016:
18015:
18007:
18005:
18002:
18001:
17999:
17972:
17969:
17968:
17966:
17919:
17915:
17913:
17910:
17909:
17892:
17870:
17862:
17858:
17857:
17853:
17845:
17831:
17823:
17812:
17809:
17808:
17781:
17778:
17777:
17775:
17746:
17739:
17738:
17730:
17728:
17725:
17724:
17722:
17687:
17680:
17679:
17671:
17669:
17666:
17665:
17663:
17627:
17620:
17619:
17617:
17614:
17613:
17611:
17588:
17584:
17569:
17565:
17553:
17549:
17548:
17544:
17527:
17523:
17522:
17518:
17516:
17513:
17512:
17491:
17488:
17487:
17485:
17471:
17469:
17432:
17425:
17424:
17422:
17419:
17418:
17414:
17405:
17396:
17386:
17377:
17369:
17346:
17345:
17336:
17332:
17323:
17319:
17310:
17309:
17292:
17288:
17273:
17269:
17260:
17256:
17241:
17237:
17236:
17232:
17230:
17227:
17226:
17205:
17202:
17201:
17199:
17180:
17174:
17173:
17172:
17162:
17155:
17154:
17148:
17147:
17145:
17142:
17141:
17139:
17112:
17105:
17104:
17102:
17099:
17098:
17079:
17075:
17073:
17070:
17069:
17067:
17051:
17048:
17047:
17029:
17026:
17025:
17023:
16997:
16994:
16993:
16991:
16988:
16963:
16938:
16911:
16907:
16879:
16872:
16871:
16852:
16845:
16844:
16830:
16827:
16826:
16805:
16802:
16801:
16799:
16768:
16761:
16760:
16741:
16734:
16733:
16731:
16728:
16727:
16711:
16708:
16707:
16668:
16661:
16660:
16641:
16634:
16633:
16625:
16622:
16621:
16591:
16587:
16585:
16582:
16581:
16579:
16563:
16560:
16559:
16525:
16521:
16490:
16483:
16482:
16470:
16466:
16456:
16449:
16448:
16446:
16443:
16442:
16421:
16418:
16417:
16415:
16378:
16371:
16370:
16368:
16365:
16364:
16341:
16337:
16328:
16324:
16309:
16305:
16303:
16300:
16299:
16260:
16253:
16252:
16233:
16226:
16225:
16213:
16209:
16199:
16192:
16191:
16189:
16186:
16185:
16164:
16161:
16160:
16158:
16142:
16139:
16138:
16135:
16129:
16104:
16100:
16091:
16087:
16075:
16071:
16062:
16058:
16056:
16053:
16052:
16028:
16024:
16015:
16011:
16009:
16006:
16005:
15968:
15964:
15943:
15939:
15909:
15905:
15878:
15874:
15872:
15869:
15868:
15845:
15841:
15839:
15836:
15835:
15816:
15812:
15804:
15801:
15800:
15798:
15776:
15772:
15770:
15767:
15766:
15764:
15748:
15745:
15744:
15707:
15703:
15701:
15698:
15697:
15695:
15666:
15659:
15658:
15653:
15650:
15649:
15647:
15631:
15628:
15627:
15611:
15608:
15607:
15567:
15560:
15559:
15540:
15533:
15532:
15523:
15519:
15517:
15514:
15513:
15470:
15466:
15464:
15461:
15460:
15441:
15437:
15429:
15426:
15425:
15423:
15405:
15402:
15401:
15399:
15392:
15358:
15351:
15350:
15348:
15345:
15344:
15342:
15326:
15323:
15322:
15296:
15292:
15290:
15287:
15286:
15254:
15250:
15235:
15231:
15222:
15218:
15188:
15184:
15169:
15165:
15156:
15152:
15137:
15133:
15128:
15125:
15124:
15102:
15098:
15096:
15093:
15092:
15090:
15068:
15065:
15064:
15062:
15052:
15045:
15021:
15017:
15002:
14998:
14989:
14985:
14983:
14980:
14979:
14977:
14968:
14950:
14947:
14946:
14944:
14927:
14923:
14921:
14918:
14917:
14901:
14898:
14897:
14868:
14861:
14860:
14852:
14849:
14848:
14846:
14825:
14822:
14821:
14819:
14793:
14790:
14789:
14787:
14744:
14737:
14736:
14717:
14710:
14709:
14700:
14696:
14694:
14691:
14690:
14658:
14656:
14653:
14652:
14650:
14649:on the algebra
14627:
14623:
14615:
14612:
14611:
14609:
14591:
14588:
14587:
14585:
14564:
14560:
14558:
14555:
14554:
14538:
14535:
14534:
14531:
14525:
14516:
14507:
14498:
14486:
14455:
14436:
14424:
14403:
14399:
14386:
14376:
14335:
14331:
14301:
14297:
14272:
14268:
14244:
14240:
14204:
14196:
14195:
14188:
14165:
14161:
14146:
14142:
14125:
14124:
14120:
14118:
14115:
14114:
14091:
14088:
14087:
14049:
14045:
14021:
14017:
13987:
13983:
13976:
13962:
13957:
13945:
13941:
13926:
13922:
13902:
13899:
13898:
13875:
13872:
13871:
13816:
13796:
13794:
13777:
13776:
13771:
13768:
13767:
13715:
13712:
13711:
13704:tensor products
13689:
13675:
13651:
13638:
13637:
13636:
13633:
13630:
13629:
13627:
13608:
13605:
13604:
13587:
13583:
13581:
13578:
13577:
13561:
13558:
13557:
13541:
13538:
13537:
13514:
13510:
13506:
13499:
13492:
13491:
13489:
13486:
13485:
13483:
13479:
13463:
13460:
13459:
13440:
13434:
13433:
13432:
13416:
13409:
13408:
13402:
13401:
13399:
13396:
13395:
13393:
13370:
13367:
13366:
13350:
13347:
13346:
13308:
13304:
13296:
13293:
13292:
13265:
13262:
13261:
13259:
13244:
13234:
13233:) to obtain an
13228:
13222:
13216:
13207:
13196:
13190:
13168:
13164:
13162:
13159:
13158:
13123:
13120:
13119:
13086:
13079:
13078:
13070:
13067:
13066:
13037:
13033:
13025:
13022:
13021:
13001:
12997:
12995:
12992:
12991:
12975:
12972:
12971:
12955:
12952:
12951:
12935:
12932:
12931:
12898:
12891:
12890:
12881:
12877:
12869:
12866:
12865:
12833:
12829:
12814:
12810:
12802:
12799:
12798:
12785:
12776:
12745:
12741:
12733:
12730:
12729:
12724:multilinear map
12696:
12691:
12647:
12643:
12634:
12630:
12626:
12622:
12616:
12612:
12606:
12595:
12571:
12567:
12558:
12554:
12553:
12549:
12540:
12536:
12531:
12528:
12527:
12500:
12497:
12496:
12494:
12472:
12468:
12466:
12463:
12462:
12460:
12450:
12432:
12429:
12428:
12426:
12401:
12397:
12388:
12384:
12378:
12374:
12373:
12369:
12361:
12358:
12357:
12319:
12315:
12300:
12296:
12295:
12291:
12283:
12279:
12270:
12266:
12262:
12258:
12241:
12237:
12228:
12224:
12223:
12219:
12202:
12201:
12190:
12187:
12186:
12173:
12142:
12138:
12137:
12131:
12130:
12129:
12112:
12108:
12107:
12101:
12100:
12099:
12088:
12084:
12083:
12077:
12076:
12075:
12052:
12048:
12024:
12020:
12019:
12015:
11998:
11994:
11976:
11972:
11971:
11967:
11938:
11932:
11931:
11930:
11923:
11897:
11892:
11872:
11871:
11863:
11860:
11859:
11804:
11800:
11799:
11793:
11792:
11791:
11774:
11770:
11769:
11763:
11762:
11761:
11750:
11746:
11745:
11739:
11738:
11737:
11728:
11724:
11715:
11711:
11705:
11701:
11700:
11696:
11688:
11685:
11684:
11654:
11647:
11638:
11632:
11621:
11589:
11586:
11585:
11583:
11558:
11555:
11554:
11514:
11494:
11491:
11490:
11461:
11459:
11456:
11455:
11453:
11426:
11423:
11422:
11420:
11395:
11394:
11392:
11389:
11388:
11370:
11367:
11366:
11364:
11339:
11336:
11335:
11308:
11291:
11288:
11287:
11264:
11263:
11261:
11258:
11257:
11240:
11239:
11237:
11234:
11233:
11200:
11183:
11180:
11179:
11149:
11147:
11144:
11143:
11141:
11125:
11122:
11121:
11096:
11093:
11092:
11076:
11073:
11072:
11014:
11013:
10993:
10990:
10989:
10964:
10963:
10961:
10958:
10957:
10932:
10929:
10928:
10912:
10909:
10908:
10890:
10887:
10886:
10884:
10867:
10866:
10864:
10861:
10860:
10842:
10839:
10838:
10836:
10811:
10807:
10805:
10802:
10801:
10799:
10776:
10770:
10769:
10768:
10766:
10763:
10762:
10735:
10733:
10730:
10729:
10727:
10700:
10698:
10695:
10694:
10692:
10665:
10662:
10661:
10659:
10627:
10618:
10617:
10615:
10612:
10611:
10588:
10584:
10582:
10579:
10578:
10548:
10542:
10541:
10540:
10539:
10517:
10511:
10510:
10509:
10508:
10492:
10486:
10485:
10484:
10483:
10481:
10478:
10477:
10444:
10442:
10439:
10438:
10436:
10413:
10412:
10410:
10407:
10406:
10389:
10388:
10386:
10383:
10382:
10346:
10343:
10342:
10340:
10337:symmetric group
10313:
10309:
10294:
10290:
10271:
10265:
10264:
10263:
10262:
10252:
10247:
10235:
10231:
10216:
10212:
10194:
10190:
10188:
10185:
10184:
10162:
10159:
10158:
10133:
10130:
10129:
10100:
10096:
10072:
10068:
10045:
10039:
10038:
10037:
10030:
10014:
10010:
9995:
9991:
9972:
9966:
9965:
9964:
9963:
9961:
9958:
9957:
9917:
9913:
9903:
9899:
9884:
9880:
9878:
9875:
9874:
9855:
9852:
9851:
9825:
9820:
9819:
9817:
9814:
9813:
9792:
9789:
9788:
9786:
9762:
9759:
9758:
9756:
9740:
9737:
9736:
9711:
9709:
9706:
9705:
9689:
9686:
9685:
9669:
9666:
9665:
9635:
9632:
9631:
9629:
9611:
9608:
9607:
9605:
9587:
9584:
9583:
9581:
9565:
9562:
9561:
9537:
9534:
9533:
9531:
9502:
9500:
9497:
9496:
9480:
9477:
9476:
9460:
9457:
9456:
9453:
9429:Bourbaki (1989)
9408:
9405:
9404:
9384:
9381:
9380:
9353:
9351:
9348:
9347:
9320:
9318:
9315:
9314:
9312:
9285:
9283:
9280:
9279:
9261:
9258:
9257:
9255:
9236:
9233:
9232:
9216:
9213:
9212:
9206:
9204:Generalizations
9172:
9170:
9167:
9166:
9164:
9137:
9130:
9129:
9127:
9124:
9123:
9096:
9089:
9088:
9086:
9083:
9082:
9055:
9053:
9050:
9049:
9011:
9009:
9006:
9005:
8999:
8990:
8963:
8961:
8958:
8957:
8928:
8926:
8923:
8922:
8920:
8916:
8893:
8861:
8859:
8856:
8855:
8825:
8823:
8820:
8819:
8813:
8807:
8797:
8786:
8780:
8762:
8753:
8747:
8741:
8730:
8700:
8698:
8695:
8694:
8692:
8686:
8680:
8674:
8668:
8645:
8612:
8604:
8601:
8600:
8587:
8581:
8538:
8535:
8534:
8506:
8503:
8502:
8493:
8487:
8481:
8444:
8442:
8439:
8438:
8436:
8430:
8424:
8418:
8391:
8389:
8386:
8385:
8375:
8347:
8344:
8343:
8340:associative law
8308:
8306:
8303:
8302:
8296:
8290:
8287:
8275:Bourbaki (1989)
8239:
8235:
8212:
8209:
8208:
8176:
8169:
8168:
8160:
8157:
8156:
8154:
8127:
8120:
8119:
8111:
8108:
8107:
8068:
8061:
8060:
8058:
8055:
8054:
8021:
8014:
8013:
8011:
8008:
8007:
7998:
7959:
7952:
7951:
7932:
7925:
7924:
7905:
7898:
7897:
7895:
7892:
7891:
7854:
7847:
7846:
7821:
7814:
7813:
7794:
7787:
7786:
7767:
7760:
7759:
7740:
7738:
7735:
7734:
7703:
7700:
7699:
7693:
7692:-vector with a
7687:
7684:
7649:
7627:
7625:
7623:
7618:
7615:
7614:
7567:
7565:
7563:
7561:
7558:
7557:
7553:if and only if
7538:
7535:
7534:
7518:
7515:
7514:
7495:
7492:
7491:
7471:
7467:
7465:
7462:
7461:
7437:
7433:
7418:
7414:
7412:
7409:
7408:
7388:
7384:
7375:
7371:
7362:
7358:
7346:
7334:
7331:
7330:
7311:
7308:
7307:
7289:
7286:
7285:
7283:
7282:is a basis for
7266:
7262:
7260:
7257:
7256:
7240:
7237:
7236:
7216:
7213:
7212:
7179:
7176:
7175:
7173:
7167:
7151:
7148:
7147:
7141:
7077:
7072:
7047:
7042:
7023:
7019:
7017:
7014:
7013:
6987:
6983:
6981:
6978:
6977:
6951:
6947:
6926:
6922:
6907:
6903:
6895:
6892:
6891:
6880:
6863:
6860:
6859:
6830:
6823:
6822:
6814:
6811:
6810:
6808:
6805:
6777:
6773:
6771:
6768:
6767:
6765:
6730:
6723:
6722:
6720:
6717:
6716:
6714:
6696:
6693:
6692:
6690:
6651:
6644:
6643:
6641:
6638:
6637:
6635:
6605:
6598:
6597:
6572:
6565:
6564:
6545:
6538:
6537:
6518:
6511:
6510:
6491:
6489:
6486:
6485:
6471:
6443:
6440:
6439:
6437:
6398:
6391:
6390:
6388:
6385:
6384:
6366:
6363:
6362:
6360:
6338:
6335:
6334:
6332:
6308:
6295:
6294:
6293:
6274:
6267:
6266:
6258:
6255:
6254:
6220:
6213:
6212:
6210:
6207:
6206:
6184:
6180:
6178:
6175:
6174:
6172:
6155:
6151:
6149:
6146:
6145:
6133:
6114:
6110:
6108:
6105:
6104:
6102:
6081:
6077:
6075:
6072:
6071:
6048:
6044:
6029:
6025:
6023:
6020:
6019:
5987:
5980:
5979:
5977:
5974:
5973:
5971:
5970:is a basis for
5941:
5937:
5922:
5918:
5909:
5905:
5887:
5886:
5875:
5871:
5870:
5866:
5849:
5845:
5844:
5840:
5829:
5825:
5824:
5820:
5814:
5811:
5810:
5806:, then the set
5791:
5788:
5787:
5763:
5759:
5744:
5740:
5735:
5732:
5731:
5715:
5712:
5711:
5695:
5692:
5691:
5684:
5659:
5655:
5646:
5642:
5633:
5629:
5620:
5616:
5608:
5605:
5604:
5579:
5575:
5566:
5562:
5553:
5549:
5540:
5536:
5531:
5528:
5527:
5525:
5506:
5501:
5500:
5498:
5495:
5494:
5492:
5465:
5458:
5457:
5455:
5452:
5451:
5422:
5415:
5414:
5412:
5409:
5408:
5406:
5388:
5385:
5384:
5382:
5354:
5351:
5350:
5332:
5329:
5328:
5326:
5310:
5307:
5306:
5290:
5287:
5286:
5277:
5259:
5256:
5255:
5226:
5219:
5218:
5210:
5207:
5206:
5204:
5145:
5141:
5131:
5127:
5112:
5108:
5099:
5095:
5093:
5090:
5089:
5054:
5052:
5049:
5048:
5046:
5043:vector subspace
5013:
5006:
5005:
5003:
5000:
4999:
4997:
4979:
4976:
4975:
4973:
4963:
4960:
4932:
4928:
4913:
4909:
4900:
4896:
4894:
4891:
4890:
4867:
4863:
4848:
4844:
4835:
4831:
4826:
4823:
4822:
4796:
4792:
4777:
4773:
4764:
4760:
4758:
4755:
4754:
4727:
4724:
4723:
4721:
4704:
4700:
4691:
4687:
4685:
4682:
4681:
4660:
4657:
4656:
4654:
4623:
4620:
4619:
4596:
4592:
4577:
4573:
4564:
4560:
4527:
4523:
4499:
4495:
4477:
4473:
4471:
4468:
4467:
4446:
4443:
4442:
4440:
4421:
4417:
4415:
4412:
4411:
4409:
4390:
4386:
4384:
4381:
4380:
4378:
4359:
4355:
4353:
4350:
4349:
4347:
4311:
4308:
4307:
4305:
4285:
4282:
4281:
4232:
4229:
4228:
4098:
4095:
4094:
4061:
4058:
4057:
4055:
4037:
4034:
4033:
4031:
4030:on elements of
4028:anticommutative
4002:
3999:
3998:
3970:
3967:
3966:
3948:
3945:
3944:
3942:
3941:on elements of
3933:
3928:
3893:
3870:
3867:
3866:
3838:
3835:
3834:
3818:
3815:
3814:
3788:
3759:
3756:
3755:
3730:
3727:
3726:
3704:
3701:
3700:
3684:
3681:
3680:
3653:
3650:
3649:
3630:
3627:
3626:
3601:
3598:
3597:
3594:
3531:
3518:
3511:
3504:
