546:
414:
1337:
219:
1218:
489:
927:
1101:). 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.
1241:
224:
1070:
701:
1448:
1007:
849:
409:{\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}}}
1143:
2359:
1550:
47:
Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In
2354:
425:
855:
1641:
1332:{\displaystyle \left({\frac {\mathrm {d} \alpha _{1}}{\mathrm {d} t}},\dots ,{\frac {\mathrm {d} \alpha _{n}}{\mathrm {d} t}}\right),}
1665:
2407:
1860:
74:
1730:
1956:
2009:
1537:
2293:
2058:
1013:
2041:
1650:
210:
2402:
36:
652:
534:
2253:
1660:
2238:
1961:
1735:
2283:
966:
808:
120:
2288:
2258:
1966:
1922:
1903:
1670:
1614:
1825:
1690:
2210:
2075:
1767:
1609:
627:
80:
1907:
1877:
1801:
1791:
1747:
1577:
1530:
2248:
1867:
1762:
1675:
1582:
782:
116:
1897:
1892:
1134:
2228:
2166:
2014:
1718:
1708:
1680:
1655:
1565:
1459:
1366:
530:
1378:
8:
2366:
2048:
1926:
1911:
1840:
1599:
2339:
1511:. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
2308:
2263:
2160:
2031:
1835:
1523:
1845:
2243:
2223:
2218:
2125:
2036:
1850:
1830:
1685:
1624:
711:
2381:
2175:
2130:
2053:
2024:
1882:
1815:
1810:
1805:
1795:
1587:
1570:
155:
94:
84:
32:
2324:
2233:
2063:
2019:
1785:
603:
960:
is a local solution to the ordinary differential equation/initial value problem
2190:
2115:
2085:
1983:
1976:
1916:
1887:
1757:
1713:
1389:
619:
66:
56:
52:
17:
2396:
2376:
2200:
2195:
2180:
2170:
2120:
2097:
1971:
1931:
1872:
1820:
1619:
1213:{\displaystyle (\mathrm {d} _{t}\alpha )(+1)\in \mathrm {T} _{\alpha (t)}M.}
2303:
2298:
2140:
2107:
2080:
1988:
1629:
574:
112:
2146:
2135:
2092:
1993:
1594:
1228:
578:
550:
24:
2371:
2329:
2155:
2068:
1700:
1604:
1515:
1504:
1425:
88:
61:
2185:
2150:
1855:
1742:
789:
494:
This equation says that the vector tangent to the curve at any point
2349:
2344:
2334:
1725:
1546:
484:{\displaystyle \mathbf {x} '(t)=\mathbf {F} (\mathbf {x} (t)).\!\,}
1447:
932:
922:{\displaystyle \alpha '(t)=X(\alpha (t)){\mbox{ for all }}t\in J.}
582:
83:, the integral curves for a differential equation that governs a
48:
1941:
1365:
The same thing may be phrased even more abstractly in terms of
1075:
It is local in the sense that it is defined only for times in
545:
588:
70:
419:
Such a system may be written as a single vector equation,
537:
implies that there exists a unique flow for small time.
901:
1244:
1146:
1016:
969:
858:
811:
655:
428:
222:
718:, i.e. an assignment to every point of the manifold
1331:
1212:
1064:
1001:
921:
843:
695:
483:
408:
1362:with respect to the usual coordinate directions.
