1972:
1125:
1496:
813:
27:
1967:{\displaystyle (1)\quad {\begin{cases}L_{1}u\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i}f_{1}^{i}(x){\frac {\partial u}{\partial x^{i}}}={\vec {f}}_{1}\cdot \nabla u=0\\L_{2}u\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i}f_{2}^{i}(x){\frac {\partial u}{\partial x^{i}}}={\vec {f}}_{2}\cdot \nabla u=0\\\qquad \cdots \\L_{r}u\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i}f_{r}^{i}(x){\frac {\partial u}{\partial x^{i}}}={\vec {f}}_{r}\cdot \nabla u=0\end{cases}}}
2438:
4795:
2693:) that this has the same family of level sets but with a possibly different choice of constants for each set. Thus, even though the independent solutions of (1) are not unique, the equation (1) nonetheless determines a unique family of level sets. Just as in the case of the example, general solutions
823:
them together to form a full surface. The main danger is that, if we quilt the little planes two at a time, we might go on a cycle and return to where we began, but shifted by a small amount. If this happens, then we would not get a 2-dimensional surface, but a 3-dimensional blob. An example is shown
5567:
Specifically, Carathéodory considered a thermodynamic system (concretely one can imagine a piston of gas) that can interact with the outside world by either heat conduction (such as setting the piston on fire) or mechanical work (pushing on the piston). He then defined "adiabatic process" as any
2309:
1416:
2621:
given by this expression is a solution of the original equation. Thus, because of the existence of a family of level surfaces, solutions of the original equation are in a one-to-one correspondence with arbitrary functions of one variable.
4492:
684:
550:
2193:
5327:
2257:
at every point. The involutivity condition is a generalization of the commutativity of partial derivatives. In fact, the strategy of proof of the
Frobenius theorem is to form linear combinations among the operators
779:
into surfaces, in which case, we can be sure that a curve starting at a certain surface must be restricted to wander within that surface. If not, then a curve starting at any point might end up at any other point in
54:-axis. It is not integrable, as can be verified by drawing an infinitesimal square in the x-y plane, and follow the path along the one-forms. The path would not return to the same z-coordinate after one circuit.
5961:
686:
then we can draw two local planes at each point, and their intersection is generically a line, allowing us to uniquely solve for the curve starting at any point. In other words, with two 1-forms, we can
5456:
4328:
2601:
3842:
6145:
5232:
3906:
6039:
4393:
3792:
860:
4239:
3961:
2916:
4118:
916:
2433:{\displaystyle {\begin{cases}{\frac {\partial f}{\partial x}}+{\frac {\partial f}{\partial y}}=0\\{\frac {\partial f}{\partial y}}+{\frac {\partial f}{\partial z}}=0\end{cases}}}
320:
2503:
a constant. The second observation is that, once the level surfaces are known, all solutions can then be given in terms of an arbitrary function. Since the value of a solution
1314:
972:
807:
777:
2443:
clearly permits multiple solutions. Nevertheless, these solutions still have enough structure that they may be completely described. The first observation is that, even if
1032:
on the subset. This is usually called Carathéodory's theorem in axiomatic thermodynamics. One can prove this intuitively by first constructing the little planes according to
3365:
827:
If the one-form is integrable, then loops exactly close upon themselves, and each surface would be 2-dimensional. Frobenius' theorem states that this happens precisely when
6343:
6077:
5090:
748:
191:
137:
Suppose we are to find the trajectory of a particle in a subset of 3D space, but we do not know its trajectory formula. Instead, we know only that its trajectory satisfies
3521:
6243:
3019:
1294:
1004:
6299:
5721:
5695:
5636:
3458:
1155:
5848:
2836:
1304:
In its most elementary form, the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneous
6185:
6165:
1242:
1222:
1175:
1050:
943:
325:
261:
3432:
223:
4790:{\displaystyle D_{1}F(x,y)\cdot (s_{1},s_{2})+D_{2}F(x,y)\cdot (F(x,y)\cdot s_{1},s_{2})=D_{1}F(x,y)\cdot (s_{2},s_{1})+D_{2}F(x,y)\cdot (F(x,y)\cdot s_{2},s_{1})}
1099:
554:
In other words, we can draw a "local plane" at each point in 3D space, and we know that the particle's trajectory must be tangent to the local plane at all times.
6211:
5669:
1030:
6273:
3045:
1202:
5875:
5824:
5804:
5784:
5764:
5744:
5610:
5590:
3584:
3564:
3544:
3478:
3406:
3228:
3208:
3188:
3168:
3113:
3093:
3073:
2984:
2964:
2944:
2856:
2800:
2780:
1262:
1119:
1070:
3148:
560:
5461:
This result holds locally in the same sense as the other versions of the
Frobenius theorem. In particular, the fact that it has been stated for domains in
5880:
6354:
3650:
2753:
5830:
Then, we can foliate the state space into subsets of states that are mutually adiabatically accessible. With mild assumptions on the smoothness of
2052:
5250:
2716:
The
Frobenius theorem can be restated more economically in modern language. Frobenius' original version of the theorem was stated in terms of
6580:
2303:
Even though the system is overdetermined there are typically infinitely many solutions. For example, the system of differential equations
1157:
is visualized as a stack of parallel planes. The planes are quilted together, but with "uneven thickness". With a scaling at each point,
6907:
6902:
3693:. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between
6364:
6085:
2625:
Frobenius' theorem allows one to establish a similar such correspondence for the more general case of solutions of (1). Suppose that
5379:
1052:, quilting them together into a foliation, then assigning each surface in the foliation with a scalar label. Now for each point
2521:
6680:
6648:
3797:
4250:
4124:
3709:
6345:. That is, entropy is preserved in reversible adiabatic processes, and increases during irreversible adiabatic processes.
6041:
are the possible ways to perform mechanical work on the system. For example, if the system is a tank of ideal gas, then
5168:
67:
20:
6389:"On the Unrestricted Theorem of Carathéodory and Its Application in the Treatment of the Second Law of Thermodynamics"
6788:
6623:
6564:
The notion of a continuously differentiable function on a family of level sets can be made rigorous by means of the
5191:
2697:
of (1) are in a one-to-one correspondence with (continuously differentiable) functions on the family of level sets.
4159:
3850:
919:
70:
for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear
5966:
5508:
3170:
as well. This notion of integrability need only be defined locally; that is, the existence of the vector fields
124:
studies 1-forms that maximally violates the assumptions of
Frobenius' theorem. An example is shown on the right.
4340:
5512:
3755:
2919:
2923:
830:
95:
6912:
6768:
3914:
1305:
71:
2861:
6359:
4070:
3116:
865:
5854:
2029:
The
Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators
1423:
1411:{\displaystyle \left\{f_{k}^{i}:\mathbf {R} ^{n}\to \mathbf {R} \ :\ 1\leq i\leq n,1\leq k\leq r\right\}}
266:
4188:
5132:
2708:. Once one of these constants of integration is known, then the corresponding solution is also known.
2641:
are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the
948:
783:
753:
98:
for ordinary differential equations, which guarantees that a single vector field always gives rise to
6565:
5496:
3346:
2318:
1515:
569:
6304:
6044:
5564:, Frobenius' theorem can be used to construct entropy and temperature in Carathéodory's formalism.
5561:
5043:
4132:
697:
140:
5484:
3483:
2989:
If the subbundle has dimension greater than one, a condition needs to be imposed. One says that a
6216:
5569:
5548:
3675:
2995:
2705:
2039:
1267:
977:
115:
83:
6278:
5700:
5674:
5615:
3437:
1131:
50:
maximally violates the assumption of
Frobenius' theorem. These planes appear to twist along the
5833:
2809:
6776:
6388:
6170:
6150:
5533:
3635:
1227:
1207:
1160:
1035:
928:
228:
111:
75:
3411:
2700:
The level sets corresponding to the maximal independent solution sets of (1) are called the
6717:
6400:
5544:
5480:
2023:
196:
6690:
6601:
2704:
because functions on the collection of all integral manifolds correspond in some sense to
1075:
94:
whose tangent bundles are spanned by the given vector fields. The theorem generalizes the
8:
6917:
6664:
6190:
5850:, each subset is a manifold of codimension 1. Call these manifolds "adiabatic surfaces".
5648:
5537:
5529:
5479:
generalize to higher degree forms, although there is a number of partial results such as
4476:
is twice continuously differentiable. Then (1) is completely integrable at each point of
4443:
3980:
1009:
263:. Thus, our only certainty is that if at some moment in time the particle is at location
6721:
6705:
6404:
6255:
5568:
process that the system may undergo without heat conduction, and defined a relation of "
5547:, Frobenius' theorem can be used to prove the existence of a solution to the problem of
5532:, the integrability of a system's constraint equations determines whether the system is
4838:) denotes the partial derivative with respect to the first (resp. second) variable; the
3027:
1184:
1124:
16:
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
6880:
6851:
6819:
6772:
5860:
5809:
5789:
5769:
5749:
5729:
5595:
5575:
3992:
3569:
3549:
3529:
3463:
3391:
3213:
3193:
3173:
3153:
3098:
3078:
3058:
2969:
2949:
2929:
2841:
2785:
2765:
2690:
1247:
1104:
1055:
5572:" thus: if the system can go from state A to state B after an adiabatic process, then
3999:
is involutive. Consequently, the
Frobenius theorem takes on the equivalent form that
3121:
6884:
6855:
6823:
6784:
6733:
6676:
6644:
6619:
6589:
6455:
6416:
6245:. These are the temperature and entropy functions, up to a multiplicative constant.
4128:
3694:
2946:
is nowhere zero then it defines a one-dimensional subbundle of the tangent bundle of
2749:
2721:
2268:
91:
6872:
6843:
6811:
6725:
6686:
6597:
6447:
6408:
5172:
3662:
1458:
121:
6249:
5516:
4960:
2717:
925:
During his development of axiomatic thermodynamics, Carathéodory proved that if
545:{\displaystyle a(x_{0},y_{0},z_{0})+b(x_{0},y_{0},z_{0})+c(x_{0},y_{0},z_{0})=0}
322:, then its velocity at that moment is restricted within the plane with equation
6668:
6187:
is integrable, so by Carathéodory's theorem, there exists two scalar functions
5500:
5182:
3698:
3022:
2741:
2686:
99:
102:; Frobenius gives compatibility conditions under which the integral curves of
6896:
6876:
6847:
6815:
6800:"Ueber die simultane Integration linearer partieller Differentialgleichungen"
6737:
6593:
6459:
6420:
5504:
3264:
2475:
must overlap. In fact, the level surfaces for this system are all planes in
2458:
2267:
so that the resulting operators do commute, and then to show that there is a
2188:{\displaystyle L_{i}L_{j}u(x)-L_{j}L_{i}u(x)=\sum _{k}c_{ij}^{k}(x)L_{k}u(x)}
812:
26:
4963:. The statement is essentially the same as the finite-dimensional version.