3476:
3471:
3470:
3461:
3456:
3455:
3446:
3441:
3440:
3428:
3424:
3418:
3414:
3408:
3404:
3395:
3391:
3385:
3381:
3375:
3371:
3362:
3358:
3352:
3348:
3342:
3338:
3329:
3325:
3319:
3315:
3309:
3305:
3296:
3292:
3286:
3282:
3276:
3272:
3263:
3259:
3253:
3249:
3243:
3239:
3228:
3220:
3212:
3210:
3207:
3206:
3180:
3175:
3174:
3168:
3164:
3155:
3150:
3149:
3143:
3139:
3130:
3125:
3124:
3118:
3114:
3106:
3104:
3101:
3100:
3080:
3073:
3066:
3059:
3052:
3045:
3017:
3012:
3011:
3002:
2997:
2996:
2984:
2980:
2974:
2970:
2961:
2957:
2951:
2947:
2932:
2927:
2926:
2917:
2912:
2911:
2899:
2895:
2889:
2885:
2876:
2872:
2866:
2862:
2847:
2842:
2841:
2832:
2827:
2826:
2814:
2810:
2804:
2800:
2791:
2787:
2781:
2777:
2766:
2758:
2756:
2753:
2752:
2729:
2724:
2723:
2717:
2713:
2704:
2699:
2698:
2692:
2688:
2679:
2674:
2673:
2667:
2663:
2655:
2653:
2650:
2649:
2626:
2621:
2620:
2614:
2610:
2601:
2596:
2595:
2589:
2585:
2576:
2571:
2570:
2564:
2560:
2552:
2550:
2547:
2546:
2539:
2532:
2525:
2518:
2505:For vectors in
2495:
2484:
2477:
2462:
2456:
2450:
2444:
2431:
2424:
2417:
2376:
2346:
2320:
2280:
2257:
2214:
2209:
2208:
2199:
2194:
2193:
2184:
2179:
2178:
2169:
2164:
2163:
2161:
2158:
2157:
2134:
2129:
2128:
2119:
2114:
2113:
2098:
2093:
2092:
2083:
2078:
2077:
2075:
2072:
2071:
2068:alternating map
2046:
2045:
2039:
2034:
2033:
2024:
2019:
2018:
1997:
1993:
1984:
1983:
1977:
1972:
1971:
1962:
1957:
1956:
1941:
1936:
1935:
1926:
1921:
1920:
1905:
1900:
1899:
1890:
1885:
1884:
1869:
1864:
1863:
1854:
1849:
1848:
1833:
1832:
1823:
1818:
1817:
1805:
1800:
1799:
1781:
1776:
1775:
1763:
1758:
1757:
1744:
1739:
1731:
1727:
1725:
1722:
1721:
1700:
1698:
1695:
1694:
1692:
1676:
1674:
1671:
1670:
1632:
1628:
1619:
1618:
1611:
1610:
1605:
1599:
1598:
1593:
1583:
1582:
1573:
1572:
1563:
1562:
1555:
1554:
1549:
1547:
1542:
1535:
1534:
1525:
1524:
1516:
1514:
1511:
1510:
1483:
1481:
1478:
1477:
1461:
1459:
1456:
1455:
1436:
1431:
1430:
1428:
1425:
1424:
1422:
1402:
1397:
1396:
1384:
1379:
1378:
1365:
1364:
1358:
1357:
1347:
1346:
1338:
1328:
1323:
1322:
1310:
1305:
1304:
1291:
1290:
1284:
1283:
1273:
1272:
1264:
1262:
1259:
1258:
1231:
1230:
1224:
1223:
1213:
1212:
1203:
1197:
1196:
1195:
1184:
1183:
1177:
1176:
1166:
1165:
1156:
1150:
1149:
1148:
1146:
1143:
1142:
1111:
1106:
1105:
1103:
1100:
1099:
1085:
1080:
1059:
1056:
1055:
1035:
1032:
1031:
976:
973:
972:
944:
941:
940:
901:
898:
897:
881:
878:
877:
861:
858:
857:
827:
824:
823:
807:
804:
803:
780:
777:
776:
756:
753:
752:
736:
733:
732:
713:
696:
693:
692:
645:
642:
641:
613:
610:
609:
585:
582:
581:
562:
559:
558:
542:
539:
538:
512:
509:
508:
503:
466:
463:
462:
446:
443:
442:
416:
412:
397:
393:
384:
380:
378:
375:
374:
358:
355:
354:
333:
330:
329:
313:
310:
309:
293:
290:
289:
266:
263:
262:
246:
243:
242:
214:
211:
210:
194:
191:
190:
163:
160:
159:
139:
136:
135:
121:
120:
119:
118:
110:
71:
63:
62:
61:
58:
50:
49:
46:
35:
28:
23:
22:
15:
12:
11:
5:
27905:
27895:
27894:
27889:
27884:
27867:
27866:
27864:
27863:
27858:
27856:Woldemar Voigt
27853:
27848:
27843:
27838:
27833:
27828:
27823:
27821:Leonhard Euler
27818:
27813:
27808:
27803:
27797:
27795:
27793:Mathematicians
27789:
27788:
27785:
27784:
27782:
27781:
27776:
27771:
27766:
27761:
27756:
27751:
27746:
27741:
27735:
27733:
27729:
27728:
27726:
27725:
27720:
27718:Torsion tensor
27715:
27710:
27705:
27700:
27695:
27690:
27684:
27682:
27675:
27671:
27670:
27668:
27667:
27662:
27657:
27652:
27647:
27642:
27637:
27632:
27627:
27622:
27617:
27612:
27607:
27602:
27597:
27592:
27587:
27582:
27577:
27571:
27569:
27563:
27562:
27560:
27559:
27553:
27551:Tensor product
27548:
27543:
27541:Symmetrization
27538:
27533:
27531:Lie derivative
27528:
27523:
27518:
27513:
27508:
27502:
27500:
27494:
27493:
27491:
27490:
27485:
27480:
27475:
27470:
27465:
27460:
27455:
27453:Tensor density
27450:
27445:
27439:
27437:
27431:
27430:
27428:
27427:
27425:Voigt notation
27422:
27417:
27412:
27410:Ricci calculus
27407:
27402:
27397:
27395:Index notation
27392:
27387:
27381:
27379:
27375:
27374:
27371:
27370:
27368:
27367:
27362:
27357:
27352:
27347:
27341:
27339:
27337:
27336:
27331:
27325:
27322:
27321:
27319:
27318:
27313:
27311:Tensor algebra
27308:
27303:
27298:
27293:
27291:Dyadic algebra
27288:
27283:
27277:
27275:
27266:
27262:
27261:
27254:
27251:
27250:
27243:
27242:
27235:
27228:
27220:
27211:
27210:
27208:
27207:
27196:
27193:
27192:
27190:
27189:
27184:
27179:
27174:
27169:
27167:Floating-point
27163:
27161:
27155:
27154:
27152:
27151:
27149:Tensor product
27146:
27141:
27136:
27134:Function space
27131:
27126:
27120:
27118:
27111:
27110:
27108:
27107:
27102:
27097:
27092:
27087:
27082:
27077:
27072:
27070:Triple product
27067:
27062:
27056:
27054:
27048:
27047:
27045:
27044:
27039:
27034:
27029:
27024:
27019:
27014:
27008:
27006:
27000:
26999:
26997:
26996:
26991:
26986:
26984:Transformation
26981:
26976:
26974:Multiplication
26971:
26966:
26961:
26956:
26950:
26948:
26942:
26941:
26934:
26932:
26930:
26929:
26924:
26919:
26914:
26909:
26904:
26899:
26894:
26889:
26884:
26879:
26874:
26869:
26864:
26859:
26854:
26849:
26844:
26839:
26833:
26831:
26830:Basic concepts
26827:
26826:
26824:
26823:
26818:
26812:
26809:
26808:
26805:Linear algebra
26801:
26800:
26793:
26786:
26778:
26771:
26770:
26764:
26758:
26752:
26751:
26750:
26741:
26719:
26718:
26717:
26705:
26688:
26670:
26669:
26668:
26660:
26644:
26643:
26642:
26630:
26625:
26621:
26615:
26608:
26585:
26580:
26573:
26542:
26537:
26530:
26503:
26487:
26486:
26485:
26456:
26454:
26451:
26449:
26448:
26405:
26399:
26393:, Birkhäuser,
26372:
26367:
26350:
26332:
26315:
26284:
26280:Bourbaki (1989
26275:
26273:
26270:
26268:
26267:
26266:
26265:
26250:
26249:
26248:
26240:
26220:
26219:
26218:
26191:
26190:
26189:
26176:
26160:
26159:
26158:
26150:
26129:
26127:
26124:
26122:
26119:
26117:
26116:
26114:, p. 661.
26104:
26080:Whitehead 1898
26067:
26058:
26038:
26031:
26023:Supermanifolds
26010:
26003:
25981:
25973:vector bundles
25964:
25960:Bourbaki (1989
25945:
25915:
25911:Bourbaki (1989
25896:
25874:
25854:
25838:
25823:
25791:
25773:
25766:
25748:
25730:
25726:Bourbaki (1989
25717:
25705:Bourbaki (1989
25696:
25683:
25670:
25641:
25628:
25608:
25582:
25576:algebras (cf.
25562:
25550:
25543:
25519:
25517:
25514:
25513:
25512:
25499:
25497:Tensor algebra
25494:
25488:
25483:
25481:Koszul complex
25478:
25473:
25471:quadratic form
25464:
25459:
25452:
25449:
25426:Gaston Darboux
25418:Henri Poincaré
25414:Giuseppe Peano
25367:
25364:
25356:Koszul complex
25351:
25348:
25325:
25299:
25296:
25293:
25288:
25266:
25243:
25225:
25224:
25213:
25208:
25205:
25202:
25198:
25194:
25191:
25188:
25183:
25176:
25173:
25166:
25163:
25160:
25155:
25148:
25145:
25138:
25135:
25132:
25127:
25123:
25119:
25116:
25111:
25107:
25103:
25098:
25094:
25090:
25085:
25082:
25079:
25076:
25073:
25069:
25065:
25062:
25059:
25054:
25051:
25048:
25044:
25037:
25034:
25031:
25027:
25022:
25019:
25014:
25011:
25008:
25004:
25000:
24997:
24994:
24989:
24985:
24981:
24978:
24964:
24963:
24952:
24949:
24946:
24941:
24934:
24928:
24925:
24922:
24919:
24914:
24911:
24908:
24901:
24895:
24892:
24867:
24843:
24815:
24793:
24781:
24778:
24763:
24707:
24704:
24682:Schur functors
24673:
24670:
24588:line integrals
24563:
24560:
24536:
24531:
24528:
24525:
24520:
24513:
24505:
24499:
24468:
24465:
24462:
24459:
24454:
24450:
24425:
24391:
24359:
24356:
24353:
24348:
24341:
24320:
24317:
24302:
24278:
24254:
24241:parallelepiped
24228:
24225:
24222:
24219:
24216:
24194:
24172:
24148:
24145:
24142:
24124:
24121:
24109:
24104:
24101:
24097:
24091:
24088:
24084:
24078:
24075:
24071:
24067:
24062:
24059:
24055:
24032:
24029:
24024:
24021:
24017:
24013:
24008:
24005:
24000:
23997:
23993:
23989:
23984:
23980:
23956:
23952:
23948:
23944:
23940:
23937:
23917:
23899:Lorentz metric
23883:
23880:
23875:
23872:
23869:
23866:
23863:
23860:
23856:
23852:
23847:
23844:
23841:
23838:
23835:
23832:
23828:
23807:
23804:
23801:
23798:
23795:
23792:
23789:
23786:
23762:
23757:
23754:
23751:
23748:
23745:
23741:
23737:
23732:
23729:
23726:
23723:
23720:
23716:
23712:
23707:
23704:
23700:
23672:
23669:
23666:
23663:
23640:
23637:
23628:Main article:
23625:
23622:
23577:linear algebra
23572:
23571:Linear algebra
23569:
23558:
23557:
23549:
23548:
23540:
23539:
23538:
23537:
23536:
23521:
23517:
23513:
23510:
23507:
23502:
23498:
23471:
23468:
23463:
23459:
23455:
23452:
23449:
23444:
23440:
23436:
23433:
23430:
23427:
23422:
23418:
23414:
23409:
23405:
23401:
23379:
23357:
23354:
23351:
23346:
23319:
23297:
23294:
23291:
23288:
23285:
23262:
23240:
23218:
23198:
23174:
23171:
23149:
23145:
23139:
23135:
23131:
23128:
23125:
23120:
23116:
23110:
23106:
23102:
23097:
23093:
23087:
23083:
23079:
23074:
23070:
23064:
23060:
23054:
23050:
23046:
23043:
23040:
23019:
23014:
23010:
23004:
23000:
22996:
22993:
22990:
22985:
22981:
22975:
22971:
22967:
22962:
22958:
22952:
22948:
22927:
22905:
22901:
22897:
22894:
22891:
22888:
22885:
22880:
22876:
22872:
22867:
22863:
22842:
22839:
22836:
22810:
22787:
22758:
22746:
22743:
22741:
22738:
22734:
22733:
22722:
22719:
22716:
22713:
22710:
22707:
22704:
22699:
22696:
22693:
22686:
22680:
22677:
22674:
22671:
22666:
22659:
22653:
22650:
22647:
22644:
22639:
22633:
22627:
22624:
22606:
22605:
22594:
22591:
22588:
22585:
22582:
22577:
22570:
22564:
22561:
22558:
22555:
22550:
22543:
22537:
22534:
22531:
22528:
22523:
22520:
22517:
22510:
22504:
22501:
22498:
22495:
22477:
22476:
22465:
22462:
22457:
22453:
22449:
22446:
22443:
22440:
22437:
22434:
22429:
22425:
22421:
22418:
22415:
22410:
22407:
22404:
22400:
22396:
22393:
22390:
22385:
22381:
22377:
22372:
22368:
22364:
22361:
22358:
22353:
22349:
22345:
22340:
22337:
22334:
22330:
22326:
22323:
22320:
22315:
22311:
22290:
22287:
22284:
22279:
22272:
22266:
22263:
22260:
22257:
22252:
22249:
22246:
22239:
22233:
22228:
22224:
22219:
22213:
22210:
22207:
22203:
22179:
22176:
22173:
22168:
22164:
22143:
22140:
22135:
22131:
22108:
22105:
22102:
22098:
22094:
22091:
22086:
22082:
22078:
22073:
22070:
22067:
22064:
22061:
22057:
22053:
22050:
22047:
22042:
22038:
22017:
22014:
22011:
21989:
21985:
21973:
21972:
21961:
21958:
21955:
21950:
21943:
21937:
21932:
21929:
21926:
21922:
21918:
21913:
21909:
21905:
21902:
21899:
21894:
21890:
21886:
21881:
21877:
21873:
21870:
21844:
21841:
21838:
21835:
21830:
21827:
21822:
21817:
21814:
21809:
21806:
21803:
21792:
21791:
21780:
21777:
21774:
21771:
21766:
21759:
21753:
21750:
21747:
21744:
21739:
21732:
21724:
21721:
21718:
21715:
21712:
21708:
21704:
21701:
21698:
21695:
21692:
21689:
21684:
21677:
21660:
21659:
21648:
21645:
21642:
21639:
21634:
21629:
21626:
21623:
21620:
21615:
21610:
21607:
21604:
21601:
21598:
21595:
21590:
21573:
21570:
21569:
21568:
21557:
21554:
21551:
21548:
21543:
21538:
21535:
21532:
21529:
21524:
21519:
21516:
21502:
21501:
21490:
21487:
21484:
21481:
21478:
21473:
21468:
21465:
21462:
21459:
21454:
21449:
21446:
21443:
21440:
21435:
21430:
21427:
21424:
21421:
21416:
21409:
21403:
21400:
21373:
21370:
21367:
21364:
21361:
21358:
21355:
21352:
21349:
21337:
21334:
21319:
21293:
21290:
21287:
21282:
21275:
21252:
21249:
21246:
21241:
21234:
21209:
21187:
21167:
21164:
21161:
21139:
21117:
21114:
21111:
21091:
21071:
21049:
21046:
21043:
21038:
21031:
21015:
21014:
21003:
21000:
20997:
20994:
20989:
20982:
20976:
20973:
20970:
20967:
20962:
20955:
20949:
20945:
20941:
20938:
20935:
20930:
20923:
20916:
20913:
20909:
20906:
20903:
20898:
20893:
20890:
20887:
20884:
20879:
20872:
20855:
20854:
20843:
20840:
20835:
20831:
20827:
20824:
20821:
20818:
20815:
20812:
20807:
20803:
20799:
20796:
20793:
20790:
20785:
20781:
20777:
20774:
20771:
20766:
20762:
20758:
20755:
20752:
20749:
20744:
20718:
20715:
20712:
20708:
20682:
20679:
20676:
20671:
20658:
20657:
20646:
20643:
20640:
20637:
20632:
20625:
20619:
20616:
20613:
20610:
20607:
20604:
20599:
20592:
20586:
20583:
20580:
20577:
20574:
20570:
20566:
20563:
20560:
20555:
20548:
20541:
20538:
20534:
20531:
20528:
20523:
20507:
20506:
20495:
20492:
20489:
20484:
20479:
20476:
20473:
20470:
20465:
20460:
20457:
20454:
20451:
20446:
20417:
20414:
20411:
20408:
20405:
20385:
20365:
20353:
20350:
20331:
20324:
20319:
20315:
20312:
20308:
20296:
20290:
20286:
20283:
20280:
20277:
20274:
20271:
20268:
20265:
20232:
20229:
20226:
20223:
20220:
20215:
20210:
20207:
20180:
20146:
20145:
20133:
20129:
20124:
20120:
20116:
20113:
20110:
20105:
20101:
20097:
20094:
20090:
20086:
20083:
20080:
20077:
20074:
20071:
20068:
20063:
20059:
20055:
20052:
20049:
20044:
20040:
20036:
20033:
20030:
20027:
20024:
20021:
20004:tensor algebra
19985:
19963:
19943:
19923:
19858:
19855:
19852:
19847:
19842:
19839:
19836:
19833:
19828:
19808:
19807:
19796:
19793:
19790:
19785:
19782:
19779:
19773:
19767:
19764:
19761:
19758:
19753:
19747:
19739:
19734:
19731:
19728:
19724:
19720:
19717:
19714:
19711:
19706:
19700:
19694:
19691:
19668:
19665:
19662:
19659:
19654:
19623:
19619:
19598:
19595:
19592:
19587:
19551:
19548:
19545:
19542:
19538:
19534:
19531:
19528:
19523:
19520:
19517:
19514:
19511:
19508:
19504:
19481:
19478:
19475:
19472:
19468:
19464:
19461:
19458:
19453:
19450:
19447:
19444:
19440:
19393:
19392:
19381:
19378:
19373:
19370:
19367:
19364:
19360:
19356:
19353:
19350:
19345:
19342:
19339:
19336:
19333:
19330:
19326:
19322:
19319:
19316:
19311:
19308:
19305:
19302:
19298:
19294:
19291:
19288:
19283:
19280:
19277:
19274:
19270:
19266:
19263:
19260:
19257:
19254:
19251:
19245:
19242:
19239:
19236:
19233:
19230:
19227:
19224:
19221:
19218:
19215:
19211:
19204:
19199:
19196:
19193:
19189:
19185:
19182:
19177:
19173:
19169:
19166:
19163:
19158:
19154:
19150:
19147:
19133:
19132:
19121:
19118:
19115:
19112:
19109:
19106:
19103:
19100:
19097:
19094:
19091:
19088:
19085:
19082:
19079:
19076:
19073:
19070:
19067:
19064:
19061:
19058:
19055:
19052:
19049:
19046:
19043:
19040:
19037:
19034:
19007:
19004:
19001:
18996:
18972:
18969:
18966:
18961:
18956:
18953:
18950:
18947:
18942:
18916:
18913:
18910:
18905:
18900:
18897:
18894:
18891:
18886:
18879:
18873:
18870:
18846:
18824:
18815:. The symbol
18802:
18799:
18796:
18783:
18782:
18771:
18768:
18765:
18762:
18759:
18756:
18753:
18750:
18747:
18744:
18741:
18738:
18713:
18710:
18707:
18702:
18697:
18694:
18691:
18688:
18683:
18678:
18675:
18672:
18669:
18664:
18659:
18656:
18624:
18621:
18618:
18613:
18587:tensor algebra
18564:
18542:
18539:
18536:
18531:
18517:
18514:
18499:
18496:
18493:
18488:
18481:
18475:
18471:
18457:
18456:
18445:
18441:
18437:
18433:
18429:
18426:
18423:
18419:
18415:
18410:
18405:
18378:
18354:
18348:
18342:
18339:
18336:
18331:
18324:
18316:
18311:
18307:
18302:
18298:
18294:
18288:
18281:
18275:
18270:
18265:
18252:
18251:
18240:
18236:
18228:
18223:
18217:
18213:
18209:
18205:
18202:
18199:
18195:
18191:
18187:
18183:
18179:
18175:
18150:
18147:
18144:
18139:
18132:
18126:
18122:
18097:
18094:
18091:
18086:
18079:
18073:
18069:
18044:
18041:
18038:
18033:
18030:
18027:
18020:
18014:
18010:
17982:
17979:
17976:
17963:
17962:
17951:
17948:
17945:
17942:
17939:
17936:
17933:
17930:
17927:
17922:
17918:
17889:
17888:
17877:
17873:
17865:
17861:
17856:
17852:
17848:
17844:
17841:
17838:
17834:
17830:
17826:
17822:
17819:
17816:
17791:
17788:
17785:
17761:
17758:
17755:
17750:
17743:
17737:
17733:
17708:
17705:
17702:
17697:
17694:
17691:
17684:
17678:
17674:
17642:
17639:
17636:
17631:
17624:
17608:
17607:
17596:
17591:
17587:
17583:
17580:
17577:
17572:
17568:
17563:
17556:
17552:
17547:
17543:
17540:
17537:
17530:
17526:
17521:
17495:
17465:
17450:
17447:
17444:
17441:
17436:
17429:
17410:
17401:
17394:Gramian matrix
17382:
17373:
17366:
17365:
17354:
17349:
17344:
17339:
17335:
17331:
17326:
17322:
17318:
17313:
17308:
17305:
17301:
17295:
17291:
17287:
17284:
17281:
17276:
17272:
17268:
17263:
17259:
17255:
17252:
17249:
17244:
17240:
17235:
17209:
17183:
17177:
17171:
17166:
17159:
17151:
17127:
17124:
17121:
17116:
17109:
17082:
17078:
17055:
17033:
17001:
16987:
16984:
16957:
16956:
16944:
16941:
16935:
16932:
16929:
16926:
16923:
16920:
16917:
16914:
16910:
16906:
16903:
16900:
16897:
16894:
16891:
16888:
16883:
16876:
16870:
16867:
16864:
16861:
16856:
16849:
16843:
16840:
16837:
16834:
16809:
16783:
16780:
16777:
16772:
16765:
16759:
16756:
16753:
16750:
16745:
16738:
16715:
16704:
16703:
16692:
16689:
16686:
16683:
16678:
16675:
16672:
16665:
16659:
16656:
16653:
16650:
16645:
16638:
16632:
16629:
16594:
16590:
16567:
16548:
16547:
16536:
16533:
16528:
16524:
16520:
16517:
16514:
16511:
16508:
16505:
16500:
16497:
16494:
16487:
16481:
16478:
16473:
16469:
16465:
16460:
16453:
16425:
16393:
16390:
16387:
16382:
16375:
16361:
16360:
16349:
16344:
16340:
16336:
16331:
16327:
16323:
16318:
16315:
16312:
16308:
16293:
16292:
16281:
16278:
16275:
16270:
16267:
16264:
16257:
16251:
16248:
16245:
16242:
16237:
16230:
16224:
16221:
16216:
16212:
16208:
16203:
16196:
16168:
16146:
16131:Main article:
16128:
16125:
16124:
16123:
16112:
16107:
16103:
16099:
16094:
16090:
16086:
16083:
16078:
16074:
16070:
16065:
16061:
16050:
16039:
16036:
16031:
16027:
16023:
16018:
16014:
15996:
15995:
15994:
15993:
15982:
15979:
15976:
15971:
15967:
15963:
15960:
15957:
15952:
15949:
15946:
15942:
15938:
15935:
15932:
15929:
15926:
15923:
15920:
15917:
15912:
15908:
15904:
15901:
15898:
15895:
15892:
15889:
15886:
15881:
15877:
15865:of degree −1:
15848:
15844:
15819:
15815:
15811:
15808:
15795:
15779:
15775:
15752:
15730:
15727:
15724:
15721:
15718:
15715:
15710:
15706:
15681:
15678:
15675:
15670:
15663:
15657:
15635:
15615:
15604:
15603:
15602:
15591:
15588:
15585:
15582:
15577:
15574:
15571:
15564:
15558:
15555:
15552:
15549:
15544:
15537:
15531:
15526:
15522:
15499:
15496:
15493:
15490:
15487:
15484:
15481:
15476:
15473:
15469:
15444:
15440:
15436:
15433:
15409:
15391:
15388:
15373:
15370:
15367:
15362:
15355:
15330:
15310:
15307:
15304:
15299:
15295:
15283:
15282:
15271:
15268:
15263:
15260:
15257:
15253:
15249:
15246:
15243:
15238:
15234:
15230:
15225:
15221:
15217:
15214:
15211:
15208:
15205:
15202:
15197:
15194:
15191:
15187:
15183:
15180:
15177:
15172:
15168:
15164:
15159:
15155:
15151:
15148:
15145:
15140:
15136:
15132:
15120:, then define
15105:
15101:
15078:
15075:
15072:
15057:
15050:
15043:
15024:
15020:
15016:
15013:
15010:
15005:
15001:
14997:
14992:
14988:
14954:
14930:
14926:
14905:
14883:
14880:
14877:
14872:
14865:
14859:
14856:
14829:
14797:
14780:
14779:
14768:
14765:
14762:
14759:
14754:
14751:
14748:
14741:
14735:
14732:
14729:
14726:
14721:
14714:
14708:
14703:
14699:
14673:
14670:
14667:
14662:
14647:antiderivation
14630:
14626:
14622:
14619:
14595:
14567:
14563:
14542:
14524:
14521:
14512:
14503:
14490:
14391:
14378:
14373:
14372:
14361:
14358:
14353:
14350:
14347:
14344:
14341:
14338:
14334:
14330:
14327:
14324:
14319:
14316:
14313:
14310:
14307:
14304:
14300:
14296:
14293:
14289:
14284:
14281:
14278:
14275:
14271:
14267:
14264:
14261:
14256:
14253:
14250:
14247:
14243:
14239:
14236:
14232:
14229:
14226:
14223:
14220:
14213:
14210:
14207:
14202:
14199:
14194:
14191:
14187:
14183:
14179:
14174:
14171:
14168:
14164:
14160:
14157:
14154:
14149:
14145:
14141:
14138:
14132:
14129:
14123:
14095:
14081:
14080:
14069:
14066:
14061:
14058:
14055:
14052:
14048:
14044:
14041:
14038:
14033:
14030:
14027:
14024:
14020:
14016:
14013:
14009:
14006:
14003:
14000:
13997:
13990:
13986:
13982:
13979:
13975:
13968:
13965:
13961:
13956:
13953:
13948:
13944:
13940:
13937:
13934:
13929:
13925:
13921:
13918:
13915:
13912:
13909:
13906:
13879:
13868:
13867:
13856:
13853:
13850:
13847:
13844:
13841:
13838:
13835:
13829:
13826:
13822:
13819:
13814:
13811:
13808:
13805:
13802:
13799:
13793:
13790:
13784:
13781:
13775:
13761:
13760:
13749:
13746:
13743:
13740:
13737:
13734:
13731:
13728:
13725:
13722:
13719:
13654:
13649:
13646:
13641:
13612:
13590:
13586:
13565:
13545:
13522:
13517:
13513:
13509:
13503:
13496:
13467:
13443:
13437:
13431:
13428:
13425:
13420:
13413:
13405:
13374:
13354:
13331:
13330:
13319:
13316:
13311:
13307:
13303:
13300:
13275:
13272:
13269:
13189:
13186:
13171:
13167:
13142:
13139:
13136:
13133:
13130:
13127:
13107:
13104:
13101:
13098:
13095:
13090:
13083:
13077:
13074:
13051:
13048:
13045:
13040:
13036:
13032:
13029:
13009:
13004:
13000:
12979:
12959:
12939:
12928:
12927:
12916:
12913:
12910:
12907:
12902:
12895:
12889:
12884:
12880:
12876:
12873:
12859:
12858:
12847:
12844:
12841:
12836:
12832:
12828:
12825:
12822:
12817:
12813:
12809:
12806:
12781:
12774:
12768:
12767:
12756:
12753:
12748:
12744:
12740:
12737:
12695:
12692:
12690:
12687:
12675:
12674:
12663:
12656:
12653:
12650:
12646:
12642:
12637:
12633:
12629:
12625:
12619:
12615:
12609:
12604:
12601:
12598:
12594:
12590:
12587:
12580:
12577:
12574:
12570:
12566:
12561:
12557:
12552:
12548:
12543:
12539:
12535:
12510:
12507:
12504:
12480:
12475:
12471:
12436:
12410:
12407:
12404:
12400:
12396:
12391:
12387:
12381:
12377:
12372:
12368:
12365:
12350:
12349:
12338:
12333:
12328:
12325:
12322:
12318:
12314:
12309:
12306:
12303:
12299:
12294:
12286:
12282:
12278:
12273:
12269:
12265:
12261:
12257:
12250:
12247:
12244:
12240:
12236:
12231:
12227:
12222:
12218:
12209:
12206:
12197:
12194:
12170:
12169:
12158:
12151:
12148:
12145:
12141:
12134:
12128:
12125:
12122:
12115:
12111:
12104:
12098:
12091:
12087:
12080:
12070:
12067:
12064:
12061:
12058:
12055:
12051:
12047:
12042:
12039:
12036:
12033:
12030:
12027:
12023:
12018:
12010:
12007:
12004:
12001:
11997:
11993:
11988:
11985:
11982:
11979:
11975:
11970:
11966:
11963:
11960:
11957:
11954:
11947:
11944:
11941:
11935:
11929:
11926:
11922:
11915:
11912:
11909:
11906:
11903:
11900:
11896:
11891:
11888:
11879:
11876:
11867:
11826:
11825:
11814:
11807:
11803:
11796:
11790:
11787:
11784:
11777:
11773:
11766:
11760:
11753:
11749:
11742:
11731:
11727:
11723:
11718:
11714:
11708:
11704:
11699:
11695:
11692:
11674:index notation
11643:
11636:
11620:
11619:Index notation
11617:
11602:
11599:
11596:
11593:
11571:
11568:
11565:
11562:
11542:
11539:
11536:
11533:
11530:
11527:
11524:
11521:
11517:
11513:
11510:
11507:
11504:
11501:
11498:
11476:
11473:
11470:
11465:
11439:
11436:
11433:
11430:
11402:
11399:
11374:
11352:
11349:
11346:
11343:
11323:
11320:
11317:
11312:
11307:
11304:
11301:
11298:
11295:
11273:
11270:
11267:
11243:
11230:
11229:
11218:
11215:
11212:
11209:
11204:
11199:
11196:
11193:
11190:
11187:
11162:
11159:
11156:
11152:
11129:
11109:
11106:
11103:
11100:
11080:
11069:
11068:
11057:
11054:
11051:
11048:
11045:
11042:
11039:
11036:
11033:
11030:
11021:
11018:
11009:
11006:
11003:
11000:
10997:
10971:
10968:
10945:
10942:
10939:
10936:
10916:
10894:
10870:
10846:
10820:
10817:
10814:
10810:
10785:
10782:
10779:
10773:
10748:
10745:
10742:
10738:
10713:
10710:
10707:
10703:
10678:
10675:
10672:
10669:
10654:is always the
10643:
10640:
10637:
10634:
10630:
10626:
10621:
10597:
10594:
10591:
10587:
10575:
10574:
10563:
10557:
10554:
10551:
10545:
10538:
10535:
10532:
10526:
10523:
10520:
10514:
10507:
10501:
10498:
10495:
10489:
10457:
10454:
10451:
10447:
10422:
10419:
10416:
10392:
10368:
10365:
10362:
10359:
10356:
10353:
10350:
10333:
10332:
10321:
10316:
10312:
10308:
10305:
10302:
10297:
10293:
10289:
10286:
10280:
10277:
10274:
10268:
10258:
10255:
10251:
10246:
10243:
10238:
10234:
10230:
10227:
10224:
10219:
10215:
10211:
10208:
10203:
10200:
10197:
10193:
10169:
10166:
10146:
10143:
10140:
10137:
10126:
10125:
10112:
10109:
10106:
10103:
10099:
10095:
10092:
10089:
10084:
10081:
10078:
10075:
10071:
10067:
10064:
10061:
10058:
10055:
10048:
10042:
10036:
10033:
10029:
10025:
10022:
10017:
10013:
10009:
10006:
10003:
9998:
9994:
9990:
9987:
9981:
9978:
9975:
9969:
9943:
9942:
9931:
9928:
9925:
9920:
9916:
9911:
9906:
9902:
9898:
9895:
9892:
9887:
9883:
9859:
9839:
9836:
9833:
9828:
9823:
9796:
9772:
9769:
9766:
9744:
9724:
9721:
9718:
9714:
9693:
9673:
9651:
9648:
9645:
9642:
9639:
9615:
9591:
9569:
9547:
9544:
9541:
9515:
9512:
9509:
9505:
9484:
9464:
9452:
9449:
9436:vector bundles
9412:
9388:
9368:
9365:
9362:
9357:
9333:
9330:
9327:
9323:
9300:
9297:
9294:
9289:
9265:
9240:
9220:
9205:
9202:
9187:
9184:
9181:
9176:
9152:
9149:
9146:
9141:
9134:
9111:
9108:
9105:
9100:
9093:
9070:
9067:
9064:
9059:
9026:
9023:
9020:
9015:
8978:
8975:
8972:
8967:
8943:
8940:
8937:
8932:
8913:
8912:
8901:
8896:
8891:
8888:
8885:
8882:
8879:
8876:
8873:
8870:
8865:
8840:
8837:
8834:
8829:
8760:tensor algebra
8728:, see above).
