557:
425:
1348:
230:
1229:
500:
938:
1112:). Thus, the problem of proving the existence and uniqueness of integral curves is the same as that of finding solutions to ordinary differential equations/initial value problems and showing that they are unique.
1252:
235:
1081:
712:
1018:
860:
420:{\displaystyle {\begin{aligned}{\frac {dx_{1}}{dt}}&=F_{1}(x_{1},\ldots ,x_{n})\\&\vdots \\{\frac {dx_{n}}{dt}}&=F_{n}(x_{1},\ldots ,x_{n}).\end{aligned}}}
2370:
1561:
2365:
1652:
1676:
1871:
2418:
1154:
85:
58:
Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In
1741:
1967:
436:
2020:
1548:
1459:
866:
2304:
1343:{\displaystyle \left({\frac {\mathrm {d} \alpha _{1}}{\mathrm {d} t}},\dots ,{\frac {\mathrm {d} \alpha _{n}}{\mathrm {d} t}}\right),}
2069:
1661:
2052:
221:
545:
2413:
47:
2264:
2249:
1972:
1746:
2294:
131:
2299:
2269:
1977:
1933:
1914:
1681:
1625:
1024:
1836:
1701:
2221:
2086:
1778:
1620:
638:
91:
663:
1918:
1888:
1812:
1802:
1758:
1588:
1541:
1686:
2259:
1878:
1773:
1593:
793:
127:
977:
819:
1908:
1903:
1145:
2239:
2177:
2025:
1729:
1719:
1691:
1666:
1576:
1470:
1377:
541:
2377:
2350:
2059:
1937:
1922:
1851:
1610:
2319:
2274:
2171:
2042:
1846:
1671:
1534:
1856:
2254:
2234:
2229:
2136:
2047:
1861:
1841:
1696:
1635:
2392:
2186:
2141:
2064:
2035:
1893:
1826:
1821:
1816:
1806:
1598:
1581:
166:
105:
95:
43:
2335:
2244:
2074:
2030:
1796:
614:
1522:. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
2201:
2126:
2096:
1994:
1987:
1927:
1898:
1768:
1724:
1400:
630:
77:
67:
63:
28:
2407:
2387:
2211:
2206:
2191:
2181:
2131:
2108:
1982:
1942:
1883:
1831:
1630:
2314:
2309:
2151:
2118:
2091:
1999:
1640:
1389:
722:
585:
123:
2157:
2146:
2103:
2004:
1605:
1239:
971:
is a local solution to the ordinary differential equation/initial value problem
589:
561:
35:
2382:
2340:
2166:
2079:
1711:
1615:
1515:
1458:
1436:
99:
72:
17:
2196:
2161:
1866:
1753:
1224:{\displaystyle (\mathrm {d} _{t}\alpha )(+1)\in \mathrm {T} _{\alpha (t)}M.}
800:
2360:
2355:
2345:
1736:
1557:
556:
1526:
593:
59:
1952:
505:
This equation says that the vector tangent to the curve at any point
495:{\displaystyle \mathbf {x} '(t)=\mathbf {F} (\mathbf {x} (t)).\!\,}
933:{\displaystyle \alpha '(t)=X(\alpha (t)){\mbox{ for all }}t\in J.}
81:
94:, the integral curves for a differential equation that governs a
1530:
1376:
The same thing may be phrased even more abstractly in terms of
1086:
It is local in the sense that it is defined only for times in
430:
Such a system may be written as a single vector equation,
548:
implies that there exists a unique flow for small time.
912:
1255:
1157:
1027:
980:
869:
822:
666:
439:
233:
729:, i.e. an assignment to every point of the manifold
2328:
2287:
2220:
2117:
2013:
1960:
1951:
1787:
1710:
1649:
1569:
1342:
1223:
1075:
1012:
932:
854:
706:
494:
419:
1373:with respect to the usual coordinate directions.