1104:
573:If the differential equation is represented as a
479:
2394:
933:Relationship to ordinary differential equations
522:) is tangent at each point to the vector field
1133:. From a more abstract viewpoint, this is the
1531:
581:, then the corresponding integral curves are
1065:{\displaystyle \alpha '(t)=X(\alpha (t)).\,}
553:corresponding to the differential equation
1538:
1524:
937:The above definition of an integral curve
589:Generalization to differentiable manifolds
502:) along the curve is precisely the vector
35:that represents a specific solution to an
1061:
998:
840:
480:
1545:
1442:so that the following diagram commutes:
696:{\displaystyle \pi _{M}:(x,v)\mapsto x.}
544:
2395:
1519:
1503:
213:of ordinary differential equations,
1002:{\displaystyle \alpha (t_{0})=p;\,}
844:{\displaystyle \alpha (t_{0})=p;\,}
13:
1486:) is its value at some point
1311:
1294:
1271:
1254:
1235:, this is the familiar derivative
1185:
1152:
14:
2419:
1446:
1369:. Note that the tangent bundle T
460:
452:
431:
2408:Ordinary differential equations
593:
1578:Differentiable/Smooth manifold
1199:
1193:
1177:
1168:
1165:
1147:
1105:Remarks on the time derivative
1079:, and not necessarily for all
1055:
1052:
1046:
1040:
1031:
1025:
986:
973:
897:
894:
888:
882:
873:
867:
828:
815:
684:
681:
669:
549:Three integral curves for the
473:
470:
464:
456:
445:
439:
396:
364:
303:
271:
65:, and integral curves for the
37:ordinary differential equation
1:
1497:
956:, is the same as saying that
102:
1404:) = 1 (or, more precisely, (
1117:) denotes the derivative of
585:to the field at each point.
158:with Cartesian coordinates (
7:
2284:Classification of manifolds
540:
529:If a given vector field is
209:if it is a solution of the
10:
2424:
15:
2360:over commutative algebras
2317:
2276:
2209:
2106:
2002:
1949:
1940:
1776:
1699:
1638:
1558:
1454:Then the time derivative
1396:of this bundle such that
1223:In the special case that
51:, integral curves for an
2076:Riemann curvature tensor
1358:are the coordinates for
39:or system of equations.
16:Not to be confused with
722:of a tangent vector to
714:of the tangent bundle T
535:Picard–Lindelöf theorem
42:
1868:Manifold with boundary
1583:Differential structure
1509:Differential manifolds
1333:
1214:
1066:
1003:
923:
845:
697:
570:
485:
410:
117:vector-valued function
2403:Differential geometry
1334:
1215:
1129:is pointing" at time
1067:
1004:
924:
846:
730:be a vector field on
698:
569: − 2.
548:
514:)), and so the curve
486:
411:
121:Cartesian coordinates
2015:Covariant derivative
1566:Topological manifold
1465:′ =
1242:
1144:
1014:
967:
856:
809:
653:
531:Lipschitz continuous
426:
220:
2049:Exterior derivative
1651:Atiyah–Singer index
1600:Riemannian manifold
941:for a vector field
903: for all
726:at that point. Let
565: −
87:are referred to as
2355:Secondary calculus
2309:Singularity theory
2264:Parallel transport
2032:De Rham cohomology
1671:Generalized Stokes
1329:
1210:
1135:Fréchet derivative
1062:
999:
945:, passing through
919:
905:
841:
706:A vector field on
693:
571:
481:
406:
404:
2390:
2389:
2272:
2271:
2037:Differential form
1691:Whitney embedding
1625:Differential form
1474:
1319:
1279:
1125:, the "direction
904:
626:with its natural
345:
252:
211:autonomous system
81:dynamical systems
2415:
2382:Stratified space
2340:Fréchet manifold
2054:Interior product
