914:
found six notations the other day!) I may be able to reconstruct a meaning for the nabla combinations used in this article, but that is not how it should be. At very least, if the
Knowledge is to be taken over by GR enthusiasts (but thank you for a formidable work anyway), there should be an easy way for the reader to know which link to click to find an explanation of the notation used. How do you google the "juxtaposition operator which sometimes is used to express multiplication of real and complex numbers, but often means functional application" to learn what it means in the case of a nabla next to a vector? Many readers don't even know the name "nabla", even if they know the three (or four) standard uses of nabla to indicate gradient, divergence, and curl (and Laplace).
84:
74:
53:
22:
508:
1153:
353:
1161:
It seems to me the left hands side shall be expressed in a more feasible form with the right hand side. So the question is how do I end at the right hand side? It might be possible that it is mathematically triggered but I think it reminds me about the derivation of
Maxwell equations. As a model one
358:
972:
1)The MAIN point is to understand that the "divergence" of the stress tensor should yield a force. But this step is skipped and instead some wordy material about volume elements is put in that looks like high-school motivation of an integral. In other words, the already-known stuff is explained and
913:
I have by chance read about tensors and general relativity, but not yet found the nabla symbol used as here. (There seems to be no end to the variants of the tensor language, in how many ways can you write the directional derivative of a scalar field in the direction of a coordinate basis vector? I
177:
There is an obvious relation between the Cauchy momentum equation and Euler's equation that should be mentioned: The latter is a special case of the former and considers the case of pressure and not the general form of a stress tensor, but it also predates Cauchy's equation by many decades: Cauchy
1892:
162:
This highly technical article would benefit from some non-technical explanation, such as a paragraph or two at the top that explains the history, importance, and uses of the equation. The lead section would benefit from some elaboration on what makes the equation notable and a few sentences that
1192:
molecules (we are dealing with fluids right?) or units cells to speak more generally on which the forces can act individually. Here instead of dealing with finite sums we will already go for the limit using integrals, that's alright. So we might break the forces per unit volume:
1006:
1754:
917:
Besides the use of clever nabla shorthands, there is a lack of context. Since the material derivative is used, I understand that the article is written from the perspective of fluid mechanics, where the system is a fixed volume through which some fluid flows. The
978:
2) The long-winded coordinate expressions are not properly part of the derivation. So I labelled them into a new section. As the article stands, these formulas take up way too much space, but if the article gets more filled out, maybe there is a place for them.
892:
I think this article is unhelpful for most readers. Does everybody learn tensor algebra nowadays? I am quite certain Cauchy did not know about tensors, and it should be possible to write something useful for readers that only know the stress tensor as a matrix.
897:
I agree that the article is bad. I inserted the component definitions of the index-less nabla expressions. The index-less ones are useful once you get used to them, but not before. (I face these things constantly and I prefer to express everything both ways.)
1265:
1963:
870:
There is, of course, no difficulty in using components of any other type, but in this particular case we can cheat and avoid explaining to the reader what a metric tensor is and why covariant derivatives of its components are zero. β
503:{\displaystyle \mathbf {s} =\rho \mathbf {g} ={\frac {\partial (\rho \mathbf {u} )}{\partial t}}+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} -\sigma )={\frac {\partial (\mathbf {j} )}{\partial t}}+\nabla \cdot (\mathbf {F} )}
999:
To me the derivation is not very convenient from the physical point of view. Perhaps it is easy to cope with for the ones really understanding covariant notation and stuff like that, but I tried to figure out the essence of what
888:
I have been unable to find any definition of the "del" of a vector or the "divergence" of a tensor -- or whatever you call the juxtaposition of the nabla symbol and a vector, and the "dot product" of the nabla symbol and a tensor.
251:
1343:
and a second part which is a little bit more tricky (and actually I am a bit uncertain concerning this part and would appreciate some encouraging discussion) is due to some surface forces excerted on the control volume's surface
256:
1790:
1544:
859:
2029:
The link on the word "continuum" misleadingly sends the reader to the
Knowledge article about the continuum as the set of real numbers. This is clearly not the right interpretation of the word in this context.
1485:
1148:{\displaystyle \rho \int \limits _{\Omega }{\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}\,\mathrm {d} V=\int \limits _{\Omega }\nabla _{j}\sigma _{i}^{j}\,\mathrm {d} V+\int \limits _{\Omega }f_{i}\,\mathrm {d} V}
1338:
1618:
730:. I replaced partial derivatives of the tension tensor with the covariant ones. I decided to use mixed components so that later stress tensor components could be decomposed in a straightforward way:
558:
779:
2093:
2129:
1672:
185:
140:
673:
1549:
The following conclusion made in the article that the control volume is arbitrary is fine, too, but I'm not sure how to end at the final formula for the whole system without subscript
631:
708:
It's true, this equation is technically correct only in the
Cartesian coordinate system. I'll edit it into something more plausible, but this article needs an expert's attention. β
584:
1782:
1392:
1199:
1362:
925:
Finally, there is an error in the derivation. The density belongs with the acceleration, not outside the integral where it would be multiplied with all the terms. The "
1190:
950:
1664:
2003:
1983:
1638:
1567:
1432:
1412:
1288:
1900:
348:{\displaystyle \nabla \cdot \sigma +\rho \mathbf {g} ={\frac {\partial (\rho \mathbf {u} )}{\partial t}}+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} )}
207:
2159:
130:
1887:{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\partial v}{\partial t}}+{\frac {\partial v}{\partial r}}{\frac {\partial r}{\partial t}}}
2154:
2056:
In this section after using assuming stress tensor as isotropic, it is mentioned that in the steady incompressible case the mass equation is simply
106:
1496:
788:
2059:
1440:
2037:
980:
899:
680:
189:
97:
58:
2098:
1296:
1578:
530:
1749:{\displaystyle \mathrm {d} v={\frac {\partial v}{\partial t}}\mathrm {d} t+{\frac {\partial v}{\partial r}}\mathrm {d} r}
742:
33:
1270:
down into a part which orginates form a global force field e.g. gravitational forces which acts on every subdivision
2136:
636:
1572:
But the very critical part is the stress tensor and its components. Can anybody shed a bit more light on that?