584:If the differential equation is represented as a
490:
944:Relationship to ordinary differential equations
533:) is tangent at each point to the vector field
1144:. From a more abstract viewpoint, this is the
1542:
592:, then the corresponding integral curves are
8:
1076:{\displaystyle \alpha '(t)=X(\alpha (t)).\,}
564:corresponding to the differential equation
1957:
1549:
1535:
1527:
948:The above definition of an integral curve
600:Generalization to differentiable manifolds
513:) along the curve is precisely the vector
46:that represents a specific solution to an
1321:
1313:
1304:
1301:
1281:
1273:
1264:
1261:
1254:
1200:
1195:
1167:
1162:
1156:
1072:
1026:
1009:
991:
979:
911:
868:
851:
833:
821:
671:
665:
491:
470:
462:
441:
438:
401:
382:
369:
341:
331:
308:
289:
276:
248:
238:
234:
232:
1453:so that the following diagram commutes:
707:{\displaystyle \pi _{M}:(x,v)\mapsto x.}
555:
7:
224:of ordinary differential equations,
1013:{\displaystyle \alpha (t_{0})=p;\,}
855:{\displaystyle \alpha (t_{0})=p;\,}
1497:) is its value at some point
1322:
1305:
1282:
1265:
1246:, this is the familiar derivative
1196:
1163:
25:
1457:
1380:. Note that the tangent bundle T
471:
463:
442:
2419:Ordinary differential equations
1589:Differentiable/Smooth manifold
1210:
1204:
1188:
1179:
1176:
1158:
1116:Remarks on the time derivative
1090:, and not necessarily for all
1066:
1063:
1057:
1051:
1042:
1036:
997:
984:
908:
905:
899:
893:
884:
878:
839:
826:
695:
692:
680:
560:Three integral curves for the
484:
481:
475:
467:
456:
450:
407:
375:
314:
282:
76:, and integral curves for the
48:ordinary differential equation
1:
967:, is the same as saying that
1415:) = 1 (or, more precisely, (
1128:) denotes the derivative of
596:to the field at each point.
169:with Cartesian coordinates (
2295:Classification of manifolds
540:If a given vector field is
220:if it is a solution of the
2435:
26:
2371:over commutative algebras
1465:Then the time derivative
1407:of this bundle such that
1234:In the special case that
62:, integral curves for an
2087:Riemann curvature tensor
1369:are the coordinates for
50:or system of equations.
27:Not to be confused with
733:of a tangent vector to
725:of the tangent bundle T
546:Picard–Lindelöf theorem
1879:Manifold with boundary
1594:Differential structure
1520:Differential manifolds
1344:
1225:
1077:
1014:
934:
856:
708:
581:
496:
421:
128:vector-valued function
2414:Differential geometry
1345:
1226:
1140:is pointing" at time
1078:
1015:
935:
857:
741:be a vector field on
709:
580: − 2.
559:
525:)), and so the curve
497:
422:
132:Cartesian coordinates
2026:Covariant derivative
1577:Topological manifold
1476:′ =
1253:
1155:
1025:
978:
867:
820:
664:
542:Lipschitz continuous
437:
231:
2060:Exterior derivative
1662:Atiyah–Singer index
1611:Riemannian manifold
952:for a vector field
914: for all
737:at that point. Let
576: −
98:are referred to as
2366:Secondary calculus
2320:Singularity theory
2275:Parallel transport
2043:De Rham cohomology
1682:Generalized Stokes
1340:
1221:
1146:Fréchet derivative
1073:
1010:
956:, passing through
930:
916:
852:
717:A vector field on
704:
582:
492:
417:
415:
2401:
2400:
2283:
2282:
2048:Differential form
1702:Whitney embedding
1636:Differential form
1485:
1330:
1290:
1136:, the "direction
915:
637:with its natural
356:
263:
222:autonomous system
