3023:
2710:
198:
3018:{\displaystyle {\vec {B}}=\operatorname {curl} {\vec {A}}=\left\{{\frac {\partial A_{3}}{\partial x_{2}}}-{\frac {\partial A_{2}}{\partial x_{3}}},{\frac {\partial A_{1}}{\partial x_{3}}}-{\frac {\partial A_{3}}{\partial x_{1}}},{\frac {\partial A_{2}}{\partial x_{1}}}-{\frac {\partial A_{1}}{\partial x_{2}}}\right\},{\text{ or }}\Phi _{B}={\rm {d}}\mathbf {A} .}
2533:
3510:
2234:
3176:
1109:
3307:
2649:
2528:{\displaystyle \mathbf {I} :=j_{1}(x_{1},x_{2},x_{3})\,{\rm {d}}x_{2}\wedge {\rm {d}}x_{3}+j_{2}(x_{1},x_{2},x_{3})\,{\rm {d}}x_{3}\wedge {\rm {d}}x_{1}+j_{3}(x_{1},x_{2},x_{3})\,{\rm {d}}x_{1}\wedge {\rm {d}}x_{2}.}
4048:
3292:
1000:
3221:
1282:
838:
283:
1567:
1807:
asserts that a 1-form is exact if and only if the line integral of the form depends only on the endpoints of the curve, or equivalently, if the integral around any smooth closed curve is zero.
1467:
3645:
3710:
2158:
2115:
1751:
1222:
626:
2033:
2194:
1963:
1352:
1323:
2702:
1644:
3960:
3827:
3774:
1388:
3877:
3742:
3580:
3055:
2680:
2565:
2224:
4113:
3848:
3540:
1176:
1026:
911:
884:
797:
770:
351:
328:
234:
4090:
3063:
1594:
1129:
859:
705:
681:
583:
469:
374:
305:
1682:
5104:
3923:
3795:
657:
561:
397:
4295:
1284:
accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a
1152:
1798:
1778:
1493:
745:
725:
535:
514:
492:
438:
417:
5099:
4052:
must be used, i.e. it is added to the argument of the current-density. Finally, as before, one integrates over the three primed space coordinates. (As usual
2062:
increases degree by 1; but the clues from topology suggest that only the zero function should be called "exact". The cohomology classes are identified with
4386:
1038:
2069:
Using contracting homotopies similar to the one used in the proof of the
Poincaré lemma, it can be shown that de Rham cohomology is homotopy-invariant.
4410:
4605:
3505:{\displaystyle A_{i}({\vec {r}})=\int {\frac {\mu _{0}j_{i}\left({\vec {r}}'\right)\,\,dx_{1}'\,dx_{2}'\,dx_{3}'}{4\pi |{\vec {r}}-{\vec {r}}'|}}\,.}
747:
only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative
2571:
4475:
4231:
4193:
4701:
3235:
1873:). The term incompressible is used because a non-zero divergence corresponds to the presence of sources and sinks in analogy with a fluid.
4282:
5038:
1884:
dimensions; curl is defined only in three dimensions, thus the concept of irrotational vector field does not generalize in this way.
4213:
3897:
If the condition of stationarity is left, on the left-hand side of the above-mentioned equation one must add, in the equations for
919:
4803:
3968:
4395:
4786:
4252:
1757:
3185:
3181:
The closedness of the magnetic-induction two-form corresponds to the property of the magnetic field that it is source-free:
1358:
of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element
1229:
4998:
802:
247:
1501:
4983:
4706:
4480:
5152:
5147:
5028:
237:
5033:
5003:
4711:
4667:
4648:
4415:
4359:
1820:
1396:
4570:
4435:
3588:
1855:
1839:
4955:
4820:
4512:
4354:
3653:
2123:
2080:
1690:
4652:
4622:
4546:
4536:
4492:
4322:
4275:
3887:
1870:
1183:
591:
172:
4420:
4993:
4612:
4507:
4327:
2003:
1285:
4642:
4637:
39:
2170:
1939:
1328:
1299:
141:
is not unique, but can be modified by the addition of any closed form of degree one less than that of
4973:
4911:
4759:
4463:
4453:
4425:
4400:
4310:
1355:
5111:
5084:
4793:
4671:
4656:
4585:
4344:
1866:
1832:
1816:
1473:
82:
54:
2685:
1606:
5053:
5008:
4905:
4776:
4580:
4405:
4268:
4185:
3800:
3747:
2051:
1851:
1573:
1361:
180:
176:
164:
4590:
3853:
3718:
3556:
3171:{\displaystyle \mathbf {A} :=A_{1}\,{\rm {d}}x_{1}+A_{2}\,{\rm {d}}x_{2}+A_{3}\,{\rm {d}}x_{3}.}
3031:
2656:
2541:
2200:
3549:
This equation is remarkable, because it corresponds completely to a well-known formula for the
1861:
Thinking of a vector field as a 2-form instead, a closed vector field is one whose derivative (
4988:
4968:
4963:
4870:
4781:
4595:
4575:
4430:
4369:
4248:
4227:
4209:
4189:
4095:
3891:
3833:
3543:
3518:
3226:
1894:
1158:
1008:
893:
866:
779:
752:
333:
310:
216:
168:
47:
4075:
1579:
1114:
844:
690:
666:
568:
445:
359:
290:
5126:
4920:
4875:
4798:
4769:
4627:
4560:
4555:
4550:
4540:
4332:
4315:
3935:
2118:
2063:
1847:
1804:
1652:
96:
3901:
3780:
635:
543:
379:
5069:
4978:
4808:
4764:
4530:
4177:
241:
35:
4115:
is not the differential of any zero-form. The discussion that follows elaborates on this.
2050:. The set of all forms cohomologous to a given form (and thus to each other) is called a
1936:
More generally, the lemma states that on a contractible open subset of a manifold (e.g.,
4130:
has more information on the mathematics of functions that are only locally well-defined.
1134:
4935:
4860:
4830:
4728:
4721:
4661:
4632:
4502:
4497:
4458:
4240:
4126:
1783:
1763:
1478:
730:
710:
520:
499:
477:
423:
402:
192:
17:
5141:
5121:
4945:
4940:
4925:
4915:
4865:
4842:
4716:
4676:
4617:
4565:
4364:
5048:
5043:
4885:
4852:
4825:
4733:
4374:
2117:
produced by a stationary electrical current is important. There one deals with the
2058:. It makes no real sense to ask whether a 0-form (smooth function) is exact, since
1835:), so there is a notion of a vector field corresponding to a closed or exact form.
1104:{\displaystyle H_{dR}^{1}(\mathbb {R} ^{2}\smallsetminus \{0\})\cong \mathbb {R} ,}
3886:
to quantities with six rsp. four nontrivial components, which is the basis of the
496:
but not in a globally consistent manner. This is because if we trace a loop from
4891:
4880:
4837:
4738:
4339:
197:
31:
5116:
5074:
4900:
4813:
4445:
4349:
2055:
1981:
When the difference of two closed forms is an exact form, they are said to be
1862:
202:
4930:
4895:
4600:
4487:
1876:
The concepts of conservative and incompressible vector fields generalize to
1838:
In 3 dimensions, an exact vector field (thought of as a 1-form) is called a
2644:{\displaystyle \Phi _{B}:=B_{1}{\rm {d}}x_{2}\wedge {\rm {d}}x_{3}+\cdots }
1850:. A closed vector field (thought of as a 1-form) is one whose derivative (
5094:
5089:
5079:
4470:
4291:
1843:
160:
4260:
1756:
The implication from 'exact' to 'closed' is then a consequence of the
213:
A simple example of a form that is closed but not exact is the 1-form
4686:
3287:{\displaystyle \operatorname {div} {\vec {A}}{~{\stackrel {!}{=}}~}0}
2046:
are cohomologous to each other. Exact forms are sometimes said to be
2567:
one has analogous results: it corresponds to the induction two-form
3927:
to the three space coordinates, as a fourth variable also the time
473:
etc. We can assign arguments in a locally consistent manner around
196:
155:, every exact form is necessarily closed. The question of whether
840:
that is not actually the derivative of any well-defined function
4264:
72:, that is the exterior derivative of another differential form
727:
differ from one another by constants. Since the derivative at
4182:
Differential forms with applications to the physical sciences
995:{\displaystyle d\theta ={\frac {-y\,dx+x\,dy}{x^{2}+y^{2}}},}
4043:{\displaystyle t':=t-{\frac {|{\vec {r}}-{\vec {r}}'|}{c}}}
1880:
dimensions, because gradient and divergence generalize to
137:. Since the exterior derivative of a closed form is zero,
4208:, Graduate Texts in Mathematics, vol. 94, Springer,
1288:) being the derivative of a globally defined function.