1947:
1946:
1644:
1540:
1533:
1526:
1517:
1516:
1512:
1472:
1450:
1338:
1336:
1335:
1330:
1325:
1321:
1320:
1318:
1314:
1308:
1307:
1306:
1297:
1291:
1280:
1278:
1274:
1268:
1267:
1266:
1257:
1251:
1219:
1217:
1216:
1211:
1203:
1202:
1188:
1161:
1160:
1155:
1071:
1069:
1068:
1063:
1024:
1008:
1006:
1005:
1000:
985:
984:
928:
926:
925:
920:
906:
902:
866:
850:
848:
847:
842:
827:
826:
781:, defined on an
754:passing through
702:
700:
699:
694:
665:
664:
614:≥ 2. As usual, T
490:
488:
487:
482:
463:
455:
438:
434:
415:
413:
412:
407:
405:
395:
394:
376:
375:
363:
362:
346:
344:
336:
335:
334:
321:
309:
302:
301:
283:
282:
270:
269:
253:
251:
243:
242:
241:
228:
156:parametric curve
33:parametric curve
2423:
2422:
2418:
2417:
2416:
2414:
2413:
2412:
2393:
2392:
2391:
2386:
2325:Banach manifold
2318:Generalizations
2313:
2268:
2205:
2102:
2064:Ricci curvature
2020:Cotangent space
1998:
1936:
1778:
1772:
1731:Exponential map
1695:
1640:
1634:
1554:
1544:
1500:
1471:
1458:′ is the
1433:
1388:and there is a
1357:
1348:
1310:
1309:
1302:
1298:
1293:
1292:
1290:
1270:
1269:
1262:
1258:
1253:
1252:
1250:
1249:
1245:
1243:
1240:
1239:
1189:
1184:
1183:
1156:
1151:
1150:
1145:
1142:
1141:
1107:
1100:
1089:
1017:
1015:
1012:
1011:
980:
976:
968:
965:
964:
955:
935:
900:
859:
857:
854:
853:
822:
818:
810:
807:
806:
801:
764:
660:
656:
654:
651:
650:
637:
604:Banach manifold
596:
591:
543:
459:
451:
430:
429:
427:
424:
423:
403:
402:
390:
386:
371:
367:
358:
354:
347:
337:
330:
326:
322:
320:
317:
316:
307:
306:
297:
293:
278:
274:
265:
261:
254:
244:
237:
233:
229:
227:
223:
221:
218:
217:
188:
175:
164:
145:
136:
129:
105:
45:
21:
12:
11:
5:
2421:
2411:
2410:
2405:
2388:
2387:
2385:
2384:
2379:
2374:
2369:
2364:
2363:
2362:
2352:
2347:
2342:
2337:
2332:
2327:
2321:
2319:
2315:
2314:
2312:
2311:
2306:
2301:
2296:
2291:
2286:
2280:
2278:
2274:
2273:
2270:
2269:
2267:
2266:
2261:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2215:
2213:
2207:
2206:
2204:
2203:
2198:
2193:
2188:
2183:
2178:
2173:
2163:
2158:
2153:
2143:
2138:
2133:
2128:
2123:
2118:
2112:
2110:
2104:
2103:
2101:
2100:
2095:
2090:
2089:
2088:
2078:
2073:
2072:
2071:
2061:
2056:
2051:
2046:
2045:
2044:
2034:
2029:
2028:
2027:
2017:
2012:
2006:
2004:
2000:
1999:
1997:
1996:
1991:
1986:
1981:
1980:
1979:
1969:
1964:
1959:
1953:
1951:
1944:
1938:
1937:
1935:
1934:
1929:
1919:
1914:
1900:
1895:
1890:
1885:
1880:
1878:Parallelizable
1875:
1870:
1865:
1864:
1863:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1818:
1813:
1808:
1798:
1788:
1782:
1780:
1774:
1773:
1771:
1770:
1765:
1760:
1758:Lie derivative
1755:
1753:Integral curve
1750:
1745:
1740:
1739:
1738:
1728:
1723:
1722:
1721:
1714:Diffeomorphism
1711:
1705:
1703:
1697:
1696:
1694:
1693:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1647:
1645:
1636:
1635:
1633:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1591:
1590:
1585:
1575:
1574:
1573:
1562:
1560:
1559:Basic concepts
1556:
1555:
1543:
1542:
1535:
1528:
1520:
1514:
1513:
1499:
1496:
1469:
1452:
1451:
1431:
1392:cross-section
1379:trivial bundle
1353:
1346:
1340:
1339:
1328:
1324:
1317:
1313:
1305:
1301:
1296:
1289:
1286:
1283:
1277:
1273:
1265:
1261:
1256:
1248:
1221:
1220:
1209:
1206:
1201:
1198:
1195:
1192:
1187:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1159:
1154:
1149:
1109:In the above,
