10843:
10035:
3768:
12137:
8353:
9410:
10536:
6742:
14855:
9659:
9767:
2981:
12967:
12731:
forms ensures that this is possible not just for products, but in more general situations as well. Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.
12730:
states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential
3563:
11874:
3525:
10492:
11751:
5635:
5401:
8090:
of differential forms, which can be viewed as a
Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra. They are studied in
15496:
be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the
9150:
8135:
4820:
10261:
9241:
16545:
12288:. But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. Moreover, it is also possible to define parametrizations of
10838:{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}\sum _{j_{1}<\cdots <j_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f){\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}\,dx^{j_{1}}\wedge \cdots \wedge dx^{j_{k}}.}
13255:
6201:
2267:
7300:
811:
under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the
6551:
16364:
14695:
9537:
6850:
14638:
7477:
8946:
10975:
824:
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to
3262:
4924:
14184:
2684:
9510:
5082:
14973:
12258:
11407:
11863:
10030:{\displaystyle {\begin{aligned}f^{*}(c\omega )&=c(f^{*}\omega ),\\f^{*}(\omega +\eta )&=f^{*}\omega +f^{*}\eta ,\\f^{*}(\omega \wedge \eta )&=f^{*}\omega \wedge f^{*}\eta ,\\f^{*}(d\omega )&=d(f^{*}\omega ).\end{aligned}}}
7880:, the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. This makes it possible to convert vector fields to covector fields and vice versa. It also enables the definition of additional operations such as the
6514:
1094:
for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order
397:
14418:
7862:
2804:
1995:
11148:
correspond to line integrals, surface integrals, volume integrals, and so on. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of
Euclidean space.
7594:
4311:
13986:
12839:
7970:
2112:
15110:
12547:
3763:{\displaystyle {\begin{aligned}(\partial _{\mathbf {v} +\mathbf {w} }f)(p)&=(\partial _{\mathbf {v} }f)(p)+(\partial _{\mathbf {w} }f)(p)\\(\partial _{c\mathbf {v} }f)(p)&=c(\partial _{\mathbf {v} }f)(p)\end{aligned}}}
2794:
12132:{\displaystyle \int _{M}\omega =\int _{D}\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}(\varphi ({\mathbf {u} })){\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\dots ,u^{n})}}\,du^{1}\cdots du^{n},}
7071:
13081:
6891:
As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the
6086:
4165:
13332:
7660:
6289:
16723:
16048:
14472:
14070:
8043:
3398:
7765:
8665:. The existence of pullbacks is one of the key features of the theory of differential forms. It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.
10333:
5936:
11595:
1599:
9729:
5456:
5233:
15642:
11271:
15332:
3019:, and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an
12628:. Here, the form has a well-defined Riemann or Lebesgue integral as before. The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of
11040:
1085:
15225:
12734:
Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let
8348:{\displaystyle 0\ \to \ \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\ \to \ \cdots \ \to \ \Omega ^{n}(M)\ \to \ 0.}
1214:
9405:{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ {\textstyle \bigwedge }^{k}T^{*}N\ {\stackrel {{\bigwedge }^{k}(df)^{*}}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}T^{*}M.}
9021:
15389:
5176:
4706:
4500:
1703:
10124:
2335:
15836:-dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare the
6994:. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).
4606:
1428:
16470:
13406:
12844:
12422:
9772:
3568:
2496:
16465:
13667:
7007:
is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product
13530:
11432:. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity.
7118:
1309:
15564:
16233:
12667:-dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before. For example, given a path
6737:{\displaystyle \operatorname {Alt} (\tau _{p})(x_{1},\dots ,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )\tau _{p}(x_{\sigma (1)},\dots ,x_{\sigma (k)}),}
587:
7160:
6106:
2128:
7214:
7186:
139:
14850:{\displaystyle (\beta _{\mathbf {v} })_{x}=\left(\alpha _{x}\wedge f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\right){\big /}\zeta _{y}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M.}
11509:
985:
16606:
13143:
9654:{\displaystyle {\textstyle \bigwedge }^{k}TM\ {\stackrel {{\bigwedge }^{k}df}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} }
7206:
2399:
9215:
16252:
11141:, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of
762:
11078:
6774:
14546:
7346:
625:
509:
8793:
16409:. However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the
10851:
3154:
16440:
16120:
16091:
4835:
696:
14085:
2582:
1809:, in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions,
9442:
6768:
elements. The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding
4970:
662:
14890:
12148:
11308:
16568:
11768:
2976:{\displaystyle d\tau =\sum _{I\in {\mathcal {J}}_{k,n}}\left(\sum _{j=1}^{n}{\frac {\partial a_{I}}{\partial x^{j}}}\,dx^{j}\right)\wedge dx^{I}\in \Omega ^{k+1}(M)}
8071:-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.
6537:
6440:
15758:-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the
240:
14338:
12962:{\displaystyle {\begin{aligned}\omega _{x}&\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,\\\eta _{y}&\in {\textstyle \bigwedge }^{n}T_{y}^{*}N,\end{aligned}}}
7780:
1872:
7534:
4197:
13893:
7886:
15044:
2003:
12457:
15795:
on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate
5740:
Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any
16776:
2697:
7023:
17:
12993:
5767:-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.
4079:
6026:
990:
which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:
7600:
6219:
1798:
in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that
18155:
16676:
16410:
3520:{\displaystyle {\frac {\partial f}{\partial x^{j}}}=\sum _{i=1}^{n}{\frac {\partial y^{i}}{\partial x^{j}}}{\frac {\partial f}{\partial y^{i}}}.}
3011:
The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in
15969:
17346:
14429:
16648:, in which the Lie group is not abelian. In that case, one gets relations which are similar to those described here. The analog of the field
13997:
10487:{\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f)\,df_{i_{1}}\wedge \cdots \wedge df_{i_{k}}.}
7978:
13266:
7707:
1346:
at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general
12342:. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a
11746:{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots ,i_{n}}({\mathbf {x} })\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{n}}.}
3008:, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.
18150:
5630:{\displaystyle \sum _{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}}
5396:{\displaystyle d\alpha =\sum _{j=1}^{n}df_{j}\wedge dx^{j}=\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}.}
11412:
Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say,
6434:
The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping
5876:
3027:
of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on
14189:
It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let
8078:, where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the
1513:
17437:
9683:
15569:
895:. An example of a 1-dimensional manifold is an interval , and intervals can be given an orientation: they are positively oriented if
17461:
11205:
2500:
This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.
15287:
1794:. The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the
18567:
17656:
16739:, but this does not hold in general. Likewise the field equations are modified by additional terms involving exterior products of
9145:{\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ T^{*}N\ {\stackrel {(df)^{*}}{\longrightarrow }}\ T^{*}M.}
6865:
6428:
996:
15175:
10980:
7017:
is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is
4815:{\displaystyle {\frac {\partial ^{2}f}{\partial x^{i}\,\partial x^{j}}}={\frac {\partial ^{2}f}{\partial x^{j}\,\partial x^{i}}},}
10256:{\displaystyle \omega =\sum _{i_{1}<\cdots <i_{k}}\omega _{i_{1}\cdots i_{k}}\,dy^{i_{1}}\wedge \cdots \wedge dy^{i_{k}},}
15337:
5097:
4420:
18716:
1627:
17526:
17199:
17062:
17041:
16941:
2274:
1161:
17752:
16540:{\displaystyle {\begin{aligned}d{\textbf {F}}&={\textbf {0}}\\d{\star {\textbf {F}}}&={\textbf {J}},\end{aligned}}}
4543:
1356:
881:-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential
839:
Die
Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics)
17805:
17333:
16792:
12360:
15779:
On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate
13614:
3875:, and returns a real-valued function whose value at each point is the derivative along the vector field of the function
2429:
18751:
18430:
18089:
13474:
7079:
15943:-forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains.
18210:
17220:
17168:
17145:
17124:
17085:
17069:
provides a brief discussion of integration on manifolds from the point of view of measure theory in the last section.
16988:
16890:
11290:. Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of
3020:
15521:
13355:
8050:
1242:
885:-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.
18632:
17854:
16198:
12558:. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly
8438:
808:
17837:
17446:
16802:
15791:-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no
15456:
12264:
10527:
3032:
765:
8639:
may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential
6196:{\displaystyle {\textstyle \bigwedge }^{k}T_{p}^{*}M\cong {\Big (}{\textstyle \bigwedge }^{k}T_{p}M{\Big )}^{*}}
2262:{\displaystyle {\mathcal {J}}_{k,n}:=\{I=(i_{1},\ldots ,i_{k}):1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n\}}
18859:
16764:
15960:
7295:{\displaystyle \alpha \wedge \beta ={\frac {(k+\ell )!}{k!\ell !}}\operatorname {Alt} (\alpha \otimes \beta ).}
3894:
is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential
12552:
This approach to defining integration does not assign a direct meaning to integration over the whole manifold
539:
18483:
18415:
18049:
17456:
16807:
15507:
over the interval . Assuming the usual distance (and thus measure) on the real line, this integral is either
13250:{\displaystyle \omega _{x}=(f^{*}\eta _{y})_{x}\wedge \sigma '_{x}\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,}
7123:
7165:
90:
18508:
18034:
17757:
17531:
16824:
16812:
16359:{\displaystyle {\textbf {J}}={\frac {1}{6}}j^{a}\,\varepsilon _{abcd}\,dx^{b}\wedge dx^{c}\wedge dx^{d}\,,}
11482:
6406:
2116:
Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length
939:
16584:
15889:-dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over
7191:
3374:. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates
2367:
807:. The algebra of differential forms along with the exterior derivative defined on it is preserved by the
18854:
18746:
18079:
7771:
1315:. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see
813:
9174:
6845:{\displaystyle \operatorname {Alt} \colon {\textstyle \bigwedge }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}
18557:
18377:
18084:
18054:
17762:
17718:
17699:
17466:
17410:
15916:
14633:{\displaystyle f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\in {\textstyle \bigwedge }^{n-k}T_{x}^{*}M.}
13699:-form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber
7877:
7472:{\displaystyle (\alpha \wedge \beta )_{p}(v,w)=\alpha _{p}(v)\beta _{p}(w)-\alpha _{p}(w)\beta _{p}(v)}
4629:
The second idea leading to differential forms arises from the following question: given a differential
3825:
2519:
1220:
781:
699:
15230:
A key consequence of this is that "the integral of a closed form over homologous chains is equal": If
8941:{\displaystyle (f^{*}\omega )_{p}(v_{1},\ldots ,v_{k})=\omega _{f(p)}(f_{*}v_{1},\ldots ,f_{*}v_{k}).}
18229:
17621:
17486:
16797:
15166:
12646:-forms, each of which is supported in a single positively oriented chart, and define the integral of
11045:
10970:{\textstyle {\frac {\partial (f_{i_{1}},\ldots ,f_{i_{k}})}{\partial (x^{j_{1}},\ldots ,x^{j_{k}})}}}
8397:
7337:
1219:
These conventions correspond to interpreting the integrand as a differential form, integrated over a
628:
18711:
3257:{\displaystyle (\partial _{\mathbf {v} }f)(p)=\left.{\frac {d}{dt}}f(p+t\mathbf {v} )\right|_{t=0}.}
592:
473:
18813:
18731:
18685:
18392:
18006:
17871:
17563:
17405:
15785:-forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold,
15009:. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.
12722:
12559:
5711:
Differential forms can be multiplied together using the exterior product, and for any differential
4919:{\displaystyle {\frac {\partial f_{j}}{\partial x^{i}}}-{\frac {\partial f_{i}}{\partial x^{j}}}=0}
4068:
709:
15951:
Differential forms arise in some important physical contexts. For example, in
Maxwell's theory of
14179:{\displaystyle \int _{M}\omega =\int _{N}{\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}
2679:{\displaystyle d\omega =\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}\wedge dx^{I}.}
18783:
18470:
18387:
18357:
17703:
17673:
17597:
17587:
17543:
17373:
17326:
17158:
15769:
9505:{\displaystyle {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R} ,}
8098:
Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of
5077:{\displaystyle \sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}=0,}
3929:
796:
16416:
16096:
16067:
14968:{\displaystyle \langle \gamma _{y},\mathbf {v} \rangle =\int _{f^{-1}(y)}\beta _{\mathbf {v} },}
12616:
is supported on a single positively oriented chart. On this chart, it may be pulled back to an
12253:{\displaystyle {\frac {\partial (x^{i_{1}},\ldots ,x^{i_{n}})}{\partial (u^{1},\ldots ,u^{n})}}}
11402:{\displaystyle \int _{U}\omega \ {\stackrel {\text{def}}{=}}\int _{U}f(x)\,dx^{1}\cdots dx^{n}.}
1322:
Making the notion of an oriented density precise, and thus of a differential form, involves the
675:
18741:
18597:
18552:
18044:
17663:
17558:
17471:
17378:
16752:
16654:
in such theories is the curvature form of the connection, which is represented in a gauge by a
16452:
15471:
11858:{\displaystyle \varphi ({\mathbf {u} })=(x^{1}({\mathbf {u} }),\ldots ,x^{I}({\mathbf {u} })).}
10321:. Using the linearity of pullback and its compatibility with exterior product, the pullback of
8579:
has two or more preimages, then the vector field may determine two or more distinct vectors in
6912:
of a differential form with respect to a vector field on a manifold with a defined connection.
5798:
3106:
892:
870:
850:
463:
15802:
6509:{\displaystyle \operatorname {Alt} \colon {\bigotimes }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.}
647:
18823:
18778:
18258:
18203:
17693:
17688:
17256:
17104:
16645:
16609:
15921:
15483:
15126:
The fundamental relationship between the exterior derivative and integration is given by the
8633:
By contrast, it is always possible to pull back a differential form. A differential form on
8099:
3016:
3012:
1841:
392:{\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).}
226:
16915:"Linear algebra – "Natural" pairings between exterior powers of a vector space and its dual"
16553:
16375:
are the four components of the current density. (Here it is a matter of convention to write
14413:{\displaystyle {\textstyle \bigwedge }^{k}T_{y}N\to {\textstyle \bigwedge }^{n-k}T_{y}^{*}N}
12634:
is independent of the chosen chart. In the general case, use a partition of unity to write
7857:{\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta .}
1990:{\displaystyle I=(i_{1},i_{2},\ldots ,i_{k}),1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}
18798:
18726:
18612:
18478:
18440:
18372:
18024:
17962:
17810:
17514:
17504:
17476:
17451:
17361:
17288:
17014:
16965:
15808:
Even in the presence of an orientation, there is in general no meaningful way to integrate
15676:
14788:
6909:
6862:
6522:
1861:
15156:
11476:
by an open subset of
Euclidean space. That is, assume that there exists a diffeomorphism
7589:{\displaystyle \alpha \wedge (\beta +\gamma )=\alpha \wedge \beta +\alpha \wedge \gamma ,}
4306:{\displaystyle df_{p}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}(p)(dx^{i})_{p}.}
8:
18675:
18498:
18488:
18337:
18322:
18278:
18162:
17844:
17722:
17707:
17636:
17395:
15460:
15121:
13981:{\displaystyle y\mapsto {\bigg (}\int _{f^{-1}(y)}\omega /\eta _{y}{\bigg )}\,\eta _{y}.}
12727:
8407:
7881:
7873:
6897:
5991:
5190:
2993:
2509:
1845:
816:
for integration becomes a simple statement that an integral is preserved under pullback.
635:
533:
18135:
17292:
17240:
16969:
7965:{\displaystyle \star \colon \Omega ^{k}(M)\ {\stackrel {\sim }{\to }}\ \Omega ^{n-k}(M)}
2107:{\textstyle dx^{I}:=dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}=\bigwedge _{i\in I}dx^{i}}
18808:
18665:
18518:
18332:
18268:
18104:
18059:
17956:
17827:
17631:
17319:
17278:
17266:
17208:
16955:
16850:
16829:
15467:
15127:
15105:{\displaystyle \alpha ^{\flat }\wedge \lambda =(\alpha \wedge f^{*}\lambda )^{\flat }.}
8363:
4183:
The meaning of this expression is given by evaluating both sides at an arbitrary point
3911:
3330:
3036:
3001:
1720:
is to be regarded as having the opposite orientation as the square whose first side is
777:
773:
769:
17641:
16574:
operator. Similar considerations describe the geometry of gauge theories in general.
12542:{\displaystyle \int _{c}\omega =\sum _{i=1}^{r}m_{i}\int _{D}\varphi _{i}^{*}\omega .}
7973:
18803:
18547:
18362:
18273:
18253:
18039:
18019:
18014:
17921:
17646:
17626:
17481:
17237:
17216:
17195:
17164:
17141:
17120:
17113:
17081:
17058:
17037:
17021:
16984:
16937:
16896:
16886:
16780:
16456:
15874:
15858:
vectors, is always positive, corresponding to a squared number. An orientation of a
15759:
8417:, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the
8092:
7674:
6385:
6339:
3314:
1350:-form is a linear combination of these differentials at every point on the manifold:
834:
17107:
introduces the exterior algebra of differential forms at the college calculus level.
17002:
8373:
1493:-forms. This may be thought of as an infinitesimal oriented square parallel to the
18818:
18493:
18460:
18445:
18327:
18196:
18177:
17971:
17926:
17849:
17820:
17678:
17611:
17606:
17601:
17591:
17383:
17366:
17187:
17029:
16862:
16819:
16175:
15952:
15837:
15722:
15452:
8418:
8087:
8075:
6901:
6893:
6207:
5812:
5808:
5776:
5752:
4961:
3024:
2789:{\textstyle \tau =\sum _{I\in {\mathcal {J}}_{k,n}}a_{I}\,dx^{I}\in \Omega ^{k}(M)}
1607:
1323:
830:
804:
407:
18788:
18736:
18680:
18660:
18562:
18450:
18317:
18288:
18120:
18029:
17859:
17815:
17581:
17299:
17094:
17073:
17010:
16768:
16183:
15762:
of the manifold, . Formally, in the presence of an orientation, one may identify
8126:
7066:{\displaystyle \alpha \wedge \beta =\operatorname {Alt} (\alpha \otimes \beta ).}
6759:
6210:
5788:
5741:
5673:-forms, makes it possible to restrict the sum to those sets of indices for which
3548:
3066:
13076:{\displaystyle \sigma _{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}(f^{-1}(y))}
12568:
may be defined to be the integral over the chain determined by a triangulation.
5848:
The definition of a differential form may be restated as follows. At any point
18828:
18793:
18690:
18523:
18513:
18503:
18425:
18397:
18382:
18367:
18283:
17986:
17911:
17881:
17779:
17772:
17712:
17683:
17553:
17548:
17509:
17154:
17050:
16179:
16166:
9223:
th exterior power of the dual map to the differential. Then the pullback of a
8377:
6905:
6410:
6081:{\textstyle \beta _{p}\colon {\textstyle \bigwedge }^{k}T_{p}M\to \mathbf {R} }
6008:
4160:{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}.}
1611:
1224:
1091:
451:
18773:
17191:
17033:
15864:-submanifold is therefore extra data not derivable from the ambient manifold.
15698:, which further reflects the issue of orientation. For example, under the map
826:
47:
18848:
18765:
18670:
18582:
18455:
18172:
17996:
17991:
17976:
17966:
17916:
17893:
17767:
17727:
17668:
17616:
17415:
16900:
16641:
16637:
15895:-dimensional subsets, providing a measure-theoretic analog to integration of
9434:
8079:
6367:
5956:
4941:
2518:. The exterior derivative of a differential form is a generalization of the
1795:
1618:
889:
16763:
Numerous minimality results for complex analytic manifolds are based on the
15824:
because there is no consistent way to use the ambient orientation to orient
7655:{\displaystyle \alpha \wedge (f\cdot \beta )=f\cdot (\alpha \wedge \beta ).}
6284:{\displaystyle \beta _{p}\colon \bigoplus _{n=1}^{k}T_{p}M\to \mathbf {R} .}
18833:
18637:
18622:
18587:
18435:
18420:
18099:
18094:
17936:
17903:
17876:
17784:
17425:
17304:
17179:
17133:
16976:
16718:{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}
16625:
15492:
differentiable manifold (without additional structure), differential forms
12610:-dimensional manifold is defined by working in charts. Suppose first that
11567:. In coordinates, this has the following expression. Fix an embedding of
8103:
6868:
5834:
3274:
792:
50:. It has many applications, especially in geometry, topology and physics.
16043:{\displaystyle {\textbf {F}}={\frac {1}{2}}f_{ab}\,dx^{a}\wedge dx^{b}\,,}
1621:
from vector calculus, in that it is an alternating product. For instance,
18721:
18695:
18617:
18306:
18245:
17942:
17931:
17888:
17789:
17390:
17305:
Visual differential geometry and forms: a mathematical drama in five acts
16655:
15792:
14467:{\displaystyle \mathbf {v} \mapsto \mathbf {v} \,\lrcorner \,\zeta _{y},}
12310:-forms. To make this precise, it is convenient to fix a standard domain
6431:, covector fields, or "dual vector fields", particularly within physics.
4659:? The above expansion reduces this question to the search for a function
525:
31:
14065:{\displaystyle {\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta .}
8038:{\displaystyle \delta \colon \Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)}
18602:
18167:
18125:
17951:
17864:
17496:
17400:
17311:
16960:
16867:
16571:
16186:
for the principal bundle is the vector potential, typically denoted by
13327:{\displaystyle \sigma '_{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M}
12808:
looks like the projection from a product onto one of its factors. Fix
12652:
to be the sum of the integrals of each term in the partition of unity.
10040:
The pullback of a form can also be written in coordinates. Assume that
8359:
7760:{\displaystyle \alpha \wedge \beta =(-1)^{k\ell }\beta \wedge \alpha .}
5986:
3821:
3794:
3039:, which is a central result in the theory of integration on manifolds.
3005:
2576:-form defined by taking the differential of the coefficient functions:
1509:-form is a linear combination of these at every point on the manifold:
854:
13823:
that is almost everywhere positive with respect to the orientation of
8618:
does not determine any tangent vector at all. Since a vector field on
7305:
This description is useful for explicit computations. For example, if
6206:
By the universal property of exterior powers, this is equivalently an
18577:
18528:
17981:
17946:
17651:
17538:
17283:
17245:
16629:
15448:
8624:
determines, by definition, a unique tangent vector at every point of
4188:
3530:
The first idea leading to differential forms is the observation that
39:
9427:
as a linear functional on tangent spaces. From this point of view,
5931:{\displaystyle \beta _{p}\in {\textstyle \bigwedge }^{k}T_{p}^{*}M,}
18607:
18592:
18145:
18140:
18130:
17521:
17342:
16914:
16851:"Sur certaines expressions différentielles et le problème de Pfaff"
3052:
3028:
2997:
1594:{\textstyle \sum _{1\leq i<j\leq n}f_{i,j}\,dx^{i}\wedge dx^{j}}
1343:
75:
43:
15772:; densities in turn define a measure, and thus can be integrated (
9724:{\textstyle {\textstyle \bigwedge }^{k}TM\to M\times \mathbf {R} }
8951:
There are several more abstract ways to view this definition. If
6997:
The antisymmetry inherent in the exterior algebra means that when
4964:, so that these equations can be combined into a single condition
18301:
18263:
17097:(1965), "Chapter 6: Exterior algebra and differential calculus",
17028:, Modern Birkhäuser Classics, Boston, Basel, Berlin: Birkhäuser,
15451:, hence homologous (a weaker condition). This case is called the
15801:-forms. One can instead identify densities with top-dimensional
15637:{\displaystyle \textstyle {\int _{1}^{0}dx=-\int _{0}^{1}dx=-1}}
13710:
is orientable. In particular, a choice of orientation forms on
10076:, and that these coordinate systems are related by the formulas
8982:. Using to denote a dual map, the dual to the differential of
1466:-form is integrated along an oriented curve as a line integral.
18627:
18219:
17737:
11266:{\displaystyle \omega =f(x)\,dx^{1}\wedge \cdots \wedge dx^{n}}
9415:
Another abstract way to view the pullback comes from viewing a
3895:
68:
15327:{\displaystyle \textstyle {\int _{M}\omega =\int _{N}\omega }}
6908:
of a differential form with respect to a vector field and the
5450:-forms, which can be expressed in terms of the coordinates as
1489:
can be used as a basis at every point on the manifold for all
933:
over the interval (with its natural positive orientation) is
17078:
Differential forms with applications to the physical sciences
11035:{\textstyle {\frac {\partial f_{i_{m}}}{\partial x^{j_{n}}}}}
8967:, then it may be viewed as a section of the cotangent bundle
4696:, such a function does not always exist: any smooth function
1606:
A fundamental operation defined on differential forms is the
1080:{\displaystyle \int _{b}^{a}f(x)\,dx=-\int _{a}^{b}f(x)\,dx.}
454:
that can be integrated over a region of space. In general, a
15220:{\displaystyle \int _{M}d\omega =\int _{\partial M}\omega .}
12685:-form on the path is simply pulling back the form to a form
12580:), which does directly assign a meaning to integration over
12445:
is defined to be the sum of the integrals over the terms of
9743:
Pullback respects all of the basic operations on forms. If
6988:
are elements of an exterior power of the cotangent space at
1130:-form can be integrated over an oriented surface, etc.) If
1118:-form is an oriented density that can be integrated over an
46:. The modern notion of differential forms was pioneered by
16758:
16621:
16171:
15907:-dimensional Hausdorff measure yields a density, as above.
15669:). Similarly, under a change of coordinates a differential
8630:, the pushforward of a vector field does not always exist.
3193:
1840:
are equal, in the same way that the "volume" enclosed by a
18188:
17235:
15937:
is the dual space to an appropriate space of differential
15404:
is the derivative of a potential function on the plane or
13731:
The analog of Fubini's theorem is as follows. As before,
12773:
is a surjective submersion. This implies that each fiber
8406:. Therefore, the complex is a resolution of the constant
8118:
One important property of the exterior derivative is that
6883:
are in one-to-one correspondence with such tensor fields.
1227:, by contrast, one interprets the integrand as a function
888:
Integration of differential forms is well-defined only on
803:-forms is extended to arbitrary differential forms by the
16448:; i.e., the same name is used for different quantities.)
