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757:{\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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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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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
1261:. 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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1648:Cartan formalism (physics)
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1904:Gregorio Ricci-Curbastro
1776:Riemann curvature tensor
1483:Van der Waerden notation
1874:Elwin Bruno Christoffel
1807:Angular momentum tensor
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1236:www.damtp.cam.ac.uk
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1371:
1366:
1361:
1356:
1354:Dyadic algebra
1351:
1346:
1340:
1338:
1329:
1325:
1324:
1317:
1314:
1313:
1308:
1306:
1305:
1298:
1291:
1283:
1275:
1274:
1267:
1247:
1222:
1221:
1219:
1216:
1215:
1214:
1208:
1202:
1193:
1185:
1182:
1169:
1166:
1161:
1157:
1153:
1149:
1146:
1124:
1120:
1116:
1095:
1088:
1084:
1079:
1058:
1055:
1050:
1046:
1025:
1022:
1002:
998:
995:
971:
967:
963:
960:
957:
954:
931:
928:
925:
922:
919:
894:
890:
887:
872:Main article:
869:
866:
849:
807:
804:
801:
790:winding number
787:
773:
749:
746:
738:
734:
730:
725:
721:
716:
711:
708:
705:
697:
693:
689:
684:
680:
675:
670:
667:
664:
662:
660:
657:
654:
650:
646:
643:
640:
637:
634:
631:
628:
624:
618:
614:
610:
607:
604:
600:
596:
593:
590:
587:
584:
581:
578:
574:
568:
564:
560:
557:
555:
553:
550:
547:
546:
518:
515:
512:
509:
506:
503:
483:
480:
477:
465:
462:
439:
435:
414:
409:
405:
401:
397:
394:
391:
386:
382:
378:
375:
372:
367:
363:
359:
355:
352:
349:
344:
340:
336:
331:
327:
323:
319:
316:
313:
308:
304:
300:
295:
291:
253:
249:
228:
223:
218:
215:
210:
206:
202:
197:
192:
188:
182:
177:
174:
169:
165:
160:
155:
150:
147:
144:
141:
138:
115:
94:
83:tangent bundle
66:
36:covector field
15:
13:
10:
9:
6:
4:
3:
2:
2839:
2828:
2825:
2823:
2820:
2819:
2817:
2802:
2799:
2797:
2796:Supermanifold
2794:
2792:
2789:
2787:
2784:
2780:
2777:
2776:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2745:
2742:
2741:
2739:
2735:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2700:
2698:
2694:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2635:
2633:
2631:
2627:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2587:
2583:
2581:
2578:
2576:
2573:
2571:
2567:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2532:
2530:
2528:
2524:
2518:
2517:Wedge product
2515:
2513:
2510:
2506:
2503:
2502:
2501:
2498:
2496:
2493:
2489:
2486:
2485:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2462:
2461:Vector-valued
2459:
2458:
2457:
2454:
2452:
2449:
2445:
2442:
2441:
2440:
2437:
2435:
2432:
2430:
2427:
2426:
2424:
2420:
2414:
2411:
2409:
2406:
2404:
2401:
2397:
2394:
2393:
2392:
2391:Tangent space
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2373:
2371:
2367:
2364:
2362:
2358:
2352:
2349:
2347:
2343:
2339:
2337:
2334:
2332:
2328:
2324:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2281:
2278:
2277:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2222:
2218:
2216:
2212:
2208:
2206:
2203:
2202:
2200:
2194:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2156:
2155:in Lie theory
2153:
2152:
2151:
2148:
2146:
2143:
2139:
2136:
2135:
2134:
2131:
2129:
2126:
2125:
2123:
2121:
2117:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2067:
2065:
2062:
2058:Main results
2056:
2050:
2047:
2045:
2042:
2040:
2039:Tangent space
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2008:
2005:
2003:
2000:
1999:
1998:
1995:
1991:
1988:
1987:
1986:
1983:
1982:
1980:
1976:
1971:
1967:
1960:
1955:
1953:
1948:
1946:
1941:
1940:
1937:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1861:
1859:
1857:
1853:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1799:
1797:
1793:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1761:Metric tensor
1759:
1757:
1754:
1752:
1749:
1748:
1746:
1742:
1739:
1735:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1673:Exterior form
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1635:
1633:
1627:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
1575:
1572:
1570:
1567:
1566:
1564:
1562:
1558:
1552:
1549:
1547:
1546:Tensor bundle
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1503:
1501:
1495:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1445:
1443:
1439:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1405:
1403:
1398:
1395:
1393:
1390:
1389:
1386:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1341:
1339:
1337:
1333:
1330:
1326:
1322:
1321:
1315:
1311:
1304:
1299:
1297:
1292:
1290:
1285:
1284:
1281:
1270:
1264:
1260:
1259:
1251:
1248:
1237:
1233:
1227:
1224:
1217:
1212:
1209:
1206:
1203:
1197:
1196:Inner product
1194:
1191:
1188:
1187:
1183:
1181:
1167:
1159:
1155:
1147:
1144:
1093:
1086:
1082:
1077:
1056:
1053:
1048:
1044:
1023:
1020:
1000:
996:
993:
985:
969:
958:
955:
952:
945:
926:
923:
920:
909:
888:
885:
875:
867:
865:
863:
847:
839:
835:
831:
827:
823:
818:
805:
802:
799:
791:
785:
771:
747:
744:
736:
732:
728:
723:
719:
714:
709:
706:
703:
695:
691:
687:
682:
678:
673:
668:
665:
663:
655:
652:
648:
641:
638:
635:
629:
626:
622:
616:
608:
605:
602:
598:
591:
588:
585:
579:
576:
572:
566:
558:
556:
551:
548:
536:
532:
513:
510:
507:
501:
481:
478:
475:
463:
461:
459:
455:
437:
433:
412:
407:
403:
399:
392:
384:
380:
376:
373:
370:
365:
361:
357:
350:
342:
338:
334:
329:
325:
321:
314:
306:
302:
298:
293:
289:
280:
279:differentials
276:
272:
267:
251:
247:
226:
213:
208:
204:
200:
195:
190:
186:
175:
172:
167:
163:
158:
145:
142:
139:
136:
128:
92:
84:
80:
64:
56:
52:
49:
45:
41:
37:
33:
29:
22:
2723:Moving frame
2718:Morse theory
2708:Gauge theory
2500:Tensor field
2429:Closed/Exact
2408:Vector field
2376:Distribution
2317:Hypercomplex
2312:Quaternionic
2049:Vector field
2007:Smooth atlas
1924:Hermann Weyl
1728:Vector space
1713:Pseudotensor
1678:Fiber bundle
1631:abstractions
1526:Mixed tensor
1511:Tensor field
1318:
1257:
1250:
1239:. Retrieved
1235:
1226:
877:
819:
467:
458:tensor field
268:
129:
35:
31:
25:
2668:Levi-Civita
2658:Generalized
2630:Connections
2580:Lie algebra
2512:Volume form
2413:Vector flow
2386:Pushforward
2381:Lie bracket
2280:Lie algebra
2245:G-structure
2034:Pushforward
2014:Submanifold
1864:Élie Cartan
1812:Spin tensor
1786:Weyl tensor
1744:Mathematics
1708:Multivector
1499:definitions
1397:Engineering
1336:Mathematics
266:is linear.