1106:
1103:
1098:
1087:
1073:
1072:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1023:
1020:
1009:
997:
994:
991:
988:
983:
979:
975:
972:
953:
934:
931:
930:
929:
918:
915:
912:
909:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
865:
862:
851:
839:
836:
833:
830:
825:
821:
817:
814:
799:
762:
748:integral curve
704:
703:
692:
689:
686:
683:
680:
677:
674:
671:
668:
663:
659:
633:
620:tangent bundle
595:
592:
590:
587:
542:
539:
492:
491:
478:
475:
472:
469:
466:
462:
458:
454:
450:
447:
444:
441:
437:
433:
417:
416:
401:
398:
393:
389:
385:
382:
379:
374:
370:
366:
361:
357:
353:
350:
348:
343:
340:
333:
329:
325:
319:
318:
315:
312:
310:
308:
305:
300:
296:
292:
289:
286:
281:
277:
273:
268:
264:
260:
257:
255:
250:
247:
240:
236:
232:
226:
225:
203:integral curve
184:
173:
162:
141:
134:
127:
104:
101:
67:velocity field
57:magnetic field
53:electric field
44:
41:
29:integral curve
18:Curve integral
9:
6:
4:
3:
2:
2420:
2409:
2406:
2404:
2401:
2400:
2398:
2383:
2380:
2378:
2377:Supermanifold
2375:
2373:
2370:
2368:
2365:
2361:
2358:
2357:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2322:
2320:
2316:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2281:
2279:
2275:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2216:
2214:
2212:
2208:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2168:
2164:
2162:
2159:
2157:
2154:
2152:
2148:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2113:
2111:
2109:
2105:
2099:
2098:Wedge product
2096:
2094:
2091:
2087:
2084:
2083:
2082:
2079:
2077:
2074:
2070:
2067:
2066:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2043:
2042:Vector-valued
2040:
2039:
2038:
2035:
2033:
2030:
2026:
2023:
2022:
2021:
2018:
2016:
2013:
2011:
2008:
2007:
2005:
2001:
1995:
1992:
1990:
1987:
1985:
1982:
1978:
1975:
1974:
1973:
1972:Tangent space
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1954:
1952:
1948:
1945:
1943:
1939:
1933:
1930:
1928:
1924:
1920:
1918:
1915:
1913:
1909:
1905:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1862:
1859:
1858:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1812:
1809:
1807:
1803:
1799:
1797:
1793:
1789:
1787:
1784:
1783:
1781:
1775:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1737:
1736:in Lie theory
1734:
1733:
1732:
1729:
1727:
1724:
1720:
1717:
1716:
1715:
1712:
1710:
1707:
1706:
1704:
1702:
1698:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1648:
1646:
1643:
1639:Main results
1637:
1631:
1628:
1626:
1623:
1621:
1620:Tangent space
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1589:
1586:
1584:
1581:
1580:
1579:
1576:
1572:
1569:
1568:
1567:
1564:
1563:
1561:
1557:
1552:
1548:
1541:
1536:
1534:
1529:
1527:
1522:
1521:
1518:
1510:
1506:
1502:
1501:
1495:
1493:
1490: ∈
1489:
1485:
1481:
1477:
1468:
1464:
1461:
1457:
1449:
1445:
1444:
1443:
1441:
1437:
1430:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1380:
1376:
1372:
1368:
1363:
1361:
1356:
1352:
1345:
1326:
1322:
1315:
1303:
1299:
1287:
1284:
1281:
1275:
1263:
1259:
1246:
1238:
1237:
1236:
1234:
1230:
1226:
1207:
1204:
1196:
1190:
1180:
1174:
1171:
1162:
1157:
1140:
1139:
1138:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1102:
1097:
1093:
1086:
1082:
1078:
1058:
1049:
1043:
1037:
1034:
1028:
1021:
1018:
1010:
995:
992:
989:
981:
977:
970:
963:
962:
961:
959:
952:
948:
944:
940:
916:
913:
910:
907:
891:
885:
879:
876:
870:
863:
860:
852:
837:
834:
831:
823:
819:
812:
805:
804:
803:
798:
794:
791:
787:
784:
783:open interval
780:
776:
772:
768:
761:
757:
753:
749:
745:
741:
737:
733:
729:
725:
721:
717:
713:
712:cross-section
709:
690:
687:
678:
675:
672:
666:
661:
657:
649:
648:
647:
645:
641:
636:
632:
629:
625:
621:
617:
613:
609:
605:
601:
586:
584:
580:
576:
568:
564:
561: =
560:
557: /
556:
552:
547:
538:
536:
532:
527:
525:
521:
517:
513:
509:
505:
501:
497:
476:
467:
448:
442:
435:
422:
421:
420:
399:
391:
387:
383:
380:
377:
372:
368:
359:
355:
351:
349:
341:
338:
331:
327:
323:
313:
311:
298:
294:
290:
287:
284:
279:
275:
266:
262:
258:
256:
248:
245:
238:
234:
230:
216:
215:
214:
212:
208:
204:
200:
196:
192:
187:
183:
179:
172:
168:
161:
157:
153:
149:
144:
140:
133:
126:
122:
118:
115:, that is, a
114:
110:
107:Suppose that
100:
98:
97:
92:
91:
86:
82:
78:
77:
73:are known as
72:
68:
64:
63:
59:are known as
58:
54:
50:
40:
38:
34:
30:
26:
19:
2304:Moving frame
2299:Morse theory
2289:Gauge theory
2081:Tensor field
2010:Closed/Exact
1989:Vector field
1957:Distribution
1898:Hypercomplex
1893:Quaternionic
1752:
1630:Vector field
1588:Smooth atlas
1508:
1491:
1487:
1483:
1479:
1475:
1466:
1462:
1455:
1453:
1439:
1435:
1428:
1421:
1420:. The curve
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1385:
1381:
1374:
1370:
1367:induced maps
1364:
1359:
1354:
1350:
1343:
1341:
1232:
1224:
1222:
1130:
1126:
1122:
1118:
1114:
1110:
1108:
1095:
1091:
1084:
1080:
1076:
1074:
957:
950:
946:
942:
938:
936:
802:, such that
796:
792:
785:
778:
774:
770:
766:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
719:
715:
707:
705:
643:
639:
634:
630:
623:
618:denotes the
615:
611:
607:
599:
597:
575:vector field
572:
566:
562:
558:
554:
528:
523:
519:
515:
511:
507:
503:
499:
495:
493:
418:
206:
202:
198:
194:
190:
185:
181:
177:
170:
166:
159:
151:
147:
146:), and that
142:
138:
131:
124:
113:vector field
111:is a static
108:
106:
95:
90:trajectories
89:
75:
60:
46:
28:
22:
2249:Levi-Civita
2239:Generalized
2211:Connections
2161:Lie algebra
2093:Volume form
1994:Vector flow
1967:Pushforward
1962:Lie bracket
1861:Lie algebra
1826:G-structure
1615:Pushforward
1595:Submanifold
1505:Lang, Serge
1460:composition
1229:open subset
1090:(let alone
795:containing
765:is a curve
579:slope field
551:slope field
533:, then the
76:streamlines
62:field lines
25:mathematics
2397:Categories
2372:Stratifold
2330:Diffeology
2126:Associated
1927:Symplectic
1912:Riemannian
1841:Hyperbolic
1768:Submersion
1676:Hopf–Rinow
1610:Submersion
1605:Smooth map
1498:References
1426:bundle map
1424:induces a
1412:) for all
628:projection
594:Definition
103:Definition
2254:Principal
2229:Ehresmann
2186:Subbundle
2176:Principal
2151:Fibration
2131:Cotangent
2003:Covectors
1856:Lie group
1836:Hermitian
1779:manifolds
1748:Immersion
1743:Foliation
1681:Noether's
1666:Frobenius
1661:De Rham's
1656:Darboux's
1547:Manifolds
1434: : T
1390:canonical
1300:α
1285:…
1260:α
1191:α
1181:∈
1163:α
1044:α
1019:α
971:α
911:∈
886:α
861:α
813:α
790:real line
777:of class
734:of class
685:↦
658:π
646:given by
638: : T
606:of class
381:…
314:⋮
288:…
193:)). Then
2350:Orbifold
2345:K-theory
2335:Diffiety
2059:Pullback
1873:Oriented
1851:Kenmotsu
1831:Hadamard
1777:Types of
1726:Geodesic
1551:Glossary
1507:(1972).
