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768:{\displaystyle {\begin{aligned}d\theta &=\partial _{x}\left(\operatorname {atan2} (y,x)\right)dx+\partial _{y}\left(\operatorname {atan2} (y,x)\right)dy\\&=-{\frac {y}{x^{2}+y^{2}}}dx+{\frac {x}{x^{2}+y^{2}}}dy\end{aligned}}}
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795:-axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local)
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in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the
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243:{\displaystyle \alpha :TM\rightarrow {\mathbb {R} },\quad \alpha _{x}=\alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },}
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of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.
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While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative
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is not a globally defined smooth function on the entire punctured plane. In fact, this form generates the first
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transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant
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whose restriction to each fibre is a linear functional on the tangent space. Symbolically,
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The most basic non-trivial differential one-form is the "change in angle" form
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First Steps in
Differential Geometry: Riemannian, Contact, Symplectic
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function. Taking the derivative yields the following formula for the
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Differential form of degree one or section of a cotangent bundle
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are smooth functions. From this perspective, a one-form has a
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1272:. Springer Science & Business Media. pp. 136–155.
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This is the simplest example of a differential (one-)form.
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This is defined as the derivative of the angle "function"
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Pages displaying short descriptions of redirect targets
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270:{\displaystyle \alpha _{x}}
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1659:Cartan formalism (physics)
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1073:{\displaystyle x_{0}\in U}
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32:One-form (linear algebra)
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1915:Gregorio Ricci-Curbastro
1787:Riemann curvature tensor
1494:Van der Waerden notation
1885:Elwin Bruno Christoffel
1818:Angular momentum tensor
1489:Tetrad (index notation)
1459:Abstract index notation
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864:{\displaystyle \theta }
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1600:Hodge star operator
1590:Exterior derivative
1439:Transport phenomena
1424:Continuum mechanics
1380:Multilinear algebra
1247:www.damtp.cam.ac.uk
1017:{\displaystyle f'.}
845:exterior derivative
831:In the language of
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2739:Singularity theory
2694:Parallel transport
2462:De Rham cohomology
2101:Generalized Stokes
1910:Tullio Levi-Civita
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1024:The differential
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109:{\displaystyle M}
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1431:
1426:
1421:
1415:
1413:
1411:
1410:
1405:
1399:
1396:
1395:
1393:
1392:
1387:
1385:Tensor algebra
1382:
1377:
1372:
1367:
1365:Dyadic algebra
1362:
1357:
1351:
1349:
1340:
1336:
1335:
1328:
1325:
1324:
1317:
1316:
1309:
1302:
1294:
1286:
1285:
1278:
1258:
1233:
1232:
1230:
1227:
1226:
1225:
1219:
1213:
1204:
1196:
1193:
1180:
1177:
1172:
1168:
1164:
1160:
1157:
1135:
1131:
1127:
1106:
1099:
1095:
1090:
1069:
1066:
1061:
1057:
1036:
1033:
1013:
1009:
1006:
982:
978:
974:
971:
968:
965:
942:
939:
936:
933:
930:
905:
901:
898:
883:Main article:
880:
877:
860:
818:
815:
812:
801:winding number
798:
784:
760:
757:
749:
745:
741:
736:
732:
727:
722:
719:
716:
708:
704:
700:
695:
691:
686:
681:
678:
675:
673:
671:
668:
665:
661:
657:
654:
651:
648:
645:
642:
639:
635:
629:
625:
621:
618:
615:
611:
607:
604:
601:
598:
595:
592:
589:
585:
579:
575:
571:
568:
566:
564:
561:
558:
557:
529:
526:
523:
520:
517:
514:
494:
491:
488:
476:
473:
450:
446:
425:
420:
416:
412:
408:
405:
402:
397:
393:
389:
386:
383:
378:
374:
370:
366:
363:
360:
355:
351:
347:
342:
338:
334:
330:
327:
324:
319:
315:
311:
306:
302:
264:
260:
239:
234:
229:
226:
221:
217:
213:
208:
203:
199:
193:
188:
185:
180:
176:
171:
166:
161:
158:
155:
152:
149:
126:
105:
94:tangent bundle
77:
47:covector field
26:
9:
6:
4:
3:
2:
2850:
2839:
2836:
2834:
2831:
2830:
2828:
2813:
2810:
2808:
2807:Supermanifold
2805:
2803:
2800:
2798:
2795:
2791:
2788:
2787:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2752:
2750:
2746:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2711:
2709:
2705:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2646:
2644:
2642:
2638:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2598:
2594:
2592:
2589:
2587:
2584:
2582:
2578:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2543:
2541:
2539:
2535:
2529:
2528:Wedge product
2526:
2524:
2521:
2517:
2514:
2513:
2512:
2509:
2507:
2504:
2500:
2497:
2496:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2473:
2472:Vector-valued
2470:
2469:
2468:
2465:
2463:
2460:
2456:
2453:
2452:
2451:
2448:
2446:
2443:
2441:
2438:
2437:
2435:
2431:
2425:
2422:
2420:
2417:
2415:
2412:
2408:
2405:
2404:
2403:
2402:Tangent space
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2384:
2382:
2378:
2375:
2373:
2369:
2363:
2360:
2358:
2354:
2350:
2348:
2345:
2343:
2339:
2335:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2292:
2289:
2288:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2233:
2229:
2227:
2223:
2219:
2217:
2214:
2213:
2211:
2205:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2167:
2166:in Lie theory
2164:
2163:
2162:
2159:
2157:
2154:
2150:
2147:
2146:
2145:
2142:
2140:
2137:
2136:
2134:
2132:
2128:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2078:
2076:
2073:
2069:Main results
2067:
2061:
2058:
2056:
2053:
2051:
2050:Tangent space
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2019:
2016:
2014:
2011:
2010:
2009:
2006:
2002:
1999:
1998:
1997:
1994:
1993:
1991:
1987:
1982:
1978:
1971:
1966:
1964:
1959:
1957:
1952:
1951:
1948:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1872:
1870:
1868:
1864:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1810:
1808:
1804:
1798:
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1772:Metric tensor
1770:
1768:
1765:
1763:
1760:
1759:
1757:
1753:
1750:
1746:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1684:Exterior form
1682:
1680:
1677:
1675:
1672:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1646:
1644:
1638:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1577:
1575:
1573:
1569:
1563:
1560:
1558:
1557:Tensor bundle
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1514:
1512:
1506:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1456:
1454:
1450:
1440:
1437:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1416:
1414:
1409:
1406:
1404:
1401:
1400:
1397:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1352:
1350:
1348:
1344:
1341:
1337:
1333:
1332:
1326:
1322:
1315:
1310:
1308:
1303:
1301:
1296:
1295:
1292:
1281:
1275:
1271:
1270:
1262:
1248:
1244:
1238:
1234:
1223:
1220:
1217:
1214:
1208:
1207:Inner product
1205:
1202:
1199:
1198:
1192:
1178:
1170:
1166:
1158:
1155:
1104:
1097:
1093:
1088:
1067:
1064:
1059:
1055:
1034:
1031:
1011:
1007:
1004:
996:
980:
969:
966:
963:
956:
937:
934:
931:
920:
899:
896:
886:
876:
874:
858:
850:
846:
842:
838:
834:
829:
816:
813:
810:
802:
796:
782:
758:
755:
747:
743:
739:
734:
730:
725:
720:
717:
714:
706:
702:
698:
693:
689:
684:
679:
676:
674:
666:
663:
659:
652:
649:
646:
640:
637:
633:
627:
619:
616:
613:
609:
602:
599:
596:
590:
587:
583:
577:
569:
567:
562:
559:
547:
543:
524:
521:
518:
512:
492:
489:
486:
472:
470:
466:
448:
444:
423:
418:
414:
410:
403:
395:
391:
387:
384:
381:
376:
372:
368:
361:
353:
349:
345:
340:
336:
332:
325:
317:
313:
309:
304:
300:
291:
290:differentials
287:
283:
278:
262:
258:
237:
224:
219:
215:
211:
206:
201:
197:
186:
183:
178:
174:
169:
156:
153:
150:
147:
139:
103:
95:
91:
75:
67:
63:
60:
56:
52:
48:
44:
40:
33:
19:
2734:Moving frame
2729:Morse theory
2719:Gauge theory
2511:Tensor field
2440:Closed/Exact
2419:Vector field
2387:Distribution
2328:Hypercomplex
2323:Quaternionic
2060:Vector field
2018:Smooth atlas
1935:Hermann Weyl
1739:Vector space
1724:Pseudotensor
1689:Fiber bundle
1642:abstractions
1537:Mixed tensor
1522:Tensor field
1329:
1268:
1261:
1250:. Retrieved
1246:
1237:
888:
830:
478:
469:tensor field
279:
140:
46:
42:
36:
2679:Levi-Civita
2669:Generalized
2641:Connections
2591:Lie algebra
2523:Volume form
2424:Vector flow
2397:Pushforward
2392:Lie bracket
2291:Lie algebra
2256:G-structure
2045:Pushforward
2025:Submanifold
1875:Élie Cartan
1823:Spin tensor
1797:Weyl tensor
1755:Mathematics
1719:Multivector
1510:definitions
1408:Engineering
1347:Mathematics
277:is linear.