4206:
Foundations of differentiable manifolds and Lie groups
3216:{\displaystyle \operatorname {div} {\vec {B}}\equiv 0}
1277:{\textstyle k={\frac {1}{2\pi }}\oint _{S^{1}}\omega }
1232:
171:. More general questions of this kind on an arbitrary
4098:
4078:
3971:
3938:
3904:
3856:
3836:
3803:
3783:
3750:
3721:
3656:
3591:
3559:
3521:
3310:
3238:
3188:
3066:
3034:
2713:
2688:
2659:
2574:
2544:
2237:
2203:
2173:
2126:
2083:
2054:
class; the general study of such classes is known as
2006:
1942:
1786:
1766:
1693:
1655:
1609:
1582:
1504:
1481:
1399:
1364:
1331:
1302:
1186:
1161:
1137:
1117:
1041:
1011:
922:
896:
869:
847:
805:
782:
755:
733:
713:
693:
669:
638:
594:
571:
546:
523:
502:
480:
448:
426:
405:
382:
362:
336:
313:
293:
250:
219:
833:{\displaystyle \mathbb {R} ^{2}\smallsetminus \{0\}}
307:
is not actually a function (see the next paragraph)
278:{\displaystyle \mathbb {R} ^{2}\smallsetminus \{0\}}
5062:
5021:
4954:
4851:
4747:
4694:
4685:
4521:
4444:
4383:
4303:
2077:In electrodynamics, the case of the magnetic field
1562:{\displaystyle d\alpha =(g_{x}-f_{y})\,dx\wedge dy}
4107:
4084:
4042:
3954:
3917:
3871:
3842:
3821:
3789:
3768:
3736:
3704:
3639:
3574:
3534:
3504:
3286:
3215:
3170:
3049:
3017:
2696:
2674:
2643:
2559:
2527:
2218:
2188:
2152:
2109:
2027:
1957:
1792:
1772:
1745:
1676:
1638:
1588:
1561:
1487:
1461:
1382:
1346:
1317:
1276:
1216:
1170:
1146:
1123:
1103:
1020:
994:
905:
878:
853:
832:
791:
764:
739:
719:
699:
675:
651:
620:
577:
555:
529:
508:
486:
463:
432:
411:
391:
368:
353:has vanishing derivative and is therefore closed.
345:
322:
299:
277:
228:
4245:Lecture Notes on Elementary Topology and Geometry
1035:This form generates the de Rham cohomology group
1005:which by inspection has derivative zero. Because
1846:) of a 0-form (smooth scalar field), called the
133:is called a "potential form" or "primitive" for
1472:that are of real interest. The formula for the
516:counterclockwise around the origin and back to
4222:Napier, Terrence; Ramachandran, Mohan (2011),
4276:
2653:and can be derived from the vector potential
1462:{\displaystyle \alpha =f(x,y)\,dx+g(x,y)\,dy}
683:is not technically a function, the different
376:is only defined up to an integer multiple of
8:
1084:
1078:
1028:has vanishing derivative, we say that it is
827:
821:
272:
266:
4072:This is an abuse of notation. The argument
3712:. At this place one can already guess that
3640:{\displaystyle \varphi (x_{1},x_{2},x_{3})}
4691:
4283:
4269:
4261:
167:domain, every closed form is exact by the
4097:
4077:
4029:
4014:
4013:
3998:
3997:
3992:
3989:
3970:
3943:
3937:
3909:
3903:
3858:
3857:
3855:
3835:
3805:
3804:
3802:
3782:
3752:
3751:
3749:
3723:
3722:
3720:
3693:
3680:
3667:
3655:
3628:
3615:
3602:
3590:
3561:
3560:
3558:
3526:
3520:
3498:
3490:
3475:
3474:
3459:
3458:
3453:
3436:
3428:
3419:
3411:
3402:
3394:
3393:
3374:
3373:
3362:
3352:
3345:
3325:
3324:
3315:
3309:
3269:
3264:
3262:
3261:
3257:
3246:
3245:
3237:
3196:
3195:
3187:
3159:
3149:
3148:
3147:
3141:
3128:
3118:
3117:
3116:
3110:
3097:
3087:
3086:
3085:
3079:
3067:
3065:
3036:
3035:
3033:
3007:
3001:
3000:
2991:
2982:
2965:
2950:
2940:
2928:
2913:
2903:
2891:
2876:
2866:
2854:
2839:
2829:
2817:
2802:
2792:
2780:
2765:
2755:
2736:
2735:
2715:
2714:
2712:
2689:
2687:
2661:
2660:
2658:
2629:
2619:
2618:
2609:
2599:
2598:
2592:
2579:
2573:
2546:
2545:
2543:
2516:
2506:
2505:
2496:
2486:
2485:
2484:
2475:
2462:
2449:
2436:
2423:
2413:
2412:
2403:
2393:
2392:
2391:
2382:
2369:
2356:
2343:
2330:
2320:
2319:
2310:
2300:
2299:
2298:
2289:
2276:
2263:
2250:
2238:
2236:
2205:
2204:
2202:
2180:
2176:
2175:
2172:
2142:
2128:
2127:
2125:
2099:
2085:
2084:
2082:
2005:
1949:
1945:
1944:
1941:
1785:
1765:
1733:
1727:
1713:
1707:
1692:
1654:
1627:
1614:
1608:
1581:
1543:
1534:
1521:
1503:
1480:
1452:
1424:
1398:
1363:
1338:
1334:
1333:
1330:
1309:
1305:
1304:
1301:
1263:
1258:
1239:
1231:
1185:
1160:
1136:
1116:
1094:
1093:
1069:
1065:
1064:
1054:
1046:
1040:
1010:
980:
967:
954:
941:
932:
921:
895:
868:
846:
812:
808:
807:
804:
781:
754:
732:
712:
692:
668:
643:
637:
604:
599:
593:
570:
545:
522:
501:
479:
447:
425:
404:
381:
361:
335:
312:
292:
257:
253:
252:
249:
218:
3705:{\displaystyle \rho (x_{1},x_{2},x_{3})}
2160:of this field. This case corresponds to
2153:{\displaystyle {\vec {A}}(\mathbf {r} )}
2110:{\displaystyle {\vec {B}}(\mathbf {r} )}
1993:are closed forms, and one can find some
183:information using differential methods.
4142:
4065:
2228:It corresponds to the current two-form
1746:{\displaystyle dh=h_{x}\,dx+h_{y}\,dy.}
631:over a counter-clockwise oriented loop
4161:
4149:
3057:corresponds to the potential one-form
2167:, and the defining region is the full
1217:{\displaystyle \omega =df+k\ d\theta }
621:{\displaystyle \oint _{S^{1}}d\theta }
3931:, whereas on the right-hand side, in
1842:, meaning that it is the derivative (
7:
4092:is not a well-defined function, and
2028:{\displaystyle \zeta -\eta =d\beta }
419:can be assigned different arguments
179:, which allows one to obtain purely
159:closed form is exact depends on the
4224:An introduction to Riemann surfaces
201:Vector field corresponding to (the
4056:is the vacuum velocity of light.)