1482:′(
1227:is some
1121:at time
1113:′(
1022:′
949:at time
864:′
769: :
758:at time
738:and let
541:Examples
436:′
201:) is an
2294:History
2277:Related
2191:Tangent
2169:)
2149:)
2116:Adjoint
2108:Bundles
2086:density
1984:Torsion
1950:Vectors
1942:Tensors
1925:)
1910:)
1906:,
1904:Pseudo−
1883:Poisson
1816:Finsler
1811:Fibered
1806:Contact
1804:)
1796:Complex
1794:)
1763:Section
1408:, 1) ∈
1384:×
1377:is the
1349:, ...,
788:of the
583:tangent
154:) is a
49:physics
2259:Vector
2244:Koszul
2224:Cartan
2219:Affine
2201:Vector
2196:Tensor
2181:Spinor
2171:Normal
2167:Stable
2121:Affine
2025:bundle
1977:bundle
1923:Almost
1846:Kähler
1802:Almost
1792:Almost
1786:Closed
1686:Sard's
1642:(list)
1478:, and
1342:where
180:),...,
96:orbits
85:system
2367:Sheaf
2141:Fiber
1917:Rizza
1888:Prime
1719:Local
1709:Curve
1571:Atlas
746:. An
710:is a
610:with
602:be a
137:,...,
119:with
79:. In
71:fluid
69:of a
31:is a
27:, an
2234:Form
2136:Dual
2069:flow
1932:Tame
1908:Sub−
1821:Flat
1701:Maps
750:for
598:Let
43:Name
2156:Jet
1438:→ T
1373:of
1231:of
622:of
577:or
205:of
93:or
55:or
23:In
2399::
2147:Co
1494:.
1416:∈
1137::
1094:≤
1083:≥
773:→
742:∈
642:→
559:dx
555:dy
526:.
169:),
99:.
2165:(
2145:(
1921:(
1902:(
1800:(
1790:(
1553:)
1549:(
1539:e
1532:t
1525:v
1492:J
1488:t
1484:t
1480:α
1476:ι
1473:o
1470:∗
1467:α
1463:α
1456:α
1440:M
1436:J
1432:∗
1429:α
1422:α
1418:J
1414:t
1410:ι
1406:t
1402:t
1400:(
1398:ι
1394:ι
1386:R
1382:J
1375:J
1371:J
1360:α
1355:n
1351:α
1347:1
1344:α
1327:,
1323:)
1316:t
1312:d
1304:n
1295:d
1288:,
1282:,
1276:t
1272:d
1264:1
1255:d
1247:(
1233:R
1225:M
1208:.
1205:M
1200:)
1197:t
1194:(
1186:T
1178:)
1175:1
1172:+
1169:(
1166:)
1158:t
1153:d
1148:(
1131:t
1127:α
1123:t
1119:α
1115:t
1111:α
1099:0
1096:t
1092:t
1088:0
1085:t
1081:t
1077:J
1059:.
1056:)
1053:)
1050:t
1047:(
1041:(
1038:X
1035:=
1032:)
1029:t
1026:(
996:;
993:p
990:=
987:)
982:0
978:t
974:(
958:α
954:0
951:t
947:p
943:X
939:α
917:.
914:J
908:t
898:)
895:)
892:t
889:(
883:(
880:X
877:=
874:)
871:t
868:(
838:;
835:p
832:=
829:)
824:0
820:t
816:(
800:0
797:t
793:R
786:J
779:C
775:M
771:J
767:α
763:0
760:t
756:p
752:X
744:M
740:p
736:C
732:M
728:X
724:M
720:M
716:M
708:M
691:.
688:x
682:)
679:v
676:,
673:x
670:(
667::
662:M
644:M
640:M
635:M
631:π
624:M
616:M
612:r
608:C
600:M
567:x
563:x
524:F
520:t
518:(
516:x
512:t
510:(
508:x
506:(
504:F
500:t
498:(
496:x
477:.
474:)
471:)
468:t
465:(
461:x
457:(
453:F
449:=
446:)
443:t
440:(
432:x
400:.
397:)
392:n
388:x
384:,
378:,
373:1
369:x
365:(
360:n
356:F
352:=
342:t
339:d
332:n
328:x
324:d
304:)
299:n
295:x
291:,
285:,
280:1
276:x
272:(
267:1
263:F
259:=
249:t
246:d
239:1
235:x
231:d
207:F
199:t
197:(
195:x
191:t
189:(
186:n
182:x
178:t
176:(
174:2
171:x
167:t
165:(
163:1
160:x
152:t
150:(
148:x
143:n
139:F
135:2
132:F
130:,
128:1
125:F
123:(
109:F
20:.
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