79:total space
2827:1 (number)
2816:Categories
2791:Stratifold
2749:Diffeology
2545:Associated
2346:Symplectic
2331:Riemannian
2260:Hyperbolic
2187:Submersion
2095:Hopf–Rinow
2029:Submersion
2024:Smooth map
1693:Linear map
1561:Operations
1241:2022-10-04
1218:References
984:derivative
425:where the
2673:Principal
2648:Ehresmann
2605:Subbundle
2595:Principal
2570:Fibration
2550:Cotangent
2422:Covectors
2275:Lie group
2255:Hermitian
2198:manifolds
2167:Immersion
2162:Foliation
2100:Noether's
2085:Frobenius
2080:De Rham's
2075:Darboux's
1966:Manifolds
1832:EM tensor
1668:Dimension
1619:Transpose
1119:→
1054:∈
962:→
889:⊆
848:θ
803:π
669:−
630:
613:∂
580:
563:∂
552:θ
502:θ
479:θ
454:covariant
374:⋯
290:α
248:α
217:→
176:α
164:α
149:→
137:α
2769:Orbifold
2764:K-theory
2754:Diffiety
2478:Pullback
2292:Oriented
2270:Kenmotsu
2250:Hadamard
2196:Types of
2145:Geodesic
1970:Glossary
1698:Manifold
1683:Geodesic
1441:Notation
1184:See also
1148:′
997:′
828:. It is
464:Examples
32:one-form
2713:History
2696:Related
2610:Tangent
2588:)
2568:)
2535:Adjoint
2527:Bundles
2505:density
2403:Torsion
2369:Vectors
2361:Tensors
2344:)
2329:)
2325:,
2323:Pseudo−
2302:Poisson
2235:Finsler
2230:Fibered
2225:Contact
2223:)
2215:Complex
2213:)
2182:Section
1795:Physics
1629:Related
1392:Physics
1310:Tensors
786:changes
271:locally
81:of the
53:of the
51:section
38:) on a
2678:Vector
2663:Koszul
2643:Cartan
2638:Affine
2620:Vector
2615:Tensor
2600:Spinor
2590:Normal
2586:Stable
2540:Affine
2444:bundle
2396:bundle
2342:Almost
2265:Kähler
2221:Almost
2211:Almost
2205:Closed
2105:Sard's
2061:(list)
1723:Vector
1718:Spinor
1703:Matrix
1497:Tensor
1265:
1211:Tensor
830:closed
792:times
239:where
48:smooth
2786:Sheaf
2560:Fiber
2336:Rizza
2307:Prime
2138:Local
2128:Curve
1990:Atlas
1643:Basis
1328:Scope
982:with
838:exact
832:(its
627:atan2
577:atan2
531:atan2
42:is a
2653:Form
2555:Dual
2488:flow
2351:Tame
2327:Sub−
2240:Flat
2120:Maps
1263:ISBN
908:open
878:Let
34:(or
30:, a
2575:Jet
906:be
105:to
85:of
26:In
2818::
2566:Co
1234:.
537::
460:.
2584:(
2564:(
2340:(
2321:(
2219:(
2209:(
1972:)
1968:(
1958:e
1951:t
1944:v
1302:e
1295:t
1288:v
1271:.
1244:.
1168:.
1165:)
1160:0
1156:x
1152:(
1145:f
1123:R
1115:R
1094:U
1087:0
1083:x
1078:T
1057:U
1049:0
1045:x
1024:f
1021:d
1001:.
994:f
970:,
966:R
959:U
956::
953:f
930:)
927:b
924:,
921:a
918:(
893:R
886:U
806:.
800:2
772:y
748:y
745:d
737:2
733:y
729:+
724:2
720:x
715:x
710:+
707:x
704:d
696:2
692:y
688:+
683:2
679:x
674:y
666:=
656:y
653:d
649:)
645:)
642:x
639:,
636:y
633:(
623:(
617:y
609:+
606:x
603:d
599:)
595:)
592:x
589:,
586:y
583:(
573:(
567:x
559:=
549:d
517:)
514:y
511:,
508:x
505:(
482:.
476:d
438:i
434:f
413:,
408:n
404:x
400:d
396:)
393:x
390:(
385:n
381:f
377:+
371:+
366:2
362:x
358:d
354:)
351:x
348:(
343:2
339:f
335:+
330:1
326:x
322:d
318:)
315:x
312:(
307:1
303:f
299:=
294:x
252:x
227:,
222:R
214:M
209:x
205:T
201::
196:M
191:x
187:T
181:|
173:=
168:x
159:,
154:R
146:M
143:T
140::
114:R
93:M
65:M
23:.
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