6756:
5322:{\displaystyle d\omega ^{j}=\sum _{i=1}^{r}\psi _{i}^{j}\wedge \omega ^{i}}
4039:
3702:
2760:
2756:
to foliation; to state the theorem, both concepts must be clearly defined.
2725:
79:
4839:
4442:
The conditions of the
Frobenius theorem depend on whether the underlying
3523:
is integrable. Frobenius' theorem states that the converse is also true:
2245:
679:{\displaystyle {\begin{cases}adx+bdy+cdz=0\\a'dx+b'dy+c'dz=0\end{cases}}}
59:
4959:
The infinite-dimensional version of the
Frobenius theorem also holds on
3526:
Given the above definitions, Frobenius' theorem states that a subbundle
1470:. Consider the following system of partial differential equations for a
1244:
might have "uneven thickness". This can be fixed by a scalar scaling by
6451:
2241:
6729:
6412:
2279:
for which these are precisely the partial derivatives with respect to
3690:
3623:
3385:
3234:
2990:
2737:
2724:. An alternative formulation, which is somewhat more intuitive, uses
2642:
688:
87:
6832:
6799:
6482:. Henceforth, when we speak of a solution, we mean a local solution.
4061:
2745:
1977:
One seeks conditions on the existence of a collection of solutions
820:
5956:{\displaystyle dU=\delta W+\delta Q=\sum _{i}X_{i}dx_{i}+\delta Q}
3720:; and the other which operates with subbundles of the graded ring
2685:
is another such collection of solutions, one can show (using some
5515:
conditions. Frobenius is responsible for applying the theorem to
3708:
There are thus two forms of the theorem: one which operates with
110:-dimensional integral manifolds. The theorem is foundational in
3701:. Frobenius' theorem is one of the basic tools for the study of
2507:
on a level surface is constant by definition, define a function
6763:. American Mathematical Society CBMS Series. Vol. 27. AMS.
6435:
6252:
is an (irreversible) adiabatic process, we can fix the sign of
3681:
Geometrically, the theorem states that an integrable module of
3611:
5519:, thus paving the way for its usage in differential topology.
3285:
of the foliation, with the following property: Every point in
2748:
is integrable (or involutive) if and only if it arises from a
1177:
would have "even thickness", and become an exact differential.
2803:
819:
One can imagine starting with a cloud of little planes, and
6783:(2nd ed.). American Mathematical Society. p. 93.
5451:{\displaystyle \omega ^{j}=\sum _{i=1}^{r}f_{i}^{j}dg^{i}.}
5238:
linearly independent holomorphic 1-forms on an open set in
4030:
One infinite-dimensional generalization is as follows. Let
3210:
and their integrability need only be defined on subsets of
2736:
In the vector field formulation, the theorem states that a
2426:
1960:
672:
5555:
2986:. Thus, one-dimensional subbundles are always integrable.
4010:
is closed under exterior differentiation if and only if
2711:
2596:{\displaystyle f(x,y,z)=C(t){\text{ whenever }}x-y+z=t.}
2046:. Specifically, they must satisfy relations of the form
3837:{\displaystyle k\in \{1,\dots ,\operatorname {dim} M\}}
1101:
to be the scalar label of the surface containing point
6436:"Untersuchungen über die Grundlagen der Thermodynamik"
5511:
conditions for the theorem, and
Clebsch developed the
2966:, and the integral curves form a regular foliation of
6639:
Lang, S. (1995). "Ch. VI: The theorem of Frobenius".
6307:
6281:
6258:
6219:
6193:
6173:
6153:
6088:
6047:
5969:
5883:
5863:
5836:
5812:
5792:
5772:
5752:
5732:
5703:
5677:
5651:
5618:
5598:
5578:
5382:
5253:
5194:
5046:
4495:
4343:
4323:{\displaystyle \forall x\in A:\quad u'(x)=F(x,u(x)).}
4253:
4191:
4073:
4022:
The theorem may be generalized in a variety of ways.
3917:
3853:
3800:
3758:
3572:
3552:
3532:
3486:
3466:
3440:
3414:
3394:
3349:
3216:
3196:
3176:
3156:
3124:
3101:
3081:
3061:
3030:
2998:
2972:
2952:
2932:
2864:
2844:
2812:
2788:
2768:
2720:, which today can be translated into the language of
2524:
2312:
2055:
1499:
1317:
1270:
1250:
1230:
1224:. However, it has "even thickness" everywhere, while
1210:
1187:
1163:
1134:
1107:
1078:
1058:
1038:
1012:
980:
951:
931:
868:
833:
786:
756:
700:
563:
328:
269:
231:
199:
143:
6863:
Frobenius, G. (1877). "Über das Pfaffsche Problem".
3991:) is closed under exterior differentiation (it is a
6248:By plugging in the ideal gas laws, and noting that
6337:
6293:
6267:
6237:
6205:
6179:
6159:
6139:
6071:
6033:
5955:
5869:
5842:
5818:
5798:
5778:
5758:
5738:
5715:
5689:
5663:
5630:
5604:
5584:
5450:
5321:
5226:
5084:
4789:
4387:
4322:
4233:
4112:
3955:
3900:
3836:
3786:
3578:
3558:
3538:
3515:
3472:
3452:
3426:
3400:
3359:
3222:
3202:
3182:
3162:
3142:
3107:
3087:
3067:
3039:
3013:
2978:
2958:
2938:
2910:
2850:
2830:
2794:
2774:
2752:. In this context, the Frobenius theorem relates
2731:
2595:
2432:
2187:
1966:
1410:
1288:
1256:
1236:
1216:
1204:is a one-form that has exactly the same planes as
1196:
1169:
1149:
1113:
1093:
1064:
1044:
1024:
998:
966:
937:
910:
854:
801:
771:
742:
678:
544:
314:
255:
217:
185:
6355:Integrability conditions for differential systems
6147:Now, since the adiabatic surfaces are tangent to
5877:("internal energy") on the state space, such that
3589:
6894:
6140:{\displaystyle \omega :=dU-\sum _{i}X_{i}dx_{i}}
6663:
5159:is integrable if and only if it is involutive.
3732:. These two forms are related by duality. If
945:is an integrable one-form on an open subset of
6767:
6675:. Wiley Classics Library. Vol. 2. Wiley.
5227:{\displaystyle \omega ^{1},\dots ,\omega ^{r}}
5167:The statement of the theorem remains true for
5155:The Frobenius theorem states that a subbundle
2759:One begins by noting that an arbitrary smooth
19:For other theorems named after Frobenius, see
6833:"Über die Bedingungen der Integrabilitat ..."
6581:Bulletin of the American Mathematical Society
3901:{\displaystyle \alpha (v_{1},\dots ,v_{k})=0}
3460:is the leaf of the foliation passing through
82:, the theorem gives necessary and sufficient
6613:
6433:
3971:) forms a subring and, in fact, an ideal in
3831:
3807:
2298:
918:. The notation is defined in the article on
106:vector fields mesh into coordinate grids on
6082:Now, define the one-form on the state space
6034:{\displaystyle X_{1}dx_{1},...,X_{n}dx_{n}}
4182:is a solution of the differential equation
3979:. Furthermore, using the definition of the
3546:is integrable if and only if the subbundle
6706:"The Concepts of Classical Thermodynamics"
6476:means inside small enough open subsets of
4869:, as well as the actions of the operators
4842:denotes the action of the linear operator
4466:is continuously differentiable. If it is
4388:{\displaystyle (x_{0},y_{0})\in A\times B}
3408:defines an integrable subbundle, since if
6862:
5373:such that, on a possibly smaller domain,
5360:, then there exist holomorphic functions
3787:{\displaystyle \alpha \in \Omega ^{k}(M)}
2922:, whose solvability is guaranteed by the
954:
789:
759:
6703:
6578:Lawson, H. Blaine (1974), "Foliations",
6386:
5507:. Deahna was the first to establish the
4978:be a subbundle of the tangent bundle of
3641:, the rank being constant in value over
1123:
855:{\displaystyle \omega \wedge d\omega =0}
811:
25:
6797:
5556:Carathéodory's axiomatic thermodynamics
5332:for some system of holomorphic 1-forms
4970:be a Banach manifold of class at least
3956:{\displaystyle v_{1},\dots ,v_{k}\in D}
3610:be the space of smooth, differentiable
6895:
6830:
6755:
6577:
5470:
4025:
2911:{\displaystyle {\dot {u}}(t)=X_{u(t)}}
6641:Differential and Riemannian manifolds
6571:
4113:{\displaystyle F:A\times B\to L(X,Y)}
3689:is the same thing as a codimension-r
3645:. The Frobenius theorem states that
3343:=constant. A foliation is denoted by
2712:Frobenius' theorem in modern language
1299:
6761:The Qualitative Theory of Foliations
6673:Foundations of Differential Geometry
6638:
6382:
6380:
5162:
4406:such that (1) has a unique solution
4125:continuously differentiable function
3736:is a smooth tangent distribution on
1296:. This is illustrated on the right.
911:{\displaystyle \omega :=adx+bdy+cdz}
5806:is adiabatically inaccessible from
5113:, there is an immersed submanifold
4954:
3566:arises from a regular foliation of
2918:, which is a system of first-order
750:, then we might be able to foliate
315:{\displaystyle (x_{0},y_{0},z_{0})}
132:
68:necessary and sufficient conditions
13:
5499:, the theorem was first proven by
4254:
4234:{\displaystyle (1)\quad y'=F(x,y)}
4017:
3766:
3352:
3237:exist. Here we use the following:
2408:
2400:
2385:
2377:
2355:
2347:
2332:
2324:
1945:
1904:
1896:
1848:
1845:
1842:
1792:
1751:
1743:
1695:
1692:
1689:
1647:
1606:
1598:
1550:
1547:
1544:
14:
6929:
6908:Theorems in differential topology
6903:Theorems in differential geometry
6614:Dieudonné, J (1969). "Ch. 10.9".