3150:
3119:
3088:
3002:
2988:
2958:
2943:
2921:
2906:
2884:
2869:
2847:
2832:
2810:
2795:
2773:
2758:
2620:
2600:
2576:
2507:
2487:
2414:
2394:
2321:
2301:
25:
4247:, University of Bangalore Press,
163:of the domain of interest. On a
3068:
3008:
2690:
2682:, or the corresponding one-form
2239:
2196:. The current-density vector is
2189:{\displaystyle \mathbb {R} ^{3}}
2143:
2100:
1958:{\displaystyle \mathbb {R} ^{n}}
1347:{\displaystyle \mathbb {R} ^{3}}
1318:{\displaystyle \mathbb {R} ^{2}}
125:of degree one less than that of
3964:the so-called "retarded time",
3584:electrostatic Coulomb potential
4323:Differentiable/Smooth manifold
4030:
4019:
4003:
3993:
3863:
3810:
3757:
3728:
3699:
3660:
3634:
3595:
3566:
3491:
3480:
3464:
3454:
3379:
3336:
3330:
3321:
3251:
3201:
3041:
2741:
2720:
2666:
2551:
2481:
2442:
2388:
2349:
2295:
2256:
2210:
2147:
2139:
2133:
2104:
2096:
2090:
2073:Application in electrodynamics
1758:symmetry of second derivatives
1671:
1659:
1576:. Therefore the condition for
1540:
1514:
1449:
1437:
1421:
1409:
1087:
1060:
1:
3028:Thereby the vector potential
1865:) vanishes, and is called an
1854:) vanishes, and is called an
1111:meaning that any closed form
330:is not an exact form. Still,
2697:{\displaystyle \mathbf {A} }
1985:to each other. That is, if
1639:{\displaystyle f_{y}=g_{x}.}
1572:where the subscripts denote
1390:, so that it is the 1-forms
1131:is the sum of an exact form
5029:Classification of manifolds
3822:{\displaystyle {\vec {j}},}
3769:{\displaystyle {\vec {B}},}
1383:{\displaystyle dx\wedge dy}
236:given by the derivative of
121:for some differential form
5169:
3872:{\displaystyle {\vec {A}}}
3737:{\displaystyle {\vec {E}}}
3575:{\displaystyle {\vec {E}}}
3050:{\displaystyle {\vec {A}}}
2675:{\displaystyle {\vec {A}}}
2560:{\displaystyle {\vec {B}}}
2219:{\displaystyle {\vec {j}}}
1918:is exact, for any integer
1821:pseudo-Riemannian manifold
1292:Examples in low dimensions
539:the argument increases by
190:
27:Concept of vector calculus
5105:over commutative algebras
4204:Warner, Frank W. (1983),
1977:Formulation as cohomology
1856:irrotational vector field
1840:conservative vector field
663:Even though the argument
4821:Riemann curvature tensor
4243:; Thorpe, J. A. (1976),
4108:{\displaystyle d\theta }
3843:{\displaystyle \varphi }
3535:{\displaystyle \mu _{0}}
3225:i.e., that there are no
1171:{\displaystyle d\theta }
1021:{\displaystyle d\theta }
906:{\displaystyle d\theta }
879:{\displaystyle d\theta }
792:{\displaystyle d\theta }
765:{\displaystyle d\theta }
565:Generally, the argument
346:{\displaystyle d\theta }
323:{\displaystyle d\theta }
229:{\displaystyle d\theta }
68:is a differential form,
4085:{\displaystyle \theta }
3888:relativistic invariance
2538:For the magnetic field
1871:solenoidal vector field
1589:{\displaystyle \alpha }
1354:were well known in the
1124:{\displaystyle \omega }
854:{\displaystyle \theta }
700:{\displaystyle \theta }
676:{\displaystyle \theta }
578:{\displaystyle \theta }
464:{\displaystyle r+2\pi }
369:{\displaystyle \theta }
356:Note that the argument
300:{\displaystyle \theta }
173:differentiable manifold
18:Exact differential form
4613:Manifold with boundary
4328:Differential structure
4109:
4086:
4044:
3956:
3955:{\displaystyle j_{i}'}
3919:
3873:
3844:
3823:
3791:
3770:
3738:
3706:
3641:
3576:
3536:
3506:
3288:
3217:
3172:
3051:
3019:
2698:
2676:
2645:
2561:
2529:
2220:
2190:
2154:
2111:
2029:
1959:
1833:duality via the metric
1819:, or more generally a
1811:Vector field analogies
1794:
1774:
1747:
1678:
1677:{\displaystyle h(x,y)}
1640:
1590:
1563:
1489:
1463:
1384:
1348:
1319:
1296:Differential forms in
1278:
1218:
1172:
1148:
1125:
1105:
1022:
996:
907:
880:
855:
834:
793:
766:
741:
721:
701:
677:
653:
622:
579:
557:
531:
510:
488:
465:
434:
413:
393:
370:
347:
324:
301:
279:
230:
210:
4110:
4087:
4045:
3957:
3920:
3918:{\displaystyle A_{i}}
3874:
3845:
3824:
3792:
3790:{\displaystyle \rho }
3771:
3739:
3707:
3642:
3577:
3537:
3507:
3289:
3218:
3173:
3052:
3020:
2699:
2677:
2646:
2562:
2530:
2221:
2191:
2155:
2112:
2030:
1960:
1827:-forms correspond to
1795:
1775:
1748:
1679:
1641:
1591:
1564:
1490:
1464:
1385:
1349:
1320:
1279:
1219:
1173:
1149:
1126:
1106:
1023:
997:
908:
881:
856:
835:
794:
767:
742:
722:
702:
678:
654:
652:{\displaystyle S^{1}}
623:
580:
558:
556:{\displaystyle 2\pi }
532:
511:
489:
466:
435:
414:
399:since a single point
394:
392:{\displaystyle 2\pi }
371:
348:
325:
302:
280:
231:
200:
191:Further information:
40:differential topology
4760:Covariant derivative
4311:Topological manifold
4096:
4076:
3969:
3936:
3902:
3854:
3834:
3801:
3781:
3748:
3719:
3654:
3589:
3557:
3519:
3308:
3236:
3232:In a special gauge,
3186:
3064:
3032:
2711:
2686:
2657:
2572:
2542:
2235:
2201:
2171:
2124:
2081:
2048:cohomologous to zero
2004:
1940:
1784:
1764:
1691:
1653:
1607:
1580:
1502:
1479:
1397:
1362:
1356:mathematical physics
1329:
1300:
1230:
1184:
1159:
1135:
1115:
1039:
1009:
920:
894:
867:
845:
803:
780:
753:
731:
711:
691:
667:
636:
592:
569:
544:
521:
500:
478:
446:
424:
403:
380:
360:
334:
311:
291:
248:
217:
4794:Exterior derivative
4396:Atiyah–Singer index
4345:Riemannian manifold
3951:
3444:
3427:
3410:
2038:then one says that
1902:is an open ball in
1867:incompressible flow
1831:-vector fields (by
1817:Riemannian manifold
1684:is a function then
1574:partial derivatives
1474:exterior derivative
1059:
776:The upshot is that
175:are the subject of
55:exterior derivative
5153:Lemmas in analysis
5148:Differential forms
5100:Secondary calculus
5054:Singularity theory
5009:Parallel transport
4777:De Rham cohomology
4416:Generalized Stokes
4186:Dover Publications
4152:, pp. 155–156
4105:
4082:
4040:
3952:
3939:
3915:
3869:
3840:
3819:
3787:
3766:
3734:
3702:
3637:
3572:
3532:
3502:
3432:
3415:
3398:
3284:
3227:magnetic monopoles
3213:
3168:
3047:
3015:
2694:
2672:
2641:
2557:
2525:
2216:
2186:
2150:
2107:
2052:de Rham cohomology
2025:
1973:> 0, is exact.