8715:
8712:
8709:
8704:
8633:
8630:
8627:
8624:
8621:
8616:
8611:
8608:
8569:
8566:
8563:
8560:
8557:
8554:
8551:
8548:
8545:
8542:
8522:
8519:
8516:
8513:
8510:
8478:
8459:
8456:
8453:
8448:
8406:
8403:
8400:
8395:
8363:
8360:
8357:
8354:
8351:
8323:
8320:
8317:
8312:
8286:
8283:
8271:
8270:
8259:
8256:
8253:
8250:
8245:
8242:
8238:
8234:
8231:
8228:
8225:
8222:
8219:
8216:
8191:
8188:
8185:
8180:
8173:
8167:
8164:
8142:
8139:
8136:
8131:
8124:
8118:
8115:
8104:
8103:
8092:
8089:
8086:
8083:
8080:
8077:
8072:
8065:
8042:
8039:
8036:
8033:
8030:
8025:
8018:
7995:
7994:
7983:
7980:
7977:
7974:
7969:
7966:
7963:
7956:
7950:
7947:
7944:
7941:
7936:
7929:
7923:
7920:
7917:
7914:
7909:
7902:
7885:graded algebra
7881:
7880:
7869:
7866:
7863:
7858:
7851:
7845:
7842:
7839:
7836:
7833:
7830:
7825:
7818:
7812:
7809:
7806:
7803:
7798:
7791:
7785:
7782:
7779:
7776:
7771:
7764:
7758:
7755:
7752:
7749:
7744:
7719:
7716:
7713:
7710:
7707:
7683:
7680:
7679:
7678:
7667:
7664:
7658:
7655:
7652:
7646:
7642:
7639:
7636:
7633:
7630:
7599:
7596:
7591:
7586:
7582:
7579:
7576:
7573:
7570:
7542:
7522:
7499:
7477:
7474:
7470:
7458:skew-symmetric
7443:
7440:
7436:
7432:
7429:
7424:
7421:
7417:
7405:
7404:
7391:
7387:
7383:
7378:
7374:
7368:
7365:
7361:
7355:
7352:
7349:
7345:
7341:
7338:
7315:
7293:
7269:
7265:
7244:
7220:
7205:Sternberg 1964
7183:
7155:
7134:
7133:
7122:
7119:
7116:
7113:
7110:
7107:
7104:
7101:
7098:
7095:
7091:
7086:
7083:
7080:
7075:
7071:
7067:
7064:
7061:
7056:
7053:
7050:
7045:
7041:
7037:
7032:
7029:
7026:
7022:
6996:
6993:
6990:
6986:
6974:
6973:
6960:
6957:
6954:
6950:
6946:
6943:
6940:
6935:
6932:
6929:
6925:
6921:
6916:
6913:
6910:
6906:
6902:
6899:
6867:
6845:
6842:
6839:
6834:
6827:
6821:
6818:
6804:
6797:
6780:
6776:
6751:
6748:
6745:
6742:
6739:
6734:
6727:
6700:
6672:
6669:
6666:
6663:
6660:
6655:
6648:
6632:
6631:
6620:
6617:
6614:
6609:
6602:
6596:
6593:
6590:
6587:
6584:
6581:
6576:
6569:
6563:
6560:
6557:
6554:
6549:
6542:
6536:
6533:
6530:
6527:
6522:
6515:
6509:
6506:
6503:
6500:
6495:
6453:
6450:
6447:
6425:
6422:
6419:
6416:
6413:
6410:
6407:
6402:
6395:
6370:
6342:
6329:
6328:
6317:
6311:
6306:
6303:
6298:
6292:
6289:
6286:
6283:
6278:
6271:
6265:
6262:
6246:is equal to a
6235:
6232:
6229:
6224:
6217:
6187:
6183:
6158:
6154:
6117:
6113:
6084:
6080:
6068:
6067:
6056:
6051:
6047:
6043:
6040:
6037:
6032:
6028:
6002:
5999:
5996:
5991:
5984:
5968:
5967:
5956:
5952:
5949:
5944:
5940:
5936:
5933:
5930:
5925:
5921:
5917:
5912:
5908:
5904:
5901:
5890:
5878:
5874:
5869:
5865:
5862:
5859:
5852:
5848:
5843:
5839:
5832:
5828:
5823:
5818:
5795:
5771:
5766:
5762:
5758:
5755:
5752:
5747:
5743:
5739:
5719:
5699:
5683:
5680:
5679:
5678:
5667:
5662:
5658:
5654:
5649:
5645:
5641:
5636:
5632:
5628:
5623:
5619:
5615:
5612:
5587:
5582:
5578:
5574:
5569:
5565:
5561:
5556:
5552:
5548:
5543:
5539:
5535:
5509:
5504:
5480:
5477:
5474:
5469:
5462:
5437:
5434:
5431:
5426:
5419:
5405:-vectors span
5392:
5369:is said to be
5358:
5336:
5314:
5294:
5263:
5241:
5238:
5235:
5230:
5223:
5217:
5214:
5201:
5200:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5168:
5165:
5162:
5159:
5156:
5153:
5148:
5144:
5139:
5134:
5130:
5126:
5123:
5120:
5115:
5111:
5107:
5102:
5098:
5069:
5066:
5063:
5058:
5028:
5025:
5022:
5017:
5010:
4983:
4970:exterior power
4959:
4958:Exterior power
4956:
4955:
4954:
4943:
4940:
4935:
4931:
4927:
4924:
4921:
4916:
4912:
4908:
4903:
4899:
4875:
4870:
4866:
4862:
4859:
4856:
4851:
4847:
4843:
4838:
4834:
4830:
4819:
4818:
4807:
4804:
4799:
4795:
4791:
4788:
4785:
4780:
4776:
4772:
4767:
4763:
4737:
4734:
4731:
4707:
4703:
4699:
4694:
4690:
4664:
4639:
4636:
4633:
4630:
4627:
4616:
4615:
4604:
4599:
4595:
4591:
4588:
4585:
4580:
4576:
4572:
4567:
4563:
4559:
4556:
4553:
4550:
4547:
4544:
4539:
4536:
4533:
4530:
4526:
4522:
4519:
4516:
4511:
4508:
4505:
4502:
4498:
4494:
4489:
4486:
4483:
4480:
4476:
4450:
4424:
4420:
4393:
4389:
4362:
4358:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4289:
4278:
4277:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4222:
4221:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4077:
4074:
4071:
4068:
4065:
4041:
4015:
4012:
4009:
4006:
3986:
3983:
3980:
3977:
3974:
3952:
3932:
3929:
3927:
3924:
3923:
3922:
3911:
3907:
3904:
3900:
3897:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3862:is defined by
3851:
3848:
3845:
3842:
3822:
3811:
3810:
3798:
3795:
3791:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3740:
3737:
3734:
3714:
3711:
3708:
3688:
3674:tensor algebra
3657:
3634:
3614:
3611:
3608:
3605:
3593:
3590:
3525:triple product
3516:
3509:
3502:
3496:
3495:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3436:
3431:
3427:
3421:
3417:
3411:
3407:
3403:
3398:
3394:
3388:
3384:
3378:
3374:
3370:
3365:
3361:
3355:
3351:
3345:
3341:
3337:
3332:
3328:
3322:
3318:
3312:
3308:
3304:
3299:
3295:
3289:
3285:
3279:
3275:
3271:
3266:
3262:
3256:
3252:
3246:
3242:
3238:
3235:
3231:
3227:
3223:
3219:
3215:
3200:
3199:
3188:
3183:
3178:
3171:
3167:
3163:
3158:
3153:
3146:
3142:
3138:
3133:
3128:
3121:
3117:
3113:
3109:
3078:
3071:
3064:
3057:
3050:
3043:
3037:
3036:
3025:
3020:
3015:
3010:
3005:
3000:
2995:
2992:
2987:
2983:
2977:
2973:
2969:
2964:
2960:
2954:
2950:
2946:
2943:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2907:
2902:
2898:
2892:
2888:
2884:
2879:
2875:
2869:
2865:
2861:
2858:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2822:
2817:
2813:
2807:
2803:
2799:
2794:
2790:
2784:
2780:
2776:
2773:
2769:
2765:
2761:
2746:
2745:
2732:
2727:
2720:
2716:
2712:
2707:
2702:
2695:
2691:
2687:
2682:
2677:
2670:
2666:
2662:
2658:
2643:
2642:
2629:
2624:
2617:
2613:
2609:
2604:
2599:
2592:
2588:
2584:
2579:
2574:
2567:
2563:
2559:
2555:
2537:
2530:
2523:
2515:triple product
2494:
2491:
2482:
2475:
2436:
2435:
2429:
2422:
2415:
2374:
2344:
2318:
2250:absolute value
2225:
2222:
2217:
2212:
2207:
2202:
2197:
2192:
2187:
2182:
2177:
2172:
2167:
2145:
2142:
2137:
2132:
2127:
2122:
2117:
2112:
2109:
2106:
2101:
2096:
2091:
2086:
2081:
2060:
2059:
2042:
2037:
2032:
2027:
2022:
2016:
2012:
2009:
2006:
2003:
2000:
1996:
1992:
1989:
1987:
1985:
1980:
1975:
1970:
1965:
1960:
1955:
1952:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1916:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1880:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1844:
1841:
1838:
1836:
1834:
1831:
1826:
1821:
1816:
1813:
1808:
1803:
1798:
1795:
1792:
1789:
1784:
1779:
1774:
1771:
1766:
1761:
1756:
1753:
1750:
1747:
1745:
1742:
1738:
1734:
1730:
1729:
1703:
1679:
1667:
1666:
1655:
1651:
1647:
1644:
1641:
1638:
1635:
1631:
1627:
1622:
1615:
1609:
1606:
1604:
1601:
1600:
1597:
1594:
1592:
1589:
1588:
1586:
1581:
1576:
1571:
1566:
1559:
1552:
1548:
1545:
1541:
1540:
1538:
1533:
1528:
1523:
1486:
1464:
1439:
1434:
1419:
1418:
1405:
1400:
1395:
1392:
1387:
1382:
1377:
1374:
1369:
1363:
1360:
1359:
1356:
1353:
1352:
1350:
1345:
1341:
1336:
1331:
1326:
1321:
1318:
1313:
1308:
1303:
1300:
1295:
1289:
1286:
1285:
1282:
1279:
1278:
1276:
1271:
1267:
1252:
1251:
1240:
1235:
1229:
1226:
1225:
1222:
1219:
1218:
1216:
1211:
1206:
1200:
1193:
1188:
1182:
1179:
1178:
1175:
1172:
1171:
1169:
1164:
1159:
1153:
1114:
1109:
1084:
1081:
1079:
1076:
1063:
1039:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
960:
957:
954:
951:
948:
937:characteristic
905:
885:
865:
847:graded algebra
834:
831:
811:
787:
784:
760:
740:
700:
673:
670:
667:
664:
661:
658:
655:
652:
649:
629:
626:
623:
620:
617:
589:
569:
566:
546:
522:
519:
516:
470:
450:
419:
415:
411:
408:
405:
400:
396:
392:
387:
383:
362:
340:
337:
317:
297:
273:
270:
250:
230:
227:
224:
221:
218:
198:
170:
167:
158:that contains
143:
65:
64:
59:
52:
51:
47:
40:
39:
38:
37:
36:
26:
18:Exterior power
9:
6:
4:
3:
2:
27904:
27893:
27890:
27888:
27885:
27883:
27880:
27879:
27877:
27862:
27859:
27857:
27854:
27852:
27849:
27847:
27844:
27842:
27839:
27837:
27834:
27832:
27829:
27827:
27824:
27822:
27819:
27817:
27814:
27812:
27809:
27807:
27804:
27802:
27799:
27798:
27796:
27794:
27790:
27780:
27777:
27775:
27772:
27770:
27767:
27765:
27762:
27760:
27757:
27755:
27752:
27750:
27747:
27745:
27742:
27740:
27737:
27736:
27734:
27730:
27724:
27721:
27719:
27716:
27714:
27711:
27709:
27706:
27704:
27701:
27699:
27698:Metric tensor
27696:
27694:
27691:
27689:
27686:
27685:
27683:
27679:
27676:
27672:
27666:
27663:
27661:
27658:
27656:
27653:
27651:
27648:
27646:
27643:
27641:
27638:
27636:
27633:
27631:
27628:
27626:
27623:
27621:
27618:
27616:
27613:
27611:
27610:Exterior form
27608:
27606:
27603:
27601:
27598:
27596:
27593:
27591:
27588:
27586:
27583:
27581:
27578:
27576:
27573:
27572:
27570:
27564:
27557:
27554:
27552:
27549:
27547:
27544:
27542:
27539:
27537:
27534:
27532:
27529:
27527:
27524:
27522:
27519:
27517:
27514:
27512:
27509:
27507:
27504:
27503:
27501:
27499:
27495:
27489:
27486:
27484:
27483:Tensor bundle
27481:
27479:
27476:
27474:
27471:
27469:
27466:
27464:
27461:
27459:
27456:
27454:
27451:
27449:
27446:
27444:
27441:
27440:
27438:
27432:
27426:
27423:
27421:
27418:
27416:
27413:
27411:
27408:
27406:
27403:
27401:
27398:
27396:
27393:
27391:
27388:
27386:
27383:
27382:
27380:
27376:
27366:
27363:
27361:
27358:
27356:
27353:
27351:
27348:
27346:
27343:
27342:
27340:
27335:
27332:
27330:
27327:
27326:
27323:
27317:
27314:
27312:
27309:
27307:
27304:
27302:
27299:
27297:
27294:
27292:
27289:
27287:
27284:
27282:
27279:
27278:
27276:
27274:
27270:
27267:
27263:
27259:
27258:
27252:
27248:
27241:
27236:
27234:
27229:
27227:
27222:
27221:
27218:
27206:
27198:
27197:
27194:
27188:
27185:
27183:
27182:Sparse matrix
27180:
27178:
27175:
27173:
27170:
27168:
27165:
27164:
27162:
27160:
27156:
27150:
27147:
27145:
27142:
27140:
27137:
27135:
27132:
27130:
27127:
27125:
27122:
27121:
27119:
27117:constructions
27116:
27112:
27106:
27105:Outermorphism
27103:
27101:
27098:
27096:
27093:
27091:
27088:
27086:
27083:
27081:
27078:
27076:
27073:
27071:
27068:
27066:
27065:Cross product
27063:
27061:
27058:
27057:
27055:
27053:
27049:
27043:
27040:
27038:
27035:
27033:
27032:Outer product
27030:
27028:
27025:
27023:
27020:
27018:
27015:
27013:
27012:Orthogonality
27010:
27009:
27007:
27005:
27001:
26995:
26992:
26990:
26989:Cramer's rule
26987:
26985:
26982:
26980:
26977:
26975:
26972:
26970:
26967:
26965:
26962:
26960:
26959:Decomposition
26957:
26955:
26952:
26951:
26949:
26947:
26943:
26938:
26928:
26925:
26923:
26920:
26918:
26915:
26913:
26910:
26908:
26905:
26903:
26900:
26898:
26895:
26893:
26890:
26888:
26885:
26883:
26880:
26878:
26875:
26873:
26870:
26868:
26865:
26863:
26860:
26858:
26855:
26853:
26850:
26848:
26845:
26843:
26840:
26838:
26835:
26834:
26832:
26828:
26822:
26819:
26817:
26814:
26813:
26810:
26806:
26799:
26794:
26792:
26787:
26785:
26780:
26779:
26776:
26768:
26765:
26762:
26759:
26756:
26753:
26748:
26747:
26744:
26738:
26734:
26730:
26729:
26724:
26720:
26715:
26711:
26710:
26708:
26702:
26698:
26694:
26689:
26686:
26682:
26681:
26676:
26671:
26666:
26665:
26663:
26657:
26653:
26649:
26645:
26628:
26623:
26619:
26613:
26606:
26583:
26578:
26571:
26560:
26556:
26540:
26535:
26528:
26517:
26513:
26509:
26508:
26506:
26500:
26496:
26492:
26488:
26483:
26479:
26478:
26470:on 2009-02-19
26469:
26465:
26464:
26458:
26457:
26446:
26442:
26438:
26434:
26430:
26426:
26422:
26418:
26414:
26410:
26406:
26402:
26396:
26391:
26390:
26381:
26377:
26373:
26370:
26368:0-8218-2031-1
26364:
26360:
26356:
26351:
26349:
26343:
26342:
26337:
26333:
26330:
26326:
26325:
26320:
26316:
26313:
26309:
26305:
26301:
26297:
26293:
26289:
26285:
26281:
26277:
26276:
26263:
26262:
26259:
26255:
26251:
26246:
26245:
26243:
26241:0-8218-1646-2
26237:
26233:
26229:
26225:
26221:
26216:
26212:
26211:
26208:
26204:
26200:
26196:
26192:
26186:
26182:
26181:
26179:
26177:3-540-64243-9
26173:
26169:
26165:
26161:
26156:
26155:
26153:
26151:0-486-64039-6
26147:
26142:
26141:
26135:
26131:
26130:
26113:
26112:Bourbaki 1989
26108:
26101:
26097:
26093:
26092:Clifford 1878
26089:
26085:
26081:
26077:
26071:
26062:
26055:
26051:
26047:
26042:
26034:
26032:0-521-42377-5
26028:
26024:
26020:
26019:DeWitt, Bryce
26014:
26006:
26004:0-12-398750-4
26000:
25996:
25992:
25985:
25978:
25974:
25968:
25961:
25943:
25932:
25926:
25919:
25912:
25894:
25872:
25852:
25842:
25821:
25812:structure on
25811:
25807:
25789:
25777:
25769:
25767:0-7167-0344-0
25763:
25759:
25752:
25744:
25739:
25734:
25727:
25721:
25714:
25710:
25706:
25700:
25693:
25687:
25680:
25674:
25668:, Chapter 5).
25667:
25663:
25659:
25655:
25651:
25645:
25626:
25606:
25598:
25597:
25592:
25586:
25579:
25578:Clifford 1878
25575:
25571:
25566:
25559:
25554:
25546:
25540:
25536:
25535:
25527:
25525:
25520:
25511:
25507:
25503:
25500:
25498:
25495:
25492:
25489:
25487:
25484:
25482:
25479:
25477:
25474:
25472:
25468:
25465:
25463:
25460:
25458:
25455:
25454:
25448:
25446:
25442:
25438:
25433:
25431:
25427:
25423:
25419:
25415:
25411:
25406:
25404:
25400:
25396:
25391:
25389:
25385:
25381:
25377:
25373:
25363:
25361:
25357:
25347:
25345:
25341:
25313:
25312:chain complex
25294:
25286:
25264:
25241:
25230:
25211:
25206:
25203:
25200:
25196:
25192:
25189:
25186:
25181:
25171:
25164:
25161:
25158:
25153:
25143:
25136:
25133:
25130:
25125:
25121:
25117:
25109:
25105:
25101:
25096:
25092:
25083:
25080:
25077:
25074:
25071:
25063:
25060:
25052:
25049:
25046:
25042:
25035:
25032:
25029:
25025:
25020:
25012:
25009:
25006:
25002:
24998:
24995:
24992:
24987:
24983:
24969:
24968:
24967:
24947:
24939:
24932:
24920:
24912:
24909:
24906:
24899:
24893:
24883:
24882:
24881:
24865:
24856:. This is a
24841:
24831:
24830:chain complex
24813:
24791:
24777:
24761:
24750:
24745:
24744:cylinder sets
24741:
24737:
24736:weak topology
24733:
24729:
24725:
24721:
24720:supersymmetry
24717:
24713:
24703:
24701:
24700:
24695:
24691:
24687:
24683:
24679:
24669:
24667:
24663:
24659:
24655:
24652:
24648:
24644:
24640:
24635:
24633:
24629:
24624:
24620:
24615:
24610:
24605:
24604:tangent space
24601:
24597:
24593:
24589:
24585:
24581:
24577:
24573:
24569:
24559:
24557:
24553:
24526:
24518:
24511:
24488:
24484:
24463:
24457:
24452:
24448:
24423:
24412:
24407:
24389:
24379:
24376:-dimensional
24374:
24354:
24346:
24339:
24327:
24323:Decomposable
24316:
24300:
24276:
24252:
24242:
24226:
24223:
24220:
24217:
24214:
24192:
24170:
24162:
24161:parallelogram
24146:
24143:
24140:
24131:
24120:
24107:
24102:
24099:
24095:
24089:
24086:
24082:
24076:
24073:
24069:
24065:
24060:
24057:
24053:
24030:
24027:
24022:
24019:
24015:
24011:
24006:
24003:
23998:
23995:
23991:
23987:
23982:
23978:
23970:
23954:
23950:
23946:
23942:
23938:
23935:
23915:
23908:
23905:provides the
23904:
23900:
23895:
23881:
23878:
23870:
23867:
23864:
23861:
23854:
23850:
23842:
23839:
23836:
23833:
23826:
23805:
23802:
23799:
23796:
23793:
23790:
23787:
23784:
23776:
23760:
23752:
23749:
23746:
23739:
23735:
23727:
23724:
23721:
23714:
23710:
23705:
23702:
23698:
23690:
23686:
23670:
23667:
23664:
23661:
23654:
23650:
23646:
23636:
23631:
23621:
23619:
23614:
23609:
23603:
23599:
23594:
23590:
23586:
23582:
23578:
23562:
23553:
23544:
23535:
23519:
23515:
23511:
23508:
23505:
23500:
23496:
23486:
23469:
23466:
23461:
23457:
23453:
23450:
23447:
23442:
23438:
23434:
23431:
23428:
23420:
23416:
23412:
23407:
23403:
23377:
23352:
23344:
23333:
23317:
23292:
23289:
23286:
23274:
23260:
23238:
23216:
23196:
23188:
23172:
23169:
23147:
23143:
23137:
23133:
23129:
23126:
23123:
23118:
23114:
23108:
23104:
23100:
23095:
23091:
23085:
23081:
23077:
23072:
23068:
23062:
23058:
23052:
23044:
23041:
23017:
23012:
23008:
23002:
22998:
22994:
22991:
22988:
22983:
22979:
22973:
22969:
22965:
22960:
22956:
22950:
22946:
22925:
22903:
22899:
22895:
22892:
22889:
22886:
22883:
22878:
22874:
22870:
22865:
22861:
22840:
22837:
22834:
22826:
22808:
22776:
22772:
22756:
22737:
22720:
22714:
22711:
22705:
22697:
22694:
22691:
22684:
22672:
22664:
22657:
22645:
22637:
22631:
22622:
22615:
22614:
22613:
22611:
22592:
22583:
22575:
22568:
22556:
22548:
22541:
22529:
22521:
22518:
22515:
22508:
22502:
22499:
22493:
22486:
22485:
22484:
22482:
22463:
22455:
22451:
22444:
22441:
22438:
22435:
22427:
22423:
22416:
22413:
22408:
22405:
22402:
22398:
22394:
22391:
22388:
22383:
22379:
22370:
22366:
22362:
22359:
22356:
22351:
22347:
22343:
22338:
22335:
22332:
22328:
22324:
22321:
22318:
22313:
22309:
22285:
22277:
22270:
22264:
22258:
22250:
22247:
22244:
22237:
22231:
22226:
22222:
22217:
22211:
22208:
22205:
22201:
22193:
22192:
22191:
22177:
22174:
22171:
22166:
22162:
22141:
22138:
22133:
22129:
22106:
22103:
22100:
22096:
22092:
22089:
22084:
22080:
22076:
22071:
22068:
22065:
22062:
22059:
22055:
22051:
22048:
22045:
22040:
22036:
22015:
22012:
22009:
21987:
21983:
21956:
21948:
21941:
21935:
21930:
21927:
21924:
21920:
21916:
21911:
21907:
21903:
21900:
21897:
21892:
21888:
21884:
21879:
21875:
21871:
21868:
21861:
21860:
21859:
21858:
21842:
21839:
21833:
21828:
21820:
21815:
21807:
21801:
21778:
21772:
21764:
21757:
21751:
21745:
21737:
21730:
21722:
21719:
21716:
21713:
21710:
21706:
21702:
21696:
21693:
21690:
21682:
21675:
21665:
21664:
21663:
21646:
21640:
21632:
21627:
21621:
21613:
21608:
21602:
21599:
21596:
21588:
21579:
21578:
21577:
21555:
21549:
21541:
21530:
21522:
21514:
21507:
21506:
21505:
21488:
21479:
21471:
21460:
21452:
21441:
21433:
21428:
21422:
21414:
21407:
21398:
21391:
21390:
21389:
21387:
21371:
21365:
21359:
21353:
21347:
21333:
21317:
21307:
21288:
21280:
21273:
21247:
21239:
21232:
21207:
21185:
21165:
21162:
21159:
21137:
21115:
21112:
21109:
21089:
21069:
21044:
21036:
21029:
21001:
20995:
20987:
20980:
20968:
20960:
20953:
20947:
20936:
20928:
20921:
20915:
20911:
20904:
20896:
20891:
20885:
20877:
20870:
20860:
20859:
20858:
20841:
20833:
20829:
20822:
20819:
20816:
20813:
20805:
20801:
20794:
20791:
20783:
20779:
20775:
20772:
20769:
20764:
20760:
20750:
20742:
20733:
20732:
20731:
20716:
20713:
20710:
20706:
20697:
20677:
20669:
20644:
20638:
20630:
20623:
20617:
20614:
20605:
20597:
20590:
20584:
20581:
20578:
20575:
20572:
20561:
20553:
20546:
20540:
20536:
20529:
20521:
20512:
20511:
20510:
20490:
20482:
20471:
20463:
20458:
20452:
20444:
20435:
20434:
20433:
20431:
20415:
20409:
20406:
20403:
20383:
20363:
20356:Suppose that
20352:Functoriality
20349:
20347:
20329:
20317:
20313:
20310:
20306:
20284:
20281:
20275:
20269:
20263:
20253:
20248:
20246:
20230:
20221:
20213:
20208:
20205:
20197:
20192:
20178:
20168:
20164:
20160:
20156:
20152:
20131:
20122:
20118:
20114:
20111:
20108:
20103:
20099:
20088:
20081:
20078:
20075:
20069:
20061:
20057:
20053:
20050:
20047:
20042:
20038:
20028:
20025:
20022:
20012:
20011:
20010:
20007:
20005:
20001:
19983:
19961:
19941:
19921:
19913:
19908:
19904:
19900:
19896:
19888:
19884:
19880:
19876:
19853:
19845:
19840:
19834:
19826:
19813:
19791:
19783:
19780:
19777:
19771:
19765:
19759:
19751:
19745:
19737:
19732:
19729:
19726:
19722:
19712:
19704:
19698:
19692:
19682:
19681:
19680:
19666:
19660:
19652:
19641:
19639:
19621:
19617:
19593:
19585:
19575:
19571:
19567:
19546:
19540:
19536:
19532:
19529:
19526:
19518:
19515:
19512:
19506:
19502:
19476:
19470:
19466:
19462:
19459:
19456:
19448:
19442:
19438:
19429:
19426:} → {1, ...,
19425:
19421:
19417:
19413:
19409:
19405:
19401:
19379:
19368:
19362:
19358:
19354:
19351:
19348:
19340:
19337:
19334:
19328:
19324:
19317:
19306:
19300:
19296:
19292:
19289:
19286:
19278:
19272:
19268:
19258:
19252:
19249:
19240:
19237:
19234:
19231:
19228:
19222:
19219:
19216:
19213:
19209:
19202:
19197:
19194:
19191:
19187:
19183:
19175:
19171:
19167:
19164:
19161:
19156:
19152:
19138:
19137:
19136:
19119:
19116:
19110:
19107:
19104:
19098:
19095:
19092:
19089:
19086:
19083:
19080:
19077:
19074:
19068:
19065:
19062:
19056:
19053:
19050:
19044:
19041:
19038:
19025:
19024:
19023:
19021:
19002:
18994:
18967:
18959:
18954:
18948:
18940:
18911:
18903:
18898:
18892:
18884:
18877:
18871:
18868:
18844:
18822:
18800:
18797:
18794:
18769:
18766:
18763:
18760:
18757:
18754:
18751:
18748:
18742:
18729:
18728:
18727:
18708:
18700:
18695:
18689:
18681:
18670:
18662:
18657:
18644:
18640:
18619:
18611:
18599:
18594:
18592:
18588:
18584:
18580:
18562:
18537:
18529:
18513:
18494:
18486:
18479:
18473:
18435:
18424:
18394:
18393:
18392:
18376:
18352:
18337:
18329:
18322:
18309:
18305:
18300:
18296:
18292:
18286:
18279:
18273:
18215:
18211:
18200:
18189:
18181:
18166:
18165:
18164:
18145:
18137:
18130:
18124:
18092:
18084:
18077:
18071:
18039:
18031:
18028:
18025:
18018:
18012:
17996:
17980:
17977:
17974:
17946:
17943:
17940:
17934:
17928:
17916:
17908:
17907:
17906:
17904:
17899:
17895:
17859:
17854:
17850:
17839:
17828:
17820:
17817:
17807:
17806:
17805:
17789:
17786:
17783:
17756:
17748:
17741:
17735:
17703:
17695:
17692:
17689:
17682:
17676:
17660:
17658:
17637:
17629:
17622:
17594:
17589:
17585:
17581:
17578:
17575:
17570:
17566:
17561:
17554:
17550:
17545:
17541:
17538:
17535:
17528:
17524:
17519:
17511:
17510:
17509:
17493:
17483:
17478:
17475:= 1, 2, ...,
17474:
17468:
17464:
17448:
17442:
17434:
17427:
17413:
17409:
17404:
17400:
17395:
17391:
17385:
17381:
17376:
17372:
17352:
17337:
17333:
17329:
17324:
17320:
17303:
17299:
17293:
17289:
17285:
17282:
17279:
17274:
17270:
17266:
17261:
17257:
17253:
17250:
17247:
17242:
17238:
17233:
17225:
17224:
17223:
17207:
17181:
17169:
17164:
17157:
17122:
17114:
17107:
17080:
17076:
17053:
17031:
17021:
17017:
17016:inner product
16999:
16986:Inner product
16983:
16981:
16977:
16971:
16967:
16962:
16933:
16930:
16924:
16921:
16918:
16912:
16904:
16901:
16895:
16889:
16881:
16874:
16862:
16854:
16847:
16841:
16838:
16835:
16832:
16825:
16824:
16823:
16807:
16797:
16778:
16770:
16763:
16751:
16743:
16736:
16713:
16690:
16684:
16676:
16673:
16670:
16663:
16651:
16643:
16636:
16630:
16627:
16620:
16619:
16618:
16616:
16612:
16592:
16588:
16565:
16557:
16556:inner product
16553:
16534:
16531:
16526:
16522:
16515:
16512:
16506:
16498:
16495:
16492:
16485:
16471:
16467:
16458:
16451:
16441:
16440:
16439:
16423:
16413:
16409:
16408:
16388:
16380:
16373:
16347:
16342:
16338:
16334:
16329:
16325:
16321:
16316:
16313:
16310:
16306:
16298:
16297:
16296:
16276:
16268:
16265:
16262:
16255:
16243:
16235:
16228:
16222:
16214:
16210:
16201:
16194:
16184:
16183:
16182:
16166:
16144:
16137:Suppose that
16134:
16127:Hodge duality
16110:
16105:
16101:
16097:
16092:
16088:
16084:
16081:
16076:
16072:
16068:
16063:
16059:
16051:
16037:
16034:
16029:
16025:
16021:
16016:
16012:
16004:
16003:
16002:
15999:
15980:
15974:
15969:
15965:
15958:
15955:
15950:
15947:
15944:
15936:
15933:
15927:
15924:
15921:
15915:
15910:
15906:
15899:
15893:
15890:
15887:
15879:
15875:
15867:
15866:
15864:
15846:
15842:
15817:
15813:
15809:
15806:
15796:
15777:
15773:
15750:
15725:
15719:
15716:
15713:
15708:
15704:
15676:
15668:
15661:
15655:
15633:
15613:
15605:
15589:
15583:
15575:
15572:
15569:
15562:
15550:
15542:
15535:
15529:
15524:
15520:
15512:
15511:
15494:
15488:
15482:
15474:
15471:
15442:
15438:
15434:
15431:
15407:
15397:
15396:
15395:
15387:
15368:
15360:
15353:
15328:
15308:
15305:
15302:
15297:
15293:
15269:
15261:
15258:
15255:
15251:
15247:
15244:
15241:
15236:
15232:
15228:
15223:
15219:
15215:
15212:
15206:
15203:
15195:
15192:
15189:
15185:
15181:
15178:
15175:
15170:
15166:
15162:
15157:
15153:
15143:
15138:
15134:
15123:
15122:
15121:
15103:
15099:
15076:
15073:
15070:
15060:
15056:
15049:
15042:
15022:
15018:
15014:
15011:
15008:
15003:
14999:
14995:
14990:
14986:
14976:
14971:
14952:
14928:
14924:
14903:
14878:
14870:
14863:
14857:
14854:
14845:Suppose that
14843:
14827:
14817:
14813:
14795:
14785:
14766:
14760:
14752:
14749:
14746:
14739:
14727:
14719:
14712:
14706:
14701:
14697:
14689:
14688:
14687:
14668:
14660:
14648:
14628:
14624:
14620:
14617:
14593:
14583:
14565:
14561:
14540:
14533:Suppose that
14530:
14520:
14515:
14511:
14506:
14502:
14497:
14493:
14489:
14482:
14478:
14474:
14470:
14466:
14462:
14458:
14451:
14447:
14443:
14439:
14432:
14428:
14422:
14419:
14415:
14411:
14407:
14398:
14394:
14390:
14385:
14381:
14359:
14348:
14345:
14342:
14336:
14332:
14328:
14325:
14322:
14314:
14311:
14308:
14302:
14298:
14291:
14279:
14273:
14269:
14265:
14262:
14259:
14251:
14245:
14241:
14234:
14227:
14221:
14218:
14211:
14208:
14205:
14192:
14189:
14185:
14181:
14172:
14169:
14166:
14162:
14158:
14155:
14152:
14147:
14143:
14136:
14130:
14127:
14121:
14113:
14112:
14111:
14109:
14093:
14086:
14067:
14056:
14050:
14046:
14042:
14039:
14036:
14028:
14022:
14018:
14011:
14004:
13998:
13995:
13988:
13984:
13980:
13977:
13973:
13966:
13963:
13959:
13954:
13946:
13942:
13938:
13935:
13932:
13927:
13923:
13913:
13907:
13904:
13897:
13896:
13895:
13893:
13877:
13854:
13848:
13845:
13842:
13836:
13833:
13827:
13824:
13820:
13817:
13812:
13806:
13803:
13800:
13791:
13788:
13782:
13779:
13773:
13766:
13765:
13764:
13744:
13741:
13738:
13732:
13729:
13726:
13723:
13720:
13717:
13710:
13709:
13708:
13705:
13700:
13696:
13692:
13686:
13682:
13678:
13672:
13647:
13644:
13626:
13610:
13588:
13584:
13563:
13543:
13520:
13515:
13511:
13507:
13501:
13494:
13465:
13441:
13426:
13418:
13411:
13392:
13388:
13372:
13352:
13343:
13340:
13336:
13333:is called an
13317:
13309:
13305:
13301:
13298:
13291:
13290:
13289:
13273:
13270:
13267:
13253:
13247:
13242:
13237:
13231:
13225:
13219:
13214:
13210:
13205:
13200:
13195:
13185:
13169:
13165:
13156:
13140:
13137:
13134:
13131:
13128:
13125:
13105:
13096:
13088:
13081:
13075:
13072:
13065:
13049:
13046:
13038:
13034:
13030:
13027:
13007:
13002:
12998:
12977:
12957:
12950:vectors from
12937:
12914:
12908:
12900:
12893:
12882:
12878:
12874:
12871:
12864:
12863:
12862:
12845:
12842:
12834:
12830:
12826:
12823:
12820:
12815:
12811:
12804:
12797:
12796:
12795:
12793:
12789:
12784:
12780:
12773:
12754:
12746:
12742:
12738:
12735:
12728:
12727:
12726:
12725:
12721:
12717:
12713:
12709:
12705:
12701:
12686:
12684:
12680:
12661:
12654:
12651:
12648:
12644:
12640:
12635:
12631:
12627:
12623:
12617:
12613:
12607:
12602:
12599:
12596:
12592:
12588:
12585:
12578:
12575:
12572:
12568:
12564:
12559:
12555:
12546:
12541:
12537:
12526:
12525:
12524:
12508:
12505:
12502:
12478:
12473:
12469:
12457:
12453:
12449:. Then, for
12434:
12408:
12405:
12402:
12398:
12394:
12389:
12385:
12379:
12375:
12370:
12366:
12363:
12355:
12336:
12326:
12323:
12320:
12316:
12312:
12307:
12304:
12301:
12297:
12292:
12284:
12280:
12276:
12271:
12267:
12259:
12255:
12248:
12245:
12242:
12238:
12234:
12229:
12225:
12216:
12207:
12204:
12195:
12185:
12184:
12183:
12180:
12176:
12156:
12149:
12146:
12143:
12139:
12126:
12123:
12120:
12113:
12109:
12096:
12089:
12085:
12065:
12062:
12059:
12053:
12049:
12045:
12037:
12034:
12031:
12025:
12021:
12016:
12005:
11999:
11995:
11991:
11983:
11977:
11973:
11968:
11961:
11955:
11952:
11945:
11942:
11939:
11927:
11924:
11920:
11913:
11907:
11904:
11901:
11894:
11889:
11886:
11877:
11874:
11865:
11858:
11857:
11856:
11854:
11850:
11846:
11842:
11837:
11835:
11831:
11812:
11805:
11801:
11788:
11785:
11782:
11775:
11771:
11758:
11751:
11747:
11729:
11725:
11721:
11716:
11712:
11706:
11702:
11697:
11693:
11690:
11683:
11682:
11681:
11679:
11675:
11669:
11665:
11661:
11657:
11652:
11646:
11642:
11635:
11630:
11626:
11623:Suppose that
11616:
11597:
11591:
11566:
11560:
11537:
11531:
11525:
11519:
11515:
11508:
11505:
11502:
11496:
11471:
11463:
11434:
11428:
11417:
11400:
11397:
11372:
11347:
11341:
11318:
11310:
11305:
11299:
11293:
11256:will do what
11216:
11210:
11202:
11197:
11191:
11185:
11178:
11177:
11176:
11157:
11127:
11104:
11098:
11078:
11055:
11049:
11046:
11043:
11037:
11034:
11031:
11028:
11019:
11016:
11007:
11004:
11001:
10998:
10995:
10988:
10987:
10986:
10969:
10966:
10940:
10934:
10914:
10892:
10844:
10815:
10808:
10798:is precisely
10780:
10743:
10708:
10673:
10667:
10657:
10635:
10592:
10585:
10561:
10536:
10533:
10530:
10505:
10476:
10475:
10474:
10471:
10452:
10363:
10360:
10357:
10354:
10351:
10338:
10314:
10310:
10306:
10303:
10300:
10295:
10291:
10284:
10256:
10253:
10249:
10244:
10236:
10232:
10228:
10225:
10222:
10217:
10213:
10206:
10198:
10191:
10183:
10182:
10181:
10180:might be 0):
10167:
10164:
10144:
10141:
10138:
10135:
10107:
10101:
10097:
10093:
10090:
10087:
10079:
10073:
10069:
10062:
10056:
10053:
10046:
10034:
10031:
10027:
10023:
10015:
10011:
10007:
10004:
10001:
9996:
9992:
9985:
9956:
9955:
9954:
9952:
9948:
9929:
9926:
9923:
9918:
9914:
9909:
9904:
9900:
9896:
9893:
9890:
9885:
9881:
9873:
9872:
9871:
9857:
9834:
9826:
9810:
9794:
9770:
9767:
9764:
9742:
9735:by the ideal
9719:
9691:
9671:
9649:
9646:
9643:
9640:
9637:
9613:
9589:
9567:
9545:
9542:
9539:
9529:
9510:
9482:
9462:
9448:
9446:
9442:
9437:
9432:
9430:
9426:
9410:
9402:
9386:
9363:
9355:
9328:
9295:
9287:
9263:
9254:
9238:
9218:
9211:
9201:
9182:
9174:
9147:
9139:
9132:
9106:
9098:
9091:
9065:
9057:
9046:
9044:
9040:
9021:
9013:
9002:
8996:
8993:
8973:
8965:
8938:
8930:
8899:
8894:
8886:
8880:
8877:
8871:
8863:
8854:
8853:
8852:
8835:
8827:
8818:, and define
8816:
8810:
8804:
8800:
8793:
8789:
8783:
8779:
8775:
8769:
8765:
8761:
8756:
8750:
8744:
8734:
8729:
8710:
8702:
8689:
8683:
8677:
8671:
8664:
8660:
8656:
8652:
8648:
8631:
8622:
8614:
8609:
8606:
8599:
8595:
8594:precisely one
8590:
8584:
8567:
8564:
8558:
8552:
8546:
8540:
8520:
8514:
8511:
8508:
8501:
8496:
8490:
8484:
8477:
8475:
8454:
8446:
8433:
8427:
8421:
8401:
8393:
8384:. Formally,
8382:
8378:
8361:
8358:
8355:
8352:
8349:
8341:
8337:
8318:
8310:
8299:
8293:
8282:
8280:
8279:graded module
8276:
8257:
8254:
8251:
8248:
8243:
8240:
8232:
8229:
8223:
8220:
8217:
8214:
8207:
8206:
8205:
8186:
8178:
8171:
8165:
8162:
8137:
8129:
8122:
8116:
8113:
8090:
8087:
8084:
8078:
8070:
8063:
8040:
8037:
8031:
8023:
8016:
8006:
8005:
8004:
8001:
7997:Moreover, if
7981:
7975:
7967:
7964:
7961:
7954:
7948:
7942:
7934:
7927:
7921:
7915:
7907:
7900:
7890:
7889:
7888:
7886:
7864:
7856:
7849:
7843:
7840:
7837:
7831:
7823:
7816:
7810:
7804:
7796:
7789:
7783:
7777:
7769:
7762:
7756:
7750:
7742:
7733:
7732:
7731:
7714:
7711:
7708:
7698:-vector is a
7696:
7690:
7665:
7662:
7656:
7653:
7650:
7644:
7640:
7637:
7634:
7631:
7628:
7597:
7594:
7589:
7584:
7580:
7577:
7574:
7571:
7568:
7556:
7555:
7554:
7540:
7520:
7511:
7497:
7475:
7472:
7468:
7459:
7441:
7438:
7434:
7430:
7427:
7422:
7419:
7415:
7389:
7385:
7381:
7376:
7372:
7366:
7363:
7359:
7353:
7350:
7347:
7343:
7339:
7336:
7329:
7328:
7327:
7313:
7291:
7267:
7263:
7242:
7234:
7218:
7210:
7206:
7201:
7199:
7181:
7170:
7153:
7144:
7139:
7120:
7117:
7114:
7111:
7108:
7105:
7102:
7099:
7096:
7093:
7089:
7081:
7073:
7069:
7065:
7062:
7059:
7051:
7043:
7039:
7035:
7027:
7020:
7012:
7011:
7010:
6991:
6984:
6955:
6948:
6944:
6941:
6938:
6930:
6923:
6919:
6911:
6904:
6900:
6897:
6890:
6889:
6888:
6886:
6883:
6865:
6840:
6832:
6825:
6819:
6816:
6802:
6796:
6778:
6774:
6749:
6746:
6740:
6732:
6725:
6698:
6688:
6670:
6667:
6661:
6653:
6646:
6615:
6607:
6600:
6594:
6591:
6588:
6582:
6574:
6567:
6561:
6555:
6547:
6540:
6534:
6528:
6520:
6513:
6507:
6501:
6493:
6484:
6483:
6482:
6481:
6477:
6474:
6467:
6451:
6448:
6445:
6420:
6414:
6408:
6400:
6393:
6368:
6358:
6340:
6315:
6304:
6301:
6290:
6284:
6276:
6269:
6263:
6260:
6253:
6252:
6251:
6249:
6230:
6222:
6215:
6203:
6185:
6181:
6156:
6152:
6143:
6139:
6115:
6111:
6100:
6082:
6078:
6070:every vector
6054:
6049:
6045:
6041:
6038:
6035:
6030:
6026:
6018:
6017:
6016:
5997:
5989:
5982:
5950:
5947:
5942:
5938:
5934:
5931:
5928:
5923:
5919:
5915:
5910:
5906:
5902:
5899:
5876:
5872:
5867:
5863:
5860:
5857:
5850:
5846:
5841:
5837:
5830:
5826:
5821:
5809:
5808:
5807:
5793:
5785:
5764:
5760:
5756:
5753:
5750:
5745:
5741:
5717:
5697:
5689:
5665:
5660:
5656:
5652:
5647:
5643:
5639:
5634:
5630:
5626:
5621:
5617:
5613:
5610:
5603:
5602:
5601:
5580:
5576:
5572:
5567:
5563:
5559:
5554:
5550:
5546:
5541:
5537:
5524:with a basis
5507:
5475:
5467:
5460:
5432:
5424:
5417:
5390:
5380:
5376:
5372:
5356:
5334:
5312:
5292:
5284:
5283:
5280:
5261:
5236:
5228:
5221:
5215:
5212:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5146:
5142:
5137:
5132:
5128:
5124:
5121:
5118:
5113:
5109:
5105:
5100:
5096:
5088:
5087:
5086:
5084:
5064:
5056:
5044:
5023:
5015:
5008:
4981:
4971:
4966:
4941:
4938:
4933:
4929:
4925:
4922:
4919:
4914:
4910:
4906:
4901:
4897:
4889:
4888:
4887:
4868:
4864:
4860:
4857:
4854:
4849:
4845:
4841:
4836:
4832:
4805:
4802:
4797:
4793:
4789:
4786:
4783:
4778:
4774:
4770:
4765:
4761:
4753:
4752:
4751:
4735:
4732:
4729:
4705:
4701:
4697:
4692:
4688:
4678:
4662:
4653:
4634:
4628:
4625:
4602:
4597:
4593:
4589:
4586:
4583:
4578:
4574:
4570:
4565:
4561:
4554:
4548:
4545:
4542:
4534:
4528:
4524:
4520:
4517:
4514:
4506:
4500:
4496:
4492:
4484:
4478:
4474:
4466:
4465:
4464:
4448:
4422:
4418:
4391:
4387:
4360:
4356:
4328:
4325:
4322:
4319:
4316:
4303:
4287:
4264:
4258:
4255:
4252:
4246:
4243:
4240:
4237:
4234:
4227:
4226:
4225:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4133:
4130:
4127:
4121:
4115:
4112:
4109:
4103:
4100:
4093:
4092:
4091:
4075:
4072:
4069:
4066:
4063:
4039:
4029:
4013:
4010:
4007:
4004:
3984:
3981:
3978:
3975:
3972:
3950:
3940:
3939:
3909:
3902:
3898:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3865:
3864:
3863:
3846:
3840:
3820:
3796:
3793:
3789:
3782:
3776:
3773:
3767:
3761:
3754:
3753:
3752:
3738:
3735:
3732:
3712:
3709:
3706:
3686:
3679:
3675:
3671:
3655:
3648:
3632:
3609:
3603:
3589:
3587:
3583:
3579:
3575:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3538:
3534:
3528:
3526:
3522:
3515:
3508:
3501:
3477:
3467:
3462:
3452:
3447:
3429:
3425:
3419:
3415:
3409:
3405:
3401:
3396:
3392:
3386:
3382:
3376:
3372:
3368:
3363:
3359:
3353:
3349:
3343:
3339:
3335:
3330:
3326:
3320:
3316:
3310:
3306:
3302:
3297:
3293:
3287:
3283:
3277:
3273:
3269:
3264:
3260:
3254:
3250:
3244:
3240:
3233:
3225:
3217:
3205:
3204:
3203:
3186:
3181:
3169:
3165:
3161:
3156:
3144:
3140:
3136:
3131:
3119:
3115:
3111:
3099:
3098:
3097:
3094:
3092:
3088:
3087:cross product
3084:
3077:
3070:
3063:
3056:
3049:
3042:
3018:
3008:
3003:
2985:
2981:
2975:
2971:
2967:
2962:
2958:
2952:
2948:
2941:
2933:
2923:
2918:
2900:
2896:
2890:
2886:
2882:
2877:
2873:
2867:
2863:
2856:
2848:
2838:
2833:
2815:
2811:
2805:
2801:
2797:
2792:
2788:
2782:
2778:
2771:
2763:
2751:
2750:
2749:
2730:
2718:
2714:
2710:
2705:
2693:
2689:
2685:
2680:
2668:
2664:
2660:
2648:
2647:
2646:
2627:
2615:
2611:
2607:
2602:
2590:
2586:
2582:
2577:
2565:
2561:
2557:
2545:
2544:
2543:
2536:
2529:
2522:
2516:
2512:
2511:cross product
2508:
2499:
2490:
2488:
2481:
2474:
2465:
2459:
2453:
2447:
2440:
2428:
2421:
2416:
2413:
2409:
2405:
2399:
2395:
2391:
2387:
2384:
2380:
2375:
2372:
2368:
2362:
2358:
2354:
2350:
2345:
2342:
2338:
2334:
2328:
2324:
2319:
2316:
2312:
2306:
2302:
2298:
2294:
2291:
2287:
2284:
2279:
2278:
2277:
2275:
2271:
2265:
2261:
2253:
2251:
2247:
2243:
2239:
2223:
2220:
2215:
2205:
2200:
2190:
2185:
2175:
2170:
2143:
2135:
2125:
2120:
2107:
2104:
2099:
2089:
2084:
2069:
2065:
2040:
2030:
2025:
2014:
2010:
2007:
2004:
2001:
1998:
1994:
1990:
1988:
1978:
1968:
1963:
1953:
1950:
1947:
1942:
1932:
1927:
1917:
1914:
1911:
1906:
1896:
1891:
1881:
1878:
1875:
1870:
1860:
1855:
1845:
1842:
1839:
1837:
1824:
1814:
1811:
1806:
1796:
1790:
1782:
1772:
1769:
1764:
1754:
1748:
1746:
1736:
1720:
1719:
1718:
1653:
1649:
1645:
1642:
1639:
1636:
1633:
1629:
1625:
1613:
1607:
1602:
1595:
1590:
1584:
1569:
1557:
1536:
1521:
1509:
1508:
1507:
1505:
1501:
1437:
1403:
1393:
1390:
1385:
1375:
1372:
1367:
1361:
1354:
1348:
1343:
1334:
1329:
1319:
1316:
1311:
1301:
1298:
1293:
1287:
1280:
1274:
1269:
1257:
1256:
1255:
1254:Suppose that
1238:
1233:
1227:
1220:
1214:
1209:
1204:
1191:
1186:
1180:
1173:
1167:
1162:
1157:
1141:
1140:
1139:
1138:
1134:
1130:
1112:
1098:
1089:
1075:
1061:
1053:
1037:
1029:
1025:
1021:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
958:
955:
952:
949:
946:
938:
934:
930:
926:
922:
921:vector fields
917:
903:
883:
863:
855:
850:
848:
832:
829:
809:
801:
785:
782:
774:
758:
738:
730:
726:
725:
720:
719:
698:
690:
685:
671:
668:
665:
662:
659:
656:
653:
650:
647:
627:
624:
621:
618:
615:
607:
603:
602:parallelotope
587:
567:
564:
544:
536:
535:parallelogram
520:
517:
514:
507:
500:
496:
492:
488:
484:
468:
448:
441:
437:
436:
417:
413:
409:
406:
403:
398:
394:
390:
385:
381:
360:
351:
338:
335:
328:is "outside"
315:
295:
287:
271:
268:
248:
228:
225:
222:
219:
216:
196:
188:
187:wedge product
184:
168:
165:
157:
141:
134:
130:
126:
114:
108:
104:
100:
96:
92:
91:parallelotope
88:
84:
80:
74:
69:
56:
44:
33:
19:
27861:Hermann Weyl
27665:Vector space
27650:Pseudotensor
27615:Fiber bundle
27568:abstractions
27463:Mixed tensor
27448:Tensor field
27255:
27115:Vector space
27084:
26847:Vector space
26727:
26696:
26678:
26651:
26558:
26554:
26494:
26472:, retrieved
26468:the original
26462:
26420:
26416:
26388:
26379:
26358:
26354:
26340:
26323:
26319:Forder, H.G.
26295:
26291:
26288:Clifford, W.
26257:
26231:
26228:Birkhoff, G.
26224:Mac Lane, S.
26206:
26195:Bryant, R.L.
26184:
26183:This is the
26167:
26139:
26107:
26095:
26087:
26075:
26070:
26061:
26053:
26049:
26041:
26022:
26013:
25994:
25984:
25967:
25930:
25924:
25918:
25841:
25776:
25757:
25751:
25742:
25737:
25733:
25720:
25699:
25686:
25673:
25666:Strang (1993
25644:
25594:
25590:
25585:
25573:
25565:
25560:, p. 83
25553:
25532:
25502:Weyl algebra
25434:
25410:multivectors
25407:
25402:
25394:
25392:
25388:Saint-Venant
25384:vector space
25379:
25375:
25369:
25353:
25226:
24965:
24783:
24748:
24723:
24712:superalgebra
24709:
24697:
24675:
24636:
24625:
24618:
24608:
24565:
24410:
24406:Grassmannian
24372:
24329:-vectors in
24325:
24322:
24129:
24126:
23896:
23642:
23633:
23607:
23601:
23597:
23592:
23574:
23484:
23334:
23275:
23185:to mean the
22771:affine space
22748:
22740:Applications
22735:
22609:
22607:
22480:
22478:
21974:
21793:
21661:
21575:
21503:
21339:
21016:
20856:
20695:
20659:
20508:
20355:
20346:Hopf algebra
20251:
20249:
20195:
20193:
20166:
20162:
20158:
20154:
20150:
20147:
20008:
19999:
19906:
19902:
19898:
19897:⊗ 1) ∧ (1 ⊗
19894:
19886:
19882:
19878:
19874:
19811:
19809:
19642:
19637:
19573:
19569:
19565:
19427:
19423:
19419:
19418:0) and Sh(0,
19415:
19407:
19403:
19399:
19394:
19134:
18785:on elements
18784:
18642:
18595:
18583:Hopf algebra
18519:
18458:
18367:is the dual
18253:
17997:
17964:
17897:
17893:
17890:
17661:
17609:
17476:
17472:
17466:
17462:
17411:
17407:
17402:
17398:
17389:
17383:
17379:
17374:
17370:
17367:
16989:
16979:
16975:
16969:
16965:
16958:
16705:
16614:
16610:
16558:identifying
16551:
16549:
16411:
16405:
16362:
16294:
16136:
16000:
15997:
15393:
15284:
15089:elements of
15058:
15054:
15047:
15040:
14969:
14844:
14815:
14811:
14783:
14781:
14580:denotes the
14532:
14513:
14509:
14504:
14500:
14495:
14491:
14487:
14480:
14476:
14472:
14468:
14464:
14460:
14456:
14449:
14445:
14441:
14437:
14430:
14426:
14425:{1, 2, ...,
14420:
14418:permutations
14409:
14405:
14396:
14392:
14388:
14383:
14379:
14374:
14082:
13892:permutations
13869:
13762:
13698:
13694:
13690:
13684:
13680:
13676:
13673:
13334:
13332:
13257:
13245:
13235:
13229:
13223:
13217:
13208:
13203:
12929:
12860:
12791:
12782:
12778:
12771:
12769:
12719:
12715:
12711:
12707:
12703:
12699:
12697:
12682:
12678:
12676:
12455:
12451:
12351:
12178:
12174:
12171:
11855:is given by
11852:
11848:
11844:
11840:
11838:
11829:
11827:
11667:
11663:
11659:
11655:
11650:
11644:
11640:
11633:
11628:
11624:
11622:
11418:
11231:
11070:
10927:is defined,
10655:
10576:
10472:
10334:
10127:
9950:
9946:
9944:
9811:
9454:
9447:of modules.
9433:
9207:
9047:
9000:
8997:
8991:
8914:
8814:
8808:
8802:
8798:
8791:
8787:
8781:
8767:
8763:
8754:
8748:
8742:
8739:
8687:
8681:
8675:
8669:
8662:
8658:
8654:
8650:
8646:
8593:
8588:
8582:
8494:
8488:
8482:
8479:
8431:
8425:
8419:
8380:
8376:
8297:
8291:
8288:
8272:
8105:
7999:
7996:
7882:
7694:
7688:
7685:
7512:
7406:
7202:
7168:
7142:
7137:
7135:
6975:
6881:
6806:
6800:
6633:
6472:
6468:
6356:
6330:
6204:
6069:
5969:
5685:
5378:
5374:
5371:decomposable
5370:
5325:elements of
5278:
5275:
5202:
4969:
4964:
4961:
4820:
4679:
4617:
4279:
4223:
3937:
3934:
3812:
3595:
3581:
3577:
3573:
3569:
3565:
3561:
3557:
3553:
3545:
3541:
3536:
3532:
3529:
3520:
3513:
3506:
3499:
3497:
3201:
3095:
3082:
3075:
3068:
3061:
3054:
3047:
3040:
3038:
2747:
2644:
2534:
2527:
2520:
2506:
2504:
2486:
2479:
2472:
2469:
2461:
2455:
2449:
2443:
2426:
2419:
2411:
2407:
2403:
2397:
2393:
2389:
2385:
2382:
2378:
2370:
2366:
2360:
2356:
2352:
2348:
2341:line segment
2336:
2326:
2322:
2314:
2310:
2304:
2300:
2296:
2292:
2289:
2285:
2282:
2273:
2269:
2263:
2259:
2254:
2241:
2237:
2061:
1668:
1499:
1420:
1253:
1137:unit vectors
1094:
918:
851:
772:
722:
717:
711:is called a
686:
482:
439:
434:
432:is called a
352:
209:, such that
186:
182:
133:vector space
128:
124:
122:
112:
94:
86:
82:
78:
72:
67:
27801:Élie Cartan
27749:Spin tensor
27723:Weyl tensor
27681:Mathematics
27645:Multivector
27436:definitions
27334:Engineering
27273:Mathematics
27095:Multivector
27060:Determinant
27017:Dot product
26862:Linear span
26512:integration
26344:(in German)
26199:Chern, S.S.
26100:Browne 2007
26084:Forder 1941
25758:Gravitation
25422:Élie Cartan
24724:supernumber
24243:with edges
24163:with sides
23903:orientation
23897:Adding the
23581:determinant
23189:from point
21572:Direct sums
21306:determinant
18600:defined on
16978:pluses and
16407:volume form
14816:contraction
14423:of the set
14375:where here
13339:alternating
13252:orientation
12790:vectors in
12523:, given by
7207:, §III.6) (
7198:tensor rank
6976:where each
6689:underlying
4302:permutation
3938:alternating
2246:signed area
1504:determinant
1074:variables.
729:linear span
724:multivector
537:defined by
107:orientation
103:hypervolume
27876:Categories
27630:Linear map
27498:Operations
27129:Direct sum
26964:Invertible
26867:Linear map
26648:Strang, G.
26557:-forms on
26474:2007-05-09
26134:Bishop, R.
26121:References
24742:being the
24732:superspace
24706:Superspace
24580:integrated
24438:, denoted
23969:divergence
23392:: we have
22736:is exact.
21857:filtration
21128:minors of
20509:such that
20430:linear map
20000:dual space
18254:where now
17480:, form an
17222:-vectors,
16615:Hodge dual
14582:dual space
14527:See also:
14463:+ 1) <
14435:such that
13192:See also:
13064:linear map
10473:Note that
10128:and, when
8644:such that
8580:for every
8533:such that
8500:linear map
7887:, that is
6799:Rank of a
6480:direct sum
4996:, denoted
3725:such that
2451:light blue
2343:) is zero.
2333:degenerate
800:direct sum
27769:EM tensor
27605:Dimension
27556:Transpose
27159:Numerical
26922:Transpose
26685:EMS Press
26624:∗
26607:⋀
26572:⋀
26529:⋀
26144:, Dover,
25944:⋀
25895:⋀
25589:The term
25403:axiomatic
25324:∂
25287:⋀
25193:∧
25190:⋯
25187:∧
25182:ℓ
25175:^
25165:∧
25162:⋯
25159:∧
25147:^
25137:∧
25134:⋯
25131:∧
25118:∧
25110:ℓ
25078:ℓ
25061:−
25053:ℓ
25043:∑
24999:∧
24996:⋯
24993:∧
24977:∂
24933:⋀
24927:→
24900:⋀
24891:∂
24740:open sets
24584:manifolds
24512:⋀
24458:
24340:⋀
24224:∧
24218:∧
24144:∧
23951:⋆
23943:⋆
23916:⋆
23512:∧
23509:⋯
23506:∧
23467:∧
23448:∧
23429:∧
23345:⋀
23290:−
23130:∧
23127:⋯
23124:∧
23101:∧
23078:∧
23042:−
22995:∧
22992:⋯
22989:∧
22966:∧
22823:, and a (
22718:→
22712:⊗
22695:−
22685:⋀
22679:→
22658:⋀
22652:→
22632:⋀
22626:→
22590:→
22569:⋀
22563:→
22542:⋀
22536:→
22519:−
22509:⋀
22503:⊗
22497:→
22442:∧
22439:…
22436:∧
22414:⊗
22406:−
22395:∧
22392:…
22389:∧
22376:↦
22363:∧
22360:…
22357:∧
22344:∧
22336:−
22325:∧
22322:…
22319:∧
22301:given by
22271:⋀
22265:⊗
22248:−
22238:⋀
22232:≅
22172:∈
22139:∈
22104:−
22093:…
22090:∧
22077:∧
22069:−
22052:∧
22049:…
22046:∧
22013:≥
21942:⋀
21917:⊆
21904:⊆
21901:⋯
21898:⊆
21885:⊆
21837:→
21826:→
21813:→
21805:→
21758:⋀
21752:⊗
21731:⋀
21707:⨁
21703:≅
21694:⊕
21676:⋀
21633:⋀
21628:⊗
21614:⋀
21609:≅
21600:⊕
21589:⋀
21542:⋀
21537:→
21523:⋀
21518:→
21486:→
21472:⋀
21467:→
21453:⋀
21448:→
21434:⋀
21429:∧
21408:⋀
21402:→
21369:→
21363:→
21357:→
21351:→
21336:Exactness
21274:⋀
21233:⋀
21113:×
21030:⋀
20981:⋀
20975:→
20954:⋀
20922:⋀
20897:⋀
20871:⋀
20820:∧
20817:⋯
20814:∧
20776:∧
20773:⋯
20770:∧
20743:⋀
20707:⋀
20670:⋀
20624:⋀
20612:→
20591:⋀
20547:⋀
20522:⋀
20483:⋀
20478:→
20464:⋀
20445:⋀
20413:→
20282:−
20245:bialgebra
20228:→
20214:⋀
20206:ε
20179:ε
20115:∧
20112:⋯
20109:∧
20093:Δ
20082:β
20079:⊗
20076:α
20054:∧
20051:⋯
20048:∧
20029:β
20026:∧
20023:α
19984:⊗
19962:⊗
19942:⊗
19922:∧
19881:) = 1 ⊗ (
19877:) ∧ (1 ⊗
19846:⋀
19841:⊗
19827:⋀
19781:−
19772:⋀
19766:⊗
19746:⋀
19723:⨁
19719:→
19699:⋀
19690:Δ
19653:⋀
19638:preserved
19586:⋀
19541:σ
19533:∧
19530:⋯
19527:∧
19507:σ
19471:σ
19463:∧
19460:⋯
19457:∧
19443:σ
19412:-shuffles
19363:σ
19355:∧
19352:⋯
19349:∧
19329:σ
19318:⊗
19301:σ
19293:∧
19290:⋯
19287:∧
19273:σ
19259:σ
19253:
19238:−
19217:∈
19214:σ
19210:∑
19188:∑
19168:∧
19165:⋯
19162:∧
19146:Δ
19117:⊗
19108:∧
19093:⊗
19087:−
19081:⊗
19066:∧
19057:⊗
19042:∧
19033:Δ
19020:coalgebra
18995:⋀
18960:⋀
18955:⊗
18941:⋀
18904:⋀
18899:⊆
18878:⋀
18872:≃
18798:∈
18767:⊗
18755:⊗
18737:Δ
18701:⋀
18696:⊗
18682:⋀
18677:→
18663:⋀
18655:Δ
18643:coproduct
18639:coalgebra
18612:⋀
18598:coproduct
18530:⋀
18480:⋀
18474:∈
18444:⟩
18428:⟨
18409:♭
18353:∗
18323:⋀
18310:≃
18301:∗
18280:⋀
18274:∈
18269:♭
18239:⟩
18227:♭
18216:ι
18204:⟨
18198:⟩
18182:∧
18174:⟨
18131:⋀
18125:∈
18078:⋀
18072:∈
18029:−
18019:⋀
18013:∈
17978:∈
17950:⟩
17938:⟨
17921:♭
17876:⟩
17864:♭
17855:ι
17843:⟨
17837:⟩
17821:∧
17815:⟨
17787:∈
17742:⋀
17736:∈
17693:−
17683:⋀
17677:∈
17623:⋀
17579:⋯
17542:∧
17539:⋯
17536:∧
17428:⋀
17415:⟩)
17397:(⟨
17343:⟩
17317:⟨
17286:∧
17283:⋯
17280:∧
17254:∧
17251:⋯
17248:∧
17182:∗
17158:⋀
17108:⋀
17081:∗
16982:minuses.