6377:
5857:, there exists a scalar function
5592:is adiabatically accessible from
5549:integrability of demand functions
4160:continuous linear transformations
3055:), if, for any two vector fields
2457:are two different solutions, the
2858:). These are the solutions of
2236:) that are allowed to depend on
1357:
1343:
967:{\displaystyle \mathbb {R} ^{n}}
802:{\displaystyle \mathbb {R} ^{3}}
772:{\displaystyle \mathbb {R} ^{3}}
6804:J. Reine. Angew. Math. (Crelle)
6697:
6497:corresponding to the locus of:
6167:at every point in state space,
5786:in the neighborhood, such that
5522:
4269:
4201:
3335:are described by the equations
2920:ordinary differential equations
2732:Formulation using vector fields
1808:
1509:
1464:when evaluated at any point of
1128:For each point p, the one-form
127:
6704:Buchdahl, H. A. (1960-03-01).
6657:
6632:
6616:Foundations of modern analysis
6607:
6558:
6485:
6466:
6427:
6387:Buchdahl, H. A. (April 1949).
6332:
6326:
6317:
6311:
5060:
5047:
4784:
4755:
4743:
4737:
4731:
4719:
4700:
4674:
4668:
4656:
4637:
4608:
4596:
4590:
4584:
4572:
4553:
4527:
4521:
4509:
4370:
4344:
4314:
4311:
4305:
4293:
4284:
4278:
4228:
4216:
4198:
4192:
4107:
4095:
4089:
3889:
3857:
3781:
3775:
3590:Differential forms formulation
3360:{\displaystyle {\mathcal {F}}}
3137:
3125:
2903:
2897:
2883:
2877:
2822:
2561:
2555:
2546:
2528:
2182:
2176:
2160:
2154:
2120:
2114:
2085:
2079:
1930:
1890:
1884:
1777:
1737:
1731:
1632:
1592:
1586:
1506:
1500:
1353:
1306:partial differential equations
1144:
1138:
1088:
1082:
862:over all of the domain, where
533:
514:
511:
472:
463:
444:
441:
402:
393:
374:
371:
332:
309:
270:
250:
232:
72:partial differential equations
1:
6748:
6338:{\displaystyle S(A)\leq S(B)}
6072:{\displaystyle \delta W=-pdV}
5085:{\displaystyle _{p}\in E_{p}}
5012:defined in a neighborhood of
3598:be an open set in a manifold
3293:and a system of local, class
2924:Picard–Lindelöf theorem
2617:is given, then each function
824:in the diagram on the right.
743:{\displaystyle adx+bdy+cdz=0}
694:If we have only one equation
186:{\displaystyle adx+bdy+cdz=0}
6360:Domain-straightening theorem
4170:. A differentiable mapping
3712:, that is smooth subbundles
3516:{\displaystyle E_{p}=T_{p}N}
2663:as functions with values in
2223:, and for some coefficients
7:
6777:"2.2.26 Frobenius' Theorem"
6710:American Journal of Physics
6491:A level set is a subset of
6393:American Journal of Physics
6365:Newlander-Nirenberg Theorem
6348:
6238:{\displaystyle \omega =TdS}
5855:first law of thermodynamics
3014:{\displaystyle E\subset TM}
1439:, and such that the matrix
1289:{\displaystyle \omega =fdg}
999:{\displaystyle \omega =fdg}
10:
6934:
6294:{\displaystyle A\succeq B}
6213:on state space, such that
5746:, and any neighborhood of
5716:{\displaystyle B\succeq A}
5690:{\displaystyle A\succeq B}
5631:{\displaystyle A\succeq B}
5490:
4395:, there is a neighborhood
3740:, then the annihilator of
3453:{\displaystyle N\subset M}
2606:Conversely, if a function
1150:{\displaystyle \omega (p)}
1006:for some scalar functions
18:
6566:implicit function theorem
6434:Carathéodory, C. (1909).
5497:Ferdinand Georg Frobenius
3653:if and only if for every
2299:From analysis to geometry
78:terms, given a family of
6877:10.1515/crll.1877.82.230
6848:10.1515/crll.1840.20.340
6816:10.1515/crll.1866.65.257
6781:Foundations of Mechanics
6370:
5843:{\displaystyle \succeq }
5562:classical thermodynamics
5495:Despite being named for
4135:from its inclusion into
4133:differentiable structure
3752:) consists of all forms
3676:exact differential forms
3317:such that for each leaf
3267:connected submanifolds {
2831:{\displaystyle u:I\to M}
2706:constants of integration
1999:such that the gradients
691:the domain into curves.
557:If we have two equations
225:are smooth functions of
84:integrability conditions
6671:(2009) . "Appendix 8".
6180:{\displaystyle \omega }
6160:{\displaystyle \omega }
5766:, there exists a state
5645:For any pair of states
5570:adiabatic accessibility
5175:— manifolds over
3983:, it can be shown that
3233:Several definitions of
2040:integrability condition
1237:{\displaystyle \omega }
1217:{\displaystyle \omega }
1170:{\displaystyle \omega }
1045:{\displaystyle \omega }
938:{\displaystyle \omega }
256:{\displaystyle (x,y,z)}
86:for the existence of a
6339:
6295:
6269:
6239:
6207:
6181:
6161:
6141:
6073:
6035:
5957:
5871:
5844:
5820:
5800:
5780:
5760:
5740:
5717:
5691:
5665:
5632:
5606:
5586:
5452:
5416:
5323:
5290:
5228:
5086:
4791:
4389:
4324:
4235:
4114:
3957:
3902:
3838:
3788:
3716:of the tangent bundle
3580:
3560:
3540:
3517:
3474:
3454:
3428:
3427:{\displaystyle p\in M}
3402:
3361:
3259:is a decomposition of
3255:-dimensional manifold
3224:
3204:
3184:
3164:
3144:
3109:
3089:
3069:
3041:
3015:
2980:
2960:
2940:
2926:. If the vector field
2912:
2852:
2832:
2806:, its integral curves
2796:
2776:
2597:
2434:
2240:. In other words, the
2189:
1968:
1412:
1290:
1258:
1238:
1218:
1198:
1178:
1171:
1151:
1115:
1095:
1066:
1046:
1026:
1000:
968:
939:
912:
856:
816:
803:
773:
744:
680:
546:
316:
257:
219:
187:
55:
6440:Mathematische Annalen
6340:
6296:
6270:
6240:
6208:
6182:
6162:
6142:
6074:
6036:
5958:
5872:
5845:
5821:
5801:
5781:
5761:
5741:
5718:
5692:
5666:
5633:
5607:
5587:
5485:Cartan-Kähler theorem
5453:
5396:
5324:
5270:
5229:
5139:is an isomorphism of
5127:whose image contains
5087:
5016:, the Lie bracket of
5000:and pair of sections
4792:
4390:
4335:completely integrable
4325:
4236:
4115:
3958:
3903:
3839:
3789:
3581:
3561:
3541:
3518:
3475:
3455:
3429:
3403:
3362:
3225:
3205:
3185:
3165:
3145:
3110:
3090:
3070:
3042:
3016:
2981:
2961:
2941:
2913:
2853:
2833:
2797:
2777:
2598:
2435:
2190:
1969:
1413:
1291:
1259:
1239:
1219:
1199:
1172:
1152:
1127:
1116:
1096:
1067:
1047:
1027:
1001:
969:
940:
913:
857:
815:
804:
774:
745:
681:
547:
317:
258:
220:
218:{\displaystyle a,b,c}
188:
116:calculus on manifolds
112:differential topology
29:
6913:Differential systems
6865:J. Reine Angew. Math
6836:J. Reine Angew. Math
6798:Clebsch, A. (1866).
6305:
6279:
6256:
6217:
6191:
6171:
6151:
6086:
6045:
5967:
5881:
5861:
5834:
5810:
5790:
5770:
5750:
5730:
5701:
5675:
5649:
5616:
5596:
5576:
5545:microeconomic theory
5467:is not restrictive.
5380:
5251:
5192:
5183:transition functions
5044:
4493:
4341:
4333:The equation (1) is
4251:
4189:
4071:
3915:
3851:
3798:
3756:
3570:
3550:
3530:
3484:
3464:
3438:
3412:
3392:
3347:
3324:, the components of
3247:-dimensional, class
3214:
3194:
3174:
3154:
3122:
3099:
3079:
3059:
3028:
2996:
2970:
2950:
2930:
2862:
2842:
2810:
2802:defines a family of
2786:
2766:
2566: whenever
2522:
2310:
2053:
2024:linearly independent
1497:
1315:
1268:
1248:
1228:
1208:
1185:
1161:
1132:
1105:
1094:{\displaystyle g(p)}
1076:
1056:
1036:
1010:
978:
949:
929:
866:
831:
784:
754:
698:
561:
326:
267:
229:
197:
141:
6831:Deahna, F. (1840).
6773:Marsden, Jerrold E.
6722:1960AmJPh..28..196B
6643:. Springer-Verlag.
6546:for some constants
6405:1949AmJPh..17..212B
6206:{\displaystyle T,S}
5664:{\displaystyle A,B}
5530:classical mechanics
5471:Higher degree forms
5431:
5305:
5181:with biholomorphic
5169:holomorphic 1-forms
5095:On the other hand,
4990:if, for each point
4026:Infinite dimensions
3981:exterior derivative
3289:has a neighborhood
2153:
1883:
1730:
1585:
1421:be a collection of
1337:
1025:{\displaystyle f,g}
6618:. Academic Press.