1955:
1790:
1770:
1760:, with respect to
1743:
1674:
1636:
1586:
1559:
1485:
1459:
1380:
1344:
1315:
1286:potential function
1274:
1214:
1168:
1154:and a multiple of
1147:{\displaystyle df}
1144:
1121:
1101:
1042:
1018:
992:
903:
876:
851:
830:
789:
762:
737:
717:
697:
673:
649:
618:
575:
553:
527:
506:
484:
461:
430:
409:
389:
366:
343:
320:
297:
275:
226:
211:
177:de Rham cohomology
107:For an exact form
5135:
5134:
5017:
5016:
4782:Differential form
4436:Whitney embedding
4370:Differential form
4233:978-0-8176-4693-6
4195:978-0-486-66169-8
4164:, p. 162–207
4038:
4022:
4006:
3892:Maxwell equations
3866:
3813:
3760:
3731:
3582:, namely for the
3569:
3544:magnetic constant
3496:
3483:
3467:
3382:
3333:
3279:
3274:
3260:
3254:
3204:
3044:
2985:
2972:
2935:
2898:
2861:
2824:
2787:
2744:
2723:
2669:
2554:
2213:
2136:
2093:
1793:{\displaystyle y}
1773:{\displaystyle x}
1488:{\displaystyle d}
1252:
1207:
987:
799:is a one-form on
740:{\displaystyle p}
720:{\displaystyle p}
530:{\displaystyle p}
509:{\displaystyle p}
487:{\displaystyle p}
433:{\displaystyle r}
412:{\displaystyle p}
48:differential form
16:(Redirected from
5160:
5127:Stratified space
5085:Fréchet manifold
4799:Interior product
4692:
4389:
4285:
4278:
4271:
4262:
4257:
4236:
4218:
4199:
4178:Flanders, Harley
4165:
4159:
4153:
4147:
4131:
4122:
4116:
4114:
4112:
4111:
4106:
4091:
4089:
4088:
4083:
4070:
4051:
4049:
4047:
4046:
4041:
4039:
4034:
4033:
4028:
4024:
4023:
4015:
4008:
4007:
3999:
3996:
3990:
3979:
3963:
3961:
3959:
3958:
3953:
3947:
3926:
3924:
3922:
3921:
3916:
3914:
3913:
3878:
3876:
3875:
3870:
3868:
3867:
3859:
3849:
3847:
3846:
3841:
3828:
3826:
3825:
3820:
3815:
3814:
3806:
3796:
3794:
3793:
3788:
3775:
3773:
3772:
3767:
3762:
3761:
3753:
3743:
3741:
3740:
3735:
3733:
3732:
3724:
3711:
3709:
3708:
3703:
3698:
3697:
3685:
3684:
3672:
3671:
3646:
3644:
3643:
3638:
3633:
3632:
3620:
3619:
3607:
3606:
3581:
3579:
3578:
3573:
3571:
3570:
3562:
3541:
3539:
3538:
3533:
3531:
3530:
3511:
3509:
3508:
3503:
3497:
3495:
3494:
3489:
3485:
3484:
3476:
3469:
3468:
3460:
3457:
3445:
3440:
3423:
3406:
3392:
3388:
3384:
3383:
3375:
3367:
3366:
3357:
3356:
3346:
3335:
3334:
3326:
3320:
3319:
3301:
3293:
3291:
3290:
3285:
3280:
3277:
3276:
3275:
3273:
3268:
3263:
3258:
3256:
3255:
3247:
3224:
3222:
3220:
3219:
3214:
3206:
3205:
3197:
3177:
3175:
3174:
3169:
3164:
3163:
3154:
3153:
3146:
3145:
3133:
3132:
3123:
3122:
3115:
3114:
3102:
3101:
3092:
3091:
3084:
3083:
3071:
3056:
3054:
3053:
3048:
3046:
3045:
3037:
3024:
3022:
3021:
3016:
3011:
3006:
3005:
2996:
2995:
2986:
2983:
2978:
2974:
2973:
2971:
2970:
2969:
2956:
2955:
2954:
2941:
2936:
2934:
2933:
2932:
2919:
2918:
2917:
2904:
2899:
2897:
2896:
2895:
2882:
2881:
2880:
2867:
2862:
2860:
2859:
2858:
2845:
2844:
2843:
2830:
2825:
2823:
2822:
2821:
2808:
2807:
2806:
2793:
2788:
2786:
2785:
2784:
2771:
2770:
2769:
2756:
2746:
2745:
2737:
2725:
2724:
2716:
2703:
2701:
2700:
2695:
2693:
2681:
2679:
2678:
2673:
2671:
2670:
2662:
2652:
2650:
2648:
2647:
2642:
2634:
2633:
2624:
2623:
2614:
2613:
2604:
2603:
2597:
2596:
2584:
2583:
2566:
2564:
2563:
2558:
2556:
2555:
2547:
2534:
2532:
2531:
2526:
2521:
2520:
2511:
2510:
2501:
2500:
2491:
2490:
2480:
2479:
2467:
2466:
2454:
2453:
2441:
2440:
2428:
2427:
2418:
2417:
2408:
2407:
2398:
2397:
2387:
2386:
2374:
2373:
2361:
2360:
2348:
2347:
2335:
2334:
2325:
2324:
2315:
2314:
2305:
2304:
2294:
2293:
2281:
2280:
2268:
2267:
2255:
2254:
2242:
2227:
2225:
2223:
2222:
2217:
2215:
2214:
2206:
2195:
2193:
2192:
2187:
2185:
2184:
2179:
2166:
2159:
2157:
2156:
2151:
2146:
2138:
2137:
2129:
2119:vector potential
2116:
2114:
2113:
2108:
2103:
2095:
2094:
2086:
2064:locally constant
2034:
2032:
2031:
2026:
1964:
1962:
1961:
1956:
1954:
1953:
1948:
1932:
1848:scalar potential
1805:gradient theorem
1799:
1797:
1796:
1791:
1779:
1777:
1776:
1771:
1752:
1750:
1749:
1744:
1732:
1731:
1712:
1711:
1683:
1681:
1680:
1675:
1649:In this case if
1645:
1643:
1642:
1637:
1632:
1631:
1619:
1618:
1595:
1593:
1592:
1587:
1568:
1566:
1565:
1560:
1539:
1538:
1526:
1525:
1494:
1492:
1491:
1486:
1468:
1466:
1465:
1460:
1389:
1387:
1386:
1381:
1353:
1351:
1350:
1345:
1343:
1342:
1337:
1324:
1322:
1321:
1316:
1314:
1313:
1308:
1283:
1281:
1280:
1275:
1270:
1269:
1268:
1267:
1253:
1251:
1240:
1225:
1223:
1221:
1220:
1215:
1205:
1179:
1177:
1175:
1174:
1169:
1153:
1151:
1150:
1145:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1097:
1074:
1073:
1068:
1058:
1053:
1027:
1025:
1024:
1019:
1001:
999:
998:
993:
988:
986:
985:
984:
972:
971:
961:
933:
912:
910:
909:
904:
885:
883:
882:
877:
862:
860:
858:
857:
852:
839:
837:
836:
831:
817:
816:
811:
798:
796:
795:
790:
773:
771:
769:
768:
763:
746:
744:
743:
738:
726:
724:
723:
718:
706:
704:
703:
698:
682:
680:
679:
674:
660:
658:
656:
655:
650:
648:
647:
627:
625:
624:
619:
611:
610:
609:
608:
584:
582:
581:
576:
564:
562:
560:
559:
554:
538:
536:
534:
533:
528:
515:
513:
512:
507:
495:
493:
491:
490:
485:
472:
470:
468:
467:
462:
441:
439:
437:
436:
431:
418:
416:
415:
410:
398:
396:
395:
390:
375:
373:
372:
367:
352:
350:
349:
344:
329:
327:
326:
321:
306:
304:
303:
298:
286:
284:
282:
281:
276:
262:
261:
256:
235:
233:
232:
227:
154:
120:
63:
21:
5168:
5167:
5163:
5162:
5161:
5159:
5158:
5157:
5138:
5137:
5136:
5131:
5070:Banach manifold
5063:Generalizations
5058:
5013:
4950:
4847:
4809:Ricci curvature
4765:Cotangent space
4743:
4681:
4523:
4517:
4476:Exponential map
4440:
4385:
4379:
4299:
4289:
4255:
4239:
4234:
4221:
4216:
4203:
4196:
4176:
4173:
4168:
4160:
4156:
4148:
4144:
4140:
4135:
4134:
4123:
4119:
4094:
4093:
4074:
4073:
4071:
4067:
4062:
4012:
3991:
3972:
3967:
3966:
3965:
3934:
3933:
3932:
3905:
3900:
3899:
3898:
3852:
3851:
3832:
3831:
3799:
3798:
3779:
3778:
3746:
3745:
3717:
3716:
3689:
3676:
3663:
3652:
3651:
3624:
3611:
3598:
3587:
3586:
3555:
3554:
3522:
3517:
3516:
3473:
3446:
3372:
3368:
3358:
3348:
3347:
3311:
3306:
3305:
3295:
3294:, this implies
3234:
3233:
3184:
3183:
3182:
3155:
3137:
3124:
3106:
3093:
3075:
3062:
3061:
3030:
3029:
2987:
2961:
2957:
2946:
2942:
2924:
2920:
2909:
2905:
2887:
2883:
2872:
2868:
2850:
2846:
2835:
2831:
2813:
2809:
2798:
2794:
2776:
2772:
2761:
2757:
2754:
2750:
2709:
2708:
2684:
2683:
2655:
2654:
2625:
2605:
2588:
2575:
2570:
2569:
2568:
2540:
2539:
2512:
2492:
2471:
2458:
2445:
2432:
2419:
2399:
2378:
2365:
2352:
2339:
2326:
2306:
2285:
2272:
2259:
2246:
2233:
2232:
2199:
2198:
2197:
2174:
2169:
2168:
2161:
2122:
2121:
2079:
2078:
2075:
2002:
2001:
1979:
1943:
1938:
1937:
1923:
1898:states that if
1890:
1813:
1782:
1781:
1762:
1761:
1723:
1703:
1689:
1688:
1651:
1650:
1623:
1610:
1605:
1604:
1578:
1577:
1530:
1517:
1500:
1499:
1477:
1476:
1395:
1394:
1360:
1359:
1332:
1327:
1326:
1303:
1298:
1297:
1294:
1259:
1254:
1244:
1228:
1227:
1182:
1181:
1180:
1157:
1156:
1155:
1133:
1132:
1113:
1112:
1063:
1037:
1036:
1007:
1006:
976:
963:
962:
934:
918:
917:
892:
891:
865:
864:
843:
842:
841:
806:
801:
800:
778:
777:
751:
750:
748:
729:
728:
709:
708:
689:
688:
687:definitions of
665:
664:
639:
634:
633:
632:
600:
595:
590:
589:
567:
566:
542:
541:
540:
519:
518:
517:
498:
497:
476:
475:
474:
444:
443:
442:
422:
421:
420:
401:
400:
378:
377:
358:
357:
332:
331:
309:
308:
289:
288:
251:
246:
245:
244:
242:punctured plane
215:
214:
195:
189:
149:
112:
94:form is in the
80:form is in the
58:
36:vector calculus
28:
23:
22:
15:
12:
11:
5:
5166:
5164:
5156:
5155:
5150:
5140:
5139:
5133:
5132:
5130:
5129:
5124:
5119:
5114:
5109:
5108:
5107:
5097:
5092:
5087:
5082:
5077:
5072:
5066:
5064:
5060:
5059:
5057:
5056:
5051:
5046:
5041:
5036:
5031:
5025:
5023:
5019:
5018:
5015:
5014:
5012:
5011:
5006:
5001:
4996:
4991:
4986:
4981:
4976:
4971:
4966:
4960:
4958:
4952:
4951:
4949:
4948:
4943:
4938:
4933:
4928:
4923:
4918:
4908:
4903:
4898:
4888:
4883:
4878:
4873:
4868:
4863:
4857:
4855:
4849:
4848:
4846:
4845:
4840:
4835:
4834:
4833:
4823:
4818:
4817:
4816:
4806:
4801:
4796:
4791:
4790:
4789:
4779:
4774:
4773:
4772:
4762:
4757:
4751:
4749:
4745:
4744:
4742:
4741:
4736:
4731:
4726:
4725:
4724:
4714:
4709:
4704:
4698:
4696:
4689:
4683:
4682:
4680:
4679:
4674:
4664:
4659:
4645:
4640:
4635:
4630:
4625:
4623:Parallelizable
4620:
4615:
4610:
4609:
4608:
4598:
4593:
4588:
4583:
4578:
4573:
4568:
4563:
4558:
4553:
4543:
4533:
4527:
4525:
4519:
4518:
4516:
4515:
4510:
4505:
4503:Lie derivative
4500:
4498:Integral curve
4495:
4490:
4485:
4484:
4483:
4473:
4468:
4467:
4466:
4459:Diffeomorphism
4456:
4450:
4448:
4442:
4441:
4439:
4438:
4433:
4428:
4423:
4418:
4413:
4408:
4403:
4398:
4392:
4390:
4381:
4380:
4378:
4377:
4372:
4367:
4362:
4357:
4352:
4347:
4342:
4337:
4336:
4335:
4330:
4320:
4319:
4318:
4307:
4305:
4304:Basic concepts
4301:
4300:
4290:
4288:
4287:
4280:
4273:
4265:
4259:
4258:
4253:
4237:
4232:
4226:, Birkhäuser,
4219:
4214:
4201:
4194:
4172:
4169:
4167:
4166:
4154:
4141:
4139:
4136:
4133:
4132:
4127:Covering space
4117:
4104:
4101:
4081:
4064:
4063:
4061:
4058:
4037:
4032:
4027:
4021:
4018:
4011:
4005:
4002:
3995:
3988:
3985:
3982:
3978:
3975:
3950:
3946:
3942:
3912:
3908:
3880:
3879:
3865:
3862:
3839:
3829:
3818:
3812:
3809:
3786:
3776:
3765:
3759:
3756:
3730:
3727:
3701:
3696:
3692:
3688:
3683:
3679:
3675:
3670:
3666:
3662:
3659:
3649:charge density
3636:
3631:
3627:
3623:
3618:
3614:
3610:
3605:
3601:
3597:
3594:
3568:
3565:
3529:
3525:
3513:
3512:
3501:
3493:
3488:
3482:
3479:
3472:
3466:
3463:
3456:
3452:
3449:
3443:
3439:
3435:
3431:
3426:
3422:
3418:
3414:
3409:
3405:
3401:
3397:
3391:
3387:
3381:
3378:
3371:
3365:
3361:
3355:
3351:
3344:
3341:
3338:
3332:
3329:
3323:
3318:
3314:
3283:
3272:
3267:
3253:
3250:
3244:
3241:
3212:
3209:
3203:
3200:
3194:
3191:
3179:
3178:
3167:
3162:
3158:
3152:
3144:
3140:
3136:
3131:
3127:
3121:
3113:
3109:
3105:
3100:
3096:
3090:
3082:
3078:
3074:
3070:
3043:
3040:
3026:
3025:
3014:
3010:
3004:
2999:
2994:
2990:
2984: or
2981:
2977:
2968:
2964:
2960:
2953:
2949:
2945:
2939:
2931:
2927:
2923:
2916:
2912:
2908:
2902:
2894:
2890:
2886:
2879:
2875:
2871:
2865:
2857:
2853:
2849:
2842:
2838:
2834:
2828:
2820:
2816:
2812:
2805:
2801:
2797:
2791:
2783:
2779:
2775:
2768:
2764:
2760:
2753:
2749:
2743:
2740:
2734:
2731:
2728:
2722:
2719:
2692:
2668:
2665:
2640:
2637:
2632:
2628:
2622:
2617:
2612:
2608:
2602:
2595:
2591:
2587:
2582:
2578:
2553:
2550:
2536:
2535:
2524:
2519:
2515:
2509:
2504:
2499:
2495:
2489:
2483:
2478:
2474:
2470:
2465:
2461:
2457:
2452:
2448:
2444:
2439:
2435:
2431:
2426:
2422:
2416:
2411:
2406:
2402:
2396:
2390:
2385:
2381:
2377:
2372:
2368:
2364:
2359:
2355:
2351:
2346:
2342:
2338:
2333:
2329:
2323:
2318:
2313:
2309:
2303:
2297:
2292:
2288:
2284:
2279:
2275:
2271:
2266:
2262:
2258:
2253:
2249:
2245:
2241:
2212:
2209:
2183:
2178:
2149:
2145:
2141:
2135:
2132:
2106:
2102:
2098:
2092:
2089:
2074:
2071:
2036:
2035:
2024:
2021:
2018:
2015:
2012:
2009:
1978:
1975:
1952:
1947:
1895:Poincaré lemma
1889:
1888:Poincaré lemma
1886:
1812:
1809:
1789:
1769:
1754:
1753:
1742:
1739:
1736:
1730:
1726:
1722:
1719:
1716:
1710:
1706:
1702:
1699:
1696:
1673:
1670:
1667:
1664:
1661:
1658:
1647:
1646:
1635:
1630:
1626:
1622:
1617:
1613:
1585:
1570:
1569:
1558:
1555:
1552:
1549:
1546:
1542:
1537:
1533:
1529:
1524:
1520:
1516:
1513:
1510:
1507:
1484:
1470:
1469:
1458:
1455:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1379:
1376:
1373:
1370:
1367:
1341:
1336:
1312:
1307:
1293:
1290:
1273:
1266:
1262:
1257:
1250:
1247:
1243:
1238:
1235:
1213:
1210:
1204:
1201:
1198:
1195:
1192:
1189:
1167:
1164:
1143:
1140:
1120:
1100:
1096:
1092:
1089:
1086:
1083:
1080:
1077:
1072:
1067:
1062:
1057:
1052:
1049:
1045:
1017:
1014:
1003:
1002:
991:
983:
979:
975:
970:
966:
960:
957:
953:
950:
947:
944:
940:
937:
931:
928:
925:
902:
899:
890:. Explicitly,
875:
872:
850:
829:
826:
823:
820:
815:
810:
788:
785:
761:
758:
736:
716:
696:
672:
646:
642:
629:
628:
617:
614:
607:
603:
598:
574:
552:
549:
526:
505:
483:
460:
457:
454:
451:
429:
408:
388:
385:
365:
342:
339:
319:
316:
296:
274:
271:
268:
265:
260:
255:
225:
222:
193:Winding number
188:
185:
169:Poincaré lemma
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5165:
5154:
5151:
5149:
5146:
5145:
5143:
5128:
5125:
5123:
5122:Supermanifold
5120:
5118:
5115:
5113:
5110:
5106:
5103:
5102:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5067:
5065:
5061:
5055:
5052:
5050:
5047:
5045:
5042:
5040:
5037:
5035:
5032:
5030:
5027:
5026:
5024:
5020:
5010:
5007:
5005:
5002:
5000:
4997:
4995:
4992:
4990:
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4965:
4962:
4961:
4959:
4957:
4953:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4917:
4913:
4909:
4907:
4904:
4902:
4899:
4897:
4893:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4862:
4859:
4858:
4856:
4854:
4850:
4844:
4843:Wedge product
4841:
4839:
4836:
4832:
4829:
4828:
4827:
4824:
4822:
4819:
4815:
4812:
4811:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4788:
4787:Vector-valued
4785:
4784:
4783:
4780:
4778:
4775:
4771:
4768:
4767:
4766:
4763:
4761:
4758:
4756:
4753:
4752:
4750:
4746:
4740:
4737:
4735:
4732:
4730:
4727:
4723:
4720:
4719:
4718:
4717:Tangent space
4715:
4713:
4710:
4708:
4705:
4703:
4700:
4699:
4697:
4693:
4690:
4688:
4684:
4678:
4675:
4673:
4669:
4665:
4663:
4660:
4658:
4654:
4650:
4646:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4607:
4604:
4603:
4602:
4599:
4597:
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4548:
4544:
4542:
4538:
4534:
4532:
4529:
4528:
4526:
4520:
4514:
4511:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4482:
4481:in Lie theory
4479:
4478:
4477:
4474:
4472:
4469:
4465:
4462:
4461:
4460:
4457:
4455:
4452:
4451:
4449:
4447:
4443:
4437:
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4393:
4391:
4388:
4384:Main results
4382:
4376:
4373:
4371:
4368:
4366:
4365:Tangent space
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4334:
4331:
4329:
4326:
4325:
4324:
4321:
4317:
4314:
4313:
4312:
4309:
4308:
4306:
4302:
4297:
4293:
4286:
4281:
4279:
4274:
4272:
4267:
4266:
4263:
4256:
4250:
4246:
4242:
4241:Singer, I. M.