15384:{\displaystyle \textstyle \int _{W}d\omega =\int _{W}0=0}
5442:-forms are special cases of differential forms. For each
5413:
is a necessary condition for the existence of a function
5171:{\displaystyle dx^{i}\wedge dx^{j}=-dx^{j}\wedge dx^{i}.}
4495:{\displaystyle \alpha _{p}=\sum _{i}g_{i}(p)(dh_{i})_{p}}
3871:
can be viewed as a function that takes a vector field on
16775:. The Wirtinger inequality is also a key ingredient in
14643:
Each of these covectors has an exterior product against
12571:
10977:
denotes the determinant of the matrix whose entries are
3267:(This notion can be extended pointwise to the case that
1698:{\displaystyle dx^{1}\wedge dx^{2}=-dx^{2}\wedge dx^{1}}
17055:
Real
Analysis: Modern Techniques and Their Applications
15428:
does not depend on the choice of path (the integral is
13535:
is a smooth section of the projection map; we say that
12702:
on , and this integral is the integral of the function
8563:. However, the same is not true of a vector field. If
8106:
and is a noncommutative ("quantum") deformation of the
2508:
In addition to the exterior product, there is also the
17009:, vol. 3, New York-London: Academic Press, Inc.,
15573:
15525:
15341:
15291:
14808:
14591:
14374:
14344:
13288:
13214:
13012:
12922:
12866:
12586:, but this approach requires fixing an orientation of
12282:-manifold cannot be parametrized by an open subset of
10983:
10854:
9689:
9686:
9670:, and therefore it factors through the trivial bundle
9600:
9543:
9448:
9374:
9297:
9177:
7085:
6786:
6154:
6112:
6045:
6029:
5895:
3989:
provides a way of encoding the partial derivatives of
2700:
2432:
2330:{\displaystyle \{dx^{I}\}_{I\in {\mathcal {J}}_{k,n}}}
2006:
1856:
A common notation for the wedge product of elementary
1516:
1462:
are functions of all the coordinates. A differential
1245:
1209:{\displaystyle \int _{M}\omega =-\int _{M'}\omega \,.}
829:
with reference to his 1899 paper. Some aspects of the
42:
over curves, surfaces, solids, and higher-dimensional
16679:
16587:
16556:
16468:
16419:
16255:
16201:
16099:
16070:
15972:
15852:-dimensional space, which, unlike the determinant of
15572:
15524:
15340:
15290:
15178:
15047:
14893:
14698:
14549:
14432:
14341:
14088:
14000:
13896:
13617:
13477:
13358:
13269:
13146:
12996:
12842:
12460:
12363:
12151:
11877:
11771:
11598:
11485:
11311:
11208:
11048:
10539:
10336:
10127:
9770:
9664:
defines a linear functional on each tangent space of
9540:
9445:
9244:
9024:
8796:
8653:
defines a linear functional on each tangent space of
8138:
7981:
7889:
7783:
7710:
7603:
7537:
7349:
7217:
7194:
7168:
7126:
7082:
7026:
6777:
6554:
6525:
6443:
6222:
6109:
5879:
5459:
5236:
5100:
4973:
4838:
4709:
4546:
4423:
4200:
4082:
3993:. It can be decoded by noticing that the coordinates
3566:
3401:
3157:
3031:. It also allows for a natural generalization of the
2807:
2585:
2370:
2277:
2131:
1875:
1630:
1359:
1164:
999:
942:
712:
678:
650:
638:
is an operation on differential forms that, given a
595:
542:
476:
243:
93:
14287:, and the value of that pairing is an integral over
12562:
in an essentially unique way, and the integral over
12304:, and this makes it possible to define integrals of
11470:. First, assume that there is a parametrization of
8358:
This complex is called the de Rham complex, and its
8125:. This means that the exterior derivative defines a
4601:{\displaystyle \alpha =\sum _{i=1}^{n}f_{i}\,dx^{i}}
1732:. This is why we only need to sum over expressions
1423:{\displaystyle f_{1}\,dx^{1}+\cdots +f_{n}\,dx^{n},}
16855:
16413:, the magnetic polarization vector has been called
12417:{\displaystyle c=\sum _{i=1}^{r}m_{i}\varphi _{i},}
11868:Then the integral may be written in coordinates as
11181:the restriction of that orientation. Every smooth
2491:{\textstyle |{\mathcal {J}}_{k,n}|={\binom {n}{k}}}
1603:and it is integrated just like a surface integral.
1147:is the same manifold with opposite orientation and
17112:
16717:
16600:
16562:
16539:
16434:
16358:
16227:
16114:
16085:
16042:
15636:
15558:
15447:), since different paths with given endpoints are
15383:
15326:
15219:
15104:
14967:
14849:
14632:
14466:
14412:
14178:
14064:
13980:
13662:{\displaystyle \sigma _{x}=\omega _{x}/\eta _{y}.}
13661:
13524:
13400:
13326:
13249:
13075:
12961:
12541:
12416:
12252:
12131:
11857:
11745:
11503:
11401:
11265:
11072:
11034:
10969:
10837:
10486:
10255:
10029:
9723:
9653:
9504:
9404:
9209:
9144:
8940:
8347:
8037:
7964:
7856:
7759:
7654:
7588:
7471:
7294:
7200:
7180:
7154:
7112:
7065:
6844:
6736:
6531:
6508:
6283:
6195:
6080:
5930:
5629:
5395:
5170:
5076:
4918:
4814:
4600:
4494:
4305:
4159:
3762:
3519:
3256:
2975:
2788:
2678:
2490:
2393:
2329:
2261:
2106:
1989:
1697:
1593:
1422:
1303:
1208:
1126:-form can be integrated over an oriented curve, a
1079:
979:
756:
690:
656:
619:
581:
503:
391:
133:
14164:
14117:
14050:
14003:
13959:
13905:
13769:is a surjective submersion. Fix orientations of
13525:{\displaystyle \omega \colon f^{-1}(y)\to T^{*}M}
11152:
7113:{\displaystyle {\textstyle \bigwedge }^{n}T^{*}M}
6182:
6147:
1122:-dimensional oriented manifold. (For example, a
780:as special cases of a single general result, the
517:-dimensional manifold, the top-dimensional form (
458:-form is an object that may be integrated over a
18846:
17026:Advanced Calculus; A Differential Forms Approach
16777:Gromov's inequality for complex projective space
16192:, when represented in some gauge. One then has
15883:(integer or real), which may be integrated over
15466:This theorem also underlies the duality between
13743:are two orientable manifolds of pure dimensions
8113:
1239:, without any notion of orientation; one writes
16728:In the abelian case, such as electromagnetism,
16411:International Union of Pure and Applied Physics
12747:be two orientable manifolds of pure dimensions
6904:of a differential form and a vector field, the
2522:, in the sense that the exterior derivative of
2409:-dimensional manifold, and in general space of
2271:Then locally (wherever the coordinates apply),
1304:{\textstyle \int _{A}f\,d\mu =\int _{}f\,d\mu }
27:Expression that may be integrated over a region
17258:Manifolds and differential forms lecture notes
15559:{\displaystyle \textstyle {\int _{0}^{1}dx=1}}
13401:{\displaystyle \sigma _{x}=j^{*}\sigma '_{x}.}
13086:which may be thought of as the fibral part of
8599:is not surjective, then there will be a point
5669:. Antisymmetry, which was already present for
4952:suggests introducing an antisymmetric product
4187:: on the right hand side, the sum is defined "
1326:. The differentials of a set of coordinates,
860:
18204:
17327:
16880:
16228:{\displaystyle {\textbf {F}}=d{\textbf {A}}.}
15748:In the presence of the additional data of an
9155:This is a section of the cotangent bundle of
8530:, there is a well-defined pushforward vector
7677:that depends on the degrees of the forms: if
2482:
2469:
1111:is negative in the direction of integration.
78:over an interval contained in the domain of
16442:for several decades, and by some publishers
16397:, i.e. to use capital letters, and to write
15661:(i.e. the integral of the constant function
14915:
14894:
12796:-dimensional and that, around each point of
8074:Firstly, each (co)tangent space generates a
6306:-tuple of tangent vectors to the same point
3975:, which are just the partial derivatives of
2295:
2278:
2256:
2155:
799:, and the pairing between vector fields and
16178:on which both electromagnetism and general
16064:are formed from the electromagnetic fields
15867:On a Riemannian manifold, one may define a
15459:. This path independence is very useful in
12716:
1316:
18211:
18197:
17334:
17320:
17275:A Geometric Approach to Differential Forms
17163:, Menlo Park, California: W. A. Benjamin,
16952:A Geometric Approach to Differential Forms
16934:A Geometric Approach to Differential Forms
15946:
11435:
8062:
7673:), meaning that it satisfies a variant of
3865:. Extended over the whole set, the object
2503:
849:Differential forms provide an approach to
17282:
17080:, Mineola, New York: Dover Publications,
17001:
16959:
16866:
16352:
16306:
16286:
16036:
16006:
15776:, Section 11.4, pp. 361–362).
15477:
15017:
14985:
14767:
14763:
14572:
14568:
14450:
14446:
14169:
14076:
14055:
13964:
13722:defines an orientation of every fiber of
12984:
12577:
12576:There is another approach, expounded in (
12096:
11690:
11366:
11227:
11094:-form can be integrated over an oriented
10782:
10431:
10200:
5554:
5360:
5035:
4825:so it will be impossible to find such an
4792:
4741:
4584:
4356:More generally, for any smooth functions
4140:
2996:, the exterior derivative corresponds to
2910:
2750:
2643:
1561:
1403:
1370:
1294:
1259:
1202:
1067:
1027:
970:
905:, and negatively oriented otherwise. If
869:-form can be integrated over an oriented
414:, of two differential forms. Likewise, a
367:
324:
281:
121:
18568:Covariance and contravariance of vectors
17341:
17072:
16759:Applications in geometric measure theory
15830:-dimensional subsets. Geometrically, a
14860:This form depends on the orientation of
14079:) proves the generalized Fubini formula
11422:must be the negative of the integral of
5770:
4347:were arbitrary, this proves the formula
3042:
2349:, when viewed as a module over the ring
1090:This gives a geometrical context to the
582:{\displaystyle dy\wedge dx=-dx\wedge dy}
17272:
17254:
17093:
17049:
17020:
16949:
16931:
16451:Using the above-mentioned definitions,
15773:
15334:, since the difference is the integral
15260:-chains that are homologous (such that
15012:Integration along fibers satisfies the
13564:. Then there is a smooth differential
8737:. To define the pullback, fix a point
7155:{\displaystyle {\bigotimes }^{n}T^{*}M}
6094:th exterior power is isomorphic to the
5791:. A smooth differential form of degree
3065:-form ("zero-form") is defined to be a
2545:. When generalized to higher forms, if
1851:
1708:because the square whose first side is
627:This alternating property reflects the
14:
18847:
17207:
17153:
17110:
16975:
16848:
15721:; orientation has reversed; while the
11128:-form can be integrated over oriented
7867:
7181:{\displaystyle n!\operatorname {Alt} }
1311:to indicate integration over a subset
915:then the integral of the differential
134:{\displaystyle \int _{a}^{b}f(x)\,dx.}
18192:
17315:
17236:
17132:
16644:. There are gauge theories, such as
12572:Integration using partitions of unity
12332:is a formal sum of smooth embeddings
11537:
11504:{\displaystyle \varphi \colon D\to M}
8700:. Then there is a differential form
8659:and therefore a differential form on
7504:The exterior product is bilinear: If
5181:This is an example of a differential
980:{\displaystyle \int _{a}^{b}f(x)\,dx}
38:provide a unified approach to define
16885:(2nd ed.). New York: Springer.
16767:. A succinct proof may be found in
16619:Electromagnetism is an example of a
16612:to the Faraday form, is also called
16601:{\displaystyle {\star }\mathbf {F} }
15919:or generalized function is called a
14884:is uniquely defined by the property
13875:. Moreover, there is an integrable
7201:{\displaystyle \operatorname {Alt} }
4519:-form arises this way, and by using
4073:
2562:-form, then its exterior derivative
2394:{\displaystyle {\mathcal {J}}_{k,n}}
1342:-forms. Each of these represents a
17308:. Princeton University Press, 2021.
17138:Principles of Mathematical Analysis
17101:, Addison-Wesley, pp. 205–238.
16793:Closed and exact differential forms
16525:
16510:
16492:
16478:
16258:
16217:
16204:
16165:This form is a special case of the
15975:
15708:on the line, the differential form
15644:. By contrast, the integral of the
15115:
14235:which is the result of integrating
9210:{\textstyle \bigwedge ^{k}(df)^{*}}
9015:may be defined to be the composite
7522:are any differential forms, and if
6915:
6900:of a single differential form, the
6300:-form may be evaluated against any
5446:, there is a space of differential
4641:, when does there exist a function
4537:may be expressed in coordinates as
702:(a function can be considered as a
24:
18431:Tensors in curvilinear coordinates
17178:
15915:The differential form analog of a
15203:
15094:
15053:
14253:is defined by specifying, at each
13687:. The same construction works if
12322:, usually a cube or a simplex. A
12209:
12155:
12055:
12001:
11009:
10987:
10912:
10858:
10727:
10673:
9525:is the trivial rank one bundle on
8315:
8275:
8234:
8193:
8152:
8011:
7989:
7938:
7897:
5733:called the exterior derivative of
5344:
5329:
5019:
5004:
4894:
4879:
4857:
4842:
4793:
4779:
4765:
4742:
4728:
4714:
4249:
4241:
4124:
4116:
3730:
3687:
3653:
3620:
3575:
3498:
3490:
3471:
3456:
3413:
3405:
3162:
2949:
2894:
2879:
2831:
2768:
2721:
2627:
2619:
2473:
2441:
2374:
2308:
2135:
25:
18:Integration of a differential form
18871:
17229:
16455:can be written very compactly in
15684:, while a measure changes by the
15657:on the interval is unambiguously
12655:It is also possible to integrate
11100:-dimensional manifold. When the
8725:, which captures the behavior of
8376:, the de Rham complex is locally
8067:On a pseudo-Riemannian manifold,
4525:it follows that any differential
4327:, the result on each side is the
833:of differential forms appears in
16765:Wirtinger inequality for 2-forms
16708:
16700:
16692:
16681:
16594:
15665:with respect to this measure is
14956:
14911:
14759:
14708:
14564:
14442:
14434:
14423:defined by the interior product
13462:varies smoothly with respect to
13137:is defined by the property that
11987:
11841:
11809:
11780:
11682:
9717:
9647:
9495:
8439:Pullback (differential geometry)
6274:
6074:
3735:
3695:
3658:
3625:
3588:
3580:
3227:
3167:
2798:then its exterior derivative is
2337:spans the space of differential
631:of the domain of integration.
470:in the coordinate differentials
16803:Vector-valued differential form
15457:fundamental theorem of calculus
15143:)-form with compact support on
12267:. The Jacobian exists because
11134:-dimensional submanifolds. If
11073:{\displaystyle 1\leq m,n\leq k}
8477:between the tangent bundles of
8457:is smooth. The differential of
6896:of two differential forms, the
6318:. For example, a differential
6100:th exterior power of the dual:
5821:. The set of all differential
5763:to be a family of differential
3120:, which is another function on
3033:fundamental theorem of calculus
2541:is exactly the differential of
2401:combinatorially, the module of
1338:can be used as a basis for all
766:fundamental theorem of calculus
706:-form, and its differential is
532:The differential forms form an
17374:Differentiable/Smooth manifold
17115:Geometry of Differential Forms
17099:Functions of Several Variables
16907:
16874:
16842:
16426:
16106:
16077:
15961:electromagnetic field strength
15752:, it is possible to integrate
15090:
15067:
14945:
14939:
14778:
14755:
14715:
14699:
14583:
14560:
14477:for any choice of volume form
14438:
14369:
14146:
14140:
14032:
14026:
13934:
13928:
13900:
13787:the induced orientation. Let
13506:
13503:
13497:
13184:
13160:
13070:
13067:
13061:
13045:
12759:, respectively. Suppose that
12244:
12212:
12204:
12158:
12090:
12058:
12050:
12004:
11995:
11992:
11982:
11976:
11849:
11846:
11836:
11814:
11804:
11791:
11785:
11775:
11687:
11677:
11495:
11363:
11357:
11224:
11218:
11153:Integration on Euclidean space
11083:
10961:
10915:
10907:
10861:
10776:
10730:
10722:
10676:
10667:
10628:
10428:
10389:
10017:
10001:
9988:
9979:
9923:
9911:
9855:
9843:
9823:
9807:
9794:
9785:
9707:
9680:. The vector bundle morphism
9625:
9568:
9473:
9356:
9346:
9329:
9280:
9255:
9198:
9188:
9110:
9100:
9095:
9060:
9035:
8932:
8880:
8875:
8869:
8855:
8823:
8814:
8797:
8569:is not injective, say because
8336:
8330:
8324:
8308:
8296:
8290:
8284:
8259:
8249:
8243:
8218:
8208:
8202:
8177:
8167:
8161:
8145:
8110:algebra in the vector fields.
8032:
8026:
8007:
8004:
7998:
7959:
7953:
7922:
7912:
7906:
7830:
7820:
7799:
7787:
7733:
7723:
7646:
7634:
7622:
7610:
7556:
7544:
7466:
7460:
7447:
7441:
7425:
7419:
7406:
7400:
7384:
7372:
7363:
7350:
7286:
7274:
7245:
7233:
7057:
7045:
6811:
6728:
6723:
6717:
6695:
6689:
6678:
6665:
6659:
6609:
6577:
6574:
6561:
6475:
6270:
6070:
5747:. One way to do this is cover
5640:for a collection of functions
4483:
4466:
4463:
4457:
4291:
4274:
4271:
4265:
3753:
3747:
3744:
3726:
3713:
3707:
3704:
3683:
3676:
3670:
3667:
3649:
3643:
3637:
3634:
3616:
3606:
3600:
3597:
3571:
3370:are the coordinate vectors in
3231:
3214:
3185:
3179:
3176:
3158:
3076:– the set of which is denoted
2970:
2964:
2783:
2777:
2459:
2434:
2417:-dimensional vector space, is
2196:
2164:
1927:
1882:
1286:
1274:
1064:
1058:
1024:
1018:
967:
961:
764:). This allows expressing the
745:
739:
725:
719:
620:{\displaystyle dx\wedge dx=0.}
504:{\displaystyle dx,dy,\ldots .}
462:-dimensional manifold, and is
383:
364:
346:
321:
303:
278:
260:
254:
225:that can be integrated over a
118:
112:
13:
1:
18484:Exterior covariant derivative
18416:Tensor (intrinsic definition)
16925:
16808:Equivariant differential form
16162:, or equivalent definitions.
15688:of the Jacobian determinant,
13854:is a well-defined integrable
12983:does not vanish. Following (
11175:its standard orientation and
10309:is a real-valued function of
8114:Exterior differential complex
8053:to the exterior differential
7528:is any smooth function, then
7330:-form whose value at a point
6886:
6294:Consequently, a differential
4382:, we define the differential
4015:, and so define differential
2693:-forms through linearity: if
2364:. By calculating the size of
1235:and integrates over a subset
791:-forms are naturally dual to
757:{\displaystyle df(x)=f'(x)dx}
53:For instance, the expression
18509:Raising and lowering indices
17184:An Introduction to Manifolds
16883:An introduction to manifolds
16825:Polynomial differential form
12802:, there is a chart on which
12348:-dimensional submanifold of
10508:can be expanded in terms of
6877:. The differential forms on
6427:-forms are sometimes called
6409:as the inner product with a
6023:is also a linear functional
4011:are themselves functions on
3341:th coordinate vector, i.e.,
7:
18747:Gluon field strength tensor
18218:
18080:Classification of manifolds
17175:standard introductory text.
16786:
15910:
14988:). This form also denoted
14322:. More precisely, at each
13126:to be the inclusion. Then
12622:-form on an open subset of
11530:the orientation induced by
11112:-dimensional manifold with
10526:-form can be written using
9171:. In full generality, let
8489:. This map is also denoted
8432:
4521:
4349:
4173:
3035:, called the (generalized)
1617:). This is similar to the
1138:-dimensional manifold, and
861:Integration and orientation
814:change of variables formula
698:This operation extends the
10:
18876:
18558:Cartan formalism (physics)
18378:Penrose graphical notation
17186:, Universitext, Springer,
17111:Morita, Shigeyuki (2001),
16435:{\displaystyle {\vec {J}}}
16115:{\displaystyle {\vec {B}}}
16086:{\displaystyle {\vec {E}}}
15481:
15119:
14332:, there is an isomorphism
14310:, and the orientations of
13829:. Then, for almost every
13108:. More precisely, define
12720:
12263:is the determinant of the
11540:) defines the integral of
10266:where, for each choice of
8787:is defined by the formula
8436:
8398:locally constant functions
7878:pseudo-Riemannian manifold
7208:, the exterior product is
6920:The exterior product of a
6001:of the dual bundle of the
5774:
5755:and define a differential
5719:, there is a differential
4663:whose partial derivatives
4611:for some smooth functions
3938:is uniquely determined by
3134:is the rate of change (at
2689:with extension to general
2520:differential of a function
2120:, in a space of dimension
1868:-dimensional context, for
1820:if any two of the indices
1231:with respect to a measure
844:
819:
782:generalized Stokes theorem
700:differential of a function
691:{\displaystyle d\varphi .}
144:Similarly, the expression
18764:
18704:
18653:
18646:
18538:
18469:
18406:
18350:
18297:
18244:
18237:
18230:Glossary of tensor theory
18226:
18156:over commutative algebras
18113:
18072:
18005:
17902:
17798:
17745:
17736:
17572:
17495:
17434:
17354:
17192:10.1007/978-1-4419-7400-6
17140:, New York: McGraw-Hill,
17034:10.1007/978-0-8176-8412-9
16798:Complex differential form
16640:, which is in particular
14195:be a compactly supported
13781:, and give each fiber of
13541:is a smooth differential
13468:. That is, suppose that
12294:-dimensional subsets for
11276:for some smooth function
10497:Each exterior derivative
9161:and hence a differential
7338:alternating bilinear form
6007:th exterior power of the
4944:of the left hand side in
4331:th partial derivative of
1726:and whose second side is
18814:Gregorio Ricci-Curbastro
18686:Riemann curvature tensor
18393:Van der Waerden notation
17872:Riemann curvature tensor
17296:, an undergraduate text.
17213:Mathematical Analysis II
16835:
16773:Geometric Measure Theory
16664:. The Yang–Mills field
15814:-forms over subsets for
12723:Integration along fibers
12717:Integration along fibers
6366:. In the presence of an
4069:Kronecker delta function
657:{\displaystyle \varphi }
18784:Elwin Bruno Christoffel
18717:Angular momentum tensor
18388:Tetrad (index notation)
18358:Abstract index notation
17273:Bachman, David (2003),
17255:Sjamaar, Reyer (2006),
16950:Bachman, David (2003),
16932:Bachman, David (2006),
16182:may be described. The
15947:Applications in physics
15770:densities on a manifold
15725:, which here we denote
15410:, then the integral of
12427:then the integral of a
11436:Integration over chains
11106:-form is defined on an
9761:is a real number, then
9731:defined in this way is
8512:and any tangent vector
8063:Vector field structures
6429:covariant vector fields
6088:, i.e. the dual of the
4316:Applying both sides to
3124:whose value at a point
2504:The exterior derivative
2360:of smooth functions on
1844:whose edge vectors are
853:that is independent of
797:differentiable manifold
410:, sometimes called the
18598:Levi-Civita connection
17664:Manifold with boundary
17379:Differential structure
16996:Formes différentielles
16881:Tu, Loring W. (2011).
16719:
16636:, the one-dimensional
16602:
16564:
16563:{\displaystyle \star }
16541:
16436:
16360:
16229:
16116:
16087:
16044:
15745:; it does not change.
15638:
15560:
15478:Relation with measures
15455:, and generalizes the
15385:
15328:
15221:
15106:
14969:
14866:but not the choice of
14851:
14634:
14483:in the orientation of
14468:
14414:
14180:
14066:
13982:
13663:
13526:
13402:
13328:
13251:
13077:
12963:
12592:. The integral of an
12543:
12497:
12418:
12390:
12254:
12133:
11859:
11747:
11552:to be the integral of
11505:
11403:
11296:to be the integral of
11267:
11074:
11036:
10971:
10839:
10488:
10257:
10031:
9725:
9655:
9506:
9406:
9211:
9146:
8942:
8349:
8100:differential operators
8039:
7966:
7876:, or more generally a
7865:
7858:
7761:
7656:
7590:
7473:
7296:
7202:
7182:
7156:
7114:
7067:
6964:)-form. At each point
6846:
6738:
6533:
6510:
6328:assigns to each point
6285:
6256:
6197:
6082:
5932:
5631:
5513:
5397:
5325:
5266:
5172:
5078:
5000:
4920:
4816:
4602:
4573:
4496:
4307:
4237:
4161:
4112:
3764:
3521:
3452:
3258:
3107:directional derivative
2977:
2875:
2790:
2680:
2615:
2492:
2395:
2331:
2263:
2108:
1991:
1699:
1595:
1424:
1305:
1210:
1081:
981:
851:multivariable calculus
758:
692:
658:
621:
583:
505:
393:
135:
18860:Differential geometry
18824:Jan Arnoldus Schouten
18779:Augustin-Louis Cauchy
18259:Differential geometry
17265:, a course taught at
17160:Calculus on Manifolds
17105:multivariate calculus
16849:Cartan, Élie (1899),
16814:Calculus on Manifolds
16720:
16603:
16565:
16542:
16437:
16361:
16230:
16117:
16088:
16045:
15675:-form changes by the
15639:
15561:
15484:Density on a manifold
15482:Further information:
15386:
15329:
15270:is the boundary of a
15222:
15107:
14970:
14852:
14635:
14469:
14415:
14298:that depends only on
14181:
14067:
13983:
13672:This form is denoted
13664:
13527:
13403:
13329:
13252:
13078:
12987:), there is a unique
12964:
12544:
12477:
12419:
12370:
12255:
12134:
11860:
11748:
11506:
11404:
11268:
11163:be an open subset of
11075:
11037:
10972:
10840:
10489:
10258:
10032:
9726:
9656:
9531:. The composite map
9507:
9407:
9212:
9147:
8943:
8362:is by definition the
8350:
8102:they generate is the
8040:
7967:
7859:
7776:
7762:
7657:
7591:
7474:
7297:
7203:
7183:
7157:
7115:
7068:
6863:totally antisymmetric
6847:
6739:
6534:
6532:{\displaystyle \tau }
6511:
6286:
6236:
6198:
6083:
5933:
5827:-forms on a manifold
5771:Intrinsic definitions
5632:
5460:
5398:
5299:
5246:
5173:
5079:
4974:
4921:
4817:
4603:
4553:
4497:
4308:
4217:
4162:
4092:
3765:
3522:
3432:
3392:are introduced, then
3259:
3043:Differential calculus
3017:differential topology
3013:differential geometry
2978:
2855:
2791:
2681:
2595:
2493:
2396:
2341:-forms in a manifold
2332:
2264:
2109:
1992:
1700:
1596:
1425:
1306:
1211:
1155:-form, then one has:
1082:
982:
759:
693:
659:
622:
584:
506:
394:
136:
18799:Carl Friedrich Gauss
18732:stress–energy tensor
18727:Cauchy stress tensor
18479:Covariant derivative
18441:Antisymmetric tensor
18373:Multi-index notation
17811:Covariant derivative
17362:Topological manifold
17007:Treatise on Analysis
16755:of the gauge group.