90:total space
2838:1 (number)
2827:Categories
2802:Stratifold
2760:Diffeology
2556:Associated
2357:Symplectic
2342:Riemannian
2271:Hyperbolic
2198:Submersion
2106:Hopf–Rinow
2040:Submersion
2035:Smooth map
1704:Linear map
1572:Operations
1252:2022-10-04
1229:References
995:derivative
436:where the
2684:Principal
2659:Ehresmann
2616:Subbundle
2606:Principal
2581:Fibration
2561:Cotangent
2433:Covectors
2286:Lie group
2266:Hermitian
2209:manifolds
2178:Immersion
2173:Foliation
2111:Noether's
2096:Frobenius
2091:De Rham's
2086:Darboux's
1977:Manifolds
1843:EM tensor
1679:Dimension
1630:Transpose
1130:→
1065:∈
973:→
900:⊆
859:θ
814:π
680:−
641:
624:∂
591:
574:∂
563:θ
513:θ
490:θ
465:covariant
385:⋯
301:α
259:α
228:→
187:α
175:α
160:→
148:α
2780:Orbifold
2775:K-theory
2765:Diffiety
2489:Pullback
2303:Oriented
2281:Kenmotsu
2261:Hadamard
2207:Types of
2156:Geodesic
1981:Glossary
1709:Manifold
1694:Geodesic
1452:Notation
1195:See also
1159:′
1008:′
839:. It is
475:Examples
43:one-form
18:One-form
2724:History
2707:Related
2621:Tangent
2599:)
2579:)
2546:Adjoint
2538:Bundles
2516:density
2414:Torsion
2380:Vectors
2372:Tensors
2355:)
2340:)
2336:,
2334:Pseudo−
2313:Poisson
2246:Finsler
2241:Fibered
2236:Contact
2234:)
2226:Complex
2224:)
2193:Section
1806:Physics
1640:Related
1403:Physics
1321:Tensors
797:changes
282:locally
92:of the
64:of the
62:section
49:) on a
2689:Vector
2674:Koszul
2654:Cartan
2649:Affine
2631:Vector
2626:Tensor
2611:Spinor
2601:Normal
2597:Stable
2551:Affine
2455:bundle
2407:bundle
2353:Almost
2276:Kähler
2232:Almost
2222:Almost
2216:Closed
2116:Sard's
2072:(list)
1734:Vector
1729:Spinor
1714:Matrix
1508:Tensor
1276:
1222:Tensor
841:closed
803:times
250:where
59:smooth
2797:Sheaf
2571:Fiber
2347:Rizza
2318:Prime
2149:Local
2139:Curve
2001:Atlas
1654:Basis
1339:Scope
993:with
849:exact
843:(its
638:atan2
588:atan2
542:atan2
53:is a
2664:Form
2566:Dual
2499:flow
2362:Tame
2338:Sub−
2251:Flat
2131:Maps
1274:ISBN
919:open
889:Let
45:(or
41:, a
2586:Jet
917:be
116:to
96:of
37:In
2829::
2577:Co
1245:.
548::
471:.
2595:(
2575:(
2351:(
2332:(
2230:(
2220:(
1983:)
1979:(
1969:e
1962:t
1955:v
1313:e
1306:t
1299:v
1282:.
1255:.
1179:.
1176:)
1171:0
1167:x
1163:(
1156:f
1134:R
1126:R
1105:U
1098:0
1094:x
1089:T
1068:U
1060:0
1056:x
1035:f
1032:d
1012:.
1005:f
981:,
977:R
970:U
967::
964:f
941:)
938:b
935:,
932:a
929:(
904:R
897:U
817:.
811:2
783:y
759:y
756:d
748:2
744:y
740:+
735:2
731:x
726:x
721:+
718:x
715:d
707:2
703:y
699:+
694:2
690:x
685:y
677:=
667:y
664:d
660:)
656:)
653:x
650:,
647:y
644:(
634:(
628:y
620:+
617:x
614:d
610:)
606:)
603:x
600:,
597:y
594:(
584:(
578:x
570:=
560:d
528:)
525:y
522:,
519:x
516:(
493:.
487:d
449:i
445:f
424:,
419:n
415:x
411:d
407:)
404:x
401:(
396:n
392:f
388:+
382:+
377:2
373:x
369:d
365:)
362:x
359:(
354:2
350:f
346:+
341:1
337:x
333:d
329:)
326:x
323:(
318:1
314:f
310:=
305:x
263:x
238:,
233:R
225:M
220:x
216:T
212::
207:M
202:x
198:T
192:|
184:=
179:x
170:,
165:R
157:M
154:T
151::
125:R
104:M
76:M
34:.
20:)
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