4238:
4235:
4229:
4225:
4220:
4217:
4215:0-387-90894-3
4211:
4207:
4202:
4197:
4191:
4187:
4183:
4179:
4175:
4174:
4170:
4163:
4158:
4155:
4151:
4146:
4143:
4137:
4129:
4128:
4121:
4118:
4102:
4099:
4079:
4069:
4066:
4059:
4057:
4055:
4035:
4025:
4016:
4009:
4000:
3986:
3983:
3980:
3976:
3973:
3948:
3944:
3940:
3930:
3910:
3906:
3895:
3893:
3889:
3885:
3860:
3837:
3830:
3816:
3807:
3784:
3777:
3763:
3754:
3725:
3715:
3714:
3713:
3694:
3690:
3686:
3681:
3677:
3673:
3668:
3664:
3657:
3650:
3629:
3625:
3621:
3616:
3612:
3608:
3603:
3599:
3592:
3585:
3563:
3552:
3547:
3545:
3527:
3523:
3499:
3486:
3477:
3470:
3461:
3450:
3447:
3441:
3437:
3433:
3429:
3424:
3420:
3416:
3412:
3407:
3403:
3399:
3395:
3389:
3385:
3376:
3369:
3363:
3359:
3353:
3349:
3342:
3339:
3327:
3316:
3312:
3304:
3303:
3302:
3299:
3281:
3270:
3265:
3248:
3242:
3239:
3230:
3228:
3210:
3207:
3198:
3192:
3189:
3165:
3160:
3156:
3142:
3138:
3134:
3129:
3125:
3111:
3107:
3103:
3098:
3094:
3080:
3076:
3072:
3060:
3059:
3058:
3038:
3012:
2997:
2992:
2979:
2975:
2966:
2962:
2951:
2947:
2937:
2929:
2925:
2914:
2910:
2900:
2892:
2888:
2877:
2873:
2863:
2855:
2851:
2840:
2836:
2826:
2818:
2814:
2803:
2799:
2789:
2781:
2777:
2766:
2762:
2751:
2747:
2738:
2732:
2729:
2726:
2717:
2707:
2706:
2705:
2663:
2638:
2635:
2630:
2626:
2615:
2610:
2606:
2593:
2589:
2585:
2580:
2548:
2522:
2517:
2513:
2502:
2497:
2493:
2476:
2472:
2468:
2463:
2459:
2455:
2450:
2446:
2437:
2433:
2429:
2424:
2420:
2409:
2404:
2400:
2383:
2379:
2375:
2370:
2366:
2362:
2357:
2353:
2344:
2340:
2336:
2331:
2327:
2316:
2311:
2307:
2290:
2286:
2282:
2277:
2273:
2269:
2264:
2260:
2251:
2247:
2243:
2231:
2230:
2229:
2207:
2181:
2164:
2130:
2120:
2087:
2072:
2070:
2067:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2022:
2019:
2016:
2013:
2010:
2007:
2000:
1999:
1998:
1996:
1992:
1988:
1984:
1976:
1974:
1972:
1968:
1950:
1934:
1931:
1927:
1921:
1917:
1913:
1909:
1906:, any closed
1905:
1901:
1897:
1896:
1887:
1885:
1883:
1879:
1874:
1872:
1868:
1864:
1859:
1857:
1853:
1849:
1845:
1841:
1836:
1834:
1830:
1826:
1822:
1818:
1810:
1808:
1806:
1801:
1787:
1767:
1759:
1740:
1737:
1734:
1728:
1724:
1720:
1717:
1714:
1708:
1704:
1700:
1697:
1694:
1687:
1686:
1685:
1668:
1665:
1662:
1656:
1633:
1628:
1624:
1620:
1615:
1611:
1603:
1602:
1601:
1599:
1583:
1575:
1556:
1553:
1550:
1547:
1544:
1535:
1531:
1527:
1522:
1518:
1511:
1508:
1505:
1498:
1497:
1496:
1482:
1475:
1456:
1453:
1446:
1443:
1440:
1434:
1431:
1428:
1425:
1418:
1415:
1412:
1406:
1403:
1400:
1393:
1392:
1391:
1377:
1374:
1371:
1368:
1365:
1357:
1339:
1310:
1291:
1289:
1287:
1271:
1264:
1260:
1255:
1248:
1245:
1241:
1236:
1233:
1211:
1208:
1202:
1199:
1196:
1193:
1190:
1187:
1165:
1162:
1141:
1138:
1118:
1098:
1090:
1081:
1075:
1070:
1055:
1050:
1047:
1043:
1033:
1031:
1015:
1012:
989:
981:
977:
973:
968:
964:
958:
955:
951:
948:
945:
942:
938:
935:
929:
926:
923:
916:
915:
914:
913:is given as:
900:
897:
889:
873:
870:
848:
824:
818:
813:
786:
783:
774:
759:
756:
734:
714:
694:
686:
670:
661:
644:
640:
615:
612:
605:
601:
596:
588:
587:
586:
572:
550:
547:
524:
503:
481:
458:
455:
452:
449:
427:
406:
386:
383:
363:
354:
340:
337:
317:
314:
294:
269:
263:
258:
243:
239:
223:
220:
208:
204:
199:
194:
186:
184:
182:
178:
174:
170:
166:
162:
158:
152:
146:
144:
140:
136:
132:
128:
124:
119:
115:
110:
105:
103:
99:
98:
93:
89:
85:
84:
79:
75:
71:
67:
61:
56:
52:
49:
45:
41:
37:
34:, especially
33:
19:
5049:Moving frame
5044:Morse theory
5034:Gauge theory
4826:Tensor field
4755:Closed/Exact
4754:
4734:Vector field
4702:Distribution
4643:Hypercomplex
4638:Quaternionic
4375:Vector field
4333:Smooth atlas
4244:
4223:
4205:
4184:. New York:
4181:
4157:
4145:
4125:
4124:The article
4120:
4068:
4053:
3928:
3896:
3883:
3881:
3648:
3583:
3550:
3548:
3514:
3297:
3231:
3180:
3027:
2537:
2162:
2076:
2068:
2059:
2047:
2043:
2039:
2037:
1994:
1990:
1986:
1983:cohomologous
1982:
1980:
1970:
1966:
1965:), a closed
1935:
1929:
1925:
1919:
1915:
1911:
1907:
1903:
1899:
1893:
1891:
1881:
1877:
1875:
1860:
1837:
1828:
1824:
1814:
1802:
1755:
1648:
1597:
1571:
1471:
1295:
1034:
1029:
1004:
887:
863:We say that
775:
684:
662:
630:
355:
212:
206:
165:contractible
156:
150:
147:
142:
138:
134:
130:
126:
122:
117:
113:
108:
106:
101:
95:
91:
87:
81:
77:
76:. Thus, an
73:
69:
65:
59:
50:
43:
29:
4994:Levi-Civita
4984:Generalized
4956:Connections
4906:Lie algebra
4838:Volume form
4739:Vector flow
4712:Pushforward
4707:Lie bracket
4606:Lie algebra
4571:G-structure
4360:Pushforward
4340:Submanifold
4162:Warner 1983
4150:Warner 1983
2066:functions.
1914:defined on
1869:(sometimes
707:at a point
585:changes by
181:topological
129:. The form
44:closed form
32:mathematics
5142:Categories
5117:Stratifold
5075:Diffeology
4871:Associated
4672:Symplectic
4657:Riemannian
4586:Hyperbolic
4513:Submersion
4421:Hopf–Rinow
4355:Submersion
4350:Smooth map
4254:0721114784
4171:References
3551:electrical
2056:cohomology
1997:such that
1863:divergence
203:Hodge dual
66:exact form
64:), and an
4999:Principal
4974:Ehresmann
4931:Subbundle
4921:Principal
4896:Fibration
4876:Cotangent
4748:Covectors
4601:Lie group
4581:Hermitian
4524:manifolds
4493:Immersion
4488:Foliation
4426:Noether's
4411:Frobenius
4406:De Rham's
4401:Darboux's
4292:Manifolds
4180:(1989) .