92:dynamical systems
16:(Redirected from
2426:
2393:Stratified space
2351:Fréchet manifold
2065:Interior product
1958:
1655:
1551:
1544:
1537:
1528:
1523:
1483:
1461:
1349:
1347:
1346:
1341:
1336:
1332:
1331:
1329:
1325:
1319:
1318:
1317:
1308:
1302:
1291:
1289:
1285:
1279:
1278:
1277:
1268:
1262:
1230:
1228:
1227:
1222:
1214:
1213:
1199:
1172:
1171:
1166:
1082:
1080:
1079:
1074:
1035:
1019:
1017:
1016:
1011:
996:
995:
939:
937:
936:
931:
917:
913:
877:
861:
859:
858:
853:
838:
837:
792:, defined on an
765:passing through
713:
711:
710:
705:
676:
675:
625:≥ 2. As usual, T
501:
499:
498:
493:
474:
466:
449:
445:
426:
424:
423:
418:
416:
406:
405:
387:
386:
374:
373:
357:
355:
347:
346:
345:
332:
320:
313:
312:
294:
293:
281:
280:
264:
262:
254:
253:
252:
239:
167:parametric curve
44:parametric curve
21:
2434:
2433:
2429:
2428:
2427:
2425:
2424:
2423:
2404:
2403:
2402:
2397:
2336:Banach manifold
2329:Generalizations
2324:
2279:
2216:
2113:
2075:Ricci curvature
2031:Cotangent space
2009:
1947:
1789:
1783:
1742:Exponential map
1706:
1651:
1645:
1565:
1555:
1514:
1511:
1482:
1469:′ is the
1444:
1399:and there is a
1368:
1359:
1320:
1309:
1303:
1280:
1269:
1263:
1260:
1256:
1251:
1250:
1194:
1161:
1153:
1152:
1118:
1111:
1100:
1028:
1023:
1022:
987:
976:
975:
966:
946:
870:
865:
864:
829:
818:
817:
812:
775:
667:
662:
661:
648:
615:Banach manifold
607:
602:
554:
440:
435:
434:
414:
413:
397:
378:
365:
358:
348:
337:
333:
328:
327:
318:
317:
304:
285:
272:
265:
255:
244:
240:
229:
228:
199:
186:
175:
156:
147:
140:
116:
56:
32:
23:
22:
15:
12:
11:
5:
2432:
2430:
2422:
2421:
2416:
2406:
2405:
2399:
2398:
2396:
2395:
2390:
2385:
2380:
2375:
2374:
2373:
2363:
2358:
2353:
2348:
2343:
2338:
2332:
2330:
2326:
2325:
2323:
2322:
2317:
2312:
2307:
2302:
2297:
2291:
2289:
2285:
2284:
2281:
2280:
2278:
2277:
2272:
2267:
2262:
2257:
2252:
2247:
2242:
2237:
2232:
2226:
2224:
2218:
2217:
2215:
2214:
2209:
2204:
2199:
2194:
2189:
2184:
2174:
2169:
2164:
2154:
2149:
2144:
2139:
2134:
2129:
2123:
2121:
2115:
2114:
2112:
2111:
2106:
2101:
2100:
2099:
2089:
2084:
2083:
2082:
2072:
2067:
2062:
2057:
2056:
2055:
2045:
2040:
2039:
2038:
2028:
2023:
2017:
2015:
2011:
2010:
2008:
2007:
2002:
1997:
1992:
1991:
1990:
1980:
1975:
1970:
1964:
1962:
1955:
1949:
1948:
1946:
1945:
1940:
1930:
1925:
1911:
1906:
1901:
1896:
1891:
1889:Parallelizable
1886:
1881:
1876:
1875:
1874:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1824:
1819:
1809:
1799:
1793:
1791:
1785:
1784:
1782:
1781:
1776:
1771:
1769:Lie derivative
1766:
1764:Integral curve
1761:
1756:
1751:
1750:
1749:
1739:
1734:
1733:
1732:
1725:Diffeomorphism
1722:
1716:
1714:
1708:
1707:
1705:
1704:
1699:
1694:
1689:
1684:
1679:
1674:
1669:
1664:
1658:
1656:
1647:
1646:
1644:
1643:
1638:
1633:
1628:
1623:
1618:
1613:
1608:
1603:
1602:
1601:
1596:
1586:
1585:
1584:
1573:
1571:
1570:Basic concepts
1567:
1566:
1556:
1554:
1553:
1546:
1539:
1531:
1525:
1524:
1510:
1507:
1480:
1463:
1462:
1442:
1403:cross-section
1390:trivial bundle
1364:
1357:
1351:
1350:
1339:
1335:
1328:
1324:
1316:
1312:
1307:
1300:
1297:
1294:
1288:
1284:
1276:
1272:
1267:
1259:
1232:
1231:
1220:
1217:
1212:
1209:
1206:
1203:
1198:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1170:
1165:
1160:
1120:In the above,
1117:
1114:
1109:
1098:
1084:
1083:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1034:
1031:
1020:
1008:
1005:
1002:
999:
994:
990:
986:
983:
964:
945:
942:
941:
940:
929:
926:
923:
920:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
876:
873:
862:
850:
847:
844:
841:
836:
832:
828:
825:
810:
773:
759:integral curve
715:
714:
703:
700:
697:
694:
691:
688:
685:
682:
679:
674:
670:
644:
631:tangent bundle
606:
603:
601:
598:
553:
550:
503:
502:
489:
486:
483:
480:
477:
473:
469:
465:
461:
458:
455:
452:
448:
444:
428:
427:
412:
409:
404:
400:
396:
393:
390:
385:
381:
377:
372:
368:
364:
361:
359:
354:
351:
344:
340:
336:
330:
329:
326:
323:
321:
319:
316:
311:
307:
303:
300:
297:
292:
288:
284:
279:
275:
271:
268:
266:
261:
258:
251:
247:
243:
237:
236:
214:integral curve
195:
184:
173:
152:
145:
138:
115:
112:
78:velocity field
68:magnetic field
64:electric field
55:
52:
40:integral curve
29:Curve integral
24:
18:Solution curve
14:
13:
10:
9:
6:
4:
3:
2:
2431:
2420:
2417:
2415:
2412:
2411:
2409:
2394:
2391:
2389:
2388:Supermanifold
2386:
2384:
2381:
2379:
2376:
2372:
2369:
2368:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2333:
2331:
2327:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2292:
2290:
2286:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2227:
2225:
2223:
2219:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2179:
2175:
2173:
2170:
2168:
2165:
2163:
2159:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2124:
2122:
2120:
2116:
2110:
2109:Wedge product
2107:
2105:
2102:
2098:
2095:
2094:
2093:
2090:
2088:
2085:
2081:
2078:
2077:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2054:
2053:Vector-valued
2051:
2050:
2049:
2046:
2044:
2041:
2037:
2034:
2033:
2032:
2029:
2027:
2024:
2022:
2019:
2018:
2016:
2012:
2006:
2003:
2001:
1998:
1996:
1993:
1989:
1986:
1985:
1984:
1983:Tangent space
1981:
1979:
1976:
1974:
1971:
1969:
1966:
1965:
1963:
1959:
1956:
1954:
1950:
1944:
1941:
1939:
1935:
1931:
1929:
1926:
1924:
1920:
1916:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1873:
1870:
1869:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1814:
1810:
1808:
1804:
1800:
1798:
1795:
1794:
1792:
1786:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1748:
1747:in Lie theory
1745:
1744:
1743:
1740:
1738:
1735:
1731:
1728:
1727:
1726:
1723:
1721:
1718:
1717:
1715:
1713:
1709:
1703:
1700:
1698:
1695:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1659:
1657:
1654:
1650:Main results
1648:
1642:
1639:
1637:
1634:
1632:
1631:Tangent space
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1600:
1597:
1595:
1592:
1591:
1590:
1587:
1583:
1580:
1579:
1578:
1575:
1574:
1572:
1568:
1563:
1559:
1552:
1547:
1545:
1540:
1538:
1533:
1532:
1529:
1521:
1517:
1513:
1512:
1508:
1506:
1504:
1501: ∈
1500:
1496:
1492:
1488:
1479:
1475:
1472:
1468:
1460:
1456:
1455:
1454:
1452:
1448:
1441:
1438:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1391:
1387:
1383:
1379:
1374:
1372:
1367:
1363:
1356:
1337:
1333:
1326:
1314:
1310:
1298:
1295:
1292:
1286:
1274:
1270:
1257:
1249:
1248:
1247:
1245:
1241:
1237:
1218:
1215:
1207:
1201:
1191:
1185:
1182:
1173:
1168:
1151:
1150:
1149:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1115:
1113:
1108:
1104:
1097:
1093:
1089:
1069:
1060:
1054:
1048:
1045:
1039:
1032:
1029:
1021:
1006:
1003:
1000:
992:
988:
981:
974:
973:
972:
970:
963:
959:
955:
951:
943:
927:
924:
921:
918:
902:
896:
890:
887:
881:
874:
871:
863:
848:
845:
842:
834:
830:
823:
816:
815:
814:
809:
805:
802:
798:
795:
794:open interval
791:
787:
783:
779:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
723:cross-section
720:
701:
698:
689:
686:
683:
677:
672:
668:
660:
659:
658:
656:
652:
647:
643:
640:
636:
632:
628:
624:
620:
616:
612:
604:
599:
597:
595:
591:
587:
579:
575:
572: =
571:
568: /
567:
563:
558:
551:
549:
547:
543:
538:
536:
532:
528:
524:
520:
516:
512:
508:
487:
478:
459:
453:
446:
433:
432:
431:
410:
402:
398:
394:
391:
388:
383:
379:
370:
366:
362:
360:
352:
349:
342:
338:
334:
324:
322:
309:
305:
301:
298:
295:
290:
286:
277:
273:
269:
267:
259:
256:
249:
245:
241:
227:
226:
225:
223:
219:
215:
211:
207:
203:
198:
194:
190:
183:
179:
172:
168:
164:
160:
155:
151:
144:
137:
133:
129:
126:, that is, a
125:
121:
118:Suppose that
113:
111:
109:
108:
103:
102:
97:
93:
89:
88:
84:are known as
83:
79:
75:
74:
70:are known as
69:
65:
61:
53:
51:
49:
45:
41:
37:
30:
19:
2315:Moving frame
2310:Morse theory
2300:Gauge theory
2092:Tensor field
2021:Closed/Exact
2000:Vector field
1968:Distribution
1909:Hypercomplex
1904:Quaternionic
1763:
1641:Vector field
1599:Smooth atlas
1519:
1502:
1498:
1494:
1490:
1486:
1477:
1473:
1466:
1464:
1450:
1446:
1439:
1432:
1431:. The curve
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1396:
1392:
1385:
1381:
1378:induced maps
1375:
1370:
1365:
1361:
1354:
1352:
1243:
1235:
1233:
1141:
1137:
1133:
1129:
1125:
1121:
1119:
1106:
1102:
1095:
1091:
1087:
1085:
968:
961:
957:
953:
949:
947:
813:, such that
807:
803:
796:
789:
785:
781:
777:
770:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
726:
718:
716:
654:
650:
645:
641:
634:
629:denotes the
626:
622:
618:
610:
608:
586:vector field
583:
577:
573:
569:
565:
539:
534:
530:
526:
522:
518:
514:
510:
506:
504:
429:
217:
213:
209:
205:
201:
196:
192:
188:
181:
177:
170:
162:
158:
157:), and that
153:
149:
142:
135:
124:vector field
122:is a static
119:
117:
106:
101:trajectories
100:
86:
71:
57:
39:
33:
2260:Levi-Civita
2250:Generalized
2222:Connections
2172:Lie algebra
2104:Volume form
2005:Vector flow
1978:Pushforward
1973:Lie bracket
1872:Lie algebra
1837:G-structure
1626:Pushforward
1606:Submanifold
1516:Lang, Serge
1471:composition
1240:open subset
1101:(let alone
806:containing
776:is a curve
590:slope field
562:slope field
544:, then the
87:streamlines
73:field lines
36:mathematics
2408:Categories
2383:Stratifold
2341:Diffeology
2137:Associated
1938:Symplectic
1923:Riemannian
1852:Hyperbolic
1779:Submersion
1687:Hopf–Rinow
1621:Submersion
1616:Smooth map
1509:References
1437:bundle map
1435:induces a
1423:) for all
639:projection
605:Definition
114:Definition
2265:Principal
2240:Ehresmann
2197:Subbundle
2187:Principal
2162:Fibration
2142:Cotangent
2014:Covectors
1867:Lie group
1847:Hermitian
1790:manifolds
1759:Immersion
1754:Foliation
1692:Noether's
1677:Frobenius
1672:De Rham's
1667:Darboux's
1558:Manifolds
1445: : T
1401:canonical
1311:α
1296:…
1271:α
1202:α
1192:∈
1174:α
1055:α
1030:α
982:α
922:∈
897:α
872:α
824:α
801:real line
788:of class
745:of class
696:↦
669:π
657:given by
649: : T
617:of class
392:…
325:⋮
299:…
204:)). Then
2361:Orbifold
2356:K-theory
2346:Diffiety
2070:Pullback
1884:Oriented
1862:Kenmotsu
1842:Hadamard
1788:Types of
1737:Geodesic
1562:Glossary
1518:(1972).