16922:−
16902:−
16875:⋀
16869:→
16848:⋀
16839:⋆
16836:∘
16833:⋆
16764:⋀
16758:→
16737:⋀
16714:⋆
16674:−
16664:⋀
16658:→
16637:⋀
16628:⋆
16593:∗
16532:σ
16527:α
16523:ι
16519:↦
16516:α
16496:−
16486:⋀
16480:→
16472:∗
16452:⋀
16424:σ
16374:⋀
16343:α
16339:ι
16335:∘
16330:β
16326:ι
16317:β
16314:∧
16311:α
16307:ι
16266:−
16256:⋀
16250:→
16229:⋀
16223:⊗
16215:∗
16195:⋀
16106:α
16102:ι
16098:∘
16093:β
16089:ι
16085:−
16077:β
16073:ι
16069:∘
16064:α
16060:ι
16030:α
16026:ι
16022:∘
16017:α
16013:ι
15970:α
15966:ι
15959:∧
15948:
15934:−
15922:∧
15911:α
15907:ι
15891:∧
15880:α
15876:ι
15847:α
15843:ι
15818:∗
15810:∈
15807:α
15797:For each
15778:∗
15720:α
15709:α
15705:ι
15662:⋀
15573:−
15563:⋀
15557:→
15536:⋀
15525:α
15521:ι
15472:−
15468:Λ
15443:∗
15435:∈
15432:α
15422:and each
15398:For each
15354:⋀
15321:whenever
15298:α
15294:ι
15259:−
15245:…
15213:α
15193:−
15179:…
15139:α
15135:ι
15104:∗
15074:−
15023:∗
15015:×
15012:⋯
15009:×
15004:∗
14996:×
14991:∗
14929:∗
14864:⋀
14858:∈
14828:α
14796:α
14750:−
14740:⋀
14734:→
14713:⋀
14702:α
14698:ι
14661:⋀
14629:∗
14621:∈
14618:α
14566:∗
14440:(1) <
14337:σ
14326:…
14303:σ
14292:η
14274:σ
14263:…
14246:σ
14235:ω
14228:σ
14222:
14193:∈
14190:σ
14186:∑
14156:…
14137:η
14131:˙
14128:∧
14122:ω
14083:When the
14051:σ
14040:…
14023:σ
14012:ω
14005:σ
13999:
13981:∈
13978:σ
13974:∑
13936:…
13914:ω
13908:
13849:η
13846:⊗
13843:ω
13837:
13789:η
13783:˙
13780:∧
13774:ω
13745:η
13742:⊗
13739:ω
13733:
13724:η
13721:∧
13718:ω
13516:∗
13495:⋀
13442:∗
13412:⋀
13387:naturally
13315:→
13248:= 1, 2, 3
13135:∘
13132:ϕ
13103:→
13082:⋀
13073:ϕ
13044:→
12894:⋀
12888:→
12824:…
12752:→
12652:−
12641:⋯
12614:α
12593:∑
12576:−
12565:⋯
12542:α
12538:ι
12506:−
12474:α
12470:ι
12406:−
12395:⋯
12313:⋯
12277:⋯
12235:⋯
12208:^
12205:⊗
12127:⊗
12124:⋯
12121:⊗
12097:⊗
12054:σ
12046:⋯
12026:σ
12000:σ
11992:⋯
11978:σ
11962:σ
11956:
11928:∈
11925:σ
11921:∑
11878:^
11875:⊗
11847:of ranks
11789:⊗
11786:⋯
11783:⊗
11759:⊗
11722:⋯
11676:with the
11464:⋀
11401:˙
11398:∧
11311:⋀
11306:≅
11203:⋀
11198:≅
11071:Assuming
11047:⊗
11038:
11020:^
11017:⊗
10999:∧
10970:^
10967:⊗
10506:
10358:…
10307:⊗
10304:⋯
10301:⊗
10285:
10229:⊗
10226:⋯
10223:⊗
10207:
10142:≠
10102:σ
10094:⊗
10091:⋯
10088:⊗
10074:σ
10063:σ
10057:
10035:∈
10032:σ
10028:∑
10008:⊗
10005:⋯
10002:⊗
9986:
9924:∈
9897:⊗
9894:⋯
9891:⊗
9768:⊗
9647:
9641:≥
9543:
9356:⋀
9288:⋀
9175:⋀
9133:⋀
9092:⋀
9058:⋀
9041:from the
9014:⋀
8989:contains
8966:⋀
8931:⋀
8915:(and use
8864:⋀
8828:⋀
8703:⋀
8629:→
8615:⋀
8518:→
8486:-algebra
8447:⋀
8394:⋀
8353:∧
8311:⋀
8255:α
8252:∧
8249:β
8230:−
8221:β
8218:∧
8215:α
8172:⋀
8166:∈
8163:β
8123:⋀
8117:∈
8114:α
8064:⋀
8017:⋀
7955:⋀
7949:⊂
7928:⋀
7922:∧
7901:⋀
7850:⋀
7844:⊕
7841:⋯
7838:⊕
7817:⋀
7811:⊕
7790:⋀
7784:⊕
7763:⋀
7743:⋀
7645:⏟
7641:α
7638:∧
7635:⋯
7632:∧
7629:α
7595:≠
7585:⏟
7581:α
7578:∧
7575:⋯
7572:∧
7569:α
7533:has rank
7521:α
7498:α
7431:−
7382:∧
7344:∑
7337:α
7314:α
7243:α
7219:α
7182:α
7154:α
7112:…
7070:α
7066:∧
7063:⋯
7060:∧
7040:α
7021:α
6985:α
6949:α
6942:⋯
6924:α
6905:α
6898:α
6866:α
6826:⋀
6820:∈
6817:α
6726:⋀
6647:⋀
6601:⋀
6595:⊕
6592:⋯
6589:⊕
6568:⋀
6562:⊕
6541:⋀
6535:⊕
6514:⋀
6494:⋀
6394:⋀
6270:⋀
6264:
6216:⋀
6042:∧
6039:⋯
6036:∧
5983:⋀
5948:≤
5932:⋯
5903:≤
5864:∧
5861:⋯
5858:∧
5838:∧
5754:…
5688:dimension
5653:∧
5627:∧
5611:α
5461:⋀
5418:⋀
5373:(or
5357:α
5293:α
5262:α
5222:⋀
5216:∈
5213:α
5179:…
5152:∈
5125:∧
5122:⋯
5119:∧
5106:∧
5057:⋀
5041:, is the
5009:⋀
4926:∧
4923:⋯
4920:∧
4907:∧
4858:…
4790:∧
4787:⋯
4784:∧
4771:∧
4733:≠
4720:for some
4663:σ
4635:σ
4629:
4590:∧
4587:⋯
4584:∧
4571:∧
4555:σ
4549:
4529:σ
4521:∧
4518:⋯
4515:∧
4501:σ
4493:∧
4479:σ
4323:…
4288:σ
4256:∧
4247:−
4238:∧
4206:∧
4194:∧
4182:∧
4170:∧
4158:∧
4146:∧
4122:∧
4073:∈
4008:∈
3976:∧
3891:β
3888:⊗
3885:α
3879:β
3876:∧
3873:α
3841:⋀
3821:∧
3762:⋀
3736:∈
3710:⊗
3604:⋀
3468:∧
3453:∧
3402:−
3369:−
3336:−
3226:∧
3218:∧
3009:∧
2968:−
2924:∧
2883:−
2839:∧
2798:−
2764:∧
2463:light red
2339:(i.e., a
2206:∧
2176:∧
2126:∧
2108:−
2090:∧
2031:∧
2005:−
1969:∧
1933:∧
1897:∧
1861:∧
1791:∧
1737:∧
1640:−
1506:formula:
994:∧
982:∧
950:∧
925:functions
916:is zero.
854:universal
666:∧
660:−
651:∧
619:∧
518:∧
499:magnitude
489:to study
410:∧
407:⋯
404:∧
391:∧
296:∧
220:∧
197:∧
99:ellipsoid
32:Wedge sum
27882:Algebras
27635:Manifold
27620:Geodesic
27378:Notation
27205:Category
27144:Subspace
27139:Quotient
27090:Bivector
27004:Bilinear
26946:Matrices
26821:Glossary
26733:Springer
26650:(1993),
26493:(1965),
26411:(1898),
26378:(1888),
26338:(1844),
26321:(1941),
26256:(1964),
26230:(1999),
26205:(1991),
26188:§III.11.
26166:(1989),
25596:4-vector
25591:k-vector
25574:extended
25451:See also
25395:calculus
25340:homology
24716:fermions
24647:pullback
23777:. Then
23583:and the
20252:antipode
20250:With an
20171:, where
19679:one has
19568:= 0 and
18459:for all
17965:for all
17300:⟩
17234:⟨
15694:), then
14896:. Then
14414:shuffles
13693: :
13679: :
12861:The map
10907:. When
9208:Given a
9043:category
8774:quotient
8667:for all
8492:and any
7146:-vector
6885:-vectors
6476:-vectors
3997:for all
3091:bivector
487:geometry
373:vectors
27732:Physics
27566:Related
27329:Physics
27247:Tensors
26816:Outline
26445:3985954
26425:Bibcode
26312:2369379
26232:Algebra
25834:
25814:
25804:is the
25802:
25782:
25366:History
25338:. The
25336:
25316:
25255:
25233:
24878:
24858:
24854:
24834:
24826:
24806:
24774:
24754:
24692:of the
24651:natural
24616:on the
24485:of the
24479:
24440:
24436:
24416:
24402:
24382:
24313:
24293:
24289:
24269:
24265:
24245:
24205:
24185:
23901:and an
23685:4-space
23624:Physics
23593:defined
23488:-blade
23390:
23370:
23330:
23310:
23251:
23231:
22853:points
22825:simplex
22821:
22801:
21330:
21310:
21222:, then
21220:
21200:
21150:
21130:
21060:
21019:
20342:
20256:
19996:
19976:
19869:
19817:
18983:
18931:
18927:
18861:
18857:
18837:
18813:
18787:
18724:
18647:
18641:. The
18635:
18602:
18575:
18555:
18510:
18461:
18389:
18369:
18161:
18112:
18108:
18059:
18055:
18000:
17993:
17967:
17901:is the
17802:
17776:
17772:
17723:
17719:
17664:
17653:
17612:
17506:
17486:
17220:
17200:
17196:
17140:
17095:
17068:
17044:
17024:
17012:
16992:
16820:
16800:
16607:
16580:
16436:
16416:
16179:
16159:
15832:
15799:
15792:
15765:
15741:
15696:
15692:
15648:
15457:
15424:
15420:
15400:
15384:
15343:
15118:
15091:
15053:, ...,
15037:
14978:
14965:
14945:
14894:
14847:
14840:
14820:
14808:
14788:
14684:
14651:
14643:
14610:
14606:
14586:
13669:
13628:
13623:is the
13534:
13484:
13456:
13394:
13286:
13260:
13213:1-forms
12794:, then
12777:, ...,
12689:Duality
12521:
12495:
12491:
12461:
12447:
12427:
11639:, ...,
11613:
11584:
11487:
11454:
11450:
11421:
11385:
11365:
11173:
11142:
10905:
10885:
10857:
10837:
10833:
10800:
10759:
10728:
10724:
10693:
10689:
10660:
10468:
10437:
10379:
10341:
9807:
9787:
9783:
9757:
9662:
9630:
9626:
9606:
9602:
9582:
9558:
9532:
9445:sheaves
9344:
9313:
9276:
9256:
9231:and an
9198:
9165:
9039:functor
8954:
8921:
8726:
8693:
8596:unital
8470:
8437:
8204:, then
8202:
8155:
7306:, then
7304:
7284:
7194:
7174:
7140:of the
6856:
6809:
6803:-vector
6793:
6766:
6762:
6715:
6711:
6691:
6683:
6636:
6464:
6438:
6381:
6361:
6357:vectors
6353:
6333:
6200:
6173:
6140:of the
6130:
6103:
6013:
5972:
5686:If the
5598:
5526:
5522:
5493:
5448:
5407:
5403:
5383:
5349:, then
5347:
5327:
5282:-vector
5254:, then
5252:
5205:
5083:spanned
5080:
5047:
5039:
4998:
4994:
4974:
4748:
4722:
4675:
4655:
4650:is the
4461:
4441:
4437:
4410:
4408:, ...,
4406:
4379:
4375:
4348:
4344:
4306:
4088:
4056:
4052:
4032:
3963:
3943:
3672:of the
3645:over a
3039:where {
2355:) = −A(
1715:
1693:
1452:
1423:
1022:over a
1020:modules
933:scalars
802:of the
731:of the
495:volumes
105:), and
27660:Vector
27655:Spinor
27640:Matrix
27434:Tensor
27100:Tensor
26912:Kernel
26842:Vector
26837:Scalar
26739:
26703:
26658:
26501:
26443:
26417:Nature
26397:
26365:
26310:
26238:
26174:
26148:
26086:), or
26029:
26001:
25764:
25746:-form.
25656:; see
25541:
25424:, and
24746:. An
24738:, the
24291:, and
24045:where
23647:, the
23589:matrix
23585:minors
21975:where
20196:counit
18110:, and
17891:where
17774:, and
17018:(or a
15039:. If
14973:-fold
14454:, and
13763:or as
13458:. If
12677:where
12214:
12199:
11884:
11869:
11828:where
11026:
11011:
9253:module
8758:, the
8679:(here
7621:
7601:
7407:where
6713:, and
6685:, the
6359:, and
6331:where
6142:matrix
6138:minors
5897:
5894:
5884:
5375:simple
4618:where
4346:, and
4224:hence
3580:, and
3568:, and
3550:minors
3498:where
2392:) = A(
929:domain
927:whose
727:. The
718:vector
506:-blade
440:degree
154:is an
27580:Basis
27265:Scope
26969:Minor
26954:Block
26892:Basis
26441:S2CID
26308:JSTOR
25516:Notes
25378:, or
25310:is a
24696:(see
24612:is a
23618:basis
23587:of a
21384:is a
20428:is a
20169:) ∘ Δ
19873:(1 ⊗
17138:with
17066:with
16578:with
15861:is a
14814:, or
14786:with
14085:field
13153:This
13118:with
12722:is a
12714:from
12710:, an
9475:over
9399:be a
9037:is a
8778:ideal
8338:, an
6687:field
5784:basis
5782:is a
5379:blade
4300:is a
3678:ideal
3647:field
2432:) = 1
2329:) = 0
1133:basis
1127:is a
608:that
501:of a
491:areas
483:blade
435:blade
131:of a
27124:Dual
26979:Rank
26737:ISBN
26701:ISBN
26656:ISBN
26514:and
26499:ISBN
26395:ISBN
26363:ISBN
26236:ISBN
26172:ISBN
26146:ISBN
26027:ISBN
25999:ISBN
25928:and
25865:and
25762:ISBN
25738:Note
25724:See
25703:See
25652:and
25539:ISBN
25504:, a
25227:The
25050:<
24784:Let
24718:and
24590:and
24183:and
23773:the
22154:and
22121:for
22002:for
21178:and
21082:and
20857:Let
20376:and
20194:The
19901:) =
19891:and
19494:and
17582:<
17576:<
16990:For
16410:(or
15063:are
14106:has
13688:and
12786:are
12702:and
12352:The
11851:and
11843:and
11662:) ⊂
11658:∈ A(
10405:and
9945:The
9812:Let
9644:char
8806:for
8653:) =
8374:for
8289:Let
8153:and
8053:and
7613:and
7138:rank
7136:The
6449:>
6436:for
5935:<
5929:<
5916:<
5786:for
5730:and
4962:The
3556:and
3544:and
2645:and
2513:and
2478:and
2445:blue
2369:and
2313:and
2295:) =
2272:and
2240:and
1691:and
1518:Area
1500:area
1476:and
1129:real
923:and
557:and
115:− 1)
26433:doi
26300:doi
26052:as
25386:.
24726:or
24702:).
24676:In
24626:An
24408:of
24380:of
23683:in
23643:In
23209:to
21340:If
21308:of
20302:deg
20161:∘ (
19812:not
19636:is
19250:sgn
17484:of
17461:If
17307:det
16798:of
15945:deg
15606:If
15510:),
15386:).
14943:to
14818:by
14499:/ (
14219:sgn
13996:sgn
13905:Alt
13834:Alt
13730:Alt
13603:to
13556:is
13482:to
13385:is
13365:on
13206:of
12718:to
11953:sgn
11832:is
11680:as
11649:of
11140:in
11035:Alt
10915:Alt
10883:is
10586:Alt
10192:Alt
10054:sgn
9540:dim
9423:be
8812:in
8785:in
8691:in
8673:in
8586:in
6807:If
6261:dim
5710:is
5690:of
5203:If
5045:of
4972:of
4968:th
4626:sgn
4546:sgn
3899:mod
2748:is
2457:red
2410:to
1580:det
1532:det
1054:in
1030:in
775:of
461:or
438:of
261:in
185:or
127:or
75:= 0
27878::
26735:.