6452:10.1007/BF01450409
6335:
6291:
6268:{\displaystyle dS}
6265:
6235:
6203:
6177:
6157:
6137:
6113:
6069:
6031:
5953:
5920:
5867:
5840:
5816:
5796:
5776:
5756:
5736:
5713:
5687:
5671:, at least one of
5661:
5628:
5602:
5582:
5448:
5417:
5319:
5291:
5224:
5082:
4787:
4385:
4320:
4231:
4131:(which inherits a
4110:
3993:differential ideal
3953:
3898:
3834:
3784:
3695:differential forms
3576:
3556:
3536:
3513:
3470:
3450:
3424:
3398:
3357:
3220:
3200:
3180:
3160:
3140:
3105:
3085:
3065:
3040:{\displaystyle TM}
3037:
3011:
2976:
2956:
2936:
2908:
2848:
2828:
2792:
2772:
2722:differential forms
2702:integral manifolds
2691:mean value theorem
2593:
2430:
2425:
2185:
2136:
2135:
2038:satisfy a certain
1964:
1959:
1869:
1868:
1716:
1715:
1571:
1570:
1408:
1323:
1300:Multiple one-forms
1286:
1254:
1234:
1214:
1197:{\displaystyle dg}
1194:
1179:
1167:
1147:
1111:
1091:
1062:
1042:
1022:
996:
964:
935:
908:
852:
817:
799:
769:
740:
676:
671:
542:
312:
253:
215:
183:
92:integral manifolds
64:Frobenius' theorem
56:
6730:10.1119/1.1935102
6682:978-0-471-15732-8
6650:978-0-387-94338-1
6413:10.1119/1.1989552
6104:
5911:
5870:{\displaystyle U}
5819:{\displaystyle A}
5799:{\displaystyle B}
5779:{\displaystyle B}
5759:{\displaystyle A}
5739:{\displaystyle A}
5605:{\displaystyle A}
5585:{\displaystyle B}
5481:Darboux's theorem
5188:Specifically, if
5173:complex manifolds
5163:Holomorphic forms
4143:) into the space
4129:Cartesian product
3995:) if and only if
3579:{\displaystyle M}
3559:{\displaystyle E}
3539:{\displaystyle E}
3473:{\displaystyle p}
3401:{\displaystyle M}
3223:{\displaystyle M}
3203:{\displaystyle Y}
3183:{\displaystyle X}
3163:{\displaystyle E}
3108:{\displaystyle E}
3095:taking values in
3088:{\displaystyle Y}
3068:{\displaystyle X}
2979:{\displaystyle M}
2959:{\displaystyle M}
2939:{\displaystyle X}
2874:
2851:{\displaystyle I}
2795:{\displaystyle M}
2775:{\displaystyle X}
2750:regular foliation
2567:
2415:
2392:
2362:
2339:
2269:coordinate system
2126:
1933:
1918:
1859:
1858:
1853:
1831:
1780:
1765:
1706:
1705:
1700:
1678:
1635:
1620:
1561:
1560:
1555:
1533:
1369:
1363:
1257:{\displaystyle f}
1114:{\displaystyle p}
1065:{\displaystyle p}
96:existence theorem
21:Frobenius theorem
6925:
6888:
6859:
6827:
6794:
6764:
6742:
6741:
6701:
6695:
6694:
6661:
6655:
6654:
6636:
6630:
6629:
6611:
6605:
6604:
6575:
6569:
6562:
6556:
6554:
6542:
6496:
6489:
6483:
6481:
6470:
6464:
6463:
6431:
6425:
6424:
6384:
6344:
6342:
6341:
6336:
6300:
6298:
6297:
6292:
6275:, and find that
6274:
6272:
6271:
6266:
6244:
6242:
6241:
6236:
6212:
6210:
6209:
6204:
6186:
6184:
6183:
6178:
6166:
6164:
6163:
6158:
6146:
6144:
6143:
6138:
6136:
6135:
6123:
6122:
6112:
6078:
6076:
6075:
6070:
6040:
6038:
6037:
6032:
6030:
6029:
6017:
6016:
5992:
5991:
5979:
5978:
5962:
5960:
5959:
5954:
5943:
5942:
5930:
5929:
5919:
5876:
5874:
5873:
5868:
5849:
5847:
5846:
5841:
5825:
5823:
5822:
5817:
5805:
5803:
5802:
5797:
5785:
5783:
5782:
5777:
5765:
5763:
5762:
5757:
5745:
5743:
5742:
5737:
5722:
5720:
5719:
5714:
5696:
5694:
5693:
5688:
5670:
5668:
5667:
5662:
5641:Now assume that
5637:
5635:
5634:
5629:
5611:
5609:
5608:
5603:
5591:
5589:
5588:
5583:
5517:Pfaffian systems
5466:
5457:
5455:
5454:
5449:
5444:
5443:
5430:
5425:
5415:
5410:
5392:
5391:
5372:
5359:
5346:
5345:
5328:
5326:
5325:
5320:
5318:
5317:
5304:
5299:
5289:
5284:
5266:
5265:
5243:
5233:
5231:
5230:
5225:
5223:
5222:
5204:
5203:
5180:
5158:
5151:
5138:
5131:, such that the
5126:
5112:
5098:
5091:
5089:
5088:
5083:
5081:
5080:
5068:
5067:
5036:
5019:
5011:
5003:
4999:
4985:
4981:
4977:
4969:
4961:Banach manifolds
4955:Banach manifolds
4950:
4909:
4868:
4837:
4828:
4819:
4796:
4794:
4793:
4788:
4783:
4782:
4770:
4769:
4715:
4714:
4699:
4698:
4686:
4685:
4652:
4651:
4636:
4635:
4623:
4622:
4568:
4567:
4552:
4551:
4539:
4538:
4505:
4504:
4485:
4471:
4457:
4451:
4416:
4394:
4392:
4391:
4386:
4369:
4368:
4356:
4355:
4329:
4327:
4326:
4321:
4277:
4240:
4238:
4237:
4232:
4209:
4165:
4157:
4119:
4117:
4116:
4111:
4059:
4037:
4033:
4009:
3978:
3962:
3960:
3959:
3954:
3946:
3945:
3927:
3926:
3907:
3905:
3904:
3899:
3888:
3887:
3869:
3868:
3843:
3841:
3840:
3835:
3793:
3791:
3790:
3785:
3774:
3773:
3739:
3728:of all forms on
3727:
3705:and foliations.
3684:
3671:is generated by
3660:
3656:
3633:
3609:
3601:
3585:
3583:
3582:
3577:
3565:
3563:
3562:
3557:
3545:
3543:
3542:
3537:
3522:
3520:
3519:
3514:
3509:
3508:
3496:
3495:
3479:
3477:
3476:
3471:
3459:
3457:
3456:
3451:
3433:
3431:
3430:
3425:
3407:
3405:
3404:
3399:
3366:
3364:
3363:
3358:
3356:
3355:
3339:=constant, ⋅⋅⋅,
3263:into a union of
3251:foliation of an
3229:
3227:
3226:
3221:
3209:
3207:
3206:
3201:
3189:
3187:
3186:
3181:
3169:
3167:
3166:
3161:
3150:takes values in
3149:
3147:
3146:
3143:{\displaystyle }
3141:
3114:
3112:
3111:
3106:
3094:
3092:
3091:
3086:
3074:
3072:
3071:
3066:
3046:
3044:
3043:
3038:
3020:
3018:
3017:
3012:
2985:
2983:
2982:
2977:
2965:
2963:
2962:
2957:
2945:
2943:
2942:
2937:
2917:
2915:
2914:
2909:
2907:
2906:
2876:
2875:
2867:
2857:
2855:
2854:
2849:
2837:
2835:
2834:
2829:
2801:
2799:
2798:
2793:
2781:
2779:
2778:
2773:
2718:Pfaffian systems
2684:
2668:
2662:
2640:
2616:
2602:
2600:
2599:
2594:
2568:
2565:
2502:
2498:
2480:
2439:
2437:
2436:
2431:
2429:
2428:
2416:
2414:
2406:
2398:
2393:
2391:
2383:
2375:
2363:
2361:
2353:
2345:
2340:
2338:
2330:
2322:
2294:
2278:
2266:
2256:
2244:must lie in the
2218:
2212:
2194:
2192:
2191:
2186:
2172:
2171:
2152:
2147:
2134:
2110:
2109:
2100:
2099:
2075:
2074:
2065:
2064:
2037:
2021:
1998:
1973:
1971:
1970:
1965:
1963:
1962:
1941:
1940:
1935:
1934:
1926:
1919:
1917:
1916:
1915:
1902:
1894:
1882:
1877:
1867:
1856:
1855:
1854:
1852:
1851:
1839:
1834:
1829:
1825:
1824:
1788:
1787:
1782:
1781:
1773:
1766:
1764:
1763:
1762:
1749:
1741:
1729:
1724:
1714:
1703:
1702:
1701:
1699:
1698:
1686:
1681:
1676:
1672:
1671:
1643:
1642:
1637:
1636:
1628:
1621:
1619:
1618:
1617:
1604:
1596:
1584:
1579:
1569:
1558:
1557:
1556:
1554:
1553:
1541:
1536:
1531:
1527:
1526:
1489:
1475:
1469:
1456:
1454:
1453:
1438:
1429:functions, with
1428:
1417:
1415:
1414:
1409:
1407:
1403:
1367:
1361:
1360:
1352:
1351:
1346:
1336:
1331:
1295:
1293:
1292:
1287:
1263:
1261:
1260:
1255:
1243:
1241:
1240:
1235:
1223:
1221:
1220:
1215:
1203:
1201:
1200:
1195:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1120:
1118:
1117:
1112:
1100:
1098:
1097:
1092:
1071:
1069:
1068:
1063:
1051:
1049:
1048:
1043:
1031:
1029:
1028:
1023:
1005:
1003:
1002:
997:
973:
971:
970:
965:
963:
962:
957:
944:
942:
941:
936:
917:
915:
914:
909:
861:
859:
858:
853:
808:
806:
805:
800:
798:
797:
792:
778:
776:
775:
770:
768:
767:
762:
749:
747:
746:
741:
685:
683:
682:
677:
675:
674:
656:
639:
622:
551:
549:
548:
543:
532:
531:
510:
509:
497:
496:
484:
483:
462:
461:
440:
439:
427:
426:
414:
413:
392:
391:
370:
369:
357:
356:
344:
343:
321:
319:
318:
313:
308:
307:
295:
294:
282:
281:
262:
260:
259:
254:
224:
222:
221:
216:
192:
190:
189:
184:
133:One-form version
122:Contact geometry
45:
6933:
6932:
6928:
6927:
6926:
6924:
6923:
6922:
6893:
6892:
6891:
6871:(82): 230–315.
6810:(65): 257–268.
6791:
6769:Abraham, Ralph.