1493:′(
1238:is some
1132:at time
1124:′(
1033:′
960:at time
875:′
780: :
769:at time
749:and let
552:Examples
447:′
212:) is an
2305:History
2288:Related
2202:Tangent
2180:)
2160:)
2127:Adjoint
2119:Bundles
2097:density
1995:Torsion
1961:Vectors
1953:Tensors
1936:)
1921:)
1917:,
1915:Pseudo−
1894:Poisson
1827:Finsler
1822:Fibered
1817:Contact
1815:)
1807:Complex
1805:)
1774:Section
1419:, 1) ∈
1395:×
1388:is the
1360:, ...,
799:of the
594:tangent
165:) is a
60:physics
2270:Vector
2255:Koszul
2235:Cartan
2230:Affine
2212:Vector
2207:Tensor
2192:Spinor
2182:Normal
2178:Stable
2132:Affine
2036:bundle
1988:bundle
1934:Almost
1857:Kähler
1813:Almost
1803:Almost
1797:Closed
1697:Sard's
1653:(list)
1489:, and
1353:where
191:),...,
107:orbits
96:system
2378:Sheaf
2152:Fiber
1928:Rizza
1899:Prime
1730:Local
1720:Curve
1582:Atlas
757:. An
721:is a
621:with
613:be a
148:,...,
130:with
90:. In
82:fluid
80:of a
42:is a
38:, an
2245:Form
2147:Dual
2080:flow
1943:Tame
1919:Sub−
1832:Flat
1712:Maps
761:for
609:Let
54:Name
2167:Jet
1449:→ T
1384:of
1242:of
633:of
588:or
216:of
104:or
66:or
34:In
2410::
2158:Co
1505:.
1427:∈
1148::
1105:≤
1094:≥
784:→
753:∈
653:→
570:dx
566:dy
537:.
180:),
110:.
2176:(
2156:(
1932:(
1913:(
1811:(
1801:(
1564:)
1560:(
1550:e
1543:t
1536:v
1503:J
1499:t
1495:t
1491:α
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1484:o
1481:∗
1478:α
1474:α
1467:α
1451:M
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1443:∗
1440:α
1433:α
1429:J
1425:t
1421:ι
1417:t
1413:t
1411:(
1409:ι
1405:ι
1397:R
1393:J
1386:J
1382:J
1371:α
1366:n
1362:α
1358:1
1355:α
1338:,
1334:)
1327:t
1323:d
1315:n
1306:d
1299:,
1293:,
1287:t
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1266:d
1258:(
1244:R
1236:M
1219:.
1216:M
1211:)
1208:t
1205:(
1197:T
1189:)
1186:1
1183:+
1180:(
1177:)
1169:t
1164:d
1159:(
1142:t
1138:α
1134:t
1130:α
1126:t
1122:α
1110:0
1107:t
1103:t
1099:0
1096:t
1092:t
1088:J
1070:.
1067:)
1064:)
1061:t
1058:(
1052:(
1049:X
1046:=
1043:)
1040:t
1037:(
1007:;
1004:p
1001:=
998:)
993:0
989:t
985:(
969:α
965:0
962:t
958:p
954:X
950:α
928:.
925:J
919:t
909:)
906:)
903:t
900:(
894:(
891:X
888:=
885:)
882:t
879:(
849:;
846:p
843:=
840:)
835:0
831:t
827:(
811:0
808:t
804:R
797:J
790:C
786:M
782:J
778:α
774:0
771:t
767:p
763:X
755:M
751:p
747:C
743:M
739:X
735:M
731:M
727:M
719:M
702:.
699:x
693:)
690:v
687:,
684:x
681:(
678::
673:M
655:M
651:M
646:M
642:π
635:M
627:M
623:r
619:C
611:M
578:x
574:x
535:F
531:t
529:(
527:x
523:t
521:(
519:x
517:(
515:F
511:t
509:(
507:x
488:.
485:)
482:)
479:t
476:(
472:x
468:(
464:F
460:=
457:)
454:t
451:(
443:x
411:.
408:)
403:n
399:x
395:,
389:,
384:1
380:x
376:(
371:n
367:F
363:=
353:t
350:d
343:n
339:x
335:d
315:)
310:n
306:x
302:,
296:,
291:1
287:x
283:(
278:1
274:F
270:=
260:t
257:d
250:1
246:x
242:d
218:F
210:t
208:(
206:x
202:t
200:(
197:n
193:x
189:t
187:(
185:2
182:x
178:t
176:(
174:1
171:x
163:t
161:(
159:x
154:n
150:F
146:2
143:F
141:,
139:1
136:F
134:(
120:F
31:.
20:)
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