16677:
16585:
16554:
16466:
16417:
16253:
16199:
16097:
16068:
15970:
15677:Jacobian determinant
15570:
15522:
15338:
15288:
15176:
15045:
15002:along the fibers of
14891:
14696:
14547:
14430:
14339:
14241:along the fibers of
14086:
13998:
13991:Denote this form by
13894:
13615:
13475:
13445:Moreover, for fixed
13422:may also be notated
13356:
13349:-covector for which
13267:
13144:
12994:
12840:
12458:
12361:
12149:
11875:
11769:
11596:
11483:
11309:
11206:
11046:
10981:
10852:
10537:
10334:
10125:
9768:
9684:
9538:
9443:
9242:
9175:
9022:
8794:
8749:and tangent vectors
8731:as seen relative to
8136:
7979:
7887:
7781:
7708:
7601:
7535:
7347:
7215:
7192:
7166:
7162:is done via the map
7124:
7080:
7076:If the embedding of
7024:
6910:covariant derivative
6775:
6552:
6523:
6441:
6220:
6107:
6027:
5992:naturally isomorphic
5877:
5457:
5234:
5189:-form is called the
5098:
5091:is defined so that:
4971:
4836:
4707:
4544:
4421:
4198:
4080:
3879:. Note that at each
3785:and any real number
3564:
3399:
3337:with respect to the
3291:in the definition.)
3155:
2805:
2698:
2583:
2430:
2368:
2275:
2129:
2004:
1873:
1862:multi-index notation
1860:-forms is so called
1852:Multi-index notation
1628:
1514:
1357:
1243:
1162:
997:
940:
710:
676:
648:
593:
540:
536:. This implies that
474:
241:
91:
18676:Nonmetricity tensor
18531:(2nd-order tensors)
18499:Hodge star operator
18489:Exterior derivative
18338:Transport phenomena
18323:Continuum mechanics
18279:Multilinear algebra
17845:Exterior derivative
17447:Atiyah–Singer index
17396:Riemannian manifold
17293:2003math......6194B
17241:"Differential form"
17209:Zorich, Vladimir A.
17095:Fleming, Wendell H.
17057:(Second ed.),
16970:2003math......6194B
16753:structure equations
16670:is then defined by
16453:Maxwell's equations
15616:
15589:
15541:
15461:contour integration
14840:
14623:
14406:
14217:. Then there is a
13593:such that, at each
13394:
13320:
13282:
13240:
13208:
13044:
12948:
12892:
12661:-forms on oriented
12532:
12354:. If the chain is
12273:is differentiable.
10070:are coordinates on
10052:are coordinates on
9009:. The pullback of
8781:. The pullback of
8682:be smooth, and let
8045:, which has degree
7882:Hodge star operator
7874:Riemannian manifold
7868:Riemannian manifold
7772:graded Leibniz rule
6898:exterior derivative
6138:
5921:
5870:defines an element
5191:exterior derivative
4515:. Any differential
3883:, the differential
3023:of degree 1 on the
2994:Hodge star operator
2510:exterior derivative
1714:and second side is
1610:(the symbol is the
1505:-plane. A general
1114:More generally, an
1054:
1014:
957:
636:exterior derivative
534:alternating algebra
523:-form) is called a
108:
67:is an example of a
18855:Differential forms
18809:Tullio Levi-Civita
18752:Metric tensor (GR)
18666:Levi-Civita symbol
18519:Tensor contraction
18333:General relativity
18269:Euclidean geometry
18151:Secondary calculus
18105:Singularity theory
18060:Parallel transport
17828:De Rham cohomology
17467:Generalized Stokes
17267:Cornell University
17238:Weisstein, Eric W.
17051:Folland, Gerald B.
17022:Edwards, Harold M.
16981:Differential Forms
16868:10.24033/asens.467
16830:Presymplectic form
16715:
16598:
16560:
16537:
16535:
16432:
16356:
16225:
16112:
16083:
16040:
15634:
15633:
15602:
15575:
15556:
15555:
15527:
15468:de Rham cohomology
15381:
15380:
15324:
15323:
15217:
15102:
15014:projection formula
14965:
14847:
14826:
14812:
14630:
14609:
14595:
14464:
14410:
14392:
14378:
14348:
14176:
14062:
13978:
13659:
13522:
13398:
13382:
13324:
13306:
13292:
13270:
13247:
13226:
13218:
13196:
13073:
13030:
13016:
12959:
12957:
12934:
12926:
12878:
12870:
12539:
12518:
12414:
12250:
12129:
11939:
11855:
11743:
11640:
11501:
11399:
11263:
11070:
11032:
10967:
10835:
10627:
10591:
10484:
10388:
10253:
10169:
10118:can be written as
10027:
10025:
9721:
9693:
9651:
9604:
9547:
9502:
9452:
9402:
9378:
9301:
9207:
9142:
8938:
8364:de Rham cohomology
8345:
8082:. This algebra is
8035:
7962:
7854:
7757:
7671:graded commutative
7652:
7586:
7469:
7292:
7198:
7178:
7152:
7110:
7089:
7063:
6855:This map exhibits
6842:
6790:
6734:
6652:
6529:
6506:
6281:
6193:
6158:
6124:
6116:
6078:
6049:
5928:
5907:
5899:
5627:
5393:
5227:. It is given by
5168:
5074:
4916:
4812:
4598:
4492:
4446:
4303:
4157:
4071:, it follows that
3912:linear combination
3760:
3758:
3517:
3331:partial derivative
3294:In particular, if
3254:
2973:
2849:
2786:
2739:
2676:
2488:
2421: choose
2391:
2327:
2259:
2104:
2090:
1987:
1846:linearly dependent
1695:
1591:
1544:
1420:
1301:
1206:
1077:
1040:
1000:
977:
943:
877:. A differential
770:divergence theorem
754:
688:
654:
617:
579:
501:
389:
131:
94:
36:differential forms
18842:
18841:
18804:Hermann Grassmann
18760:
18759:
18712:Moment of inertia
18573:Differential form
18548:Affine connection
18363:Einstein notation
18346:
18345:
18274:Exterior calculus
18254:Coordinate system
18186:
18185:
18068:
18067:
17833:Differential form
17487:Whitney embedding
17421:Differential form
17201:978-0-387-48098-5
17103:This textbook in
17064:978-0-471-31716-6
17043:978-0-8176-8411-2
16943:978-0-8176-4499-4
16781:systolic geometry
16658:-valued one-form
16646:Yang–Mills theory
16527:
16512:
16494:
16480:
16457:geometrized units
16429:
16274:
16260:
16219:
16206:
16109:
16080:
15991:
15977:
15875:Hausdorff measure
15760:fundamental class
15416:over a path from
15165:with its induced
14649:, so there is an
12713:on the interval.
12248:
12094:
11904:
11605:
11579:with coordinates
11341:
11339:
11327:
11030:
10965:
10780:
10592:
10556:
10353:
10134:
9639:
9634:
9620:
9597:
9592:
9563:
9487:
9482:
9468:
9433:is a morphism of
9371:
9366:
9324:
9294:
9289:
9275:
9269:
9264:
9250:
9235:is the composite
9187:
9125:
9120:
9090:
9074:
9069:
9055:
9049:
9044:
9030:
8388:. The kernel at
8341:
8335:
8313:
8307:
8301:
8295:
8273:
8268:
8254:
8232:
8227:
8213:
8191:
8186:
8172:
8150:
8144:
8093:geometric algebra
7936:
7931:
7917:
7770:One also has the
7675:anticommutativity
7266:
6630:
6628:
6386:Riemannian metric
6340:linear functional
5989:. This space is
5753:coordinate charts
5358:
5033:
4908:
4871:
4807:
4756:
4437:
4263:
4181:
4180:
4138:
3904:Since any vector
3512:
3485:
3427:
3315:coordinate vector
3209:
3093:is any vector in
3061:. A differential
2908:
2817:
2707:
2641:
2480:
2413:-covectors on an
2075:
1517:
1105:), the increment
835:Hermann Grassmann
16:(Redirected from
18867:
18819:Bernhard Riemann
18651:
18650:
18494:Exterior product
18461:Two-point tensor
18446:Symmetric tensor
18328:Electromagnetism
18242:
18241:
18213:
18206:
18199:
18190:
18189:
18178:Stratified space
18136:Fréchet manifold
17850:Interior product
17743:
17742:
17440:
17336:
17329:
17322:
17313:
17312:
17300:Needham, Tristan
17295:
17286:
17264:
17263:
17251:
17250:
17225:
17204:
17174:
17150:
17129:
17118:
17102:
17090:
17074:Flanders, Harley
17068:
17046:
17017:
16994:—Translation of
16993:
16972:
16963:
16946:
16919:
16918:
16911:
16905:
16904:
16878:
16872:
16871:
16870:
16846:
16820:Multilinear form
16771:'s classic text
16750:
16744:
16738:
16724:
16722:
16721:
16716:
16711:
16703:
16695:
16684:
16669:
16663:
16653:
16635:
16624:
16607:
16605:
16604:
16599:
16597:
16592:
16580:
16569:
16567:
16566:
16561:
16546:
16544:
16543:
16538:
16536:
16529:
16528:
16515:
16514:
16513:
16496:
16495:
16482:
16481:
16447:
16441:
16439:
16438:
16433:
16431:
16430:
16422:
16408:
16402:
16396:
16385:
16374:
16365:
16363:
16362:
16357:
16351:
16350:
16335:
16334:
16319:
16318:
16305:
16304:
16285:
16284:
16275:
16267:
16262:
16261:
16243:
16234:
16232:
16231:
16226:
16221:
16220:
16208:
16207:
16191:
16176:principal bundle
16174:
16161:
16143:
16121:
16119:
16118:
16113:
16111:
16110:
16102:
16092:
16090:
16089:
16084:
16082:
16081:
16073:
16063:
16049:
16047:
16046:
16041:
16035:
16034:
16019:
16018:
16005:
16004:
15992:
15984:
15979:
15978:
15953:electromagnetism
15942:
15936:
15930:
15925:. The space of
15906:
15900:
15894:
15888:
15882:
15872:
15863:
15857:
15851:
15845:
15838:Gram determinant
15835:
15829:
15823:
15813:
15800:
15790:
15784:
15767:
15757:
15744:
15742:
15735:, pulls back to
15734:
15732:
15723:Lebesgue measure
15720:
15713:
15707:
15697:
15695:
15683:
15674:
15668:
15664:
15660:
15656:
15654:
15643:
15641:
15640:
15635:
15632:
15615:
15610:
15588:
15583:
15565:
15563:
15562:
15557:
15554:
15540:
15535:
15514:
15510:
15506:
15500:
15453:gradient theorem
15446:
15427:
15421:
15415:
15409:
15403:
15394:For example, if
15390:
15388:
15387:
15382:
15370:
15369:
15351:
15350:
15333:
15331:
15330:
15325:
15322:
15318:
15317:
15302:
15301:
15283:
15277:
15269:
15259:
15253:
15247:
15241:
15235:
15226:
15224:
15223:
15218:
15210:
15209:
15188:
15187:
15164:
15154:
15148:
15142:
15135:
15122:Stokes's theorem
15116:Stokes's theorem
15111:
15109:
15108:
15103:
15098:
15097:
15085:
15084:
15057:
15056:
15037:
15031:
15025:
15007:
15001:
14993:
14983:
14974:
14972:
14971:
14966:
14961:
14960:
14959:
14949:
14948:
14938:
14937:
14914:
14906:
14905:
14883:
14877:
14871:
14865:
14856:
14854:
14853:
14848:
14839:
14834:
14825:
14824:
14813:
14802:
14801:
14792:
14791:
14785:
14781:
14777:
14776:
14762:
14754:
14753:
14741:
14740:
14723:
14722:
14713:
14712:
14711:
14688:
14677:
14671:
14660:
14648:
14639:
14637:
14636:
14631:
14622:
14617:
14608:
14607:
14596:
14582:
14581:
14567:
14559:
14558:
14539:
14533:
14521:
14515:
14509:
14503:
14488:
14482:
14473:
14471:
14470:
14465:
14460:
14459:
14445:
14437:
14419:
14417:
14416:
14411:
14405:
14400:
14391:
14390:
14379:
14365:
14364:
14355:
14354:
14349:
14331:
14321:
14315:
14309:
14303:
14297:
14286:
14280:
14274:
14269:pairs with each
14268:
14262:
14252:
14246:
14240:
14234:
14228:
14222:
14216:
14210:
14194:
14185:
14183:
14182:
14177:
14168:
14167:
14158:
14150:
14149:
14139:
14138:
14121:
14120:
14114:
14113:
14098:
14097:
14071:
14069:
14068:
14063:
14054:
14053:
14044:
14036:
14035:
14025:
14024:
14007:
14006:
13987:
13985:
13984:
13979:
13974:
13973:
13963:
13962:
13956:
13955:
13946:
13938:
13937:
13927:
13926:
13909:
13908:
13886:
13880:
13874:
13863:
13853:
13838:
13828:
13822:
13816:
13810:
13804:
13798:
13792:
13786:
13780:
13774:
13768:
13754:
13748:
13742:
13736:
13727:
13721:
13715:
13709:
13698:
13692:
13686:
13668:
13666:
13665:
13660:
13655:
13654:
13645:
13640:
13639:
13627:
13626:
13607:
13592:
13581:
13575:
13563:
13552:
13546:
13540:
13531:
13529:
13528:
13523:
13518:
13517:
13496:
13495:
13467:
13461:
13450:
13441:
13421:
13407:
13405:
13404:
13399:
13390:
13381:
13380:
13368:
13367:
13348:
13333:
13331:
13330:
13325:
13319:
13314:
13305:
13304:
13293:
13278:
13256:
13254:
13253:
13248:
13239:
13234:
13225:
13224:
13219:
13204:
13192:
13191:
13182:
13181:
13172:
13171:
13156:
13155:
13136:
13125:
13107:
13097:with respect to
13096:
13082:
13080:
13079:
13074:
13060:
13059:
13043:
13038:
13029:
13028:
13017:
13006:
13005:
12982:
12968:
12966:
12965:
12960:
12958:
12947:
12942:
12933:
12932:
12927:
12912:
12911:
12891:
12886:
12877:
12876:
12871:
12856:
12855:
12833:. Suppose that
12832:
12817:
12807:
12801:
12795:
12783:
12772:
12758:
12752:
12746:
12740:
12728:Fubini's theorem
12712:
12701:
12697:
12684:
12681:, integrating a
12680:
12666:
12660:
12651:
12645:
12639:
12633:
12627:
12621:
12615:
12609:
12603:
12597:
12591:
12585:
12567:
12557:
12548:
12546:
12545:
12540:
12531:
12526:
12517:
12516:
12507:
12506:
12496:
12491:
12470:
12469:
12450:
12444:
12438:
12432:
12423:
12421:
12420:
12415:
12410:
12409:
12400:
12399:
12389:
12384:
12353:
12347:
12341:
12327:
12321:
12315:
12309:
12303:
12293:
12287:
12281:
12272:
12259:
12257:
12256:
12251:
12249:
12247:
12243:
12242:
12224:
12223:
12207:
12203:
12202:
12201:
12200:
12177:
12176:
12175:
12174:
12153:
12138:
12136:
12135:
12130:
12125:
12124:
12109:
12108:
12095:
12093:
12089:
12088:
12070:
12069:
12053:
12049:
12048:
12047:
12046:
12023:
12022:
12021:
12020:
11999:
11991:
11990:
11975:
11974:
11973:
11972:
11954:
11953:
11938:
11937:
11936:
11918:
11917:
11903:
11902:
11887:
11886:
11864:
11862:
11861:
11856:
11845:
11844:
11835:
11834:
11813:
11812:
11803:
11802:
11784:
11783:
11761:
11752:
11750:
11749:
11744:
11739:
11738:
11737:
11736:
11710:
11709:
11708:
11707:
11686:
11685:
11676:
11675:
11674:
11673:
11655:
11654:
11639:
11638:
11637:
11619:
11618:
11588:
11578:
11572:
11566:
11560:
11551:
11545:
11535:
11529:
11523:
11510:
11508:
11507:
11502:
11475:
11469:
11463:
11457:
11451:
11445:
11431:
11421:
11408:
11406:
11405:
11400:
11395:
11394:
11379:
11378:
11353:
11352:
11343:
11342:
11340:
11337:
11335:
11330:
11325:
11321:
11320:
11301:
11295:
11289:
11272:
11270:
11269:
11264:
11262:
11261:
11240:
11239:
11198:
11192:
11186:
11180:
11174:
11168:
11162:
11147:
11140:
11133:
11127:
11121:
11111:
11105:
11099:
11093:
11079:
11077:
11076:
11071:
11041:
11039:
11038:
11033:
11031:
11029:
11028:
11027:
11026:
11025:
11007:
11006:
11005:
11004:
11003:
10985:
10976:
10974:
10973:
10968:
10966:
10964:
10960:
10959:
10958:
10957:
10934:
10933:
10932:
10931:
10910:
10906:
10905:
10904:
10903:
10880:
10879:
10878:
10877:
10856:
10844:
10842:
10841:
10836:
10831:
10830:
10829:
10828:
10802:
10801:
10800:
10799:
10781:
10779:
10775:
10774:
10773:
10772:
10749:
10748:
10747:
10746:
10725:
10721:
10720:
10719:
10718:
10695:
10694:
10693:
10692:
10671:
10660:
10659:
10658:
10657:
10645:
10644:
10626:
10625:
10624:
10606:
10605:
10590:
10589:
10588:
10570:
10569:
10549:
10548:
10525:
10520:. The resulting
10519:
10513:
10507:
10493:
10491:
10490:
10485:
10480:
10479:
10478:
10477:
10451:
10450:
10449:
10448:
10421:
10420:
10419:
10418:
10406:
10405:
10387:
10386:
10385:
10367:
10366:
10346:
10345:
10327:has the formula
10326:
10320:
10314:
10308:
10285:
10274:
10262:
10260:
10259:
10254:
10249:
10248:
10247:
10246:
10220:
10219:
10218:
10217:
10199:
10198:
10197:
10196:
10184:
10183:
10168:
10167:
10166:
10148:
10147:
10117:
10111:
10105:
10099:
10075:
10069:
10063:
10057:
10051:
10045:
10036:
10034:
10033:
10028:
10026:
10013:
10012:
9978:
9977:
9958:
9957:
9942:
9941:
9910:
9909:
9890:
9889:
9874:
9873:
9842:
9841:
9819:
9818:
9784:
9783:
9760:
9754:
9748:
9739:
9730:
9728:
9727:
9722:
9720:
9700:
9699:
9694:
9679:
9669:
9660:
9658:
9657:
9652:
9650:
9637:
9636:
9635:
9633:
9628:
9623:
9618:
9611:
9610:
9605:
9595:
9594:
9593:
9591:
9584:
9583:
9578:
9571:
9566:
9561:
9554:
9553:
9548:
9530:
9524:
9511:
9509:
9508:
9503:
9498:
9485:
9484:
9483:
9481:
9476:
9471:
9466:
9459:
9458:
9453:
9432:
9426:
9420:
9411:
9409:
9408:
9403:
9395:
9394:
9385:
9384:
9379:
9369:
9368:
9367:
9365:
9364:
9363:
9345:
9344:
9339:
9332:
9327:
9322:
9318:
9317:
9308:
9307:
9302:
9292:
9291:
9290:
9288:
9283:
9278:
9273:
9267:
9266:
9265:
9263:
9258:
9253:
9248:
9234:
9228:
9222:
9216:
9214:
9213:
9208:
9206:
9205:
9186:
9178:
9170:
9164:
9160:
9151:
9149:
9148:
9143:
9135:
9134:
9123:
9122:
9121:
9119:
9118:
9117:
9098:
9093:
9088:
9084:
9083:
9072:
9071:
9070:
9068:
9063:
9058:
9053:
9047:
9046:
9045:
9043:
9038:
9033:
9028:
9014:
9008:
8987:
8981:
8975:
8966:
8960:
8956:
8947:
8945:
8944:
8939:
8931:
8930:
8921:
8920:
8902:
8901:
8892:
8891:
8879:
8878:
8854:
8853:
8835:
8834:
8822:
8821:
8809:
8808:
8786:
8780:
8774:
8768:
8757:
8748:
8742:
8736:
8730:
8724:
8714:
8708:
8699:
8693:
8687:
8681:
8664:
8658:
8652:
8638:
8629:
8623:
8617:
8608:
8598:
8592:
8578:
8568:
8562:
8543:
8529:
8511:
8502:. For any point
8497:
8488:
8482:
8476:
8463:is a smooth map
8462:
8456:
8428:
8427:
8419:sheaf cohomology
8416:
8415:
8405:
8396:is the space of
8395:
8387:
8371:
8354:
8352:
8351:
8346:
8339:
8333:
8323:
8322:
8311:
8305:
8299:
8293:
8283:
8282:
8271:
8270:
8269:
8267:
8262:
8257:
8252:
8242:
8241:
8230:
8229:
8228:
8226:
8221:
8216:
8211:
8201:
8200:
8189:
8188:
8187:
8185:
8180:
8175:
8170:
8160:
8159:
8148:
8142:
8124:
8088:exterior algebra
8076:Clifford algebra
8070:
8058:
8048:
8044:
8042:
8041:
8036:
8025:
8024:
7997:
7996:
7971:
7969:
7968:
7963:
7952:
7951:
7934:
7933:
7932:
7930:
7925:
7920:
7915:
7905:
7904:
7863:
7861:
7860:
7855:
7838:
7837:
7766:
7764:
7763:
7758:
7744:
7743:
7700:
7694:
7688:
7682:
7667:skew commutative
7661:
7659:
7658:
7653:
7595:
7593:
7592:
7587:
7527:
7521:
7515:
7509:
7500:
7478:
7476:
7475:
7470:
7459:
7458:
7440:
7439:
7418:
7417:
7399:
7398:
7371:
7370:
7335:
7329:
7325:
7315:
7301:
7299:
7298:
7293:
7267:
7265:
7251:
7231:
7207:
7205:
7204:
7199:
7187:
7185:
7184:
7179:
7161:
7159:
7158:
7153:
7148:
7147:
7138:
7137:
7132:
7119:
7117:
7116:
7111:
7106:
7105:
7096:
7095:
7090:
7072:
7070:
7069:
7064:
7016:
7006:
6993:
6987:
6981:
6975:
6970:of the manifold
6969:
6963:
6953:
6943:
6937:
6931:
6925:
6916:Exterior product
6902:interior product
6894:exterior product
6882:
6876:
6860:
6851:
6849:
6848:
6843:
6835:
6834:
6825:
6824:
6819:
6807:
6806:
6797:
6796:
6791:
6767:
6757:
6743:
6741:
6740:
6735:
6727:
6726:
6699:
6698:
6677:
6676:
6651:
6650:
6649:
6629:
6627:
6616:
6608:
6607:
6589:
6588:
6573:
6572:
6544:
6538:
6536:
6535:
6530:
6515:
6513:
6512:
6507:
6499:
6498:
6489:
6488:
6483:
6471:
6470:
6461:
6460:
6455:
6426:
6422:
6404:
6393:
6383:
6365:
6351:
6337:
6327:
6321:
6317:
6311:
6305:
6299:
6290:
6288:
6287:
6282:
6277:
6266:
6265:
6255:
6250:
6232:
6231:
6202:
6200:
6199:
6194:
6192:
6191:
6186:
6185:
6175:
6174:
6165:
6164:
6159:
6151:
6150:
6137:
6132:
6123:
6122:
6117:
6099:
6093:
6087:
6085:
6084:
6079:
6077:
6066:
6065:
6056:
6055:
6050:
6039:
6038:
6022:
6016:
6006:
6000:
5995:to the fiber at
5994:
5984:
5970:
5964:
5954:
5937:
5935:
5934:
5929:
5920:
5915:
5906:
5905:
5900:
5889:
5888:
5869:
5863:
5857:
5844:
5837:, often denoted
5832:
5826:
5820:
5813:cotangent bundle
5806:
5796:
5786:
5777:Exterior algebra
5766:
5762:
5758:
5750:
5746:
5736:
5732:
5726:
5718:
5714:
5707:
5672:
5668:
5636:
5634:
5633:
5628:
5626:
5625:
5624:
5623:
5597:
5596:
5595:
5594:
5574:
5573:
5572:
5571:
5553:
5552:
5551:
5550:
5538:
5537:
5528:
5527:
5512:
5507:
5500:
5499:
5487:
5486:
5474:
5473:
5449:
5445:
5441:
5437:
5433:
5426:
5416:
5412:
5402:
5400:
5399:
5394:
5389:
5388:
5373:
5372:
5359:
5357:
5356:
5355:
5342:
5341:
5340:
5327:
5324:
5319:
5295:
5294:
5279:
5278:
5265:
5260:
5226:
5214:
5213:
5197:
5188:
5184:
5177:
5175:
5174:
5169:
5164:
5163:
5148:
5147:
5129:
5128:
5113:
5112:
5090:
5083:
5081:
5080:
5075:
5064:
5063:
5048:
5047:
5034:
5032:
5031:
5030:
5017:
5016:
5015:
5002:
4999:
4994:
4962:exterior product
4959:
4956:on differential
4955:
4951:
4947:
4936:
4932:
4925:
4923:
4922:
4917:
4909:
4907:
4906:
4905:
4892:
4891:
4890:
4877:
4872:
4870:
4869:
4868:
4855:
4854:
4853:
4840:
4828:
4821:
4819:
4818:
4813:
4808:
4806:
4805:
4804:
4791:
4790:
4777:
4773:
4772:
4762:
4757:
4755:
4754:
4753:
4740:
4739:
4726:
4722:
4721:
4711:
4699:
4695:
4688:
4678:given functions
4677:
4673:
4662:
4658:
4648:
4644:
4640:
4636:
4632:
4625:
4621:
4607:
4605:
4604:
4599:
4597:
4596:
4583:
4582:
4572:
4567:
4536:
4532:
4528:
4518:
4514:
4501:
4499:
4498:
4493:
4491:
4490:
4481:
4480:
4456:
4455:
4445:
4433:
4432:
4413:
4385:
4381:
4377:
4366:
4346:
4342:
4338:
4334:
4330:
4326:
4312:
4310:
4309:
4304:
4299:
4298:
4289:
4288:
4264:
4262:
4261:
4260:
4247:
4239:
4236:
4231:
4213:
4212:
4186:
4175:
4166:
4164:
4163:
4158:
4153:
4152:
4139:
4137:
4136:
4135:
4122:
4114:
4111:
4106:
4074:
4066:
4046:
4036:
4030:
4024:
4018:
4014:
4010:
4004:
3998:
3992:
3988:
3982:
3978:
3974:
3964:
3958:
3937:
3927:
3909:
3898:
3893:
3882:
3878:
3874:
3870:
3864:
3835:
3831:
3819:
3808:
3802:
3789:. At each point
3788:
3784:
3778:
3773:for any vectors
3769:
3767:
3766:
3761:
3759:
3740:
3739:
3738:
3700:
3699:
3698:
3663:
3662:
3661:
3630:
3629:
3628:
3593:
3592:
3591:
3583:
3556:
3546:
3526:
3524:
3523:
3518:
3513:
3511:
3510:
3509:
3496:
3488:
3486:
3484:
3483:
3482:
3469:
3468:
3467:
3454:
3451:
3446:
3428:
3426:
3425:
3424:
3411:
3403:
3391:
3385:
3379:
3373:
3369:
3363:
3357:
3351:
3340:
3336:
3328:
3312:
3308:
3290:
3286:
3280:
3272:
3263:
3261:
3260:
3255:
3250:
3249:
3238:
3234:
3230:
3210:
3208:
3197:
3172:
3171:
3170:
3147:
3141:
3137:
3133:
3123:
3119:
3104:
3098:
3092:
3086:
3075:
3071:
3064:
3060:
3050:
3025:exterior algebra
2991:
2982:
2980:
2979:
2974:
2963:
2962:
2944:
2943:
2928:
2924:
2923:
2922:
2909:
2907:
2906:
2905:
2892:
2891:
2890:
2877:
2874:
2869:
2848:
2847:
2846:
2835:
2834:
2797:
2795:
2793:
2792:
2787:
2776:
2775:
2763:
2762:
2749:
2748:
2738:
2737:
2736:
2725:
2724:
2692:
2685:
2683:
2682:
2677:
2672:
2671:
2656:
2655:
2642:
2640:
2639:
2638:
2625:
2617:
2614:
2609:
2575:
2567:
2561:
2557:
2544:
2540:
2517:
2499:
2497:
2495:
2494:
2489:
2487:
2486:
2485:
2472:
2462:
2457:
2456:
2445:
2444:
2437:
2424:
2420:
2416:
2412:
2408:
2404:
2400:
2398:
2397:
2392:
2390:
2389:
2378:
2377:
2363:
2359:
2348:
2344:
2340:
2336:
2334:
2333:
2328:
2326:
2325:
2324:
2323:
2312:
2311:
2293:
2292:
2270:
2268:
2266:
2265:
2260:
2249:
2248:
2230:
2229:
2217:
2216:
2195:
2194:
2176:
2175:
2151:
2150:
2139:
2138:
2123:
2119:
2115:
2113:
2111:
2110:
2105:
2103:
2102:
2089:
2071:
2070:
2069:
2068:
2042:
2041:
2040:
2039:
2019:
2018:
1998:
1996:
1994:
1993:
1988:
1980:
1979:
1961:
1960:
1948:
1947:
1926:
1925:
1907:
1906:
1894:
1893:
1867:
1859:
1839:
1828:
1819:
1808:
1793:
1751:
1741:
1731:
1725:
1719:
1713:
1704:
1702:
1701:
1696:
1694:
1693:
1678:
1677:
1659:
1658:
1643:
1642:
1616:
1608:exterior product
1602:
1600:
1598:
1597:
1592:
1590:
1589:
1574:
1573:
1560:
1559:
1543:
1508:
1504:
1498:
1492:
1488:
1478:
1469:The expressions
1465:
1461:
1429:
1427:
1426:
1421:
1416:
1415:
1402:
1401:
1383:
1382:
1369:
1368:
1349:
1341:
1337:
1331:
1324:exterior algebra
1314:
1310:
1308:
1307:
1302:
1290:
1289:
1255:
1254:
1238:
1234:
1230:
1215:
1213:
1212:
1207:
1198:
1197:
1196:
1174:
1173:
1154:
1150:
1146:
1145:
1137:
1133:
1129:
1125:
1121:
1117:
1110:
1104:
1086:
1084:
1083:
1078:
1053:
1048:
1013:
1008:
986:
984:
983:
978:
956:
951:
932:
918:
914:
904:
884:
880:
876:
868:
831:exterior algebra
805:interior product
802:
790:
763:
761:
760:
755:
738:
705:
697:
695:
694:
689:
671:
663:
661:
660:
655:
643:
626:
624:
623:
618:
588:
586:
585:
580:
522:
516:
510:
508:
507:
502:
469:
461:
457:
449:
418:
408:exterior product
405:
398:
396:
395:
390:
253:
252:
233:
222:
217:
140:
138:
137:
132:
107:
102:
83:
71:
66:
21:
18875:
18874:
18870:
18869:
18868:
18866:
18865:
18864:
18845:
18844:
18843:
18838:
18789:Albert Einstein
18756:
18737:Einstein tensor
18700:
18681:Ricci curvature
18661:Kronecker delta
18647:Notable tensors
18642:
18563:Connection form
18540:
18534:
18465:
18451:Tensor operator
18408:
18402:
18342:
18318:Computer vision
18311:
18293:
18289:Tensor calculus
18233:
18222:
18217:
18187:
18182:
18121:Banach manifold
18114:Generalizations
18109:
18064:
18001:
17898:
17860:Ricci curvature
17816:Cotangent space
17794:
17732:
17574:
17568:
17527:Exponential map
17491:
17436:
17430:
17350:
17340:
17261:
17232:
17223:
17202:
17171:
17155:Spivak, Michael
17148:
17127:
17088:
17065:
17044:
17003:Dieudonné, Jean
16991:
16944:
16928:
16923:
16922:
16913:
16912:
16908:
16893:
16879:
16875:
16847:
16843:
16838:
16789:
16769:Herbert Federer
16761:
16751:, owing to the
16746:
16740:
16729:
16707:
16699:
16691:
16680:
16678:
16675:
16674:
16665:
16659:
16649:
16633:
16620:
16593:
16588:
16586:
16583:
16582:
16578:
16555:
16552:
16551:
16534:
16533:
16524:
16523:
16516:
16509:
16508:
16504:
16498:
16497:
16491:
16490:
16483:
16477:
16476:
16469:
16467:
16464:
16463:
16443:
16421:
16420:
16418:
16415:
16414:
16404:
16398:
16395:
16387:
16384:
16376:
16370:
16346:
16342:
16330:
16326:
16314:
16310:
16291:
16287:
16280:
16276:
16266:
16257:
16256:
16254:
16251:
16250:
16241:
16216:
16215:
16203:
16202:
16200:
16197:
16196:
16187:
16184:connection form
16170:
16160:
16151:
16145:
16138:
16129:
16123:
16101:
16100:
16098:
16095:
16094:
16072:
16071:
16069:
16066:
16065:
16062:
16054:
16030:
16026:
16014:
16010:
15997:
15993:
15983:
15974:
15973:
15971:
15968:
15967:
15949:
15938:
15932:
15926:
15913:
15902:
15896:
15890:
15884:
15878:
15868:
15859:
15853:
15847:
15841:
15831:
15825:
15815:
15809:
15796:
15786:
15780:
15763:
15753:
15738:
15736:
15728:
15726:
15715:
15709:
15699:
15691:
15689:
15679:
15670:
15666:
15662:
15658:
15650:
15648:
15611:
15606:
15584:
15579:
15574:
15571:
15568:
15567:
15536:
15531:
15526:
15523:
15520:
15519:
15515:, depending on
15512:
15508:
15502:
15498:
15486:
15480:
15429:
15423:
15417:
15411:
15405:
15395:
15365:
15361:
15346:
15342:
15339:
15336:
15335:
15313:
15309:
15297:
15293:
15292:
15289:
15286:
15285:
15279:
15271:
15261:
15255:
15249:
15243:
15237:
15231:
15202:
15198:
15183:
15179:
15177:
15174:
15173:
15160:
15150:
15144:
15137:
15131:
15128:Stokes' theorem
15124:
15118:
15093:
15089:
15080:
15076:
15052:
15048:
15046:
15043:
15042:
15033:
15027:
15021:
15003:
14997:
14994:and called the
14989:
14979:
14955:
14954:
14950:
14930:
14926:
14925:
14921:
14910:
14901:
14897:
14892:
14889:
14888:
14879:
14873:
14867:
14861:
14835:
14830:
14814:
14807:
14806:
14797:
14793:
14787:
14786:
14772:
14768:
14758:
14749:
14745:
14736:
14732:
14731:
14727:
14718:
14714:
14707:
14706:
14702:
14697:
14694:
14693:
14679:
14673:
14670:
14662:
14650:
14644:
14618:
14613:
14597:
14590:
14589:
14577:
14573:
14563:
14554:
14550:
14548:
14545:
14544:
14535:
14523:
14517:
14511:
14505:
14490:
14484:
14478:
14455:
14451:
14441:
14433:
14431:
14428:
14427:
14401:
14396:
14380:
14373:
14372:
14360:
14356:
14350:
14343:
14342:
14340:
14337:
14336:
14323:
14317:
14311:
14305:
14299:
14288:
14282:
14276:
14270:
14264:
14254:
14248:
14242:
14236:
14230:
14224:
14218:
14212:
14196:
14190:
14163:
14162:
14154:
14131:
14127:
14126:
14122:
14116:
14115:
14109:
14105:
14093:
14089:
14087:
14084:
14083:
14049:
14048:
14040:
14017:
14013:
14012:
14008:
14002:
14001:
13999:
13996:
13995:
13969:
13965:
13958:
13957:
13951:
13947:
13942:
13919:
13915:
13914:
13910:
13904:
13903:
13895:
13892:
13891:
13882:
13876:
13865:
13855:
13852:
13840:
13830:
13824:
13818:
13812:
13806:
13800:
13794:
13788:
13782:
13776:
13770:
13756:
13750:
13744:
13738:
13732:
13723:
13717:
13711:
13700:
13694:
13688:
13685:
13673:
13650:
13646:
13641:
13635:
13631:
13622:
13618:
13616:
13613:
13612:
13594:
13583:
13577:
13565:
13554:
13548:
13542:
13536:
13513:
13509:
13488:
13484:
13476:
13473:
13472:
13463:
13460:
13452:
13446:
13440:
13431:
13423:
13420:
13412:
13386:
13376:
13372:
13363:
13359:
13357:
13354:
13353:
13338:
13315:
13310:
13294:
13287:
13286:
13274:
13268:
13265:
13264:
13235:
13230:
13220:
13213:
13212:
13200:
13187:
13183:
13177:
13173:
13167:
13163:
13151:
13147:
13145:
13142:
13141:
13135:
13127:
13109:
13106:
13098:
13095:
13087:
13052:
13048:
13039:
13034:
13018:
13011:
13010:
13001:
12997:
12995:
12992:
12991:
12981:
12973:
12956:
12955:
12943:
12938:
12928:
12921:
12920:
12913:
12907:
12903:
12900:
12899:
12887:
12882:
12872:
12865:
12864:
12857:
12851:
12847:
12843:
12841:
12838:
12837:
12819:
12809:
12803:
12797:
12785:
12774:
12760:
12754:
12748:
12742:
12736:
12725:
12719:
12703:
12695:
12686:
12682:
12668:
12662:
12656:
12647:
12641:
12635:
12629:
12623:
12617:
12611:
12605:
12599:
12593:
12587:
12581:
12574:
12563:
12553:
12527:
12522:
12512:
12508:
12502:
12498:
12492:
12481:
12465:
12461:
12459:
12456:
12455:
12446:
12440:
12434:
12428:
12405:
12401:
12395:
12391:
12385:
12374:
12362:
12359:
12358:
12349:
12343:
12333:
12323:
12317:
12311:
12305:
12295:
12289:
12283:
12277:
12276:In general, an
12268:
12238:
12234:
12219:
12215:
12208:
12196:
12192:
12191:
12187:
12170:
12166:
12165:
12161:
12154:
12152:
12150:
12147:
12146:
12120:
12116:
12104:
12100:
12084:
12080:
12065:
12061:
12054:
12042:
12038:
12037:
12033:
12016:
12012:
12011:
12007:
12000:
11998:
11986:
11985:
11968:
11964:
11949:
11945:
11944:
11940:
11932:
11928:
11913:
11909:
11908:
11898:
11894:
11882:
11878:
11876:
11873:
11872:
11840:
11839:
11830:
11826:
11808:
11807:
11798:
11794:
11779:
11778:
11770:
11767:
11766:
11757:
11732:
11728:
11727:
11723:
11703:
11699:
11698:
11694:
11681:
11680:
11669:
11665:
11650:
11646:
11645:
11641:
11633:
11629:
11614:
11610:
11609:
11597:
11594:
11593:
11580:
11574:
11568:
11562:
11553:
11547:
11541:
11531:
11525:
11515:
11484:
11481:
11480:
11471:
11465:
11459:
11453:
11447:
11441:
11438:
11423:
11413:
11390:
11386:
11374:
11370:
11348:
11344:
11336:
11331:
11329:
11328:
11316:
11312:
11310:
11307:
11306:
11297:
11291:
11277:
11257:
11253:
11235:
11231:
11207:
11204:
11203:
11194:
11188:
11182:
11176:
11170:
11164:
11158:
11155:
11142:
11135:
11129:
11123:
11113:
11107:
11101:
11095:
11089:
11088:A differential
11086:
11047:
11044:
11043:
11021:
11017:
11016:
11012:
11008:
10999:
10995:
10994:
10990:
10986:
10984:
10982:
10979:
10978:
10953:
10949:
10948:
10944:
10927:
10923:
10922:
10918:
10911:
10899:
10895:
10894:
10890:
10873:
10869:
10868:
10864:
10857:
10855:
10853:
10850:
10849:
10824:
10820:
10819:
10815:
10795:
10791:
10790:
10786:
10768:
10764:
10763:
10759:
10742:
10738:
10737:
10733:
10726:
10714:
10710:
10709:
10705:
10688:
10684:
10683:
10679:
10672:
10670:
10653:
10649:
10640:
10636:
10635:
10631:
10620:
10616:
10601:
10597:
10596:
10584:
10580:
10565:
10561:
10560:
10544:
10540:
10538:
10535:
10534:
10521:
10515:
10509:
10506:
10498:
10473:
10469:
10468:
10464:
10444:
10440:
10439:
10435:
10414:
10410:
10401:
10397:
10396:
10392:
10381:
10377:
10362:
10358:
10357:
10341:
10337:
10335:
10332:
10331:
10322:
10316:
10310:
10307:
10306:
10297:
10287:
10284:
10276:
10273:
10267:
10242:
10238:
10237:
10233:
10213:
10209:
10208:
10204:
10192:
10188:
10179:
10175:
10174:
10170:
10162:
10158:
10143:
10139:
10138:
10126:
10123:
10122:
10113:
10107:
10101:
10089:
10077:
10071:
10065:
10059:
10053:
10047:
10041:
10024:
10023:
10008:
10004:
9991:
9973:
9969:
9966:
9965:
9953:
9949:
9937:
9933:
9926:
9905:
9901:
9898:
9897:
9885:
9881:
9869:
9865:
9858:
9837:
9833:
9830:
9829:
9814:
9810:
9797:
9779:
9775:
9771:
9769:
9766:
9765:
9756:
9750:
9744:
9732:
9716:
9695:
9688:
9687:
9685:
9682:
9681:
9671:
9665:
9646:
9629:
9624:
9622:
9621:
9606:
9599:
9598:
9579:
9574:
9573:
9572:
9567:
9565:
9564:
9549:
9542:
9541:
9539:
9536:
9535:
9526:
9516:
9494:
9477:
9472:
9470:
9469:
9454:
9447:
9446:
9444:
9441:
9440:
9428:
9422:
9416:
9390:
9386:
9380:
9373:
9372:
9359:
9355:
9340:
9335:
9334:
9333:
9328:
9326:
9325:
9313:
9309:
9303:
9296:
9295:
9284:
9279:
9277:
9276:
9259:
9254:
9252:
9251:
9243:
9240:
9239:
9230:
9224:
9218:
9201:
9197:
9182:
9176:
9173:
9172:
9166:
9162:
9156:
9130:
9126:
9113:
9109:
9099:
9094:
9092:
9091:
9079:
9075:
9064:
9059:
9057:
9056:
9039:
9034:
9032:
9031:
9023:
9020:
9019:
9010:
8989:
8983:
8977:
8968:
8962:
8958:
8952:
8926:
8922:
8916:
8912:
8897:
8893:
8887:
8883:
8865:
8861:
8849:
8845:
8830:
8826:
8817:
8813:
8804:
8800:
8795:
8792:
8791:
8782:
8776:
8770:
8767:
8759:
8756:
8750:
8744:
8738:
8732:
8726:
8720:
8710:
8701:
8695:
8689:
8683:
8669:
8660:
8654:
8640:
8634:
8625:
8619:
8616:
8610:
8600:
8594:
8588:
8580:
8570:
8564:
8558:
8545:
8537:
8531:
8525:
8513:
8503:
8498:and called the
8496:
8490:
8484:
8478:
8464:
8458:
8444:
8441:
8435:
8423:
8422:
8411:
8410:
8401:
8389:
8381:
8367:
8318:
8314:
8278:
8274:
8263:
8258:
8256:
8255:
8237:
8233:
8222:
8217:
8215:
8214:
8196:
8192:
8181:
8176:
8174:
8173:
8155:
8151:
8137:
8134:
8133:
8127:cochain complex
8119:
8116:
8068:
8065:
8054:
8046:
8014:
8010:
7992:
7988:
7980:
7977:
7976:
7941:
7937:
7926:
7921:
7919:
7918:
7900:
7896:
7888:
7885:
7884:
7870:
7833:
7829:
7782:
7779:
7778:
7736:
7732:
7709:
7706:
7705:
7696:
7690:
7684:
7678:
7669:(also known as
7602:
7599:
7598:
7536:
7533:
7532:
7523:
7517:
7511:
7505:
7496:
7483:
7454:
7450:
7435:
7431:
7413:
7409:
7394:
7390:
7366:
7362:
7348:
7345:
7344:
7331:
7327:
7317:
7306:
7252:
7232:
7230:
7216:
7213:
7212:
7193:
7190:
7189:
7167:
7164:
7163:
7143:
7139:
7133:
7128:
7127:
7125:
7122:
7121:
7101:
7097:
7091:
7084:
7083:
7081:
7078:
7077:
7025:
7022:
7021:
7008:
6998:
6989:
6983:
6977:
6971:
6965:
6955:
6945:
6939:
6933:
6927:
6921:
6918:
6889:
6878:
6872:
6856:
6830:
6826:
6820:
6815:
6814:
6802:
6798:
6792:
6785:
6784:
6776:
6773:
6772:
6763:
6760:symmetric group
6756:
6748:
6713:
6709:
6685:
6681:
6672:
6668:
6645:
6641:
6634:
6620:
6615:
6603:
6599:
6584:
6580:
6568:
6564:
6553:
6550:
6549:
6540:
6524:
6521:
6520:
6494:
6490:
6484:
6479:
6478:
6466:
6462:
6456:
6451:
6450:
6442:
6439:
6438:
6424:
6423:. Differential
6421:
6413:
6403:
6395:
6389:
6379:
6371:
6361:
6353:
6350:
6342:
6329:
6323:
6319:
6313:
6307:
6301:
6295:
6273:
6261:
6257:
6251:
6240:
6227:
6223:
6221:
6218:
6217:
6211:multilinear map
6187:
6181:
6180:
6179:
6170:
6166:
6160:
6153:
6152:
6146:
6145:
6133:
6128:
6118:
6111:
6110:
6108:
6105:
6104:
6095:
6089:
6073:
6061:
6057:
6051:
6044:
6043:
6034:
6030:
6028:
6025:
6024:
6018:
6012:
6002:
5996:
5990:
5980:
5972:
5966:
5960:
5950:
5942:
5916:
5911:
5901:
5894:
5893:
5884:
5880:
5878:
5875:
5874:
5865:
5859:
5849:
5838:
5828:
5822:
5816:
5802:
5792:
5789:smooth manifold
5782:
5779:
5773:
5764:
5760:
5756:
5748:
5744:
5742:smooth manifold
5734:
5728:
5720:
5716:
5712:
5706:
5697:
5687:
5680:
5674:
5670:
5667:
5666:
5657:
5651:
5641:
5619:
5615:
5614:
5610:
5590:
5586:
5585:
5581:
5567:
5563:
5562:
5558:
5546:
5542:
5533:
5529:
5523:
5519:
5518:
5514:
5508:
5495:
5491:
5482:
5478:
5469:
5465:
5464:
5458:
5455:
5454:
5447:
5443:
5439:
5435:
5431:
5418:
5414:
5407:
5384:
5380:
5368:
5364:
5351:
5347:
5343:
5336:
5332:
5328:
5326:
5320:
5303:
5290:
5286:
5274:
5270:
5261:
5250:
5235:
5232:
5231:
5222:
5212:
5206:
5205:
5204:
5199:
5193:
5186:
5182:
5159:
5155:
5143:
5139:
5124:
5120:
5108:
5104:
5099:
5096:
5095:
5088:
5059:
5055:
5043:
5039:
5026:
5022:
5018:
5011:
5007:
5003:
5001:
4995:
4978:
4972:
4969:
4968:
4957:
4953:
4949:
4945:
4934:
4930:
4901:
4897:
4893:
4886:
4882:
4878:
4876:
4864:
4860:
4856:
4849:
4845:
4841:
4839:
4837:
4834:
4833:
4826:
4800:
4796:
4786:
4782:
4778:
4768:
4764:
4763:
4761:
4749:
4745:
4735:
4731:
4727:
4717:
4713:
4712:
4710:
4708:
4705:
4704:
4697:
4690:
4687:
4679:
4675:
4664:
4660:
4650:
4646:
4642:
4638:
4634:
4630:
4623:
4620:
4612:
4592:
4588:
4578:
4574:
4568:
4557:
4545:
4542:
4541:
4534:
4530:
4526:
4516:
4506:
4486:
4482:
4476:
4472:
4451:
4447:
4441:
4428:
4424:
4422:
4419:
4418:
4412:
4404:
4396:
4387:
4383:
4379:
4376:
4368:
4365:
4357:
4344:
4340:
4336:
4332:
4328:
4325:
4317:
4294:
4290:
4284:
4280:
4256:
4252:
4248:
4240:
4238:
4232:
4221:
4208:
4204:
4199:
4196:
4195:
4184:
4148:
4144:
4131:
4127:
4123:
4115:
4113:
4107:
4096:
4081:
4078:
4077:
4065:
4048:
4038:
4032:
4026:
4020:
4016:
4012:
4006:
4000:
3994:
3990:
3984:
3980:
3976:
3966:
3960:
3956:
3947:
3939:
3933:
3926:
3914:
3905:
3896:
3892:
3884:
3880:
3876:
3872:
3866:
3855:
3845:
3837:
3833:
3829:
3820:and called the
3818:
3810:
3804:
3798:
3786:
3780:
3774:
3757:
3756:
3734:
3733:
3729:
3716:
3694:
3690:
3686:
3680:
3679:
3657:
3656:
3652:
3624:
3623:
3619:
3609:
3587:
3579:
3578:
3574:
3567:
3565:
3562:
3561:
3552:
3549:linear function
3537:
3531:
3505:
3501:
3497:
3489:
3487:
3478:
3474:
3470:
3463:
3459:
3455:
3453:
3447:
3436:
3420:
3416:
3412:
3404:
3402:
3400:
3397:
3396:
3387:
3381:
3375:
3371:
3365:
3359:
3353:
3342:
3338:
3334:
3324:
3318:
3310:
3307:
3295:
3288:
3282:
3278:
3268:
3239:
3226:
3201:
3196:
3195:
3192:
3191:
3166:
3165:
3161:
3156:
3153:
3152:
3143:
3139:
3135:
3125:
3121:
3115:
3109:
3100:
3094:
3088:
3077:
3073:
3069:
3067:smooth function
3062:
3056:
3048:
3045:
3037:Stokes' theorem
2987:
2952:
2948:
2939:
2935:
2918:
2914:
2901:
2897:
2893:
2886:
2882:
2878:
2876:
2870:
2859:
2854:
2850:
2836:
2830:
2829:
2828:
2821:
2806:
2803:
2802:
2771:
2767:
2758:
2754:
2744:
2740:
2726:
2720:
2719:
2718:
2711:
2699:
2696:
2695:
2694:
2690:
2667:
2663:
2651:
2647:
2634:
2630:
2626:
2618:
2616:
2610:
2599:
2584:
2581:
2580:
2569:
2563:
2559:
2546:
2542:
2523:
2513:
2506:
2481:
2468:
2467:
2466:
2458:
2446:
2440:
2439:
2438:
2433:
2431:
2428:
2427:
2426:
2422:
2418:
2414:
2410:
2406:
2402:
2379:
2373:
2372:
2371:
2369:
2366:
2365:
2361:
2350:
2346:
2342:
2338:
2313:
2307:
2306:
2305:
2298:
2294:
2288:
2284:
2276:
2273:
2272:
2244:
2240:
2225:
2221:
2212:
2208:
2190:
2186:
2171:
2167:
2140:
2134:
2133:
2132:
2130:
2127:
2126:
2125:
2121:
2117:
2098:
2094:
2079:
2064:
2060:
2059:
2055:
2035:
2031:
2030:
2026:
2014:
2010:
2005:
2002:
2001:
2000:
1975:
1971:
1956:
1952:
1943:
1939:
1921:
1917:
1902:
1898:
1889:
1885:
1874:
1871:
1870:
1869:
1865:
1857:
1854:
1838:
1830:
1827:
1821:
1810:
1799:
1753:
1752:; for example:
1743:
1733:
1727:
1721:
1715:
1709:
1689:
1685:
1673:
1669:
1654:
1650:
1638:
1634:
1629:
1626:
1625:
1614:
1585:
1581:
1569:
1565:
1549:
1545:
1521:
1515:
1512:
1511:
1510:
1506:
1500:
1494:
1490:
1480:
1470:
1463:
1451:
1442:
1434:
1411:
1407:
1397:
1393:
1378:
1374:
1364:
1360:
1358:
1355:
1354:
1347:
1339:
1333:
1327:
1312:
1273:
1269:
1250:
1246:
1244:
1241:
1240:
1236:
1232:
1228:
1189:
1188:
1184:
1169:
1165:
1163:
1160:
1159:
1152:
1148:
1143:
1139:
1135:
1134:is an oriented
1131:
1127:
1123:
1119:
1115:
1106:
1096:
1049:
1044:
1009:
1004:
998:
995:
994:
952:
947:
941:
938:
937:
920:
916:
906:
896:
882:
878:
874:
866:
865:A differential
863:
847:
822:
800:
788:
778:Stokes' theorem
774:Green's theorem
731:
711:
708:
707:
703:
677:
674:
673:
665:
649:
646:
645:
639:
594:
591:
590:
541:
538:
537:
518:
512:
475:
472:
471:
467:
459:
455:
421:
416:
403:
248:
244:
242:
239:
238:
229:
220:
145:
103:
98:
92:
89:
88:
79:
69:
54:
28:
23:
22:
15:
12:
11:
5:
18873:
18863:
18862:
18857:
18840:
18839:
18837:
18836:
18831:
18829:Woldemar Voigt
18826:
18821:
18816:
18811:
18806:
18801:
18796:
18794:Leonhard Euler
18791:
18786:
18781:
18776:
18770:
18768:
18766:Mathematicians
18762:
18761:
18758:
18757:
18755:
18754:
18749:
18744:
18739:
18734:
18729:
18724:
18719:
18714:
18708:
18706:
18702:
18701:
18699:
18698:
18693:
18691:Torsion tensor
18688:
18683:
18678:
18673:
18668:
18663:
18657:
18655:
18648:
18644:
18643:
18641:
18640:
18635:
18630:
18625:
18620:
18615:
18610:
18605:
18600:
18595:
18590:
18585:
18580:
18575:
18570:
18565:
18560:
18555:
18550:
18544:
18542:
18536:
18535:
18533:
18532:
18526:
18524:Tensor product
18521:
18516:
18514:Symmetrization
18511:
18506:
18504:Lie derivative
18501:
18496:
18491:
18486:
18481:
18475:
18473:
18467:
18466:
18464:
18463:
18458:
18453:
18448:
18443:
18438:
18433:
18428:
18426:Tensor density
18423:
18418:
18412:
18410:
18404:
18403:
18401:
18400:
18398:Voigt notation
18395:
18390:
18385:
18383:Ricci calculus
18380:
18375:
18370:
18368:Index notation
18365:
18360:
18354:
18352:
18348:
18347:
18344:
18343:
18341:
18340:
18335:
18330:
18325:
18320:
18314:
18312:
18310:
18309:
18304:
18298:
18295:
18294:
18292:
18291:
18286:
18284:Tensor algebra
18281:
18276:
18271:
18266:
18264:Dyadic algebra
18261:
18256:
18250:
18248:
18239:
18235:
18234:
18227:
18224:
18223:
18216:
18215:
18208:
18201:
18193:
18184:
18183:
18181:
18180:
18175:
18170:
18165:
18160:
18159:
18158:
18148:
18143:
18138:
18133:
18128:
18123:
18117:
18115:
18111:
18110:
18108:
18107:
18102:
18097:
18092:
18087:
18082:
18076:
18074:
18070:
18069:
18066:
18065:
18063:
18062:
18057:
18052:
18047:
18042:
18037:
18032:
18027:
18022:
18017:
18011:
18009:
18003:
18002:
18000:
17999:
17994:
17989:
17984:
17979:
17974:
17969:
17959:
17954:
17949:
17939:
17934:
17929:
17924:
17919:
17914:
17908:
17906:
17900:
17899:
17897:
17896:
17891:
17886:
17885:
17884:
17874:
17869:
17868:
17867:
17857:
17852:
17847:
17842:
17841:
17840:
17830:
17825:
17824:
17823:
17813:
17808:
17802:
17800:
17796:
17795:
17793:
17792:
17787:
17782:
17777:
17776:
17775:
17765:
17760:
17755:
17749:
17747:
17740:
17734:
17733:
17731:
17730:
17725:
17715:
17710:
17696:
17691:
17686:
17681:
17676:
17674:Parallelizable
17671:
17666:
17661:
17660:
17659:
17649:
17644:
17639:
17634:
17629:
17624:
17619:
17614:
17609:
17604:
17594:
17584:
17578:
17576:
17570:
17569:
17567:
17566:
17561:
17556:
17554:Lie derivative
17551:
17549:Integral curve
17546:
17541:
17536:
17535:
17534:
17524:
17519:
17518:
17517:
17510:Diffeomorphism
17507:
17501:
17499:
17493:
17492:
17490:
17489:
17484:
17479:
17474:
17469:
17464:
17459:
17454:
17449:
17443:
17441:
17432:
17431:
17429:
17428:
17423:
17418:
17413:
17408:
17403:
17398:
17393:
17388:
17387:
17386:
17381:
17371:
17370:
17369:
17358:
17356:
17355:Basic concepts
17352:
17351:
17339:
17338:
17331:
17324:
17316:
17310:
17309:
17297:
17270:
17252:
17231:
17230:External links
17228:
17227:
17226:
17221:
17205:
17200:
17176:
17169:
17151:
17146:
17130:
17125:
17108:
17091:
17086:
17070:
17063:
17047:
17042:
17018:
16999:
16989:
16973:
16961:math/0306194v1
16947:
16942:
16936:, Birkhäuser,
16927:
16924:
16921:
16920:
16906:
16891:
16873:
16840:
16839:
16837:
16834:
16833:
16832:
16827:
16822:
16817:
16810:
16805:
16800:
16795:
16788:
16785:
16760:
16757:
16726:
16725:
16714:
16710:
16706:
16702:
16698:
16694:
16690:
16687:
16683:
16614:Maxwell 2-form
16596:
16591:
16559:
16548:
16547:
16532:
16522:
16519:
16517:
16507:
16503:
16500:
16499:
16489:
16486:
16484:
16475:
16472:
16471:
16428:
16425:
16391:
16380:
16367:
16366:
16355:
16349:
16345:
16341:
16338:
16333:
16329:
16325:
16322:
16317:
16313:
16309:
16303:
16300:
16297:
16294:
16290:
16283:
16279:
16273:
16270:
16265:
16236:
16235:
16224:
16214:
16211:
16180:gauge theories
16167:curvature form
16156:
16149:
16134:
16127:
16108:
16105:
16079:
16076:
16058:
16051:
16050:
16039:
16033:
16029:
16025:
16022:
16017:
16013:
16009:
16003:
16000:
15996:
15990:
15987:
15982:
15957:Faraday 2-form
15948:
15945:
15912:
15909:
15846:vectors in an
15714:pulls back to
15686:absolute value
15631:
15628:
15625:
15622:
15619:
15614:
15609:
15605:
15601:
15598:
15595:
15592:
15587:
15582:
15578:
15553:
15550:
15547:
15544:
15539:
15534:
15530:
15479:
15476:
15379:
15376:
15373:
15368:
15364:
15360:
15357:
15354:
15349:
15345:
15321:
15316:
15312:
15308:
15305:
15300:
15296:
15228:
15227:
15216:
15213:
15208:
15205:
15201:
15197:
15194:
15191:
15186:
15182:
15120:Main article:
15117:
15114:
15113:
15112:
15101:
15096:
15092:
15088:
15083:
15079:
15075:
15072:
15069:
15066:
15063:
15060:
15055:
15051:
15018:Dieudonné 1972
14986:Dieudonné 1972
14976:
14975:
14964:
14958:
14953:
14947:
14944:
14941:
14936:
14933:
14929:
14924:
14920:
14917:
14913:
14909:
14904:
14900:
14896:
14858:
14857:
14846:
14843:
14838:
14833:
14829:
14823:
14820:
14817:
14811:
14805:
14800:
14796:
14790:
14784:
14780:
14775:
14771:
14766:
14761:
14757:
14752:
14748:
14744:
14739:
14735:
14730:
14726:
14721:
14717:
14710:
14705:
14701:
14666:
14641:
14640:
14629:
14626:
14621:
14616:
14612:
14606:
14603:
14600:
14594:
14588:
14585:
14580:
14576:
14571:
14566:
14562:
14557:
14553:
14522:determines an
14475:
14474:
14463:
14458:
14454:
14449:
14444:
14440:
14436:
14421:
14420:
14409:
14404:
14399:
14395:
14389:
14386:
14383:
14377:
14371:
14368:
14363:
14359:
14353:
14347:
14187:
14186:
14175:
14172:
14166:
14161:
14157:
14153:
14148:
14145:
14142:
14137:
14134:
14130:
14125:
14119:
14112:
14108:
14104:
14101:
14096:
14092:
14077:Dieudonné 1972
14073:
14072:
14061:
14058:
14052:
14047:
14043:
14039:
14034:
14031:
14028:
14023:
14020:
14016:
14011:
14005:
13989:
13988:
13977:
13972:
13968:
13961:
13954:
13950:
13945:
13941:
13936:
13933:
13930:
13925:
13922:
13918:
13913:
13907:
13902:
13899:
13848:
13681:
13670:
13669:
13658:
13653:
13649:
13644:
13638:
13634:
13630:
13625:
13621:
13533:
13532:
13521:
13516:
13512:
13508:
13505:
13502:
13499:
13494:
13491:
13487:
13483:
13480:
13456:
13436:
13427:
13416:
13409:
13408:
13397:
13393:
13389:
13385:
13379:
13375:
13371:
13366:
13362:
13335:
13334:
13323:
13318:
13313:
13309:
13303:
13300:
13297:
13291:
13285:
13281:
13277:
13273:
13258:
13257:
13246:
13243:
13238:
13233:
13229:
13223:
13217:
13211:
13207:
13203:
13199:
13195:
13190:
13186:
13180:
13176:
13170:
13166:
13162:
13159:
13154:
13150:
13131:
13102:
13091:
13084:
13083:
13072:
13069:
13066:
13063:
13058:
13055:
13051:
13047:
13042:
13037:
13033:
13027:
13024:
13021:
13015:
13009:
13004:
13000:
12985:Dieudonné 1972
12977:
12970:
12969:
12954:
12951:
12946:
12941:
12937:
12931:
12925:
12919:
12916:
12914:
12910:
12906:
12902:
12901:
12898:
12895:
12890:
12885:
12881:
12875:
12869:
12863:
12860:
12858:
12854:
12850:
12846:
12845:
12721:Main article:
12718:
12715:
12578:Dieudonné 1972
12573:
12570:
12550:
12549:
12538:
12535:
12530:
12525:
12521:
12515:
12511:
12505:
12501:
12495:
12490:
12487:
12484:
12480:
12476:
12473:
12468:
12464:
12425:
12424:
12413:
12408:
12404:
12398:
12394:
12388:
12383:
12380:
12377:
12373:
12369:
12366:
12261:
12260:
12246:
12241:
12237:
12233:
12230:
12227:
12222:
12218:
12214:
12211:
12206:
12199:
12195:
12190:
12186:
12183:
12180:
12173:
12169:
12164:
12160:
12157:
12140:
12139:
12128:
12123:
12119:
12115:
12112:
12107:
12103:
12099:
12092:
12087:
12083:
12079:
12076:
12073:
12068:
12064:
12060:
12057:
12052:
12045:
12041:
12036:
12032:
12029:
12026:
12019:
12015:
12010:
12006:
12003:
11997:
11994:
11989:
11984:
11981:
11978:
11971:
11967:
11963:
11960:
11957:
11952:
11948:
11943:
11935:
11931:
11927:
11924:
11921:
11916:
11912:
11907:
11901:
11897:
11893:
11890:
11885:
11881:
11866:
11865:
11854:
11851:
11848:
11843:
11838:
11833:
11829:
11825:
11822:
11819:
11816:
11811:
11806:
11801:
11797:
11793:
11790:
11787:
11782:
11777:
11774:
11762:is defined by
11754:
11753:
11742:
11735:
11731:
11726:
11722:
11719:
11716:
11713:
11706:
11702:
11697:
11693:
11689:
11684:
11679:
11672:
11668:
11664:
11661:
11658:
11653:
11649:
11644:
11636:
11632:
11628:
11625:
11622:
11617:
11613:
11608:
11604:
11601:
11512:
11511:
11500:
11497:
11494:
11491:
11488:
11452:-manifold and
11437:
11434:
11410:
11409:
11398:
11393:
11389:
11385:
11382:
11377:
11373:
11369:
11365:
11362:
11359:
11356:
11351:
11347:
11334:
11324:
11319:
11315:
11274:
11273:
11260:
11256:
11252:
11249:
11246:
11243:
11238:
11234:
11230:
11226:
11223:
11220:
11217:
11214:
11211:
11154:
11151:
11146:= 1, 2, 3, ...
11085:
11082:
11069:
11066:
11063:
11060:
11057:
11054:
11051:
11024:
11020:
11015:
11011:
11002:
10998:
10993:
10989:
10963:
10956:
10952:
10947:
10943:
10940:
10937:
10930:
10926:
10921:
10917:
10914:
10909:
10902:
10898:
10893:
10889:
10886:
10883:
10876:
10872:
10867:
10863:
10860:
10846:
10845:
10834:
10827:
10823:
10818:
10814:
10811:
10808:
10805:
10798:
10794:
10789:
10785:
10778:
10771:
10767:
10762:
10758:
10755:
10752:
10745:
10741:
10736:
10732:
10729:
10724:
10717:
10713:
10708:
10704:
10701:
10698:
10691:
10687:
10682:
10678:
10675:
10669:
10666:
10663:
10656:
10652:
10648:
10643:
10639:
10634:
10630:
10623:
10619:
10615:
10612:
10609:
10604:
10600:
10595:
10587:
10583:
10579:
10576:
10573:
10568:
10564:
10559:
10555:
10552:
10547:
10543:
10502:
10495:
10494:
10483:
10476:
10472:
10467:
10463:
10460:
10457:
10454:
10447:
10443:
10438:
10434:
10430:
10427:
10424:
10417:
10413:
10409:
10404:
10400:
10395:
10391:
10384:
10380:
10376:
10373:
10370:
10365:
10361:
10356:
10352:
10349:
10344:
10340:
10302:
10295:
10291:
10280:
10271:
10264:
10263:
10252:
10245:
10241:
10236:
10232:
10229:
10226:
10223:
10216:
10212:
10207:
10203:
10195:
10191:
10187:
10182:
10178:
10173:
10165:
10161:
10157:
10154:
10151:
10146:
10142:
10137:
10133:
10130:
10085:
10038:
10037:
10022:
10019:
10016:
10011:
10007:
10003:
10000:
9997:
9994:
9992:
9990:
9987:
9984:
9981:
9976:
9972:
9968:
9967:
9964:
9961:
9956:
9952:
9948:
9945:
9940:
9936:
9932:
9929:
9927:
9925:
9922:
9919:
9916:
9913:
9908:
9904:
9900:
9899:
9896:
9893:
9888:
9884:
9880:
9877:
9872:
9868:
9864:
9861:
9859:
9857:
9854:
9851:
9848:
9845:
9840:
9836:
9832:
9831:
9828:
9825:
9822:
9817:
9813:
9809:
9806:
9803:
9800:
9798:
9796:
9793:
9790:
9787:
9782:
9778:
9774:
9773:
9755:are forms and
9719:
9715:
9712:
9709:
9706:
9703:
9698:
9692:
9662:
9661:
9649:
9645:
9642:
9632:
9627:
9617:
9614:
9609:
9603:
9590:
9587:
9582:
9577:
9570:
9560:
9557:
9552:
9546:
9513:
9512:
9501:
9497:
9493:
9490:
9480:
9475:
9465:
9462:
9457:
9451:
9435:vector bundles
9413:
9412:
9401:
9398:
9393:
9389:
9383:
9377:
9362:
9358:
9354:
9351:
9348:
9343:
9338:
9331:
9321:
9316:
9312:
9306:
9300:
9287:
9282:
9272:
9262:
9257:
9247:
9204:
9200:
9196:
9193:
9190:
9185:
9181:
9153:
9152:
9141:
9138:
9133:
9129:
9116:
9112:
9108:
9105:
9102:
9097:
9087:
9082:
9078:
9067:
9062:
9052:
9042:
9037:
9027:
8949:
8948:
8937:
8934:
8929:
8925:
8919:
8915:
8911:
8908:
8905:
8900:
8896:
8890:
8886:
8882:
8877:
8874:
8871:
8868:
8864:
8860:
8857:
8852:
8848:
8844:
8841:
8838:
8833:
8829:
8825:
8820:
8816:
8812:
8807:
8803:
8799:
8763:
8754:
8668:Formally, let
8614:
8584:
8549:
8535:
8521:
8494:
8434:
8431:
8374:Poincaré lemma
8356:
8355:
8344:
8338:
8332:
8329:
8326:
8321:
8317:
8310:
8304:
8298:
8292:
8289:
8286:
8281:
8277:
8266:
8261:
8251:
8248:
8245:
8240:
8236:
8225:
8220:
8210:
8207:
8204:
8199:
8195:
8184:
8179:
8169:
8166:
8163:
8158:
8154:
8147:
8141:
8115:
8112:
8064:
8061:
8034:
8031:
8028:
8023:
8020:
8017:
8013:
8009:
8006:
8003:
8000:
7995:
7991:
7987:
7984:
7974:codifferential
7961:
7958:
7955:
7950:
7947:
7944:
7940:
7929:
7924:
7914:
7911:
7908:
7903:
7899:
7895:
7892:
7869:
7866:
7853:
7850:
7847:
7844:
7841:
7836:
7832:
7828:
7825:
7822:
7819:
7816:
7813:
7810:
7807:
7804:
7801:
7798:
7795:
7792:
7789:
7786:
7768:
7767:
7756:
7753:
7750:
7747:
7742:
7739:
7735:
7731:
7728:
7725:
7722:
7719:
7716:
7713:
7663:
7662:
7651:
7648:
7645:
7642:
7639:
7636:
7633:
7630:
7627:
7624:
7621:
7618:
7615:
7612:
7609:
7606:
7596:
7585:
7582:
7579:
7576:
7573:
7570:
7567:
7564:
7561:
7558:
7555:
7552:
7549:
7546:
7543:
7540:
7492:
7480:
7479:
7468:
7465:
7462:
7457:
7453:
7449:
7446:
7443:
7438:
7434:
7430:
7427:
7424:
7421:
7416:
7412:
7408:
7405:
7402:
7397:
7393:
7389:
7386:
7383:
7380:
7377:
7374:
7369:
7365:
7361:
7358:
7355:
7352:
7303:
7302:
7291:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7264:
7261:
7258:
7255:
7250:
7247:
7244:
7241:
7238:
7235:
7229:
7226:
7223:
7220:
7197:
7177:
7174:
7171:
7151:
7146:
7142:
7136:
7131:
7109:
7104:
7100:
7094:
7088:
7074:
7073:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
6917:
6914:
6906:Lie derivative
6888:
6885:
6853:
6852:
6841:
6838:
6833:
6829:
6823:
6818:
6813:
6810:
6805:
6801:
6795:
6789:
6783:
6780:
6752:
6745:
6744:
6733:
6730:
6725:
6722:
6719:
6716:
6712:
6708:
6705:
6702:
6697:
6694:
6691:
6688:
6684:
6680:
6675:
6671:
6667:
6664:
6661:
6658:
6655:
6648:
6644:
6640:
6637:
6633:
6626:
6623:
6619:
6614:
6611:
6606:
6602:
6598:
6595:
6592:
6587:
6583:
6579:
6576:
6571:
6567:
6563:
6560:
6557:
6528:
6517:
6516:
6505:
6502:
6497:
6493:
6487:
6482:
6477:
6474:
6469:
6465:
6459:
6454:
6449:
6446:
6417:
6411:tangent vector
6399:
6384:(induced by a
6375:
6357:
6346:
6292:
6291:
6280:
6276:
6272:
6269:
6264:
6260:
6254:
6249:
6246:
6243:
6239:
6235:
6230:
6226:
6204:
6203:
6190:
6184:
6178:
6173:
6169:
6163:
6157:
6149:
6144:
6141:
6136:
6131:
6127:
6121:
6115:
6076:
6072:
6069:
6064:
6060:
6054:
6048:
6042:
6037:
6033:
6009:tangent bundle
5976:
5946:
5939:
5938:
5927:
5924:
5919:
5914:
5910:
5904:
5898:
5892:
5887:
5883:
5809:exterior power
5799:smooth section
5772:
5769:
5702:
5692:
5688:< ... <
5685:
5678:
5662:
5655:
5649:
5645:
5638:
5637:
5622:
5618:
5613:
5609:
5606:
5603:
5600:
5593:
5589:
5584:
5580:
5577:
5570:
5566:
5561:
5557:
5549:
5545:
5541:
5536:
5532:
5526:
5522:
5517:
5511:
5506:
5503:
5498:
5494:
5490:
5485:
5481:
5477:
5472:
5468:
5463:
5406:To summarize:
5404:
5403:
5392:
5387:
5383:
5379:
5376:
5371:
5367:
5363:
5354:
5350:
5346:
5339:
5335:
5331:
5323:
5318:
5315:
5312:
5309:
5306:
5302:
5298:
5293:
5289:
5285:
5282:
5277:
5273:
5269:
5264:
5259:
5256:
5253:
5249:
5245:
5242:
5239:
5218:
5207:
5179:
5178:
5167:
5162:
5158:
5154:
5151:
5146:
5142:
5138:
5135:
5132:
5127:
5123:
5119:
5116:
5111:
5107:
5103:
5085:
5084:
5073:
5070:
5067:
5062:
5058:
5054:
5051:
5046:
5042:
5038:
5029:
5025:
5021:
5014:
5010:
5006:
4998:
4993:
4990:
4987:
4984:
4981:
4977:
4927:
4926:
4915:
4912:
4904:
4900:
4896:
4889:
4885:
4881:
4875:
4867:
4863:
4859:
4852:
4848:
4844:
4823:
4822:
4811:
4803:
4799:
4795:
4789:
4785:
4781:
4776:
4771:
4767:
4760:
4752:
4748:
4744:
4738:
4734:
4730:
4725:
4720:
4716:
4683:
4616:
4609:
4608:
4595:
4591:
4587:
4581:
4577:
4571:
4566:
4563:
4560:
4556:
4552:
4549:
4503:
4502:
4489:
4485:
4479:
4475:
4471:
4468:
4465:
4462:
4459:
4454:
4450:
4444:
4440:
4436:
4431:
4427:
4408:
4400:
4392:
4372:
4361:
4321:
4314:
4313:
4302:
4297:
4293:
4287:
4283:
4279:
4276:
4273:
4270:
4267:
4259:
4255:
4251:
4246:
4243:
4235:
4230:
4227:
4224:
4220:
4216:
4211:
4207:
4203:
4179:
4178:
4169:
4167:
4156:
4151:
4147:
4143:
4134:
4130:
4126:
4121:
4118:
4110:
4105:
4102:
4099:
4095:
4091:
4088:
4085:
4061:
3952:
3943:
3922:
3888:
3851:
3841:
3814:
3771:
3770:
3755:
3752:
3749:
3746:
3743:
3737:
3732:
3728:
3725:
3722:
3719:
3717:
3715:
3712:
3709:
3706:
3703:
3697:
3693:
3689:
3685:
3682:
3681:
3678:
3675:
3672:
3669:
3666:
3660:
3655:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3627:
3622:
3618:
3615:
3612:
3610:
3608:
3605:
3602:
3599:
3596:
3590:
3586:
3582:
3577:
3573:
3570:
3569:
3533:
3528:
3527:
3516:
3508:
3504:
3500:
3495:
3492:
3481:
3477:
3473:
3466:
3462:
3458:
3450:
3445:
3442:
3439:
3435:
3431:
3423:
3419:
3415:
3410:
3407:
3320:
3303:
3281:by evaluating
3265:
3264:
3253:
3248:
3245:
3242:
3237:
3233:
3229:
3225:
3222:
3219:
3216:
3213:
3207:
3204:
3200:
3194:
3190:
3187:
3184:
3181:
3178:
3175:
3169:
3164:
3160:
3111:
3044:
3041:
3021:antiderivation
2984:
2983:
2972:
2969:
2966:
2961:
2958:
2955:
2951:
2947:
2942:
2938:
2934:
2931:
2927:
2921:
2917:
2913:
2904:
2900:
2896:
2889:
2885:
2881:
2873:
2868:
2865:
2862:
2858:
2853:
2845:
2842:
2839:
2833:
2827:
2824:
2820:
2816:
2813:
2810:
2785:
2782:
2779:
2774:
2770:
2766:
2761:
2757:
2753:
2747:
2743:
2735:
2732:
2729:
2723:
2717:
2714:
2710:
2706:
2703:
2687:
2686:
2675:
2670:
2666:
2662:
2659:
2654:
2650:
2646:
2637:
2633:
2629:
2624:
2621:
2613:
2608:
2605:
2602:
2598:
2594:
2591:
2588:
2505:
2502:
2484:
2479:
2476:
2471:
2465:
2461:
2455:
2452:
2449:
2443:
2436:
2388:
2385:
2382:
2376:
2322:
2319:
2316:
2310:
2304:
2301:
2297:
2291:
2287:
2283:
2280:
2258:
2255:
2252:
2247:
2243:
2239:
2236:
2233:
2228:
2224:
2220:
2215:
2211:
2207:
2204:
2201:
2198:
2193:
2189:
2185:
2182:
2179:
2174:
2170:
2166:
2163:
2160:
2157:
2154:
2149:
2146:
2143:
2137:
2101:
2097:
2093:
2088:
2085:
2082:
2078:
2074:
2067:
2063:
2058:
2054:
2051:
2048:
2045:
2038:
2034:
2029:
2025:
2022:
2017:
2013:
2009:
1986:
1983:
1978:
1974:
1970:
1967:
1964:
1959:
1955:
1951:
1946:
1942:
1938:
1935:
1932:
1929:
1924:
1920:
1916:
1913:
1910:
1905:
1901:
1897:
1892:
1888:
1884:
1881:
1878:
1853:
1850:
1834:
1825:
1706:
1705:
1692:
1688:
1684:
1681:
1676:
1672:
1668:
1665:
1662:
1657:
1653:
1649:
1646:
1641:
1637:
1633:
1588:
1584:
1580:
1577:
1572:
1568:
1564:
1558:
1555:
1552:
1548:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1520:
1447:
1438:
1431:
1430:
1419:
1414:
1410:
1406:
1400:
1396:
1392:
1389:
1386:
1381:
1377:
1373:
1367:
1363:
1300:
1297:
1293:
1288:
1285:
1282:
1279:
1276:
1272:
1268:
1265:
1262:
1258:
1253:
1249:
1225:measure theory
1217:
1216:
1205:
1201:
1195:
1192:
1187:
1183:
1180:
1177:
1172:
1168:
1088:
1087:
1076:
1073:
1070:
1066:
1063:
1060:
1057:
1052:
1047:
1043:
1039:
1036:
1033:
1030:
1026:
1023:
1020:
1017:
1012:
1007:
1003:
988:
987:
976:
973:
969:
966:
963:
960:
955:
950:
946:
862:
859:
846:
843:
837:'s 1844 work,
821:
818:
753:
750:
747:
744:
741:
737:
734:
730:
727:
724:
721:
718:
715:
687:
684:
681:
653:
616:
613:
610:
607:
604:
601:
598:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
500:
497:
494:
491:
488:
485:
482:
479:
452:volume element
400:
399:
388:
385:
382:
379:
376:
373:
370:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
280:
277:
274:
271:
268:
265:
262:
259:
256:
251:
247:
142:
141:
130:
127:
124:
120:
117:
114:
111:
106:
101:
97:
26:
9:
6:
4:
3:
2:
18872:
18861:
18858:
18856:
18853:
18852:
18850:
18835:
18832:
18830:
18827:
18825:
18822:
18820:
18817:
18815:
18812:
18810:
18807:
18805:
18802:
18800:
18797:
18795:
18792:
18790:
18787:
18785:
18782:
18780:
18777:
18775:
18772:
18771:
18769:
18767:
18763:
18753:
18750:
18748:
18745:
18743:
18740:
18738:
18735:
18733:
18730:
18728:
18725:
18723:
18720:
18718:
18715:
18713:
18710:
18709:
18707:
18703:
18697:
18694:
18692:
18689:
18687:
18684:
18682:
18679:
18677:
18674:
18672:
18671:Metric tensor
18669:
18667:
18664:
18662:
18659:
18658:
18656:
18652:
18649:
18645:
18639:
18636:
18634:
18631:
18629:
18626:
18624:
18621:
18619:
18616:
18614:
18611:
18609:
18606:
18604:
18601:
18599:
18596:
18594:
18591:
18589:
18586:
18584:
18583:Exterior form
18581:
18579:
18576:
18574:
18571:
18569:
18566:
18564:
18561:
18559:
18556:
18554:
18551:
18549:
18546:
18545:
18543:
18537:
18530:
18527:
18525:
18522:
18520:
18517:
18515:
18512:
18510:
18507:
18505:
18502:
18500:
18497:
18495:
18492:
18490:
18487:
18485:
18482:
18480:
18477:
18476:
18474:
18472:
18468:
18462:
18459:
18457:
18456:Tensor bundle
18454:
18452:
18449:
18447:
18444:
18442:
18439:
18437:
18434:
18432:
18429:
18427:
18424:
18422:
18419:
18417:
18414:
18413:
18411:
18405:
18399:
18396:
18394:
18391:
18389:
18386:
18384:
18381:
18379:
18376:
18374:
18371:
18369:
18366:
18364:
18361:
18359:
18356:
18355:
18353:
18349:
18339:
18336:
18334:
18331:
18329:
18326:
18324:
18321:
18319:
18316:
18315:
18313:
18308:
18305:
18303:
18300:
18299:
18296:
18290:
18287:
18285:
18282:
18280:
18277:
18275:
18272:
18270:
18267:
18265:
18262:
18260:
18257:
18255:
18252:
18251:
18249:
18247:
18243:
18240:
18236:
18232:
18231:
18225:
18221:
18214:
18209:
18207:
18202:
18200:
18195:
18194:
18191:
18179:
18176:
18174:
18173:Supermanifold
18171:
18169:
18166:
18164:
18161:
18157:
18154:
18153:
18152:
18149:
18147:
18144:
18142:
18139:
18137:
18134:
18132:
18129:
18127:
18124:
18122:
18119:
18118:
18116:
18112:
18106:
18103:
18101:
18098:
18096:
18093:
18091:
18088:
18086:
18083:
18081:
18078:
18077:
18075:
18071:
18061:
18058:
18056:
18053:
18051:
18048:
18046:
18043:
18041:
18038:
18036:
18033:
18031:
18028:
18026:
18023:
18021:
18018:
18016:
18013:
18012:
18010:
18008:
18004:
17998:
17995:
17993:
17990:
17988:
17985:
17983:
17980:
17978:
17975:
17973:
17970:
17968:
17964:
17960:
17958:
17955:
17953:
17950:
17948:
17944:
17940:
17938:
17935:
17933:
17930:
17928:
17925:
17923:
17920:
17918:
17915:
17913:
17910:
17909:
17907:
17905:
17901:
17895:
17894:Wedge product
17892:
17890:
17887:
17883:
17880:
17879:
17878:
17875:
17873:
17870:
17866:
17863:
17862:
17861:
17858:
17856:
17853:
17851:
17848:
17846:
17843:
17839:
17838:Vector-valued
17836:
17835:
17834:
17831:
17829:
17826:
17822:
17819:
17818:
17817:
17814:
17812:
17809:
17807:
17804:
17803:
17801:
17797:
17791:
17788:
17786:
17783:
17781:
17778:
17774:
17771:
17770:
17769:
17768:Tangent space
17766:
17764:
17761:
17759:
17756:
17754:
17751:
17750:
17748:
17744:
17741:
17739:
17735:
17729:
17726:
17724:
17720:
17716:
17714:
17711:
17709:
17705:
17701:
17697:
17695:
17692:
17690:
17687:
17685:
17682:
17680:
17677:
17675:
17672:
17670:
17667:
17665:
17662:
17658:
17655:
17654:
17653:
17650:
17648:
17645:
17643:
17640:
17638:
17635:
17633:
17630:
17628:
17625:
17623:
17620:
17618:
17615:
17613:
17610:
17608:
17605:
17603:
17599:
17595:
17593:
17589:
17585:
17583:
17580:
17579:
17577:
17571:
17565:
17562:
17560:
17557:
17555:
17552:
17550:
17547:
17545:
17542:
17540:
17537:
17533:
17532:in Lie theory
17530:
17529:
17528:
17525:
17523:
17520:
17516:
17513:
17512:
17511:
17508:
17506:
17503:
17502:
17500:
17498:
17494:
17488:
17485:
17483:
17480:
17478:
17475:
17473:
17470:
17468:
17465:
17463:
17460:
17458:
17455:
17453:
17450:
17448:
17445:
17444:
17442:
17439:
17435:Main results
17433:
17427:
17424:
17422:
17419:
17417:
17416:Tangent space
17414:
17412:
17409:
17407:
17404:
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17399:
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17385:
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17233:
17224:
17222:3-540-40633-6
17218:
17214:
17210:
17206:
17203:
17197:
17193:
17189:
17185:
17181:
17180:Tu, Loring W.