4138:Citations
4103:θ
4080:θ
4020:→
4010:−
4004:→
3987:−
3864:→
3838:φ
3811:→
3785:ρ
3758:→
3729:→
3658:ρ
3593:φ
3567:→
3524:μ
3481:→
3471:−
3465:→
3451:π
3380:→
3350:μ
3343:∫
3331:→
3300:= 1, 2, 3
3252:→
3243:
3208:≡
3202:→
3193:
3042:→
2989:Φ
2959:∂
2944:∂
2938:−
2922:∂
2907:∂
2885:∂
2870:∂
2864:−
2848:∂
2833:∂
2811:∂
2796:∂
2790:−
2774:∂
2759:∂
2742:→
2733:
2721:→
2667:→
2639:⋯
2616:∧
2577:Φ
2552:→
2503:∧
2410:∧
2317:∧
2211:→
2134:→
2091:→
2023:β
2014:η
2011:−
2008:ζ
1584:α
1551:∧
1528:−
1509:α
1401:α
1372:∧
1272:ω
1256:∮
1249:π
1212:θ
1188:ω
1166:θ
1119:ω
1091:≅
1076:∖
1016:θ
936:−
927:θ
901:θ
874:θ
849:θ
819:∖
787:θ
760:θ
695:θ
671:θ
616:θ
597:∮
573:θ
551:π
459:π
387:π
364:θ
341:θ
318:θ
295:θ
264:∖
224:θ
57:is zero (
5095:Orbifold
5090:K-theory
5080:Diffiety
4804:Pullback
4618:Oriented
4596:Kenmotsu
4576:Hadamard
4522:Types of
4471:Geodesic
4296:Glossary
4026:′
3977:′
3949:′
3487:′
3442:′
3425:′
3408:′
3386:′
1844:gradient
1495:here is
238:argument
187:Examples
161:topology
148:Because
90:, and a
5039:History
5022:Related
4936:Tangent
4914:)
4894:)
4861:Adjoint
4853:Bundles
4831:density
4729:Torsion
4695:Vectors
4687:Tensors
4670:)
4655:)
4651:,
4649:Pseudo−
4628:Poisson
4561:Finsler
4556:Fibered
4551:Contact
4549:)
4541:Complex
4539:)
4508:Section
3890:of the
3884:unified
3882:can be
3542:is the
1969:-form,
886:is not
240:on the
5004:Vector
4989:Koszul
4969:Cartan
4964:Affine
4946:Vector
4941:Tensor
4926:Spinor
4916:Normal
4912:Stable
4866:Affine
4770:bundle
4722:bundle
4668:Almost
4591:Kähler
4547:Almost
4537:Almost
4531:Closed
4431:Sard's
4387:(list)
4251:
4230:
4212:
4192:
3553:field
3515:(Here
3278:
3259:
1910:-form
1598:closed
1596:to be
1226:where
1206:
1030:closed
287:Since
97:kernel
92:closed
53:whose
5112:Sheaf
4886:Fiber
4662:Rizza
4633:Prime
4464:Local
4454:Curve
4316:Atlas
4060:Notes
3647:of a
1922:with
1815:On a
888:exact
685:local
157:every
83:image
78:exact
46:is a
4979:Form
4881:Dual
4814:flow
4677:Tame
4653:Sub−
4566:Flat
4446:Maps
4249:ISBN
4228:ISBN
4210:ISBN
4190:ISBN
3850:and
3797:and
3744:and
3296:for
2730:curl
2042:and
1989:and
1924:1 ≤
1892:The
1852:curl
1803:The
1780:and
1325:and
205:of)
42:, a
38:and
4901:Jet
3546:.)
3240:div
3190:div
2165:= 2
1600:is
153:= 0
100:of
86:of
62:= 0
30:In
5144::
4892:Co
4188:.
3981::=
3894:.
3229:.
3073::=
2704:,
2586::=
2244::=
1933:.
1928:≤
1858:.
1823:,
1800:.
1032:.
772:".
207:dθ
145:.
118:dβ
116:=
111:,
104:.
60:dα
4910:(
4890:(
4666:(
4647:(
4545:(
4535:(
4298:)
4294:(
4284:e
4277:t
4270:v
4200:.
4198:.
4100:d
4054:c
4050:,
4036:c
4031:|
4017:r
4001:r
3994:|
3984:t
3974:t
3962:,
3945:i
3941:j
3929:t
3925:,
3911:i
3907:A
3861:A
3817:,
3808:j
3764:,
3755:B
3726:E
3700:)
3695:3
3691:x
3687:,
3682:2
3678:x
3674:,
3669:1
3665:x
3661:(
3635:)
3630:3
3626:x
3622:,
3617:2
3613:x
3609:,
3604:1
3600:x
3596:(
3564:E
3528:0
3500:.
3492:|
3478:r
3462:r
3455:|
3448:4
3438:3
3434:x
3430:d
3421:2
3417:x
3413:d
3404:1
3400:x
3396:d
3390:)
3377:r
3370:(
3364:i
3360:j
3354:0
3340:=
3337:)
3328:r
3322:(
3317:i
3313:A
3298:i
3282:0
3271:!
3266:=
3249:A
3223:,
3211:0
3199:B
3166:.
3161:3
3157:x
3151:d
3143:3
3139:A
3135:+
3130:2
3126:x
3120:d
3112:2
3108:A
3104:+
3099:1
3095:x
3089:d
3081:1
3077:A
3069:A
3039:A
3013:.
3009:A
3003:d
2998:=
2993:B
2980:,
2976:}
2967:2
2963:x
2952:1
2948:A
2930:1
2926:x
2915:2
2911:A
2901:,
2893:1
2889:x
2878:3
2874:A
2856:3
2852:x
2841:1
2837:A
2827:,
2819:3
2815:x
2804:2
2800:A
2782:2
2778:x
2767:3
2763:A
2752:{
2748:=
2739:A
2727:=
2718:B
2691:A
2664:A
2651:,
2636:+
2631:3
2627:x
2621:d
2611:2
2607:x
2601:d
2594:1
2590:B
2581:B
2549:B
2523:.
2518:2
2514:x
2508:d
2498:1
2494:x
2488:d
2482:)
2477:3
2473:x
2469:,
2464:2
2460:x
2456:,
2451:1
2447:x
2443:(
2438:3
2434:j
2430:+
2425:1
2421:x
2415:d
2405:3
2401:x
2395:d
2389:)
2384:3
2380:x
2376:,
2371:2
2367:x
2363:,
2358:1
2354:x
2350:(
2345:2
2341:j
2337:+
2332:3
2328:x
2322:d
2312:2
2308:x
2302:d
2296:)
2291:3
2287:x
2283:,
2278:2
2274:x
2270:,
2265:1
2261:x
2257:(
2252:1
2248:j
2240:I
2226:.
2208:j
2182:3
2177:R
2163:k
2148:)
2144:r
2140:(
2131:A
2105:)
2101:r
2097:(
2088:B
2060:d
2044:η
2040:ζ
2020:d
2017:=
1995:β
1991:η
1987:ζ
1971:p
1967:p
1951:n
1946:R
1930:n
1926:p
1920:p
1916:B
1912:ω
1908:p
1904:R
1900:B
1882:n
1878:n
1829:k
1825:k
1788:y
1768:x
1741:.
1738:y
1735:d
1729:y
1725:h
1721:+
1718:x
1715:d
1709:x
1705:h
1701:=
1698:h
1695:d
1672:)
1669:y
1666:,
1663:x
1660:(
1657:h
1634:.
1629:x
1625:g
1621:=
1616:y
1612:f
1557:y
1554:d
1548:x
1545:d
1541:)
1536:y
1532:f
1523:x
1519:g
1515:(
1512:=
1506:d
1483:d
1457:y
1454:d
1450:)
1447:y
1444:,
1441:x
1438:(
1435:g
1432:+
1429:x
1426:d
1422:)
1419:y
1416:,
1413:x
1410:(
1407:f
1404:=
1378:y
1375:d
1369:x
1366:d
1340:3
1335:R
1311:2
1306:R
1265:1
1261:S
1246:2
1242:1
1237:=
1234:k
1224:,
1209:d
1203:k
1200:+
1197:f
1194:d
1191:=
1178::
1163:d
1142:f
1139:d
1099:,
1095:R
1088:)
1085:}
1082:0
1079:{
1071:2
1066:R
1061:(
1056:1
1051:R
1048:d
1044:H
1013:d
990:,
982:2
978:y
974:+
969:2
965:x
959:y
956:d
952:x
949:+
946:x
943:d
939:y
930:=
924:d
898:d
871:d
861:.
828:}
825:0
822:{
814:2
809:R
784:d
757:d
749:"
735:p
715:p
659:.
645:1
641:S
613:d
606:1
602:S
563:.
548:2
537:,
525:p
504:p
494:,
482:p
471:,
456:2
453:+
450:r
440:,
428:r
407:p
384:2
338:d
315:d
285:.
273:}
270:0
267:{
259:2
254:R
221:d
209:.
151:d
143:α
139:β
135:α
131:β
127:α
123:β
114:α
109:α
102:d
88:d
74:β
70:α
51:α
20:)
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