6751:
6746:
6745:
6702:
6698:
6683:
6662:
6658:
6651:
6637:
6633:
6626:
6612:
6608:
6576:
6572:
6563:
6559:
6552:
6547:
6540:
6527:
6520:
6507:
6500:
6492:
6490:
6486:
6477:
6471:
6467:
6432:
6428:
6385:
6378:
6373:
6351:
6306:
6303:
6302:
6280:
6277:
6276:
6257:
6254:
6253:
6250:Joule expansion
6218:
6215:
6214:
6192:
6189:
6188:
6172:
6169:
6168:
6152:
6149:
6148:
6131:
6127:
6118:
6114:
6108:
6087:
6084:
6083:
6046:
6043:
6042:
6025:
6021:
6012:
6008:
5987:
5983:
5974:
5970:
5968:
5965:
5964:
5938:
5934:
5925:
5921:
5915:
5882:
5879:
5878:
5862:
5859:
5858:
5835:
5832:
5831:
5811:
5808:
5807:
5791:
5788:
5787:
5771:
5768:
5767:
5751:
5748:
5747:
5731:
5728:
5727:
5702:
5699:
5698:
5676:
5673:
5672:
5650:
5647:
5646:
5617:
5614:
5613:
5597:
5594:
5593:
5577:
5574:
5573:
5558:
5525:
5493:
5473:
5462:
5439:
5435:
5426:
5421:
5411:
5400:
5387:
5383:
5381:
5378:
5377:
5368:
5366:
5344:
5339:
5338:
5337:
5333:
5313:
5309:
5300:
5295:
5285:
5274:
5261:
5257:
5252:
5249:
5248:
5239:
5218:
5214:
5199:
5195:
5193:
5190:
5189:
5176:
5165:
5156:
5144:
5136:
5114:
5104:
5096:
5076:
5072:
5063:
5059:
5045:
5042:
5041:
5034:
5029:
5017:
5009:
5001:
4991:
4983:
4979:
4975:
4967:
4957:
4917:
4911:
4876:
4870:
4843:
4836:
4830:
4827:
4821:
4814:
4807:
4801:
4778:
4774:
4765:
4761:
4710:
4706:
4694:
4690:
4681:
4677:
4647:
4643:
4631:
4627:
4618:
4614:
4563:
4559:
4547:
4543:
4534:
4530:
4500:
4496:
4494:
4491:
4490:
4486:if and only if
4477:
4467:
4453:
4447:
4438:
4431:
4407:
4405:
4364:
4360:
4351:
4347:
4342:
4339:
4338:
4270:
4252:
4249:
4248:
4202:
4190:
4187:
4186:
4163:
4144:
4072:
4069:
4068:
4043:
4035:
4031:
4028:
4020:
4018:Generalizations
4014:is integrable.
4000:
3972:
3941:
3937:
3922:
3918:
3916:
3913:
3912:
3883:
3879:
3864:
3860:
3852:
3849:
3848:
3799:
3796:
3795:
3769:
3765:
3757:
3754:
3753:
3737:
3721:
3699:Lie derivatives
3685:-forms of rank
3682:
3669:
3658:
3654:
3627:
3603:
3599:
3592:
3571:
3568:
3567:
3551:
3548:
3547:
3531:
3528:
3527:
3504:
3500:
3491:
3487:
3485:
3482:
3481:
3465:
3462:
3461:
3439:
3436:
3435:
3413:
3410:
3409:
3393:
3390:
3389:
3384:Trivially, any
3380:
3373:
3351:
3350:
3348:
3345:
3344:
3334:
3323:
3280:
3273:
3215:
3212:
3211:
3195:
3192:
3191:
3175:
3172:
3171:
3155:
3152:
3151:
3123:
3120:
3119:
3100:
3097:
3096:
3080:
3077:
3076:
3060:
3057:
3056:
3029:
3026:
3025:
2997:
2994:
2993:
2971:
2968:
2967:
2951:
2948:
2947:
2931:
2928:
2927:
2893:
2889:
2866:
2865:
2863:
2860:
2859:
2843:
2840:
2839:
2838:(for intervals
2811:
2808:
2807:
2787:
2784:
2783:
2767:
2764:
2763:
2734:
2714:
2682:
2676:
2670:
2664:
2659:
2653:
2646:
2638:
2632:
2626:
2607:
2564:
2523:
2520:
2519:
2500:
2482:
2476:
2474:
2467:
2456:
2449:
2424:
2423:
2407:
2399:
2397:
2384:
2376:
2374:
2371:
2370:
2354:
2346:
2344:
2331:
2323:
2321:
2314:
2313:
2311:
2308:
2307:
2301:
2292:
2286:
2280:
2276:
2271:
2264:
2259:
2254:
2249:
2231:
2214:
2199:
2167:
2163:
2148:
2140:
2130:
2105:
2101:
2095:
2091:
2070:
2066:
2060:
2056:
2054:
2051:
2050:
2035:
2030:
2020:
2007:
2000:
1997:
1984:
1978:
1958:
1957:
1936:
1925:
1924:
1923:
1911:
1907:
1903:
1895:
1893:
1878:
1873:
1863:
1841:
1840:
1835:
1833:
1832:
1820:
1816:
1813:
1812:
1805:
1804:
1783:
1772:
1771:
1770:
1758:
1754:
1750:
1742:
1740:
1725:
1720:
1710:
1688:
1687:
1682:
1680:
1679:
1667:
1663:
1660:
1659:
1638:
1627:
1626:
1625:
1613:
1609:
1605:
1597:
1595:
1580:
1575:
1565:
1543:
1542:
1537:
1535:
1534:
1522:
1518:
1511:
1510:
1498:
1495:
1494:
1477:
1471:
1465:
1452:
1447:
1446:
1445:
1440:
1430:
1422:
1356:
1347:
1342:
1341:
1332:
1327:
1322:
1318:
1316:
1313:
1312:
1302:
1269:
1266:
1265:
1249:
1246:
1245:
1229:
1226:
1225:
1209:
1206:
1205:
1186:
1183:
1182:
1162:
1159:
1158:
1133:
1130:
1129:
1106:
1103:
1102:
1077:
1074:
1073:
1057:
1054:
1053:
1037:
1034:
1033:
1011:
1008:
1007:
979:
976:
975:
958:
953:
952:
950:
947:
946:
930:
927:
926:
867:
864:
863:
832:
829:
828:
793:
788:
787:
785:
782:
781:
763:
758:
757:
755:
752:
751:
699:
696:
695:
670:
669:
649:
632:
615:
612:
611:
565:
564:
562:
559:
558:
527:
523:
505:
501:
492:
488:
479:
475:
457:
453:
435:
431:
422:
418:
409:
405:
387:
383:
365:
361:
352:
348:
339:
335:
327:
324:
323:
303:
299:
290:
286:
277:
273:
268:
265:
264:
230:
227:
226:
198:
195:
194:
142:
139:
138:
135:
130:
100:integral curves
31:
24:
17:
12:
11:
5:
6931:
6921:
6920:
6915:
6910:
6905:
6890:
6889:
6860:
6828:
6795:
6789:
6765:
6752:
6750:
6747:
6744:
6743:
6716:(3): 196–201.
6696:
6681:
6656:
6649:
6631:
6624:
6606:
6588:(3): 369–418,
6570:
6557:
6550:
6545:
6544:
6532:
6525:
6512:
6505:
6484:
6465:
6446:(3): 355–386.
6426:
6399:(4): 212–218.