17177:
17172:
17170:0-8053-9021-9
17166:
17162:
17161:
17156:
17152:
17149:
17147:0-07-054235-X
17143:
17139:
17135:
17134:Rudin, Walter
17131:
17128:
17126:0-8218-1045-6
17122:
17117:
17116:
17109:
17106:
17100:
17096:
17092:
17089:
17087:0-486-66169-5
17083:
17079:
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17045:
17039:
17035:
17031:
17027:
17023:
17019:
17016:
17012:
17008:
17004:
17000:
16997:
16992:
16990:0-486-45010-4
16986:
16982:
16978:
16977:Cartan, Henri
16974:
16971:
16967:
16962:
16957:
16953:
16948:
16945:
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16892:9781441974006
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16638:unitary group
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15931:-currents on
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15873:-dimensional
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14541:
14540:by pullback:
14538:
14534:-covector at
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14514:
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12867:
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12848:
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12280:
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11899:
11895:
11891:
11888:
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11879:
11871:
11870:
11869:
11852:
11831:
11827:
11823:
11820:
11817:
11799:
11795:
11788:
11772:
11765:
11764:
11763:
11760:
11756:Suppose that
11740:
11733:
11729:
11724:
11720:
11717:
11714:
11711:
11704:
11700:
11695:
11691:
11670:
11666:
11662:
11659:
11656:
11651:
11647:
11642:
11634:
11630:
11626:
11623:
11620:
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11611:
11606:
11602:
11599:
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11590:
11587:
11583:
11577:
11571:
11565:
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11556:
11550:
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11528:
11522:
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11492:
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11477:
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11468:
11462:
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11450:
11444:
11433:
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11420:
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11396:
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11380:
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11367:
11360:
11354:
11349:
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11332:
11322:
11317:
11313:
11305:
11304:
11303:
11300:
11294:
11288:
11284:
11280:
11258:
11254:
11250:
11247:
11244:
11241:
11236:
11232:
11228:
11221:
11215:
11212:
11209:
11202:
11201:
11200:
11199:has the form
11197:
11191:
11185:
11179:
11173:
11167:
11161:
11150:
11145:
11138:
11132:
11126:
11120:
11116:
11110:
11104:
11098:
11092:
11081:
11067:
11064:
11061:
11058:
11055:
11052:
11049:
11022:
11018:
11013:
11000:
10996:
10991:
10954:
10950:
10945:
10941:
10938:
10935:
10928:
10924:
10919:
10900:
10896:
10891:
10887:
10884:
10881:
10874:
10870:
10865:
10832:
10825:
10821:
10816:
10812:
10809:
10806:
10803:
10796:
10792:
10787:
10783:
10769:
10765:
10760:
10756:
10753:
10750:
10743:
10739:
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10715:
10711:
10706:
10702:
10699:
10696:
10689:
10685:
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10664:
10661:
10654:
10650:
10646:
10641:
10637:
10632:
10621:
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10613:
10610:
10607:
10602:
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10593:
10585:
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10571:
10566:
10562:
10557:
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10550:
10545:
10541:
10533:
10532:
10531:
10529:
10524:
10518:
10512:
10505:
10501:
10481:
10474:
10470:
10465:
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10458:
10455:
10452:
10445:
10441:
10436:
10432:
10425:
10422:
10415:
10411:
10407:
10402:
10398:
10393:
10382:
10378:
10374:
10371:
10368:
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10359:
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10330:
10329:
10328:
10325:
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10313:
10305:
10301:
10294:
10290:
10283:
10279:
10270:
10250:
10243:
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10230:
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10224:
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10205:
10201:
10193:
10189:
10185:
10180:
10176:
10171:
10163:
10159:
10155:
10152:
10149:
10144:
10140:
10135:
10131:
10128:
10121:
10120:
10119:
10116:
10110:
10106:. Locally on
10104:
10097:
10093:
10088:
10084:
10080:
10074:
10068:
10062:
10056:
10050:
10044:
10020:
10014:
10009:
10005:
9998:
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9974:
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9950:
9946:
9943:
9938:
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9930:
9928:
9920:
9917:
9914:
9906:
9902:
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9891:
9886:
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9875:
9870:
9866:
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9846:
9838:
9834:
9826:
9820:
9815:
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9788:
9780:
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9764:
9763:
9762:
9759:
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9747:
9741:
9738:
9735:
9713:
9710:
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9701:
9696:
9690:
9678:
9674:
9668:
9643:
9640:
9630:
9615:
9612:
9607:
9601:
9588:
9585:
9580:
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9558:
9555:
9550:
9544:
9534:
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9529:
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9439:
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9437:
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9431:
9425:
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9381:
9375:
9360:
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9341:
9336:
9319:
9314:
9310:
9304:
9298:
9285:
9270:
9260:
9245:
9238:
9237:
9236:
9233:
9227:
9221:
9202:
9194:
9191:
9183:
9179:
9169:
9159:
9139:
9136:
9131:
9127:
9114:
9106:
9103:
9085:
9080:
9076:
9065:
9050:
9040:
9025:
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9017:
9016:
9013:
9007:
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8986:
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8971:
8965:
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8935:
8927:
8923:
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8913:
8909:
8906:
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8898:
8894:
8888:
8884:
8872:
8866:
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8858:
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8831:
8827:
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8779:
8773:
8766:
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8753:
8747:
8741:
8735:
8729:
8723:
8718:
8715:, called the
8713:
8707:
8704:
8698:
8692:
8686:
8680:
8676:
8672:
8666:
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8657:
8651:
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8637:
8631:
8628:
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8591:
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8573:
8567:
8561:
8556:
8552:
8548:
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8534:
8528:
8524:
8520:
8516:
8510:
8506:
8501:
8493:
8487:
8481:
8475:
8471:
8467:
8461:
8455:
8451:
8447:
8443:Suppose that
8440:
8430:
8426:
8420:
8414:
8409:
8404:
8399:
8393:
8385:
8379:
8375:
8370:
8365:
8361:
8342:
8327:
8319:
8302:
8287:
8279:
8264:
8246:
8238:
8223:
8205:
8197:
8182:
8164:
8156:
8139:
8132:
8131:
8130:
8128:
8122:
8111:
8109:
8105:
8101:
8096:
8094:
8089:
8085:
8081:
8077:
8072:
8060:
8057:
8052:
8029:
8021:
8018:
8015:
8001:
7993:
7985:
7982:
7975:
7956:
7948:
7945:
7942:
7927:
7909:
7901:
7893:
7890:
7883:
7879:
7875:
7864:
7851:
7848:
7845:
7842:
7839:
7834:
7826:
7823:
7817:
7814:
7811:
7808:
7805:
7802:
7796:
7793:
7790:
7784:
7775:
7773:
7754:
7751:
7748:
7745:
7740:
7737:
7729:
7726:
7720:
7717:
7714:
7711:
7704:
7703:
7702:
7699:
7693:
7687:
7681:
7676:
7672:
7668:
7649:
7643:
7640:
7637:
7631:
7628:
7625:
7619:
7616:
7613:
7607:
7604:
7597:
7583:
7580:
7577:
7574:
7571:
7568:
7565:
7562:
7559:
7553:
7550:
7547:
7541:
7538:
7531:
7530:
7529:
7526:
7520:
7514:
7508:
7502:
7499:
7495:
7490:
7486:
7463:
7455:
7451:
7444:
7436:
7432:
7428:
7422:
7414:
7410:
7403:
7395:
7391:
7387:
7381:
7378:
7375:
7367:
7359:
7356:
7353:
7343:
7342:
7341:
7339:
7334:
7324:
7320:
7313:
7309:
7289:
7283:
7280:
7277:
7271:
7268:
7262:
7259:
7256:
7253:
7248:
7242:
7239:
7236:
7227:
7224:
7221:
7218:
7211:
7210:
7209:
7195:
7175:
7172:
7169:
7149:
7144:
7140:
7134:
7129:
7107:
7102:
7098:
7092:
7086:
7060:
7054:
7051:
7048:
7042:
7039:
7036:
7033:
7030:
7027:
7020:
7019:
7018:
7015:
7011:
7005:
7001:
6995:
6992:
6986:
6980:
6974:
6968:
6962:
6958:
6952:
6948:
6942:
6936:
6930:
6924:
6913:
6911:
6907:
6903:
6899:
6895:
6884:
6881:
6875:
6870:
6867:
6864:
6859:
6839:
6836:
6831:
6827:
6821:
6816:
6808:
6803:
6799:
6793:
6787:
6781:
6778:
6771:
6770:
6769:
6766:
6761:
6755:
6751:
6731:
6720:
6714:
6710:
6706:
6703:
6700:
6692:
6686:
6682:
6673:
6669:
6662:
6656:
6653:
6646:
6642:
6638:
6635:
6631:
6624:
6621:
6617:
6612:
6604:
6600:
6596:
6593:
6590:
6585:
6581:
6569:
6565:
6558:
6555:
6548:
6547:
6546:
6543:
6526:
6519:For a tensor
6503:
6500:
6495:
6491:
6485:
6480:
6472:
6467:
6463:
6457:
6452:
6447:
6444:
6437:
6436:
6435:
6432:
6430:
6420:
6416:
6412:
6408:
6402:
6398:
6392:
6387:
6382:
6378:
6374:
6369:
6368:inner product
6364:
6360:
6356:
6349:
6345:
6341:
6336:
6332:
6326:
6316:
6310:
6304:
6298:
6278:
6267:
6262:
6258:
6252:
6247:
6244:
6241:
6237:
6233:
6228:
6224:
6216:
6215:
6214:
6212:
6209:
6188:
6176:
6171:
6167:
6161:
6155:
6142:
6139:
6134:
6129:
6125:
6119:
6113:
6103:
6102:
6101:
6098:
6092:
6067:
6062:
6058:
6052:
6046:
6040:
6035:
6031:
6021:
6015:
6010:
6005:
5999:
5993:
5988:
5983:
5979:
5975:
5969:
5963:
5958:
5957:tangent space
5953:
5949:
5945:
5925:
5922:
5917:
5912:
5908:
5902:
5896:
5890:
5885:
5881:
5873:
5872:
5871:
5868:
5862:
5856:
5852:
5846:
5842:
5836:
5831:
5825:
5819:
5814:
5810:
5805:
5800:
5795:
5790:
5785:
5778:
5768:
5754:
5743:
5738:
5731:
5724:
5709:
5705:
5701:
5695:
5691:
5684:
5677:
5665:
5661:
5654:
5648:
5644:
5620:
5616:
5611:
5607:
5604:
5601:
5598:
5591:
5587:
5582:
5578:
5575:
5568:
5564:
5559:
5555:
5547:
5543:
5539:
5534:
5530:
5524:
5520:
5515:
5509:
5504:
5501:
5496:
5492:
5488:
5483:
5479:
5475:
5470:
5466:
5461:
5453:
5452:
5451:
5430:Differential
5428:
5425:
5421:
5410:
5390:
5385:
5381:
5377:
5374:
5369:
5365:
5361:
5352:
5348:
5337:
5333:
5321:
5316:
5313:
5310:
5307:
5304:
5300:
5296:
5291:
5287:
5283:
5280:
5275:
5271:
5267:
5262:
5257:
5254:
5251:
5247:
5243:
5240:
5237:
5230:
5229:
5228:
5225:
5221:
5217:
5210:
5202:
5196:
5192:
5185:-form. This
5165:
5160:
5156:
5152:
5149:
5144:
5140:
5136:
5133:
5130:
5125:
5121:
5117:
5114:
5109:
5105:
5101:
5094:
5093:
5092:
5071:
5068:
5065:
5060:
5056:
5052:
5049:
5044:
5040:
5036:
5027:
5023:
5012:
5008:
4996:
4991:
4988:
4985:
4982:
4979:
4975:
4967:
4966:
4965:
4963:
4943:
4942:skew-symmetry
4938:
4913:
4910:
4902:
4898:
4887:
4883:
4873:
4865:
4861:
4850:
4846:
4832:
4831:
4830:
4809:
4801:
4797:
4787:
4783:
4774:
4769:
4758:
4750:
4746:
4736:
4732:
4723:
4718:
4703:
4702:
4701:
4693:
4686:
4682:
4674:are equal to
4672:
4668:
4657:
4653:
4627:
4619:
4615:
4593:
4589:
4585:
4579:
4575:
4569:
4564:
4561:
4558:
4554:
4550:
4547:
4540:
4539:
4538:
4524:
4523:
4513:
4509:
4487:
4477:
4473:
4469:
4460:
4452:
4448:
4442:
4438:
4434:
4429:
4425:
4417:
4416:
4415:
4414:pointwise by
4411:
4407:
4403:
4399:
4395:
4390:
4375:
4371:
4364:
4360:
4354:
4352:
4351:
4324:
4320:
4300:
4295:
4285:
4281:
4277:
4268:
4257:
4253:
4244:
4233:
4228:
4225:
4222:
4218:
4214:
4209:
4205:
4201:
4194:
4193:
4192:
4190:
4177:
4170:
4168:
4154:
4149:
4145:
4141:
4132:
4128:
4119:
4108:
4103:
4100:
4097:
4093:
4089:
4086:
4083:
4076:
4075:
4072:
4070:
4064:
4060:
4056:
4052:
4045:
4041:
4035:
4029:
4023:
4009:
4003:
3997:
3987:
3973:
3969:
3963:
3955:
3951:
3946:
3942:
3936:
3931:
3925:
3921:
3918:
3913:
3908:
3902:
3900:
3891:
3887:
3869:
3862:
3858:
3854:
3849:
3844:
3840:
3827:
3823:
3817:
3813:
3807:
3801:
3796:
3792:
3783:
3777:
3750:
3741:
3723:
3720:
3718:
3710:
3701:
3691:
3673:
3664:
3646:
3640:
3631:
3613:
3611:
3603:
3594:
3584:
3560:
3559:
3558:
3555:
3550:
3544:
3540:
3536:
3514:
3506:
3502:
3493:
3479:
3475:
3464:
3460:
3448:
3443:
3440:
3437:
3433:
3429:
3421:
3417:
3408:
3395:
3394:
3393:
3390:
3384:
3378:
3368:
3362:
3356:
3350:
3346:
3332:
3327:
3323:
3316:
3306:
3302:
3298:
3292:
3287:at the point
3285:
3276:
3271:
3251:
3246:
3243:
3240:
3235:
3223:
3220:
3217:
3211:
3205:
3202:
3198:
3188:
3182:
3173:
3151:
3150:
3149:
3146:
3132:
3128:
3118:
3114:
3108:
3103:
3097:
3091:
3084:
3080:
3068:
3059:
3054:
3040:
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3009:
3007:
3003:
2999:
2995:
2990:
2967:
2959:
2956:
2953:
2945:
2940:
2936:
2932:
2929:
2925:
2919:
2915:
2911:
2902:
2898:
2887:
2883:
2871:
2866:
2863:
2860:
2856:
2851:
2843:
2840:
2837:
2825:
2822:
2818:
2814:
2811:
2808:
2801:
2800:
2799:
2780:
2772:
2764:
2759:
2755:
2751:
2745:
2741:
2733:
2730:
2727:
2715:
2712:
2708:
2704:
2701:
2673:
2668:
2664:
2660:
2657:
2652:
2648:
2644:
2635:
2631:
2622:
2611:
2606:
2603:
2600:
2596:
2592:
2589:
2586:
2579:
2578:
2577:
2573:
2566:
2556:
2553:
2549:
2538:
2534:
2530:
2526:
2521:
2516:
2511:
2501:
2477:
2474:
2463:
2453:
2450:
2447:
2405:-forms on an
2386:
2383:
2380:
2357:
2353:
2345:of dimension
2320:
2317:
2314:
2302:
2299:
2289:
2285:
2281:
2253:
2250:
2245:
2241:
2237:
2234:
2231:
2226:
2222:
2218:
2213:
2209:
2205:
2202:
2199:
2191:
2187:
2183:
2180:
2177:
2172:
2168:
2161:
2158:
2152:
2147:
2144:
2141:
2099:
2095:
2091:
2086:
2083:
2080:
2076:
2072:
2065:
2061:
2056:
2052:
2049:
2046:
2043:
2036:
2032:
2027:
2023:
2020:
2015:
2011:
2007:
1984:
1981:
1976:
1972:
1968:
1965:
1962:
1957:
1953:
1949:
1944:
1940:
1936:
1933:
1930:
1922:
1918:
1914:
1911:
1908:
1903:
1899:
1895:
1890:
1886:
1879:
1876:
1863:
1849:
1847:
1843:
1842:parallelotope
1837:
1833:
1824:
1817:
1813:
1806:
1802:
1797:
1796:cross product
1792:
1788:
1784:
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1750:
1746:
1740:
1736:
1730:
1724:
1718:
1712:
1690:
1686:
1682:
1679:
1674:
1670:
1666:
1663:
1660:
1655:
1651:
1647:
1644:
1639:
1635:
1631:
1624:
1623:
1622:
1620:
1619:cross product
1613:
1609:
1604:
1586:
1582:
1578:
1575:
1570:
1566:
1562:
1556:
1553:
1550:
1546:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1518:
1503:
1497:
1487:
1483:
1477:
1473:
1467:
1459:
1455:
1450:
1446:
1441:
1437:
1417:
1412:
1408:
1404:
1398:
1394:
1390:
1387:
1384:
1379:
1375:
1371:
1365:
1361:
1353:
1352:
1351:
1345:
1336:
1330:
1325:
1320:
1319:for details.