26731:.
26695:,
26683:,
26677:,
26439:,
26431:,
26421:58
26419:,
26415:,
26383:;
26327:,
26306:,
26294:,
26226:;
26197:;
26102:).
26082:;
25580:).
25523:^
25432:.
25420:,
25362:.
25346:.
24634:.
24558:.
24449:Gr
24315:.
24267:,
23882:0.
23600:×
21332:.
20348:.
20165:⊗
20157:=
20153:∧
19905:⊗
19885:∧
19572:=
19416:k,
19402:,
19120:1.
18512:.
18057:,
17896:∈
17804:,
17721:,
17659:.
17470:,
17406:,
17378:=
16974:—
16968:,
16617::
16038:0.
15834:,
15061:−1
15046:,
14842:.
14686:,
14519:.
14508:×
14479:+
14429:+
14416::
14408:,
14387:⊂
14377:Sh
13697:→
13683:→
13671:.
13227:,
13221:,
12846:0.
12685:.
12459:,
12454:∈
12177:⊗
10470:.
9431:.
8801:⊗
8665:))
8476::
8379:∈
7666:0.
7510:.
7200:.
6887::
6795:.
6466:.
6250::
6202:.
4942:0.
4806:0.
4677:.
4377:,
4090:,
3774::=
3576:,
3564:,
3535:×
3512:∧
3505:∧
3093:.
3074:∧
3067:,
3060:∧
3053:,
3046:∧
2533:,
2526:,
2466:).
2425:,
2418:A(
2396:,
2388:,
2381:+
2377:A(
2359:,
2351:,
2347:A(
2325:,
2321:A(
2303:,
2299:A(
2297:rs
2288:,
2281:A(
2262:,
2258:A(
2224:0.
1717::
849:.
493:,
93:,
27239:e
27232:t
27225:v
26797:e
26790:t
26783:v
26745:.
26629:.
26620:V
26614:k
26584:V
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26559:V
26555:k
26541:V
26536:k
26435::
26427::
26404:.
26359:)
26302::
26296:1
26098:(
26090:(
26078:(
26056:.
26035:.
26007:.
25979:.
25931:W
25925:V
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25822:V
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25770:.
25743:k
25694:.
25681:.
25627:k
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25547:.
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25075:+
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25064:1
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24988:1
24984:x
24980:(
24951:)
24948:L
24945:(
24940:p
24924:)
24921:L
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24910:+
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24894::
24866:K
24842:L
24814:K
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24762:n
24749:n
24619:k
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24530:)
24527:V
24524:(
24519:k
24504:(
24498:P
24467:)
24464:V
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24453:k
24424:V
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24347:k
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24108:.
24103:l
24100:k
24096:F
24090:l
24087:j
24083:g
24077:k
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24070:g
24066:=
24061:j
24058:i
24054:F
24031:j
24028:i
24023:j
24020:;
24016:F
24012:=
24007:j
24004:i
23999:j
23996:,
23992:F
23988:=
23983:i
23979:J
23955:F
23947:d
23939:=
23936:J
23879:=
23874:]
23871:k
23868:;
23865:j
23862:i
23859:[
23855:F
23851:=
23846:]
23843:k
23840:,
23837:j
23834:i
23831:[
23827:F
23806:0
23803:=
23800:A
23797:d
23794:d
23791:=
23788:F
23785:d
23761:,
23756:]
23753:j
23750:;
23747:i
23744:[
23740:A
23736:=
23731:]
23728:j
23725:,
23722:i
23719:[
23715:A
23711:=
23706:j
23703:i
23699:F
23671:A
23668:d
23665:=
23662:F
23608:k
23602:k
23598:k
23520:k
23516:v
23501:1
23497:v
23485:k
23470:v
23462:2
23458:u
23454:+
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23443:1
23439:u
23435:=
23432:v
23426:)
23421:2
23417:u
23413:+
23408:1
23404:u
23400:(
23378:V
23356:)
23353:V
23350:(
23318:k
23296:)
23293:1
23287:k
23284:(
23261:n
23239:n
23217:Q
23197:P
23173:Q
23170:P
23148:k
23144:A
23138:j
23134:A
23119:2
23115:A
23109:j
23105:A
23096:1
23092:A
23086:j
23082:A
23073:0
23069:A
23063:j
23059:A
23053:j
23049:)
23045:1
23039:(
23018:=
23013:k
23009:A
23003:0
22999:A
22984:2
22980:A
22974:0
22970:A
22961:1
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22951:0
22947:A
22926:k
22904:k
22900:A
22896:,
22893:.
22890:.
22887:.
22884:,
22879:1
22875:A
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22866:0
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22835:k
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22786:A
22757:k
22721:0
22715:W
22709:)
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22670:(
22665:k
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22638:k
22623:0
22610:W
22593:0
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22584:W
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22576:k
22560:)
22557:V
22554:(
22549:k
22533:)
22530:W
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22522:1
22516:k
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22481:U
22464:.
22461:)
22456:p
22452:v
22448:(
22445:g
22433:)
22428:1
22424:v
22420:(
22417:g
22409:p
22403:k
22399:u
22384:1
22380:u
22371:p
22367:v
22352:1
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22339:p
22333:k
22329:u
22314:1
22310:u
22289:)
22286:W
22283:(
22278:p
22262:)
22259:U
22256:(
22251:p
22245:k
22227:p
22223:F
22218:/
22212:1
22209:+
22206:p
22202:F
22178:.
22175:V
22167:i
22163:v
22142:U
22134:i
22130:u
22107:1
22101:p
22097:v
22085:1
22081:v
22072:p
22066:1
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22060:k
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22041:1
22037:u
22016:1
22010:p
21988:p
21984:F
21960:)
21957:V
21954:(
21949:k
21936:=
21931:1
21928:+
21925:k
21921:F
21912:k
21908:F
21893:1
21889:F
21880:0
21876:F
21872:=
21869:0
21843:,
21840:0
21834:W
21829:g
21821:V
21816:f
21808:U
21802:0
21779:.
21776:)
21773:W
21770:(
21765:q
21749:)
21746:V
21743:(
21738:p
21723:k
21720:=
21717:q
21714:+
21711:p
21700:)
21697:W
21691:V
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21683:k
21647:.
21644:)
21641:W
21638:(
21625:)
21622:V
21619:(
21606:)
21603:W
21597:V
21594:(
21556:.
21553:)
21550:V
21547:(
21534:)
21531:U
21528:(
21515:0
21489:0
21483:)
21480:W
21477:(
21464:)
21461:V
21458:(
21445:)
21442:V
21439:(
21426:)
21423:U
21420:(
21415:1
21399:0
21372:0
21366:W
21360:V
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21348:0
21318:f
21292:)
21289:V
21286:(
21281:n
21251:)
21248:f
21245:(
21240:n
21208:n
21186:V
21166:W
21163:=
21160:V
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21116:k
21110:k
21090:W
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21048:)
21045:f
21042:(
21037:k
21002:.
20999:)
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20948::
20940:)
20937:V
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20929:k
20912:|
20908:)
20905:f
20902:(
20892:=
20889:)
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20878:k
20842:.
20839:)
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20826:(
20823:f
20811:)
20806:1
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20795:f
20792:=
20789:)
20784:k
20780:x
20765:1
20761:x
20757:(
20754:)
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20748:(
20717:)
20714:f
20711:(
20696:k
20681:)
20678:f
20675:(
20645:.
20642:)
20639:W
20636:(
20631:1
20618:=
20615:W
20609:)
20606:V
20603:(
20598:1
20585:=
20582:V
20579::
20576:f
20573:=
20565:)
20562:V
20559:(
20554:1
20537:|
20533:)
20530:f
20527:(
20494:)
20491:W
20488:(
20475:)
20472:V
20469:(
20459::
20456:)
20453:f
20450:(
20416:W
20410:V
20407::
20404:f
20384:W
20364:V
20330:x
20323:)
20318:2
20314:1
20311:+
20307:x
20295:(
20289:)
20285:1
20279:(
20276:=
20273:)
20270:x
20267:(
20264:S
20231:K
20225:)
20222:V
20219:(
20209::
20167:β
20163:α
20159:ε
20155:β
20151:α
20132:)
20128:)
20123:k
20119:x
20104:1
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20096:(
20089:(
20085:)
20073:(
20070:=
20067:)
20062:k
20058:x
20043:1
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20035:(
20032:)
20020:(
19907:w
19903:v
19899:w
19895:v
19893:(
19889:)
19887:w
19883:v
19879:w
19875:v
19857:)
19854:V
19851:(
19838:)
19835:V
19832:(
19795:)
19792:V
19789:(
19784:p
19778:k
19763:)
19760:V
19757:(
19752:p
19738:k
19733:0
19730:=
19727:p
19716:)
19713:V
19710:(
19705:k
19693::
19667:,
19664:)
19661:V
19658:(
19622:k
19618:x
19597:)
19594:V
19591:(
19574:k
19570:p
19566:p
19550:)
19547:k
19544:(
19537:v
19522:)
19519:1
19516:+
19513:p
19510:(
19503:v
19480:)
19477:p
19474:(
19467:v
19452:)
19449:1
19446:(
19439:v
19428:k
19424:k
19420:k
19410:)
19408:p
19406:−
19404:k
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19398:(
19380:.
19377:)
19372:)
19369:k
19366:(
19359:x
19344:)
19341:1
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19335:p
19332:(
19325:x
19321:(
19315:)
19310:)
19307:p
19304:(
19297:x
19282:)
19279:1
19276:(
19269:x
19265:(
19262:)
19256:(
19244:)
19241:p
19235:k
19232:,
19229:p
19226:(
19223:h
19220:S
19203:k
19198:0
19195:=
19192:p
19184:=
19181:)
19176:k
19172:x
19157:1
19153:x
19149:(
19114:)
19111:w
19105:v
19102:(
19099:+
19096:v
19090:w
19084:w
19078:v
19075:+
19072:)
19069:w
19063:v
19060:(
19054:1
19051:=
19048:)
19045:w
19039:v
19036:(
19006:)
19003:V
19000:(
18971:)
18968:V
18965:(
18952:)
18949:V
18946:(
18915:)
18912:V
18909:(
18896:)
18893:V
18890:(
18885:0
18869:K
18845:K
18823:1
18801:V
18795:v
18770:1
18764:v
18761:+
18758:v
18752:1
18749:=
18746:)
18743:v
18740:(
18712:)
18709:V
18706:(
18693:)
18690:V
18687:(
18674:)
18671:V
18668:(
18658::
18623:)
18620:V
18617:(
18563:V
18541:)
18538:V
18535:(
18498:)
18495:V
18492:(
18487:l
18470:y
18440:y
18436:,
18432:x
18425:=
18422:)
18418:y
18414:(
18404:x
18377:l
18347:)
18341:)
18338:V
18335:(
18330:l
18315:(
18306:)
18297:V
18293:(
18287:l
18264:x
18235:w
18222:x
18212:,
18208:v
18201:=
18194:w
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9110:)
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9025:)
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8971:(
8942:)
8939:V
8936:(
8917:∧
8900:I
8895:/
8890:)
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8881:T
8878:=
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8872:V
8869:(
8839:)
8836:V
8833:(
8815:V
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8803:v
8799:v
8794:)
8792:V
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8768:V
8766:(
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8623:V
8620:(
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8565:=
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8426:V
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8319:V
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8091:.
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7982:.
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7428:=
7423:j
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7340:=
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7143:k
7121:.
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7103:,
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7055:)
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7036:=
7031:)
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6959:)
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6953:(
6945:+
6939:+
6934:)
6931:2
6928:(
6920:+
6915:)
6912:1
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6901:=
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6844:)
6841:V
6838:(
6833:k
6801:k
6779:n
6775:2
6750:V
6747:=
6744:)
6741:V
6738:(
6733:1
6699:V
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6668:=
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6616:V
6613:(
6608:n
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6575:2
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6548:1
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6446:k
6424:}
6421:0
6418:{
6415:=
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6291:=
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5770:}
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5738:{
5718:n
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4865:x
4861:,
4855:,
4850:2
4846:x
4842:,
4837:1
4833:x
4829:{
4803:=
4798:k
4794:x
4779:2
4775:x
4766:1
4762:x
4736:j
4730:i
4706:j
4702:x
4698:=
4693:i
4689:x
4638:)
4632:(
4603:,
4598:k
4594:x
4579:2
4575:x
4566:1
4562:x
4558:)
4552:(
4543:=
4538:)
4535:k
4532:(
4525:x
4510:)
4507:2
4504:(
4497:x
4488:)
4485:1
4482:(
4475:x
4449:V
4423:k
4419:x
4392:2
4388:x
4361:1
4357:x
4332:]
4329:k
4326:,
4320:,
4317:1
4314:[
4265:.
4262:)
4259:x
4253:y
4250:(
4244:=
4241:y
4235:x
4209:x
4203:y
4200:+
4197:y
4191:x
4188:=
4185:y
4179:y
4176:+
4173:x
4167:y
4164:+
4161:y
4155:x
4152:+
4149:x
4143:x
4140:=
4137:)
4134:y
4131:+
4128:x
4125:(
4119:)
4116:y
4113:+
4110:x
4107:(
4104:=
4101:0
4076:V
4070:y
4067:,
4064:x
4040:V
4014:,
4011:V
4005:x
3985:0
3982:=
3979:x
3973:x
3951:V
3910:.
3906:)
3903:I
3896:(
3882:=
3850:)
3847:V
3844:(
3797:.
3794:I
3790:/
3786:)
3783:V
3780:(
3777:T
3771:)
3768:V
3765:(
3739:V
3733:x
3713:x
3707:x
3687:I
3656:K
3633:V
3613:)
3610:V
3607:(
3582:w
3578:v
3574:u
3570:w
3566:v
3562:u
3558:v
3554:u
3546:v
3542:u
3537:v
3533:u
3521:R
3517:3
3514:e
3510:2
3507:e
3503:1
3500:e
3483:)
3478:3
3473:e
3463:2
3458:e
3448:1
3443:e
3438:(
3435:)
3430:1
3426:w
3420:2
3416:v
3410:3
3406:u
3397:3
3393:w
3387:1
3383:v
3377:2
3373:u
3364:2
3360:w
3354:3
3350:v
3344:1
3340:u
3331:2
3327:w
3321:1
3317:v
3311:3
3307:u
3303:+
3298:1
3294:w
3288:3
3284:v
3278:2
3274:u
3270:+
3265:3
3261:w
3255:2
3251:v
3245:1
3241:u
3237:(
3234:=
3230:w
3222:v
3214:u
3187:,
3182:3
3177:e
3170:3
3166:w
3162:+
3157:2
3152:e
3145:2
3141:w
3137:+
3132:1
3127:e
3120:1
3116:w
3112:=
3108:w
3083:R
3079:3
3076:e
3072:2
3069:e
3065:1
3062:e
3058:3
3055:e
3051:2
3048:e
3044:1
3041:e
3024:)
3019:3
3014:e
3004:2
2999:e
2994:(
2991:)
2986:2
2982:v
2976:3
2972:u
2963:3
2959:v
2953:2
2949:u
2945:(
2942:+
2939:)
2934:1
2929:e
2919:3
2914:e
2909:(
2906:)
2901:3
2897:v
2891:1
2887:u
2878:1
2874:v
2868:3
2864:u
2860:(
2857:+
2854:)
2849:2
2844:e
2834:1
2829:e
2824:(
2821:)
2816:1
2812:v
2806:2
2802:u
2793:2
2789:v
2783:1
2779:u
2775:(
2772:=
2768:v
2760:u
2731:3
2726:e
2719:3
2715:v
2711:+
2706:2
2701:e
2694:2
2690:v
2686:+
2681:1
2676:e
2669:1
2665:v
2661:=
2657:v
2628:3
2623:e
2616:3
2612:u
2608:+
2603:2
2598:e
2591:2
2587:u
2583:+
2578:1
2573:e
2566:1
2562:u
2558:=
2554:u
2540:}
2538:3
2535:e
2531:2
2528:e
2524:1
2521:e
2519:{
2507:R
2483:2
2480:e
2476:1
2473:e
2430:2
2427:e
2423:1
2420:e
2412:v
2408:w
2404:r
2400:)
2398:w
2394:v
2390:w
2386:w
2383:r
2379:v
2371:w
2367:v
2363:)
2361:w
2357:v
2353:v
2349:w
2337:v
2327:v
2323:v
2315:s
2311:r
2307:)
2305:w
2301:v
2293:w
2290:s
2286:v
2283:r
2274:w
2270:v
2266:)
2264:w
2260:v
2242:w
2238:v
2221:=
2216:2
2211:e
2201:2
2196:e
2191:=
2186:1
2181:e
2171:1
2166:e
2144:.
2141:)
2136:2
2131:e
2121:1
2116:e
2111:(
2105:=
2100:1
2095:e
2085:2
2080:e
2041:2
2036:e
2026:1
2021:e
2015:)
2011:c
2008:b
2002:d
1999:a
1995:(
1991:=
1979:2
1974:e
1964:2
1959:e
1954:d
1951:b
1948:+
1943:1
1938:e
1928:2
1923:e
1918:c
1915:b
1912:+
1907:2
1902:e
1892:1
1887:e
1882:d
1879:a
1876:+
1871:1
1866:e
1856:1
1851:e
1846:c
1843:a
1840:=
1830:)
1825:2
1820:e
1815:d
1812:+
1807:1
1802:e
1797:c
1794:(
1788:)
1783:2
1778:e
1773:b
1770:+
1765:1
1760:e
1755:a
1752:(
1749:=
1741:w
1733:v
1702:w
1678:v
1654:.
1650:|
1646:c
1643:b
1637:d
1634:a
1630:|
1626:=
1621:|
1614:]
1608:d
1603:b
1596:c
1591:a
1585:[
1575:|
1570:=
1565:|
1558:]
1551:w
1544:v
1537:[
1527:|
1522:=
1485:w
1463:v
1438:2
1433:R
1404:2
1399:e
1394:d
1391:+
1386:1
1381:e
1376:c
1373:=
1368:]
1362:d
1355:c
1349:[
1344:=
1340:w
1335:,
1330:2
1325:e
1320:b
1317:+
1312:1
1307:e
1302:a
1299:=
1294:]
1288:b
1281:a
1275:[
1270:=
1266:v
1239:.
1234:]
1228:1
1221:0
1215:[
1210:=
1205:2
1199:e
1192:,
1187:]
1181:0
1174:1
1168:[
1163:=
1158:1
1152:e
1113:2
1108:R
1062:k
1038:k
1006:,
1003:0
1000:=
997:v
991:w
988:+
985:w
979:v
959:0
956:=
953:v
947:v
904:V
884:V
864:V
833:,
830:V
810:k
786:.
783:V
771:-
759:k
739:k
716:-
714:k
699:k
672:,
669:v
663:w
657:=
654:w
648:v
628:0
625:=
622:v
616:v
588:k
568:,
565:w
545:v
521:w
515:v
504:2
481:-
469:k
449:k
418:k
414:v
399:2
395:v
386:1
382:v
361:k
339:.
336:V
316:V
272:.
269:V
249:v
229:0
226:=
223:v
217:v
169:,
166:V
142:V
113:n
111:(
97:-
95:n
89:-
87:n
83:n
79:n
73:n
68:n
34:.
20:)
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