6375:
6374:
6372:
6369:
6368:
6367:
6362:
6357:
6350:
6347:
6334:
6331:
6328:
6325:
6322:
6319:
6316:
6313:
6310:
6290:
6287:
6284:
6264:
6261:
6234:
6231:
6228:
6225:
6222:
6202:
6199:
6196:
6176:
6156:
6134:
6130:
6126:
6121:
6117:
6111:
6107:
6103:
6100:
6097:
6094:
6091:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6028:
6024:
6020:
6015:
6011:
6007:
6004:
6001:
5998:
5995:
5990:
5986:
5982:
5977:
5973:
5952:
5949:
5946:
5941:
5937:
5933:
5928:
5924:
5918:
5914:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5866:
5839:
5828:
5827:
5815:
5795:
5775:
5755:
5735:
5726:For any state
5724:
5712:
5709:
5706:
5686:
5683:
5680:
5660:
5657:
5654:
5627:
5624:
5621:
5612:. Write it as
5601:
5581:
5557:
5554:
5553:
5552:
5541:
5524:
5521:
5501:Alfred Clebsch
5492:
5489:
5475:The statement
5472:
5469:
5459:
5458:
5447:
5442:
5438:
5434:
5429:
5424:
5420:
5414:
5409:
5406:
5403:
5399:
5395:
5390:
5386:
5364:
5340:
5330:
5329:
5316:
5312:
5308:
5303:
5298:
5294:
5288:
5283:
5280:
5277:
5273:
5269:
5264:
5260:
5256:
5221:
5217:
5213:
5210:
5207:
5202:
5198:
5164:
5161:
5093:
5092:
5079:
5075:
5071:
5066:
5062:
5058:
5055:
5052:
5049:
5032:
4982:. The bundle
4956:
4953:
4915:
4874:
4834:
4825:
4812:
4805:
4798:
4797:
4786:
4781:
4777:
4773:
4768:
4764:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4713:
4709:
4705:
4702:
4697:
4693:
4689:
4684:
4680:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4650:
4646:
4642:
4639:
4634:
4630:
4626:
4621:
4617:
4613:
4610:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4566:
4562:
4558:
4555:
4550:
4546:
4542:
4537:
4533:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4503:
4499:
4472:, then assume
4462:, then assume
4436:
4429:
4403:
4384:
4381:
4378:
4375:
4372:
4367:
4363:
4359:
4354:
4350:
4346:
4331:
4330:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4276:
4273:
4268:
4265:
4262:
4259:
4256:
4242:
4241:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4208:
4205:
4200:
4197:
4194:
4121:
4120:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4027:
4024:
4019:
4016:
3952:
3949:
3944:
3940:
3936:
3933:
3930:
3925:
3921:
3909:
3908:
3897:
3894:
3891:
3886:
3882:
3878:
3875:
3872:
3867:
3863:
3859:
3856:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3783:
3780:
3777:
3772:
3768:
3764:
3761:
3667:
3591:
3588:
3575:
3555:
3535:
3512:
3507:
3503:
3499:
3494:
3490:
3469:
3449:
3446:
3443:
3423:
3420:
3417:
3397:
3375:
3371:
3354:
3332:
3321:
3275:
3271:
3219:
3199:
3179:
3159:
3139:
3136:
3133:
3130:
3127:
3104:
3084:
3064:
3036:
3033:
3023:tangent bundle
3010:
3007:
3004:
3001:
2975:
2955:
2935:
2905:
2902:
2899:
2896:
2892:
2888:
2885:
2882:
2879:
2873:
2870:
2847:
2827:
2824:
2821:
2818:
2815:
2791:
2782:on a manifold
2771:
2742:tangent bundle
2733:
2730:
2713:
2710:
2687:linear algebra
2680:
2674:
2657:
2651:
2636:
2630:
2604:
2603:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2472:
2465:
2459:level surfaces
2454:
2447:
2441:
2440:
2427:
2422:
2419:
2413:
2410:
2405:
2402:
2396:
2390:
2387:
2382:
2379:
2373:
2372:
2369:
2366:
2360:
2357:
2352:
2349:
2343:
2337:
2334:
2329:
2326:
2320:
2319:
2317:
2300:
2297:
2290:
2284:
2274:
2262:
2252:
2227:
2196:
2195:
2184:
2181:
2178:
2175:
2170:
2166:
2162:
2159:
2156:
2151:
2146:
2143:
2139:
2133:
2129:
2125:
2122:
2119:
2116:
2113:
2108:
2104:
2098:
2094:
2090:
2087:
2084:
2081:
2078:
2073:
2069:
2063:
2059:
2033:
2012:
2005:
1989:
1982:
1975:
1974:
1961:
1956:
1953:
1950:
1947:
1944:
1939:
1932:
1929:
1922:
1914:
1910:
1906:
1901:
1898:
1892:
1889:
1886:
1881:
1876:
1872:
1866:
1862:
1850:
1847:
1844:
1838:
1828:
1823:
1819:
1815:
1814:
1811:
1807:
1806:
1803:
1800:
1797:
1794:
1791:
1786:
1779:
1776:
1769:
1761:
1757:
1753:
1748:
1745:
1739:
1736:
1733:
1728:
1723:
1719:
1713:
1709:
1697:
1694:
1691:
1685:
1675:
1670:
1666:
1662:
1661:
1658:
1655:
1652:
1649:
1646:
1641:
1634:
1631:
1624:
1616:
1612:
1608:
1603:
1600:
1594:
1591:
1588:
1583:
1578:
1574:
1568:
1564:
1552:
1549:
1546:
1540:
1530:
1525:
1521:
1517:
1516:
1514:
1508:
1505:
1502:
1448:
1419:
1418:
1406:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1366:
1359:
1355:
1350:
1345:
1340:
1335:
1330:
1326:
1321:
1301:
1298:
1285:
1282:
1279:
1276:
1273:
1253:
1233:
1213:
1193:
1190:
1166:
1146:
1143:
1140:
1137:
1110:
1090:
1087:
1084:
1081:
1061:
1041:
1021:
1018:
1015:
995:
992:
989:
986:
983:
961:
956:
934:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
851:
848:
845:
842:
839:
836:
796:
791:
766:
761:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
673:
668:
665:
662:
659:
655:
652:
648:
645:
642:
638:
635:
631:
628:
625:
621:
618:
614:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
570:
568:
541:
538:
535:
530:
526:
522:
519:
516:
513:
508:
504:
500:
495:
491:
487:
482:
478:
474:
471:
468:
465:
460:
456:
452:
449:
446:
443:
438:
434:
430:
425:
421:
417:
412:
408:
404:
401:
398:
395:
390:
386:
382:
379:
376:
373:
368:
364:
360:
355:
351:
347:
342:
338:
334:
331:
311:
306:
302:
298:
293:
289:
285:
280:
276:
272:
252:
249:
246:
243:
240:
237:
234:
214:
211:
208:
205:
202:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
134:
131:
129:
126:
15:
9:
6:
4:
3:
2:
6930:
6919:
6916:
6914:
6911:
6909:
6906:
6904:
6901:
6900:
6898:
6886:
6882:
6878:
6874:
6870:
6866:
6861:
6857:
6853:
6849:
6845:
6841:
6837:
6834:
6829:
6825:
6821:
6817:
6813:
6809:
6805:
6801:
6796:
6792:
6790:9780821844380
6786:
6782:
6778:
6774:
6770:
6766:
6762:
6758:
6754:
6753:
6739:
6735:
6731:
6727:
6723:
6719:
6715:
6711:
6707:
6700:
6692:
6688:
6684:
6678:
6674:
6670:
6666:
6665:Kobayashi, S.
6660:
6652:
6646:
6642:
6635:
6627:
6625:9780122155307
6621:
6617:
6610:
6603:
6599:
6595:
6591:
6587:
6583:
6582:
6574:
6567:
6561:
6553:
6539:
6535:
6531:
6524:
6519:
6515:
6511:
6504:
6499:
6498:
6495:
6488:
6480:
6475:
6469:
6461:
6457:
6453:
6449:
6445:
6441:
6437:
6430:
6422:
6418:
6414:
6410:
6406:
6402:
6398:
6394:
6390:
6383:
6381:
6376:
6366:
6363:
6361:
6358:
6356:
6353:
6352:
6346:
6329:
6323:
6320:
6314:
6308:
6288:
6285:
6282:
6262:
6259:
6251:
6246:
6232:
6229:
6226:
6223:
6220:
6200:
6197:
6194:
6174:
6154:
6132:
6128:
6124:
6119:
6115:
6109:
6105:
6101:
6098:
6095:
6092:
6089:
6080:
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6026:
6022:
6018:
6013:
6009:
6005:
6002:
5999:
5996:
5993:
5988:
5984:
5980:
5975:
5971:
5950:
5947:
5944:
5939:
5935:
5931:
5926:
5922:
5916:
5912:
5908:
5905:
5902:
5899:
5896:
5893:
5890:
5887:
5884:
5864:
5856:
5851:
5837:
5813:
5793:
5773:
5753:
5733:
5725:
5710:
5707:
5704:
5684:
5681:
5678:
5658:
5655:
5652:
5644:
5643:
5642:
5639:
5625:
5622:
5619:
5599:
5579:
5571:
5565:
5563:
5550:
5546:
5542:
5539:
5535:
5531:
5527:
5526:
5520:
5518:
5514:
5510:
5506:
5505:Feodor Deahna
5502:
5498:
5488:
5486:
5482:
5478:
5468:
5465:
5445:
5440:
5436:
5432:
5427:
5422:
5418:
5412:
5407:
5404:
5401:
5397:
5393:
5388:
5384:
5376:
5375:
5374:
5371:
5363:
5358:
5354:
5350:
5343:
5336:
5314:
5310:
5306:
5301:
5296:
5292:
5286:
5281:
5278:
5275:
5271:
5267:
5262:
5258:
5254:
5247:
5246:
5245:
5242:
5237:
5219:
5215:
5211:
5208:
5205:
5200:
5196:
5186:
5184:
5179:
5174:
5170:
5160:
5153:
5150:
5147:
5142:
5134:
5130:
5125:
5121:
5117:
5111:
5107:
5103:if, for each
5102:
5077:
5073:
5069:
5064:
5056:
5053:
5050:
5040:
5039:
5038:
5035:
5027:
5024:evaluated at
5023:
5015:
5007:
4998:
4994:
4989:
4973:
4964:
4962:
4952:
4948:
4944:
4940:
4936:
4932:
4928:
4924:
4920:
4914:
4907:
4903:
4899:
4895:
4891:
4887:
4883:
4879:
4873:
4866:
4862:
4858:
4854:
4850:
4846:
4841:
4833:
4824:
4818:
4811:
4804:
4779:
4775:
4771:
4766:
4762:
4758:
4752:
4749:
4746:
4740:
4734:
4728:
4725:
4722:
4716:
4711:
4707:
4703:
4695:
4691:
4687:
4682:
4678:
4671:
4665:
4662:
4659:
4653:
4648:
4644:
4640:
4632:
4628:
4624:
4619:
4615:
4611:
4605:
4602:
4599:
4593:
4587:
4581:
4578:
4575:
4569:
4564:
4560:
4556:
4548:
4544:
4540:
4535:
4531:
4524:
4518:
4515:
4512:
4506:
4501:
4497:
4489:
4488:
4487:
4484:
4480:
4475:
4470:
4465:
4461:
4456:
4450:
4445:
4440:
4435:
4428:
4424:
4420:
4414:
4410:
4402:
4398:
4382:
4379:
4376:
4373:
4365:
4361:
4357:
4352:
4348:
4336:
4317:
4308:
4302:
4299:
4296:
4290:
4287:
4281:
4274:
4271:
4266:
4263:
4260:
4257:
4247:
4246:
4245:
4225:
4222:
4219:
4213:
4210:
4206:
4203:
4195:
4185:
4184:
4183:
4181:
4177:
4173:
4169:
4161:
4155:
4151:
4147:
4142:
4138:
4134:
4130:
4126:
4104:
4101:
4098:
4092:
4086:
4083:
4080:
4077:
4074:
4067:
4066:
4065:
4063:
4058:
4054:
4050:
4046:
4041:
4040:Banach spaces
4023:
4015:
4013:
4007:
4003:
3998:
3994:
3990:
3986:
3982:
3976:
3970:
3966:
3950:
3947:
3942:
3938:
3934:
3931:
3928:
3923:
3919:
3895:
3892:
3884:
3880:
3876:
3873:
3870:
3865:
3861:
3854:
3847:
3846:
3845:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3804:
3801:
3778:
3770:
3762:
3759:
3751:
3747:
3743:
3735:
3731:
3725:
3719:
3715:
3711:
3710:distributions
3706:
3704:
3703:vector fields
3700:
3696:
3692:
3688:
3679:
3677:
3674:
3670:
3664:
3652:
3648:
3644:
3640:
3637:
3631:
3625:
3621:
3617:
3613:
3607:
3597:
3587:
3573:
3553:
3533:
3524:
3510:
3505:
3501:
3497:
3492:
3488:
3467:
3447:
3444:
3441:
3421:
3418:
3415:
3395:
3387:
3382:
3379:
3370:
3342:
3338:
3331:
3327:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3281:, called the
3279:
3270:
3266:
3262:
3258:
3254:
3250:
3246:
3242:
3238:
3236:
3231:
3217:
3197:
3177:
3157:
3134:
3131:
3128:
3118:
3102:
3082:
3062:
3054:
3050:
3034:
3031:
3024:
3008:
3005:
3002:
2999:
2992:
2987:
2973:
2953:
2933:
2925:
2921:
2900:
2894:
2890:
2886:
2880:
2871:
2868:
2845:
2825:
2819:
2816:
2813:
2805:
2789:
2769:
2762:
2757:
2755:
2754:integrability
2751:
2747:
2743:
2739:
2729:
2727:
2726:vector fields
2723:
2719:
2709:
2707:
2703:
2698:
2696:
2692:
2688:
2683:
2673:
2667:
2660:
2650:
2644:
2639:
2629:
2623:
2620:
2614:
2610:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2558:
2552:
2549:
2543:
2540:
2537:
2534:
2531:
2525:
2518:
2517:
2516:
2514:
2510:
2506:
2497:
2493:
2489:
2485:
2479:
2471:
2464:
2460:
2453:
2446:
2420:
2417:
2411:
2403:
2394:
2388:
2380:
2367:
2364:
2358:
2350:
2341:
2335:
2327:
2315:
2306:
2305:
2304:
2296:
2293:
2283:
2277:
2270:
2265:
2255:
2247:
2243:
2239:
2235:
2230:
2226:
2222:
2217:
2211:
2207:
2203:
2179:
2173:
2168:
2164:
2157:
2149:
2144:
2141:
2137:
2131:
2127:
2123:
2117:
2111:
2106:
2102:
2096:
2092:
2088:
2082:
2076:
2071:
2067:
2061:
2057:
2049:
2048:
2047:
2045:
2041:
2036:
2027:
2025:
2019:
2015:
2011:
2004:
1996:
1992:
1988:
1981:
1954:
1951:
1948:
1942:
1937:
1927:
1920:
1912:
1908:
1899:
1887:
1879:
1874:
1870:
1864:
1860:
1836:
1826:
1821:
1817:
1809:
1801:
1798:
1795:
1789:
1784:
1774:
1767:
1759:
1755:
1746:
1734:
1726:
1721:
1717:
1711:
1707:
1683:
1673:
1668:
1664:
1656:
1653:
1650:
1644:
1639:
1629:
1622:
1614:
1610:
1601:
1589:
1581:
1576:
1572:
1566:
1562:
1538:
1528:
1523:
1519:
1512:
1503:
1493:
1492:
1491:
1488:
1484:
1480:
1474:
1468:
1463:
1460:
1451:
1444:
1437:
1433:
1427:
1426:
1404:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1364:
1348:
1338:
1333:
1328:
1324:
1319:
1311:
1310:
1309:
1307:
1297:
1283:
1280:
1277:
1274:
1271:
1251:
1231:
1211:
1191:
1188:
1164:
1141:
1135:
1126:
1122:
1108:
1085:
1079:
1059:
1039:
1019:
1016:
1013:
993:
990:
987:
984:
981:
959:
932:
923:
921:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
849:
846:
843:
840:
837:
834:
825:
822:
814:
810:
794:
764:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
692:
690:
666:
663:
660:
657:
653:
650:
646:
643:
640:
636:
633:
629:
626:
623:
619:
616:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
566:
555:
552:
539:
536:
528:
524:
520:
517:
506:
502:
498:
493:
489:
485:
480:
476:
469:
466:
458:
454:
450:
447:
436:
432:
428:
423:
419:
415:
410:
406:
399:
396:
388:
384:
380:
377:
366:
362:
358:
353:
349:
345:
340:
336:
329:
304:
300:
296:
291:
287:
283:
278:
274:
247:
244:
241:
238:
235:
212:
209:
206:
203:
200:
180:
177:
174:
171:
168:
165:
162:
159:
156:
153:
150:
147:
144:
125:
123:
119:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
80:vector fields
77:
73:
69:
65:
61:
53:
49:
43:
39:
35:
28:
22:
6868:
6864:
6839:
6835:
6807:
6803:
6780:
6760:
6757:Lawson, H.B.
6713:
6709:
6699:
6672:
6659:
6640:
6634:
6615:
6609:
6585:
6579:
6573:
6560:
6548:
6537:
6533:
6529:
6522:
6517:
6513:
6509:
6502:
6493:
6487:
6478:
6473:
6468:
6443:
6439:
6429:
6396:
6392:
6247:
6081:
5852:
5829:
5640:
5566:
5559:
5538:nonholonomic
5523:Applications
5494:
5476:
5474:
5463:
5460:
5369:
5361:
5356:
5352:
5348:
5341:
5334:
5331:
5240:
5235:
5187:
5177:
5166:
5154:
5148:
5145:
5140:
5133:differential
5128:
5123:
5119:
5115:
5109:
5105:
5100:
5094:
5030:
5025:
5021:
5013:
5005:
4996:
4992:
4987:
4971:
4965:
4958:
4946:
4942:
4938:
4934:
4930:
4926:
4922:
4918:
4912:
4905:
4901:
4897:
4893:
4889:
4885:
4881:
4877:
4871:
4864:
4860:
4856:
4852:
4848:
4844:
4831:
4822:
4816:
4809:
4802:
4799:
4482:
4478:
4473:
4468:
4463:
4459:
4458:. If it is
4454:
4448:
4441:
4433:
4426:
4422:
4418:
4412:
4408:
4400:
4396:
4337:if for each
4334:
4332:
4243:
4179:
4175:
4171:
4167:
4153:
4149:
4145:
4140:
4136:
4122:
4056:
4052:
4048:
4044:
4029:
4021:
4011:
4005:
4001:
3996:
3988:
3984:
3974:
3968:
3964:
3910:
3844:) such that
3749:
3745:
3741:
3733:
3729:
3723:
3717:
3713:
3707:
3686:
3680:
3672:
3665:
3646:
3642:
3638:
3629:
3619:
3615:
3605:
3595:
3593:
3525:
3383:
3377:
3368:
3340:
3336:
3329:
3325:
3318:
3314:
3310:
3306:
3302:
3298:
3297:coordinates
3294:
3290:
3286:
3282:
3277:
3268:
3260:
3256:
3252:
3248:
3244:
3240:
3239:
3232:
3052:
3048:
2988:
2761:vector field
2758:
2735:
2715:
2701:
2699:
2694:
2678:
2671:
2665:
2655:
2648:
2634:
2627:
2624:
2618:
2612:
2608:
2605:
2512:
2508:
2504:
2495:
2491:
2487:
2483:
2481:of the form
2477:
2469:
2462:
2451:
2444:
2442:
2302:
2288:
2281:
2272:
2260:
2250:
2237:
2233:
2228:
2224:
2220:
2215:
2209:
2205:
2201:
2197:
2044:involutivity
2043:
2031:
2028:
2017:
2013:
2009:
2002:
1994:
1990:
1986:
1979:
1976:
1486:
1482:
1478:
1472:
1466:
1461:
1449:
1442:
1435:
1431:
1424:
1420:
1303:
1180:
924:
826:
818:
693:
556:
553:
136:
128:Introduction
120:
107:
103:
74:. In modern
63:
57:
51:
47:
41:
37:
33:
6842:: 340–350.
4840:dot product
4417:defined on
3963:. The set
3241:Definition.
3117:Lie bracket
2246:linear span
2242:commutators
90:by maximal
60:mathematics
30:The 1-form
6918:Foliations
6897:Categories
6749:References
6691:0175.48504
6669:Nomizu, K.
6602:0293.57014
5509:sufficient
5244:such that
5101:integrable
5028:, lies in
4988:involutive
4421:such that
4060:a pair of
3651:integrable
3053:involutive
3049:integrable
2643:level sets
2219:functions
2213:, and all
6885:119848431
6856:120057555
6824:122439486
6775:(2008) .
6738:0002-9505
6594:0040-9383
6460:0025-5831
6421:0002-9505
6321:≤
6286:⪰
6221:ω
6175:ω
6155:ω
6106:∑
6102:−
6090:ω
6058:−
6049:δ
5948:δ
5913:∑
5903:δ
5894:δ
5838:⪰
5708:⪰
5682:⪰
5623:⪰
5534:holonomic
5513:necessary
5398:∑
5385:ω
5311:ω
5307:∧
5293:ψ
5272:∑
5259:ω
5216:ω
5209:…
5197:ω
5070:∈
4759:⋅
4735:⋅
4672:⋅
4612:⋅
4588:⋅
4525:⋅
4380:×
4374:∈
4261:∈
4255:∀
4090:→
4084:×
4062:open sets
3948:∈
3932:…
3874:…
3855:α
3826:
3817:…
3805:∈
3794:(for any
3767:Ω
3763:∈
3760:α
3691:foliation
3624:submodule
3445:⊂
3419:∈
3386:foliation
3309:) :
3235:foliation
3003:⊂
2991:subbundle
2872:˙
2823:→
2738:subbundle
2573:−
2409:∂
2401:∂
2386:∂
2378:∂
2356:∂
2348:∂
2333:∂
2325:∂
2128:∑
2089:−
2042:known as
1946:∇
1943:⋅
1931:→
1905:∂
1897:∂
1861:∑
1810:⋯
1793:∇
1790:⋅
1778:→
1752:∂
1744:∂
1708:∑
1648:∇
1645:⋅
1633:→
1607:∂
1599:∂
1563:∑
1476:function
1398:≤
1392:≤
1380:≤
1374:≤
1354:→
1272:ω
1264:, giving
1232:ω
1212:ω
1165:ω
1136:ω
1072:, define
1040:ω
982:ω
933:ω
920:one-forms
870:ω
844:ω
838:∧
835:ω
521:−
451:−
381:−
88:foliation
76:geometric
6759:(1977).
6349:See also
5483:and the
5477:does not
5118: :
4800:for all
4275:′
4207:′
4174: :
3911:for all
3265:disjoint
2746:manifold
2689:and the
2008:, ..., ∇
1481: :
1455: )
821:quilting
654:′
637:′
620:′
193:, where
6718:Bibcode
6528:, ...,
6508:, ...,
6474:locally
6401:Bibcode
5853:By the
5491:History
4974:. Let
4829:(resp.