1318:
1298:
1295:
1291:
1283:
1280:
1277:
1270:
1266:
1263:
1260:
1256:
1251:
1247:
1226:
1222:
1203:
1199:
1193:
1190:
1185:
1181:
1178:
1175:
1170:
1166:
1158:
1157:
1156:
1142:
1112:
1109:
1103:
1099:
1093:
1074:
1071:
1068:
1061:
1055:
1050:
1045:
1041:
1037:
1034:
1031:
1028:
1021:
1015:
1010:
1005:
1001:
993:
992:
991:
974:
971:
964:
958:
953:
948:
944:
936:
935:
934:
931:
927:
923:
913:
909:
903:
899:
894:
891:
886:
873:of dimension
872:
858:
856:
852:
842:
840:
836:
832:
828:
817:
815:
810:
806:
798:
794:
793:vector fields
787:Differential
785:
783:
779:
775:
771:
767:
751:
748:
742:
735:
732:
728:
722:
716:
713:
701:
685:
682:
679:
669:
664:, produces a
651:
642:
637:
632:
630:
614:
611:
608:
605:
602:
599:
596:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
535:
530:
528:
527:
521:
515:
498:
495:
492:
489:
486:
483:
480:
477:
465:
453:
450:represents a
448:
444:
440:
436:
432:
428:
424:
420:
413:
412:wedge product
409:
386:
380:
377:
374:
371:
368:
361:
358:
355:
352:
349:
343:
340:
337:
334:
331:
328:
325:
318:
315:
312:
309:
306:
300:
297:
294:
291:
288:
285:
282:
275:
272:
269:
266:
263:
257:
249:
245:
237:
236:
235:
232:
228:
224:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
148:
128:
125:
122:
115:
109:
104:
99:
95:
87:
86:
85:
82:
77:
74:, and can be
73:
65:
61:
57:
51:
49:
45:
41:
37:
33:
19:
18834:Hermann Weyl
18638:Vector space
18623:Pseudotensor
18588:Fiber bundle
18572:
18541:abstractions
18436:Mixed tensor
18421:Tensor field
18228:
18100:Moving frame
18095:Morse theory
18085:Gauge theory
17877:Tensor field
17832:
17806:Closed/Exact
17785:Vector field
17753:Distribution
17694:Hypercomplex
17689:Quaternionic
17426:Vector field
17420:
17384:Smooth atlas
17303:
17284:math/0306194
17274:
17257:
17244:
17215:, Springer,
17212:
17183:
17159:
17137:
17114:
17098:
17077:
17054:
17025:
17006:
16995:
16980:
16951:
16933:
16909:
16882:
16876:
16858:
16854:
16844:
16813:
16772:
16762:
16747:
16741:
16734:
16730:
16727:
16666:
16660:
16650:
16628:. Here the
16626:gauge theory
16618:
16613:
16576:
16570:denotes the
16549:
16450:
16444:
16405:
16399:
16392:
16388:
16381:
16377:
16371:
16368:
16239:
16237:
16188:
16164:
16157:
16153:
16146:
16140:
16135:
16131:
16124:
16059:
16055:
16052:
15956:
15950:
15939:
15933:
15927:
15920:
15917:distribution
15914:
15903:
15901:-forms. The
15897:
15891:
15885:
15879:
15869:
15866:
15860:
15854:
15848:
15842:
15840:of a set of
15832:
15826:
15820:
15816:
15810:
15807:
15797:
15793:volume forms
15787:
15781:
15778:
15774:Folland 1999
15768:-forms with
15764:
15754:
15749:
15747:
15739:
15729:
15717:
15710:
15704:
15700:
15692:
15685:
15680:
15671:
15651:
15645:
15517:orientation:
15516:
15503:
15493:
15489:
15487:
15465:
15442:
15438:
15434:
15430:
15424:
15418:
15412:
15406:
15400:
15396:
15393:
15280:
15273:
15266:
15262:
15256:
15250:
15244:
15238:
15236:is a closed
15232:
15229:
15161:
15155:denotes the
15151:
15145:
15138:
15132:
15125:
15034:
15028:
15022:
15013:
15011:
15004:
14998:
14996:integral of
14995:
14990:
14980:
14977:
14880:
14874:
14872:. Then the
14868:
14862:
14859:
14684:
14680:
14674:
14667:
14663:
14656:
14652:
14645:
14642:
14536:
14529:
14525:
14518:
14512:
14506:
14499:
14495:
14491:
14485:
14479:
14476:
14422:
14328:
14324:
14318:
14312:
14306:
14300:
14293:
14289:
14283:
14277:
14271:
14265:
14259:
14255:
14249:
14247:. The form
14243:
14237:
14231:
14225:
14219:
14213:
14206:
14202:
14198:
14191:
14188:
14074:
13990:
13883:
13877:
13870:
13866:
13860:
13856:
13849:
13845:
13841:
13835:
13831:
13825:
13819:
13813:
13807:
13801:
13795:
13789:
13783:
13777:
13771:
13765:
13761:
13757:
13751:
13745:
13739:
13733:
13730:
13724:
13718:
13712:
13705:
13701:
13695:
13689:
13682:
13678:
13674:
13671:
13603:
13599:
13595:
13588:
13584:
13578:
13571:
13567:
13559:
13555:
13549:
13543:
13537:
13534:
13464:
13457:
13453:
13447:
13444:
13437:
13433:
13428:
13424:
13417:
13413:
13410:
13344:
13340:
13336:
13259:
13132:
13128:
13122:
13118:
13114:
13110:
13103:
13099:
13092:
13088:
13085:
12978:
12974:
12971:
12828:
12824:
12820:
12814:
12810:
12804:
12798:
12791:
12787:
12779:
12775:
12769:
12765:
12761:
12755:
12749:
12743:
12737:
12733:
12726:
12708:
12704:
12698:
12691:
12687:
12677:
12676:) : →
12673:
12669:
12663:
12657:
12654:
12648:
12642:
12640:as a sum of
12636:
12630:
12624:
12618:
12612:
12606:
12600:
12594:
12588:
12582:
12575:
12564:
12560:triangulated
12554:
12551:
12447:
12441:
12435:
12429:
12426:
12350:
12344:
12338:
12334:
12329:
12324:
12318:
12312:
12306:
12300:
12296:
12290:
12284:
12278:
12275:
12269:
12262:
12141:
11867:
11758:
11755:
11585:
11581:
11575:
11569:
11563:
11557:
11554:
11548:
11542:
11532:
11526:
11520:
11516:
11513:
11472:
11466:
11460:
11454:
11448:
11442:
11439:
11428:
11424:
11418:
11414:
11411:
11298:
11292:
11286:
11282:
11278:
11275:
11195:
11189:
11183:
11177:
11171:
11165:
11159:
11156:
11143:
11136:
11130:
11124:
11118:
11114:
11108:
11102:
11096:
11090:
11087:
10847:
10522:
10516:
10510:
10503:
10499:
10496:
10323:
10317:
10311:
10303:
10299:
10292:
10288:
10281:
10277:
10268:
10265:
10114:
10108:
10102:
10095:
10091:
10086:
10082:
10078:
10072:
10066:
10060:
10054:
10048:
10042:
10039:
9757:
9751:
9745:
9742:
9736:
9733:
9676:
9672:
9666:
9663:
9527:
9521:
9517:
9514:
9429:
9423:
9417:
9414:
9231:
9225:
9219:
9167:
9157:
9154:
9011:
9005:
9002:
8998:
8995:
8991:
8984:
8978:
8972:
8969:
8963:
8953:
8950:
8783:
8777:
8771:
8764:
8760:
8751:
8745:
8739:
8733:
8727:
8721:
8716:
8711:
8705:
8702:
8696:
8690:
8688:be a smooth
8684:
8678:
8674:
8670:
8667:
8661:
8655:
8649:
8645:
8641:
8635:
8632:
8626:
8620:
8611:
8605:
8601:
8595:
8589:
8585:
8581:
8575:
8571:
8565:
8559:
8554:
8550:
8546:
8539:
8532:
8526:
8522:
8518:
8514:
8508:
8504:
8499:
8491:
8485:
8479:
8473:
8469:
8465:
8459:
8453:
8449:
8445:
8442:
8424:
8412:
8402:
8391:
8383:
8368:
8357:
8120:
8117:
8107:
8104:Weyl algebra
8097:
8083:
8073:
8066:
8055:
7871:
7777:
7769:
7701:-form, then
7697:
7691:
7685:
7679:
7670:
7666:
7664:
7524:
7518:
7512:
7506:
7503:
7497:
7493:
7488:
7484:
7481:
7332:
7322:
7318:
7311:
7307:
7304:
7075:
7013:
7009:
7003:
6999:
6996:
6990:
6984:
6978:
6976:, the forms
6972:
6966:
6960:
6956:
6950:
6946:
6940:
6934:
6928:
6922:
6919:
6890:
6879:
6873:
6869:tensor field
6857:
6854:
6764:
6753:
6749:
6746:
6541:
6518:
6433:
6418:
6414:
6400:
6396:
6390:
6380:
6376:
6372:
6362:
6358:
6354:
6347:
6343:
6334:
6330:
6324:
6314:
6308:
6302:
6296:
6293:
6205:
6096:
6090:
6019:
6013:
6003:
5997:
5981:
5977:
5973:
5967:
5961:
5951:
5947:
5943:
5940:
5866:
5860:
5854:
5850:
5847:
5840:
5835:vector space
5829:
5823:
5817:
5803:
5793:
5783:
5780:
5739:
5729:
5722:
5710:
5703:
5699:
5693:
5689:
5682:
5675:
5663:
5659:
5652:
5646:
5642:
5639:
5438:-forms, and
5429:
5423:
5419:
5408:
5405:
5223:
5219:
5215:
5208:
5200:
5194:
5180:
5086:
4960:-forms, the
4939:
4928:
4824:
4691:
4684:
4680:
4670:
4666:
4655:
4651:
4628:
4617:
4613:
4610:
4520:
4511:
4507:
4504:
4409:
4405:
4401:
4397:
4393:
4388:
4373:
4369:
4362:
4358:
4355:
4348:
4322:
4318:
4315:
4182:
4171:
4062:
4058:
4054:
4050:
4043:
4039:
4033:
4027:
4021:
4007:
4001:
3995:
3985:
3971:
3967:
3961:
3953:
3949:
3944:
3940:
3934:
3923:
3919:
3916:
3906:
3903:
3889:
3885:
3867:
3860:
3856:
3852:
3847:
3842:
3838:
3826:differential
3815:
3811:
3805:
3799:
3790:
3781:
3775:
3772:
3553:
3542:
3538:
3534:
3529:
3388:
3382:
3376:
3366:
3360:
3354:
3348:
3344:
3325:
3321:
3304:
3300:
3296:
3293:
3283:
3275:vector field
3269:
3266:
3144:
3130:
3126:
3116:
3112:
3101:
3095:
3089:
3082:
3078:
3057:
3046:
3010:
2988:
2985:
2688:
2571:
2564:
2558:is a simple
2554:
2551:
2547:
2536:
2532:
2528:
2524:
2514:
2507:
2355:
2351:
1855:
1835:
1831:
1822:
1815:
1811:
1804:
1800:
1790:
1786:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1748:
1744:
1738:
1734:
1728:
1722:
1716:
1710:
1707:
1605:
1501:
1495:
1485:
1481:
1475:
1471:
1468:
1457:
1453:
1448:
1444:
1439:
1435:
1432:
1334:
1328:
1321:
1218:
1140:
1113:
1107:
1101:
1097:
1089:
989:
929:
925:
921:
911:
907:
901:
897:
887:
864:
848:
838:
823:
786:
667:
640:
633:
531:
524:
519:
513:
446:
442:
438:
434:
430:
426:
422:
415:
411:
406:denotes the
401:
230:
219:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
143:
80:
63:
59:
55:
52:
35:
29:
18774:Élie Cartan
18722:Spin tensor
18696:Weyl tensor
18654:Mathematics
18618:Multivector
18409:definitions
18307:Engineering
18246:Mathematics
18045:Levi-Civita
18035:Generalized
18007:Connections
17957:Lie algebra
17889:Volume form
17790:Vector flow
17763:Pushforward
17758:Lie bracket
17657:Lie algebra
17622:G-structure
17411:Pushforward
17391:Submanifold
16861:: 239–332,
16656:Lie algebra
16608:, which is
16403:instead of
16386:instead of
15803:pseudoforms
15750:orientation
15474:of chains.
15167:orientation
14984:is smooth (
14689:defined by
13887:defined by
13839:, the form
11122:, then the
11084:Integration
9217:denote the
8500:pushforward
7340:defined by
7188:instead of
6539:at a point
6407:represented
6208:alternating
6017:. That is,
4191:", so that
3809:is denoted
3148:direction:
2992:, with the
1092:conventions
855:coordinates
827:Élie Cartan
629:orientation
526:volume form
464:homogeneous
402:The symbol
48:Élie Cartan
32:mathematics
18849:Categories
18603:Linear map
18471:Operations
18168:Stratifold
18126:Diffeology
17922:Associated
17723:Symplectic
17708:Riemannian
17637:Hyperbolic
17564:Submersion
17472:Hopf–Rinow
17406:Submersion
17401:Smooth map
16926:References
16572:Hodge star
16053:where the
15242:-form and
13805:, and let
11538:Rudin 1976
10530:matrices:
8437:See also:
8380:except at
8372:. By the
8360:cohomology
7689:-form and
6944:, denoted
6887:Operations
5987:dual space
5775:See also:
4700:satisfies
4649:such that
3930:components
3822:derivative
3795:linear map
3006:divergence
2124:, denoted
1999:we define
1433:where the
466:of degree
76:integrated
40:integrands
18742:EM tensor
18578:Dimension
18529:Transpose
18050:Principal
18025:Ehresmann
17982:Subbundle
17972:Principal
17947:Fibration
17927:Cotangent
17799:Covectors
17652:Lie group
17632:Hermitian
17575:manifolds
17544:Immersion
17539:Foliation
17477:Noether's
17462:Frobenius
17457:De Rham's
17452:Darboux's
17343:Manifolds
17246:MathWorld
17076:(1989) ,
16983:, Dover,
16901:682907530
16705:∧
16630:Lie group
16590:⋆
16558:⋆
16506:⋆
16427:→
16337:∧
16321:∧
16289:ε
16107:→
16078:→
16021:∧
15627:−
15604:∫
15600:−
15577:∫
15529:∫
15449:homotopic
15363:∫
15356:ω
15344:∫
15320:ω
15311:∫
15304:ω
15295:∫
15212:ω
15204:∂
15200:∫
15193:ω
15181:∫
15095:♭
15087:λ
15082:∗
15074:∧
15071:α
15062:λ
15059:∧
15054:♭
15050:α
15032:-form on
14952:β
14932:−
14923:∫
14916:⟩
14899:γ
14895:⟨
14837:∗
14819:−
14810:⋀
14804:∈
14795:ζ
14770:ζ
14765:⌟
14751:∗
14743:∧
14734:α
14704:β
14620:∗
14602:−
14593:⋀
14587:∈
14575:ζ
14570:⌟
14556:∗
14504:, then a
14453:ζ
14448:⌟
14439:↦
14403:∗
14385:−
14376:⋀
14370:→
14346:⋀
14211:-form on
14171:η
14160:η
14152:ω
14133:−
14124:∫
14107:∫
14100:ω
14091:∫
14057:η
14046:η
14038:ω
14019:−
14010:∫
13967:η
13949:η
13940:ω
13921:−
13912:∫
13901:↦
13881:-form on
13817:-form on
13799:-form on
13648:η
13633:ω
13620:σ
13547:-form on
13515:∗
13507:→
13490:−
13482::
13479:ω
13411:The form
13384:σ
13378:∗
13361:σ
13317:∗
13299:−
13290:⋀
13284:∈
13272:σ
13237:∗
13216:⋀
13210:∈
13198:σ
13194:∧
13175:η
13169:∗
13149:ω
13054:−
13041:∗
13023:−
13014:⋀
13008:∈
12999:σ
12972:and that
12945:∗
12924:⋀
12918:∈
12905:η
12889:∗
12868:⋀
12862:∈
12849:ω
12534:ω
12529:∗
12520:φ
12510:∫
12479:∑
12472:ω
12463:∫
12403:φ
12372:∑
12229:…
12210:∂
12182:…
12156:∂
12111:⋯
12075:…
12056:∂
12028:…
12002:∂
11980:φ
11959:…
11923:⋯
11906:∑
11896:∫
11889:ω
11880:∫
11821:…
11773:φ
11718:∧
11715:⋯
11712:∧
11660:…
11624:⋯
11607:∑
11600:ω
11536:. Then (
11496:→
11490::
11487:φ
11464:-form on
11381:⋯
11346:∫
11323:ω
11314:∫
11248:∧
11245:⋯
11242:∧
11210:ω
11065:≤
11053:≤
11010:∂
10988:∂
10939:…
10913:∂
10885:…
10859:∂
10810:∧
10807:⋯
10804:∧
10754:…
10728:∂
10700:…
10674:∂
10662:∘
10647:⋯
10633:ω
10611:⋯
10594:∑
10575:⋯
10558:∑
10551:ω
10546:∗
10459:∧
10456:⋯
10453:∧
10423:∘
10408:⋯
10394:ω
10372:⋯
10355:∑
10348:ω
10343:∗
10228:∧
10225:⋯
10222:∧
10186:⋯
10172:ω
10153:⋯
10136:∑
10129:ω
10015:ω
10010:∗
9986:ω
9975:∗
9960:η
9955:∗
9947:∧
9944:ω
9939:∗
9921:η
9918:∧
9915:ω
9907:∗
9892:η
9887:∗
9876:ω
9871:∗
9853:η
9847:ω
9839:∗
9821:ω
9816:∗
9792:ω
9781:∗
9714:×
9708:→
9691:⋀
9644:×
9631:ω
9626:→
9602:⋀
9576:⋀
9569:⟶
9545:⋀
9492:×
9479:ω
9474:→
9450:⋀
9392:∗
9376:⋀
9361:∗
9337:⋀
9330:⟶
9315:∗
9299:⋀
9286:ω
9281:→
9256:→
9203:∗
9180:⋀
9165:-form on
9132:∗
9115:∗
9096:⟶
9081:∗
9066:ω
9061:→
9036:→
8994:) :
8961:-form on
8918:∗
8907:…
8889:∗
8863:ω
8840:…
8811:ω
8806:∗
8694:-form on
8609:at which
8337:→
8316:Ω
8309:→
8303:⋯
8297:→
8276:Ω
8260:→
8235:Ω
8219:→
8194:Ω
8178:→
8153:Ω
8146:→
8108:symmetric
8086:from the
8019:−
8012:Ω
8008:→
7990:Ω
7986::
7983:δ
7946:−
7939:Ω
7928:∼
7923:→
7898:Ω
7894::
7891:⋆
7849:β
7843:∧
7840:α
7824:−
7815:β
7812:∧
7809:α
7797:β
7794:∧
7791:α
7752:α
7749:∧
7746:β
7741:ℓ
7727:−
7718:β
7715:∧
7712:α
7644:β
7641:∧
7638:α
7632:⋅
7620:β
7617:⋅
7608:∧
7605:α
7581:γ
7578:∧
7575:α
7569:β
7566:∧
7563:α
7554:γ
7548:β
7542:∧
7539:α
7452:β
7433:α
7429:−
7411:β
7392:α
7360:β
7357:∧
7354:α
7284:β
7281:⊗
7278:α
7272:
7260:ℓ
7243:ℓ
7225:β
7222:∧
7219:α
7145:∗
7130:⨂
7103:∗
7087:⋀
7055:β
7052:⊗
7049:α
7043:
7034:β
7031:∧
7028:α
6866:covariant
6832:∗
6817:⨂
6812:→
6804:∗
6788:⋀
6782::
6715:σ
6704:…
6687:σ
6670:τ
6663:σ
6657:
6639:∈
6636:σ
6632:∑
6594:…
6566:τ
6559:
6527:τ
6496:∗
6481:⨂
6476:→
6468:∗
6453:⨂
6448::
6271:→
6238:⨁
6234::
6225:β
6189:∗
6156:⋀
6143:≅
6135:∗
6114:⋀
6071:→
6047:⋀
6041::
6032:β
5918:∗
5897:⋀
5891:∈
5882:β
5759:-form on
5605:∧
5602:⋯
5599:∧
5576:∧
5540:…
5489:…
5462:∑
5375:∧
5345:∂
5330:∂
5301:∑
5281:∧
5248:∑
5241:α
5150:∧
5134:−
5115:∧
5050:∧
5020:∂
5005:∂
4976:∑
4895:∂
4880:∂
4874:−
4858:∂
4843:∂
4794:∂
4780:∂
4766:∂
4743:∂
4729:∂
4715:∂
4555:∑
4548:α
4505:for each
4439:∑
4426:α
4250:∂
4242:∂
4219:∑
4189:pointwise
4125:∂
4117:∂
4094:∑
3965:and each
3959:for each
3731:∂
3688:∂
3654:∂
3621:∂
3576:∂
3499:∂
3491:∂
3472:∂
3457:∂
3434:∑
3414:∂
3406:∂
3163:∂
3029:manifolds
2950:Ω
2946:∈
2930:∧
2895:∂
2880:∂
2857:∑
2826:∈
2819:∑
2812:τ
2769:Ω
2765:∈
2716:∈
2709:∑
2702:τ
2658:∧
2628:∂
2620:∂
2597:∑
2590:ω
2512:operator
2303:∈
2251:≤
2235:⋯
2206:≤
2181:…
2084:∈
2077:⋀
2050:∧
2047:⋯
2044:∧
1982:≤
1966:⋯
1937:≤
1912:…
1848:is zero.
1680:∧
1664:−
1645:∧
1576:∧
1538:≤
1526:≤
1519:∑
1388:⋯
1299:μ
1271:∫
1264:μ
1248:∫
1200:ω
1186:∫
1182:−
1176:ω
1167:∫
1042:∫
1038:−
1002:∫
945:∫
893:manifolds
683:φ
652:φ
603:∧
571:∧
562:−
550:∧
496:…
375:∧
332:∧
289:∧
246:∫
96:∫
44:manifolds
18608:Manifold
18593:Geodesic
18351:Notation
18146:Orbifold
18141:K-theory
18131:Diffiety
17855:Pullback
17669:Oriented
17647:Kenmotsu
17627:Hadamard
17573:Types of
17522:Geodesic
17347:Glossary
17211:(2004),
17182:(2008),
17157:(1965),
17136:(1976),
17053:(1999),
17024:(1994),
17005:(1972),
16979:(2006),
16787:See also
16240:current
16122:; e.g.,
15911:Currents
15877:for any
15566:, while
15513:−1
15472:homology
15470:and the
15284:), then
15157:boundary
14655:−
14528:−
14510:-vector
14275:-vector
14201:−
13864:form on
13859:−
13760: :
13570:−
13392:′
13343:−
13280:′
13206:′
13113: :
12818:and set
12790:−
12764: :
12265:Jacobian
11589:. Then
11524:. Give
11281: :
11169:. Give
10528:Jacobian
10100:for all
8717:pullback
8673: :
8644: :
8604:∈
8517:∈
8468: :
8448: :
8433:Pullback
8084:distinct
7972:and the
6954:, is a (
6871:of rank
5434:-forms,
4929:for all
4339:. Since
4047:. Since
3352:, where
3053:open set
2998:gradient
1864:: in an
1814:∧ ⋅⋅⋅ ∧
1479:, where
1456:, ... ,
1344:covector
1194:′
890:oriented
871:manifold
809:pullback
736:′
18705:Physics
18539:Related
18302:Physics
18220:Tensors
18090:History
18073:Related
17987:Tangent
17965:)
17945:)
17912:Adjoint
17904:Bundles
17882:density
17780:Torsion
17746:Vectors
17738:Tensors
17721:)
17706:)
17702:,
17700:Pseudo−
17679:Poisson
17612:Finsler
17607:Fibered
17602:Contact
17600:)
17592:Complex
17590:)
17559:Section
17289:Bibcode
17119:, AMS,
17015:0350769
16966:Bibcode
16642:abelian
16169:on the
15922:current
15646:measure
15490:general
15278:-chain
15169:, then
15136:is an (
15038:, then
15026:is any
15020:). If
13337:is any
12696:
11584:, ...,
10514:, ...,
10315:, ...,
10275:, ...,
10094:, ...,
10064:, ...,
10058:, that
10046:, ...,
8758:, ...,
8051:adjoint
8049:and is
7336:is the
7326:is the
7316:, then
6932:and an
6758:is the
6405:may be
5985:is its
5955:is the
5811:of the
5801:of the
4829:unless
4031:, ...,
4019:-forms
4005:, ...,
3983:. Thus
3928:of its
3836:. Thus
3793:, this
3386:, ...,
3364:, ...,
3329:is the
3309:is the
3142:in the
3099:, then
1829:, ...,
1742:, with
1332:, ...,
845:Concept
820:History
227:surface
18633:Vector
18628:Spinor
18613:Matrix
18407:Tensor
18055:Vector
18040:Koszul
18020:Cartan
18015:Affine
17997:Vector
17992:Tensor
17977:Spinor
17967:Normal
17963:Stable
17917:Affine
17821:bundle
17773:bundle
17719:Almost
17642:Kähler
17598:Almost
17588:Almost
17582:Closed
17482:Sard's
17438:(list)
17219:
17198:
17167:
17144:
17123:
17084:
17061:
17040:
17013:
16998:(1967)
16987:
16940:
16899:
16889:
16581:-form
16550:where
16369:where
15955:, the
15743:|
15737:|
15733:|
15727:|
15696:|
15690:|
15655:|
15649:|
15501:-form
15494:cannot
14878:-form
14678:along
14661:-form
14263:, how
14223:-form
14075:Then (
13811:be an
13793:be an
13755:, and
13693:is an
13576:-form
13553:along
13260:where
12604:on an
12598:-form
12433:-form
12142:where
11514:where
11446:be an
11326:
11187:-form
10848:Here,
9638:
9619:
9596:
9562:
9515:where
9486:
9467:
9421:-form
9370:
9323:
9293:
9274:
9268:
9249:
9229:-form
9124:
9089:
9073:
9054:
9048:
9029:
8340:
8334:
8312:
8306:
8300:
8294:
8272:
8253:
8231:
8212:
8190:
8171:
8149:
8143:
8080:metric
7935:
7916:
7695:is an
7665:It is
7516:, and
6938:-form
6926:-form
6747:where
6322:-form
5941:where
5864:-form
5727:-form
5715:-form
5087:where
4694:> 1
4689:. For
4633:-form
4529:-form
4386:-form
4067:, the
4037:. Let
3105:has a
3051:be an
3004:, and
2535:) = Ω(
1151:is an
919:-form
776:, and
768:, the
672:-form
644:-form
511:On an
18553:Basis
18238:Scope
18163:Sheaf
17937:Fiber
17713:Rizza
17684:Prime
17515:Local
17505:Curve
17367:Atlas
17279:arXiv
17262:(PDF)
16956:arXiv
16836:Notes
16244:-form
15963:, is
15959:, or
15819:<
15488:On a
15130:: If
14489:. If
12439:over
12330:chain
12299:<
11561:over
11546:over
11117:>
8957:is a
8593:. If
8408:sheaf
8378:exact
7872:On a
7683:is a
7120:into
6861:as a
5833:is a
5797:is a
5787:be a
5751:with
5698:<
5681:<
5417:with
3910:is a
3899:-form
3850:) = ∂
3797:from
3547:is a
3317:then
3273:is a
3138:) of
3087:. If
2568:is a
1777:) = (
1747:<
1612:wedge
1484:<
1317:below
1223:. In
1221:chain
1100:<
910:<
900:<
795:on a
419:-form
223:-form
218:is a
72:-form
18030:Form
17932:Dual
17865:flow
17728:Tame
17704:Sub−
17617:Flat
17497:Maps
17217:ISBN
17196:ISBN
17165:ISBN
17142:ISBN
17121:ISBN
17082:ISBN
17059:ISBN
17038:ISBN
16985:ISBN
16938:ISBN
16897:OCLC
16887:ISBN
16745:and
16634:U(1)
16622:U(1)
16610:dual
16577:The
16238:The
16172:U(1)
16093:and
15437:) −
15276:+ 1)
15254:are
15248:and
15149:and
14978:and
14316:and
13775:and
13749:and
13737:and
13716:and
13121:) →
12753:and
12741:and
11926:<
11920:<
11627:<
11621:<
11440:Let
11157:Let
10614:<
10608:<
10578:<
10572:<
10375:<
10369:<
10156:<
10150:<
9749:and
8483:and
7482:for
6982:and
5971:and
5858:, a
5781:Let
5725:+ 1)
4948:and
4940:The
4933:and
4367:and
4343:and
3047:Let
3002:curl
2574:+ 1)
2238:<
2232:<
2219:<
1969:<
1963:<
1950:<
1765:) +
1532:<
634:The
589:and
17952:Jet
17188:doi
17030:doi
16863:doi
16779:in
16737:= 0
16632:is
16459:as
16246:is
16152:= −
15703:↦ −
15511:or
15422:to
15159:of
15141:− 1
14672:on
14516:at
14281:at
14229:on
13582:on
12784:is
12316:in
11573:in
11458:an
11338:def
11193:on
11139:= 0
10298:⋅⋅⋅
8988:is
8976:of
8775:at
8769:to
8743:of
8719:of
8709:on
8544:in
8421:of
8400:on
8366:of
8123:= 0
7491:∈ T
7314:= 1
7269:Alt
7196:Alt
7176:Alt
7040:Alt
6779:Alt
6762:on
6654:sgn
6556:Alt
6445:Alt
6394:),
6388:on
6370:on
6352:on
6312:of
6011:of
5965:at
5959:to
5815:of
5807:th
5658:⋅⋅⋅
5411:= 0
5203:= Σ
5198:of
4669:/ ∂
4645:on
4637:on
4622:on
4533:on
4522:(*)
4391:= Σ
4378:on
4350:(*)
4335:at
4053:/ ∂
3979:on
3832:at
3828:of
3824:or
3803:to
3551:of
3347:/ ∂
3333:of
3313:th
3277:on
3072:on
3055:in
2986:In
1818:= 0
1807:= 0
670:+1)
30:In
18851::
17943:Co
17302:.