4820:. Here
4127:of the
3612:1-forms
3305:, ⋅⋅⋅,
3021:of the
2740:of the
2677:, ...,
2654:, ...,
2633:, ...,
2287:, ...,
2248:of the
1985:, ...,
1441:(
974:, then
689:foliate
6883:
6854:
6822:
6787:
6736:
6689:
6679:
6647:
6622:
6600:
6592:
6458:
6419:
6301:means
5963:where
5723:holds.
5347:, 1 ≤
4064:. Let
4042:, and
3618:, and
3480:then
3283:leaves
3115:, the
2804:curves
2515:) by:
2499:, for
1857:
1830:
1704:
1677:
1559:
1532:
1368:
1362:
1308:. Let
66:gives
6881:S2CID
6852:S2CID
6820:S2CID
6521:) = (
6472:Here
6371:Notes
5143:with
4444:field
4166:into
4123:be a
3663:stalk
3622:be a
2744:of a
2669:. If
1434:<
1181:Now,
6869:1877
6808:1866
6785:ISBN
6734:ISSN
6677:ISBN
6645:ISBN
6620:ISBN
6590:ISSN
6456:ISSN
6417:ISSN
5697:and
5503:and
5367:and
5234:are
5020:and
5004:and
4966:Let
4929:) ∈
4910:and
4888:) ∈
4855:) ∈
4034:and
3697:and
3661:the
3636:rank
3594:Let
3434:and
3190:and
3075:and
3051:(or
2468:and
2450:and
2200:1 ≤
2198:for
2022:are
1459:rank
1457:has
114:and
6873:doi
6844:doi
6812:doi
6726:doi
6687:Zbl
6598:Zbl
6448:doi
6409:doi
5560:In
5543:In
5536:or
5528:In
5171:on
5135:of
5099:is
5008:of
4986:is
4452:or
4446:is
4399:of
4244:if
4162:of
4158:of
4038:be
3823:dim
3657:in
3649:is
3634:of
3626:of
3614:on
3388:of
3047:is
2681:n−r
2658:n−r
2645:of
2637:n−r
2461:of
1121:.
58:In
46:on
6899::
6879:.
6867:.
6850:.
6840:20
6838:.
6818:.
6806:.
6802:.
6779:.
6771:;
6732:.
6724:.
6714:28
6712:.
6708:.
6685:.
6667:;
6596:,
6586:80
6584:,
6454:.
6444:67
6442:.
6438:.
6415:.
6407:.
6397:17
6395:.
6391:.
6379:^
6093::=
6079:.
5638:.
5487:.
5355:≤
5351:,
5185:.
5152:.
5141:TN
5122:→
5108:∈
5037::
4995:∈
4951:.
4949:))
4945:,
4937:,
4925:,
4908:))
4904:,
4896:,
4884:,
4863:,
4851:,
4815:∈
4808:,
4481:×
4439:.
4432:)=
4178:→
4139:×
4055:⊂
4051:,
4047:⊂
3973:Ω(
3744:,
3722:Ω(
3718:TM
3678:.
3628:Ω(
3604:Ω(
3602:,
3586:.
3381:.
3376:α∈
3367:={
3328:∩
3301:=(
3276:α∈
3243:A
3230:.
2728:.
2494:=
2490:+
2486:−
2295:.
2229:ij
2208:≤
2204:,
2026:.
1490::
1485:→
922:.
873::=
809:.
118:.
62:,
36:−
6887:.
6875::
6858:.
6846::
6826:.
6814::
6793:.
6740:.
6728::
6720::
6693:.
6653:.
6628:.
6568:.
6555:.
6551:i
6549:c
6543:,
6541:)
6538:r
6536:−
6534:n
6530:c
6526:1
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6514:n
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6503:u
6501:(
6494:R
6479:R
6462:.
6450::
6423:.
6411::
6403::
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6330:B
6327:(
6324:S
6318:)
6315:A
6312:(
6309:S
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6260:d
6233:S
6230:d
6227:T
6224:=
6201:S
6198:,
6195:T
6133:i
6129:x
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6120:i
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6110:i
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6055:=
6052:W
6027:n
6023:x
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6006:,
6003:.
6000:.
5997:.
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5989:1
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5940:i
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5909:=
5906:Q
5900:+
5897:W
5891:=
5888:U
5885:d
5865:U
5826:.
5814:A
5794:B
5774:B
5754:A
5734:A
5711:A
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5685:B
5679:A
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5656:,
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5626:B
5620:A
5600:A
5580:B
5551:.
5540:.
5464:C
5446:.
5441:i
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5402:i
5394:=
5389:j
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5349:i
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5315:i
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5282:1
5279:=
5276:i
5268:=
5263:j
5255:d
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5236:r
5220:r
5212:,
5206:,
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5157:E
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5065:p
5061:]
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5054:,
5051:X
5048:[
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4972:C
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4941:(
4939:L
4935:Y
4933:(
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4916:2
4913:D
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4892:(
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4882:x
4880:(
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4872:D
4867:)
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4859:(
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4806:1
4803:s
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4780:1
4776:s
4772:,
4767:2
4763:s
4756:)
4753:y
4750:,
4747:x
4744:(
4741:F
4738:(
4732:)
4729:y
4726:,
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4720:(
4717:F
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4708:D
4704:+
4701:)
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4688:,
4683:2
4679:s
4675:(
4669:)
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4663:,
4660:x
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4641:=
4638:)
4633:2
4629:s
4625:,
4620:1
4616:s
4609:)
4606:y
4603:,
4600:x
4597:(
4594:F
4591:(
4585:)
4582:y
4579:,
4576:x
4573:(
4570:F
4565:2
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4557:+
4554:)
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4545:s
4541:,
4536:1
4532:s
4528:(
4522:)
4519:y
4516:,
4513:x
4510:(
4507:F
4502:1
4498:D
4483:B
4479:A
4474:F
4469:C
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4460:R
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4449:R
4437:0
4434:y
4430:0
4427:x
4425:(
4423:u
4419:U
4415:)
4413:x
4411:(
4409:u
4404:0
4401:x
4397:U
4383:B
4377:A
4371:)
4366:0
4362:y
4358:,
4353:0
4349:x
4345:(
4318:.
4315:)
4312:)
4309:x
4306:(
4303:u
4300:,
4297:x
4294:(
4291:F
4288:=
4285:)
4282:x
4279:(
4272:u
4267::
4264:A
4258:x
4229:)
4226:y
4223:,
4220:x
4217:(
4214:F
4211:=
4204:y
4199:)
4196:1
4193:(
4180:B
4176:A
4172:u
4168:Y
4164:X
4156:)
4154:Y
4152:,
4150:X
4148:(
4146:L
4141:Y
4137:X
4108:)
4105:Y
4102:,
4099:X
4096:(
4093:L
4087:B
4081:A
4078::
4075:F
4057:Y
4053:B
4049:X
4045:A
4036:Y
4032:X
4012:D
4008:)
4006:D
4004:(
4002:I
3997:D
3989:D
3987:(
3985:I
3977:)
3975:M
3969:D
3967:(
3965:I
3951:D
3943:k
3939:v
3935:,
3929:,
3924:1
3920:v
3896:0
3893:=
3890:)
3885:k
3881:v
3877:,
3871:,
3866:1
3862:v
3858:(
3832:}
3829:M
3820:,
3814:,
3811:1
3808:{
3802:k
3782:)
3779:M
3776:(
3771:k
3750:D
3748:(
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3726:)
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3673:r
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3666:F
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3655:p
3647:F
3643:U
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3632:)
3630:U
3620:F
3616:U
3608:)
3606:U
3600:M
3596:U
3574:M
3554:E
3534:E
3511:N
3506:p
3502:T
3498:=
3493:p
3489:E
3468:p
3448:M
3442:N
3422:M
3416:p
3396:M
3378:A
3374:}
3372:α
3369:L
3353:F
3341:x
3337:x
3333:α
3330:L
3326:U
3322:α
3319:L
3315:R
3313:→
3311:U
3307:x
3303:x
3299:x
3295:C
3291:U
3287:M
3278:A
3274:}
3272:α
3269:L
3261:M
3257:M
3253:n
3249:C
3245:p
3218:M
3198:Y
3178:X
3158:E
3138:]
3135:Y
3132:,
3129:X
3126:[
3103:E
3083:Y
3063:X
3035:M
3032:T
3009:M
3006:T
3000:E
2974:M
2954:M
2934:X
2904:)
2901:t
2898:(
2895:u
2891:X
2887:=
2884:)
2881:t
2878:(
2869:u
2846:I
2826:M
2820:I
2817::
2814:u
2790:M
2770:X
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2679:v
2675:1
2672:v
2666:R
2661:)
2656:u
2652:1
2649:u
2647:(
2635:u
2631:1
2628:u
2619:f
2615:)
2613:t
2611:(
2609:C
2591:.
2588:t
2585:=
2582:z
2579:+
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2570:x
2562:)
2559:t
2556:(
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2550:=
2547:)
2544:z
2541:,
2538:y
2535:,
2532:x
2529:(
2526:f
2513:t
2511:(
2509:C
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2478:R
2473:2
2470:f
2466:1
2463:f
2455:2
2452:f
2448:1
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2421:0
2418:=
2412:z
2404:f
2395:+
2389:y
2381:f
2368:0
2365:=
2359:y
2351:f
2342:+
2336:x
2328:f
2316:{
2291:r
2289:y
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2169:k
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2161:)
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2124:=
2121:)
2118:x
2115:(
2112:u
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2018:r
2016:−
2014:n
2010:u
2006:1
2003:u
2001:∇
1995:r
1993:−
1991:n
1987:u
1983:1
1980:u
1955:0
1952:=
1949:u
1938:r
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1921:=
1913:i
1909:x
1900:u
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1888:x
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1880:i
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1768:=
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1623:=
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1080:g
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985:=
960:n
955:R
906:z
903:d
900:c
897:+
894:y
891:d
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876:a
850:0
847:=
841:d
795:3
790:R
765:3
760:R
738:0
735:=
732:z
729:d
726:c
723:+
720:y
717:d
714:b
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708:x
705:d
702:a
667:0
664:=
661:z
658:d
651:c
647:+
644:y
641:d
634:b
630:+
627:x
624:d
617:a
609:0
606:=
603:z
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597:c
594:+
591:y
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573:a
567:{
540:0
537:=
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529:0
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213:c
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178:=
175:z
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