17287:,
17277:,
17243:.
17194:,
17036:,
17011:MR
16964:,
16954:,
16895:.
16859:16
16857:,
16853:,
16783:.
16733:∧
16616:.
16393:ab
16382:ab
16150:23
16144:,
16130:=
16128:12
16060:ab
15805:.
15740:dx
15730:dx
15718:dx
15711:dx
15652:dx
15504:dx
15463:.
15401:df
15399:=
15391:.
15265:−
15152:∂M
14494:∈
14327:∈
14304:,
14258:∈
14205:+
13844:/
13834:∈
13764:→
13728:.
13677:/
13608:,
13598:∈
13451:,
13442:.
13432:/
12823:=
12813:∈
12768:→
12699:dt
12451::
12337:→
11519:⊆
11429:dx
11427:∧
11425:dx
11419:dx
11417:∧
11415:dx
11302::
11285:→
11080:.
11042:,
10517:dx
10511:dx
10500:df
10286:,
10112:,
10081:=
9740:.
9675:×
9520:×
9001:→
8992:df
8677:→
8650:TN
8648:→
8646:TM
8642:df
8574:∈
8507:∈
8474:TN
8472:→
8470:TM
8466:df
8452:→
8429:.
8390:Ω(
8382:Ω(
8343:0.
8129::
8095:.
8059:.
8047:−1
7510:,
7501:.
7487:,
7321:∧
7310:=
7012:⊗
7002:∧
6959:+
6949:∧
6545:,
6338:a
6333:∈
6213::
5853:∈
5845:.
5839:Ω(
5737:.
5730:dα
5708:.
5696:−1
5427:.
5424:df
5422:=
5409:dα
5224:dx
5211:=1
5195:dα
4937:.
4656:df
4654:=
4626:.
4510:∈
4406:dh
4353:.
4063:ij
4057:=
4042:=
4034:dx
4028:dx
4025:,
4022:dx
3999:,
3986:df
3970:∈
3941:df
3935:df
3932:,
3915:Σ
3901:.
3886:df
3868:df
3839:df
3812:df
3779:,
3557::
3380:,
3358:,
3299:=
3129:∈
3015:,
3000:,
2565:dω
2555:dx
2550:=
2527:∈
2425::
2153::=
2021::=
1816:dx
1812:dx
1805:dx
1803:∧
1801:dx
1791:dx
1789:∧
1787:dx
1785:)
1781:−
1775:dx
1773:∧
1771:dx
1763:dx
1761:∧
1759:dx
1739:dx
1737:∧
1735:dx
1729:dx
1723:dx
1717:dx
1711:dx
1476:dx
1474:∧
1472:dx
1443:=
1335:dx
1329:dx
1108:dx
930:dx
928:)
857:.
841:.
784:.
772:,
615:0.
529:.
447:dz
445:∧
443:dy
441:∧
439:dx
437:)
433:,
429:,
234::
215:dz
213:∧
211:dy
209:)
205:,
201:,
193:+
191:dx
189:∧
187:dz
185:)
181:,
177:,
169:+
167:dy
165:∧
163:dx
161:)
157:,
153:,
84::
64:dx
62:)
34:,
18212:e
18205:t
18198:v
17961:(
17941:(
17717:(
17698:(
17596:(
17586:(
17349:)
17345:(
17335:e
17328:t
17321:v
17291::
17281::
17269:.
17249:.
17190::
17173:,
17067:,
17032::
16968::
16958::
16917:.
16903:.
16865::
16748:F
16742:A
16735:A
16731:A
16713:.
16709:A
16701:A
16697:+
16693:A
16689:d
16686:=
16682:F
16667:F
16661:A
16651:F
16595:F
16579:2
16531:,
16526:J
16521:=
16511:F
16502:d
16493:0
16488:=
16479:F
16474:d
16445:J
16424:J
16406:j
16400:J
16389:f
16378:F
16372:j
16354:,
16348:d
16344:x
16340:d
16332:c
16328:x
16324:d
16316:b
16312:x
16308:d
16302:d
16299:c
16296:b
16293:a
16282:a
16278:j
16272:6
16269:1
16264:=
16259:J
16242:3
16223:.
16218:A
16213:d
16210:=
16205:F
16189:A
16158:z
16154:B
16147:f
16141:c
16139:/
16136:z
16132:E
16125:f
16104:B
16075:E
16056:f
16038:,
16032:b
16028:x
16024:d
16016:a
16012:x
16008:d
16002:b
15999:a
15995:f
15989:2
15986:1
15981:=
15976:F
15940:k
15934:M
15928:k
15904:n
15898:k
15892:k
15886:k
15880:k
15870:k
15861:k
15855:n
15849:n
15843:k
15833:k
15827:k
15821:n
15817:k
15811:k
15798:n
15788:n
15782:n
15765:n
15755:n
15716:−
15705:x
15701:x
15693:J
15681:J
15672:n
15667:1
15663:1
15659:1
15630:1
15624:=
15621:x
15618:d
15613:1
15608:0
15597:=
15594:x
15591:d
15586:0
15581:1
15552:1
15549:=
15546:x
15543:d
15538:1
15533:0
15509:1
15499:1
15445:)
15443:a
15441:(
15439:f
15435:b
15433:(
15431:f
15425:b
15419:a
15413:ω
15407:R
15397:ω
15378:0
15375:=
15372:0
15367:W
15359:=
15353:d
15348:W
15315:N
15307:=
15299:M
15281:W
15274:k
15272:(
15267:N
15263:M
15257:k
15251:N
15245:M
15239:k
15233:ω
15215:.
15207:M
15196:=
15190:d
15185:M
15162:M
15146:M
15139:n
15133:ω
15100:.
15091:)
15078:f
15068:(
15065:=
15035:N
15029:ℓ
15023:λ
15016:(
15005:f
14999:α
14991:α
14981:γ
14963:,
14957:v
14946:)
14943:y
14940:(
14935:1
14928:f
14919:=
14912:v
14908:,
14903:y
14881:γ
14875:k
14869:ζ
14863:N
14845:.
14842:M
14832:x
14828:T
14822:n
14816:m
14799:y
14789:/
14783:)
14779:)
14774:y
14760:v
14756:(
14747:f
14738:x
14729:(
14725:=
14720:x
14716:)
14709:v
14700:(
14687:)
14685:y
14683:(
14681:f
14675:M
14668:v
14664:β
14659:)
14657:n
14653:m
14651:(
14646:α
14628:.
14625:M
14615:x
14611:T
14605:k
14599:n
14584:)
14579:y
14565:v
14561:(
14552:f
14537:x
14532:)
14530:k
14526:n
14524:(
14519:y
14513:v
14507:k
14502:)
14500:y
14498:(
14496:f
14492:x
14486:N
14480:ζ
14462:,
14457:y
14443:v
14435:v
14408:N
14398:y
14394:T
14388:k
14382:n
14367:N
14362:y
14358:T
14352:k
14329:N
14325:y
14319:N
14313:M
14307:v
14301:α
14296:)
14294:y
14292:(
14290:f
14284:y
14278:v
14272:k
14266:γ
14260:N
14256:y
14250:α
14244:f
14238:α
14232:N
14226:γ
14220:k
14214:M
14209:)
14207:k
14203:n
14199:m
14197:(
14192:α
14174:.
14165:)
14156:/
14147:)
14144:y
14141:(
14136:1
14129:f
14118:(
14111:N
14103:=
14095:M
14060:.
14051:)
14042:/
14033:)
14030:y
14027:(
14022:1
14015:f
14004:(
13976:.
13971:y
13960:)
13953:y
13944:/
13935:)
13932:y
13929:(
13924:1
13917:f
13906:(
13898:y
13884:N
13878:n
13873:)
13871:y
13869:(
13867:f
13861:n
13857:m
13850:y
13846:η
13842:ω
13836:N
13832:y
13826:N
13820:N
13814:n
13808:η
13802:M
13796:m
13790:ω
13784:f
13778:N
13772:M
13766:N
13762:M
13758:f
13752:n
13746:m
13740:N
13734:M
13725:f
13719:N
13713:M
13708:)
13706:y
13704:(
13702:f
13696:m
13690:ω
13683:y
13679:η
13675:ω
13657:.
13652:y
13643:/
13637:x
13629:=
13624:x
13606:)
13604:y
13602:(
13600:f
13596:x
13591:)
13589:y
13587:(
13585:f
13579:σ
13574:)
13572:n
13568:m
13566:(
13562:)
13560:y
13558:(
13556:f
13550:M
13544:m
13538:ω
13520:M
13511:T
13504:)
13501:y
13498:(
13493:1
13486:f
13465:x
13458:x
13454:σ
13448:y
13438:y
13434:η
13429:x
13425:ω
13418:x
13414:σ
13396:.
13388:x
13374:j
13370:=
13365:x
13347:)
13345:n
13341:m
13339:(
13322:M
13312:x
13308:T
13302:n
13296:m
13276:x
13245:,
13242:M
13232:x
13228:T
13222:m
13202:x
13189:x
13185:)
13179:y
13165:f
13161:(
13158:=
13153:x
13133:x
13129:σ
13123:M
13119:y
13117:(
13115:f
13111:j
13104:y
13100:η
13093:x
13089:ω
13071:)
13068:)
13065:y
13062:(
13057:1
13050:f
13046:(
13036:x
13032:T
13026:n
13020:m
13003:x
12979:y
12975:η
12953:,
12950:N
12940:y
12936:T
12930:n
12909:y
12897:,
12894:M
12884:x
12880:T
12874:m
12853:x
12831:)
12829:x
12827:(
12825:f
12821:y
12815:M
12811:x
12805:f
12799:M
12794:)
12792:n
12788:m
12786:(
12782:)
12780:y
12778:(
12776:f
12770:N
12766:M
12762:f
12756:n
12750:m
12744:N
12738:M
12711:)
12709:t
12707:(
12705:f
12694:)
12692:t
12690:(
12688:f
12683:1
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12670:γ
12664:k
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12649:ω
12643:n
12637:ω
12631:ω
12625:R
12619:n
12613:ω
12607:n
12601:ω
12595:n
12589:M
12583:M
12565:M
12555:M
12537:.
12524:i
12514:D
12504:i
12500:m
12494:r
12489:1
12486:=
12483:i
12475:=
12467:c
12448:c
12442:c
12436:ω
12430:k
12412:,
12407:i
12397:i
12393:m
12387:r
12382:1
12379:=
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12368:=
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12351:M
12345:k
12339:M
12335:D
12328:-
12325:k
12319:R
12313:D
12307:k
12301:n
12297:k
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12285:R
12279:n
12270:φ
12245:)
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12232:,
12226:,
12221:1
12217:u
12213:(
12205:)
12198:n
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12179:,
12172:1
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12159:(
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12122:n
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12091:)
12086:n
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12078:,
12072:,
12067:1
12063:u
12059:(
12051:)
12044:n
12040:i
12035:x
12031:,
12025:,
12018:1
12014:i
12009:x
12005:(
11996:)
11993:)
11988:u
11983:(
11977:(
11970:n
11966:i
11962:,
11956:,
11951:1
11947:i
11942:a
11934:n
11930:i
11915:1
11911:i
11900:D
11892:=
11884:M
11853:.
11850:)
11847:)
11842:u
11837:(
11832:I
11828:x
11824:,
11818:,
11815:)
11810:u
11805:(
11800:1
11796:x
11792:(
11789:=
11786:)
11781:u
11776:(
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11741:.
11734:n
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11725:x
11721:d
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11701:i
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11688:)
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11678:(
11671:n
11667:i
11663:,
11657:,
11652:1
11648:i
11643:a
11635:n
11631:i
11616:1
11612:i
11603:=
11586:x
11582:x
11576:R
11570:M
11564:D
11558:ω
11555:φ
11549:M
11543:ω
11533:φ
11527:M
11521:R
11517:D
11499:M
11493:D
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11467:M
11461:n
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11397:.
11392:n
11388:x
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11376:1
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11368:d
11364:)
11361:x
11358:(
11355:f
11350:U
11333:=
11318:U
11299:f
11293:ω
11287:R
11283:R
11279:f
11259:n
11255:x
11251:d
11237:1
11233:x
11229:d
11225:)
11222:x
11219:(
11216:f
11213:=
11196:U
11190:ω
11184:n
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11172:R
11166:R
11160:U
11144:k
11137:k
11131:k
11125:k
11119:k
11115:n
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11091:k
11068:k
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10946:x
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10936:,
10929:1
10925:j
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10916:(
10908:)
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10875:1
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10862:(
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10826:k
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10777:)
10770:k
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10751:,
10744:1
10740:j
10735:x
10731:(
10723:)
10716:k
10712:i
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10677:(
10668:)
10665:f
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10638:i
10629:(
10622:k
10618:j
10603:1
10599:j
10586:k
10582:i
10567:1
10563:i
10554:=
10542:f
10523:k
10504:i
10482:.
10475:k
10471:i
10466:f
10462:d
10446:1
10442:i
10437:f
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10429:)
10426:f
10416:k
10412:i
10403:1
10399:i
10390:(
10383:k
10379:i
10364:1
10360:i
10351:=
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10324:ω
10318:y
10312:y
10304:k
10300:i
10296:1
10293:i
10289:ω
10282:k
10278:i
10272:1
10269:i
10251:,
10244:k
10240:i
10235:y
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10215:1
10211:i
10206:y
10202:d
10194:k
10190:i
10181:1
10177:i
10164:k
10160:i
10145:1
10141:i
10132:=
10115:ω
10109:N
10103:i
10098:)
10096:x
10092:x
10090:(
10087:i
10083:f
10079:y
10073:N
10067:y
10061:y
10055:M
10049:x
10043:x
10021:.
10018:)
10006:f
10002:(
9999:d
9996:=
9989:)
9983:d
9980:(
9971:f
9963:,
9951:f
9935:f
9931:=
9924:)
9912:(
9903:f
9895:,
9883:f
9879:+
9867:f
9863:=
9856:)
9850:+
9844:(
9835:f
9827:,
9824:)
9812:f
9808:(
9805:c
9802:=
9795:)
9789:c
9786:(
9777:f
9758:c
9752:η
9746:ω
9737:ω
9734:f
9718:R
9711:M
9705:M
9702:T
9697:k
9677:R
9673:M
9667:M
9648:R
9641:N
9616:N
9613:T
9608:k
9589:f
9586:d
9581:k
9559:M
9556:T
9551:k
9528:N
9522:R
9518:N
9500:,
9496:R
9489:N
9464:N
9461:T
9456:k
9430:ω
9424:ω
9418:k
9400:.
9397:M
9388:T
9382:k
9357:)
9353:f
9350:d
9347:(
9342:k
9320:N
9311:T
9305:k
9271:N
9261:f
9246:M
9232:ω
9226:k
9220:k
9199:)
9195:f
9192:d
9189:(
9184:k
9168:M
9163:1
9158:M
9140:.
9137:M
9128:T
9111:)
9107:f
9104:d
9101:(
9086:N
9077:T
9051:N
9041:f
9026:M
9012:ω
9006:M
9003:T
8999:N
8996:T
8990:(
8985:f
8979:N
8973:N
8970:T
8964:N
8959:1
8954:ω
8936:.
8933:)
8928:k
8924:v
8914:f
8910:,
8904:,
8899:1
8895:v
8885:f
8881:(
8876:)
8873:p
8870:(
8867:f
8859:=
8856:)
8851:k
8847:v
8843:,
8837:,
8832:1
8828:v
8824:(
8819:p
8815:)
8802:f
8798:(
8784:ω
8778:p
8772:M
8765:k
8761:v
8755:1
8752:v
8746:M
8740:p
8734:f
8728:ω
8722:ω
8712:M
8706:ω
8703:f
8697:N
8691:k
8685:ω
8679:N
8675:M
8671:f
8662:M
8656:M
8636:N
8627:N
8621:N
8615:∗
8612:f
8606:N
8602:q
8596:f
8590:N
8586:q
8582:T
8576:N
8572:q
8566:f
8560:N
8557:)
8555:p
8553:(
8551:f
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8542:)
8540:v
8538:(
8536:∗
8533:f
8527:M
8523:p
8519:T
8515:v
8509:M
8505:p
8495:∗
8492:f
8486:N
8480:M
8460:f
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8450:M
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8425:R
8413:R
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8394:)
8392:M
8386:)
8384:M
8369:M
8331:)
8328:M
8325:(
8320:n
8291:)
8288:M
8285:(
8280:3
8265:d
8250:)
8247:M
8244:(
8239:2
8224:d
8209:)
8206:M
8203:(
8198:1
8183:d
8168:)
8165:M
8162:(
8157:0
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8121:d
8069:1
8056:d
8033:)
8030:M
8027:(
8022:1
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8005:)
8002:M
7999:(
7994:k
7960:)
7957:M
7954:(
7949:k
7943:n
7913:)
7910:M
7907:(
7902:k
7852:.
7846:d
7835:k
7831:)
7827:1
7821:(
7818:+
7806:d
7803:=
7800:)
7788:(
7785:d
7774::
7755:.
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7734:)
7730:1
7724:(
7721:=
7698:ℓ
7692:β
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7680:α
7650:.
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7635:(
7629:f
7626:=
7623:)
7614:f
7611:(
7584:,
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7560:=
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7551:+
7545:(
7525:f
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7513:β
7507:α
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7401:(
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7388:=
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7364:)
7351:(
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7319:α
7312:ℓ
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7228:=
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7099:T
7093:n
7061:.
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7046:(
7037:=
7014:β
7010:α
7004:β
7000:α
6991:p
6985:β
6979:α
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6961:ℓ
6957:k
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6858:β
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6822:k
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6794:k
6765:k
6754:k
6750:S
6732:,
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6724:)
6721:k
6718:(
6711:x
6707:,
6701:,
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6693:1
6690:(
6683:x
6679:(
6674:p
6666:)
6660:(
6647:k
6643:S
6625:!
6622:k
6618:1
6613:=
6610:)
6605:k
6601:x
6597:,
6591:,
6586:1
6582:x
6578:(
6575:)
6570:p
6562:(
6542:p
6504:.
6501:M
6492:T
6486:k
6473:M
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6458:k
6425:1
6419:p
6415:X
6401:p
6397:α
6391:M
6381:M
6377:p
6373:T
6363:M
6359:p
6355:T
6348:p
6344:α
6335:M
6331:p
6325:α
6320:1
6315:M
6309:p
6303:k
6297:k
6279:.
6275:R
6268:M
6263:p
6259:T
6253:k
6248:1
6245:=
6242:n
6229:p
6183:)
6177:M
6172:p
6168:T
6162:k
6148:(
6140:M
6130:p
6126:T
6120:k
6097:k
6091:k
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6063:p
6059:T
6053:k
6036:p
6020:β
6014:M
6004:k
5998:p
5982:M
5978:p
5974:T
5968:p
5962:M
5952:M
5948:p
5944:T
5926:,
5923:M
5913:p
5909:T
5903:k
5886:p
5867:β
5861:k
5855:M
5851:p
5843:)
5841:M
5830:M
5824:k
5818:M
5804:k
5794:k
5784:M
5765:k
5761:M
5757:k
5749:M
5745:M
5735:α
5723:k
5721:(
5717:α
5713:k
5704:k
5700:i
5694:k
5690:i
5686:2
5683:i
5679:1
5676:i
5671:2
5664:k
5660:i
5656:2
5653:i
5650:1
5647:i
5643:f
5621:k
5617:i
5612:x
5608:d
5592:2
5588:i
5583:x
5579:d
5569:1
5565:i
5560:x
5556:d
5548:k
5544:i
5535:2
5531:i
5525:1
5521:i
5516:f
5510:n
5505:1
5502:=
5497:k
5493:i
5484:2
5480:i
5476:,
5471:1
5467:i
5448:k
5444:k
5440:2
5436:1
5432:0
5420:α
5415:f
5391:.
5386:j
5382:x
5378:d
5370:i
5366:x
5362:d
5353:i
5349:x
5338:j
5334:f
5322:n
5317:1
5314:=
5311:j
5308:,
5305:i
5297:=
5292:j
5288:x
5284:d
5276:j
5272:f
5268:d
5263:n
5258:1
5255:=
5252:j
5244:=
5238:d
5220:j
5216:f
5209:j
5201:α
5187:2
5183:2
5166:.
5161:i
5157:x
5153:d
5145:j
5141:x
5137:d
5131:=
5126:j
5122:x
5118:d
5110:i
5106:x
5102:d
5089:∧
5072:,
5069:0
5066:=
5061:j
5057:x
5053:d
5045:i
5041:x
5037:d
5028:i
5024:x
5013:j
5009:f
4997:n
4992:1
4989:=
4986:j
4983:,
4980:i
4958:1
4954:∧
4950:j
4946:i
4935:j
4931:i
4914:0
4911:=
4903:j
4899:x
4888:i
4884:f
4866:i
4862:x
4851:j
4847:f
4827:f
4810:,
4802:i
4798:x
4788:j
4784:x
4775:f
4770:2
4759:=
4751:j
4747:x
4737:i
4733:x
4724:f
4719:2
4698:f
4692:n
4685:i
4681:f
4676:n
4671:x
4667:f
4665:∂
4661:f
4652:α
4647:U
4643:f
4639:U
4635:α
4631:1
4624:U
4618:i
4614:f
4594:i
4590:x
4586:d
4580:i
4576:f
4570:n
4565:1
4562:=
4559:i
4551:=
4535:U
4531:α
4527:1
4517:1
4512:U
4508:p
4488:p
4484:)
4478:i
4474:h
4470:d
4467:(
4464:)
4461:p
4458:(
4453:i
4449:g
4443:i
4435:=
4430:p
4410:i
4402:i
4398:g
4394:i
4389:α
4384:1
4380:U
4374:i
4370:h
4363:i
4359:g
4345:j
4341:p
4337:p
4333:f
4329:j
4323:j
4319:e
4301:.
4296:p
4292:)
4286:i
4282:x
4278:d
4275:(
4272:)
4269:p
4266:(
4258:i
4254:x
4245:f
4234:n
4229:1
4226:=
4223:i
4215:=
4210:p
4206:f
4202:d
4185:p
4176:)
4174:*
4172:(
4155:.
4150:i
4146:x
4142:d
4133:i
4129:x
4120:f
4109:n
4104:1
4101:=
4098:i
4090:=
4087:f
4084:d
4059:δ
4055:x
4051:x
4049:∂
4044:x
4040:f
4017:1
4013:U
4008:x
4002:x
3996:x
3991:f
3981:U
3977:f
3972:U
3968:p
3962:j
3957:)
3954:j
3950:e
3948:(
3945:p
3924:j
3920:e
3917:v
3907:v
3897:1
3890:p
3881:p
3877:f
3873:U
3863:)
3861:p
3859:(
3857:f
3853:v
3848:v
3846:(
3843:p
3834:p
3830:f
3816:p
3806:R
3800:R
3791:p
3787:c
3782:w
3776:v
3754:)
3751:p
3748:(
3745:)
3742:f
3736:v
3727:(
3724:c
3721:=
3714:)
3711:p
3708:(
3705:)
3702:f
3696:v
3692:c
3684:(
3677:)
3674:p
3671:(
3668:)
3665:f
3659:w
3650:(
3647:+
3644:)
3641:p
3638:(
3635:)
3632:f
3626:v
3617:(
3614:=
3607:)
3604:p
3601:(
3598:)
3595:f
3589:w
3585:+
3581:v
3572:(
3554:v
3545:)
3543:p
3541:(
3539:f
3535:v
3532:∂
3515:.
3507:i
3503:y
3494:f
3480:j
3476:x
3465:i
3461:y
3449:n
3444:1
3441:=
3438:i
3430:=
3422:j
3418:x
3409:f
3389:y
3383:y
3377:y
3372:U
3367:x
3361:x
3355:x
3349:x
3345:f
3343:∂
3339:j
3335:f
3326:f
3322:v
3319:∂
3311:j
3305:j
3301:e
3297:v
3289:p
3284:v
3279:U
3270:v
3252:.
3247:0
3244:=
3241:t
3236:|
3232:)
3228:v
3224:t
3221:+
3218:p
3215:(
3212:f
3206:t
3203:d
3199:d
3189:=
3186:)
3183:p
3180:(
3177:)
3174:f
3168:v
3159:(
3145:v
3140:f
3136:p
3131:U
3127:p
3122:U
3117:f
3113:v
3110:∂
3102:f
3096:R
3090:v
3085:)
3083:U
3081:(
3079:C
3074:U
3070:f
3063:0
3058:R
3049:U
2989:R
2971:)
2968:M
2965:(
2960:1
2957:+
2954:k
2941:I
2937:x
2933:d
2926:)
2920:j
2916:x
2912:d
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2899:x
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2872:n
2867:1
2864:=
2861:j
2852:(
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2809:d
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2649:x
2645:d
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2632:x
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2601:i
2593:=
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2012:x
2008:d
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