21:
2041:
876:
713:
589:
515:
563:
2132:
984:
903:
684:
39:
511:
83:
2033:
1260:{\displaystyle \rho \int \limits _{\Omega }{\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}\,\mathrm {d} V}
952:" that is mentioned in the very last statement has not been named or explained anywhere. What is it?
181:
168:
1762:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
2009:
1367:
957:
872:
709:
89:
204:
Perhaps I've made a mistake in reading this article, but it seems as if the flux term should be
73:
52:
698:
2013:
1347:
1165:
928:
164:
1958:{\displaystyle {\tfrac {\partial r}{\partial t}}={\tfrac {\mathrm {d} r}{\mathrm {d} t}}}
1643:
1988:
1968:
1623:
1552:
1417:
1397:
1273:
2148:
953:
246:{\displaystyle \mathbf {F} =\rho \mathbf {u} \otimes \mathbf {u} -\mathbf {\sigma } }
2140:
2045:
2017:
988:
961:
907:
880:
717:
702:
688:
519:
193:
172:
694:
1620:? I think I figured out the mathematics behind that since the total derivative of
2131:
It should be written that for a incompressible case gradient of density is zero.
102:
79:
1575:
And can anyone explain me why one uses the partial derivatives instead of
524:
The tensor derivative needs a direction. (no signature. Who posted this?)
1539:{\displaystyle \int \limits _{\Omega }\nabla \sigma _{ij}\,\mathrm {d} V}
253:, which comes from the more general form of the linear momentum equation
854:{\displaystyle \sigma _{i}^{j}=-p\delta _{i}^{j}+\mathbb {T} _{i}^{j}}
178:
was born 35 years after Euler wrote his version of the equation.
1480:{\displaystyle \int \limits _{\Sigma }\sigma _{ij}\,\mathrm {d} S}
2095:
however for a steady incompressible case the mass equation is
15:
1333:{\displaystyle \int \limits _{\Omega }f_{i}\,\mathrm {d} V}
922:
is not just body force, it is body force per unit volume.
726:
I edited the
Derivation section, it should make sense now,
1613:{\displaystyle {\tfrac {\mathrm {d} v}{\mathrm {d} t}}\,}
163:
summarize the significance of the extensive mathematics.
1930:
1905:
1583:
2101:
2062:
1991:
1971:
1903:
1793:
1765:
1675:
1646:
1626:
1581:
1555:
1499:
1443:
1434:
th direction (<- uncertain!) in the following way
1420:
1400:
1370:
1350:
1299:
1276:
1202:
1168:
1009:
931:
791:
745:
639:
592:
566:
553:{\displaystyle \mathbf {v} \cdot \nabla \mathbf {v} }
533:
361:
259:
210:
101:, a collaborative effort to improve the coverage of
774:{\displaystyle \sigma =-p\mathbb {I} +\mathbb {T} }
2123:
2088:{\displaystyle \mathbf {u} \cdot \nabla \rho =0\,}
2087:
1997:
1977:
1957:
1886:
1776:
1748:
1658:
1632:
1612:
1561:
1538:
1479:
1426:
1406:
1386:
1356:
1332:
1282:
1259:
1184:
1162:has this control volume comprising of N different
1147:
944:
853:
773:
667:
625:
578:
552:
502:
347:
245:
1490:and using GaussβOstrogradsky theorem one finds
1364:which might be described by the stress tensor
2124:{\displaystyle \mathbf {\nabla } \cdot u=0\,}
8:
668:{\displaystyle \partial _{j}\sigma _{j}^{i}}
19:
179:
47:
2102:
2100:
2063:
2061:
1990:
1970:
1943:
1933:
1929:
1904:
1902:
1864:
1844:
1821:
1807:
1797:
1794:
1792:
1766:
1764:
1738:
1718:
1707:
1687:
1676:
1674:
1645:
1625:
1596:
1586:
1582:
1580:
1554:
1528:
1517:
1504:
1498:
1469:
1458:
1448:
1442:
1419:
1399:
1375:
1369:
1349:
1322:
1314:
1304:
1298:
1275:
1249:
1236:
1228:
1219:
1216:
1210:
1201:
1173:
1167:
1137:
1129:
1119:
1104:
1096:
1091:
1081:
1071:
1056:
1043:
1035:
1026:
1023:
1017:
1008:
936:
930:
845:
840:
836:
835:
825:
820:
801:
796:
790:
767:
766:
759:
758:
744:
659:
654:
644:
638:
617:
607:
597:
591:
565:
545:
534:
532:
492:
461:
452:
435:
427:
393:
381:
373:
362:
360:
337:
329:
295:
283:
275:
258:
238:
230:
222:
211:
209:
2119:
2083:
1608:
1526:
1467:
1320:
1247:
1135:
1102:
1054:
626:{\displaystyle \partial _{j}v^{i}v^{j}}
49:
7:
579:{\displaystyle \nabla \cdot \sigma }
186:2003:CB:727:B300:C62B:44FF:FE44:451F
95:This article is within the scope of
38:It is of interest to the following
2103:
2071:
1944:
1934:
1916:
1908:
1875:
1867:
1855:
1847:
1832:
1824:
1808:
1798:
1767:
1739:
1729:
1721:
1708:
1698:
1690:
1677:
1597:
1587:
1529:
1510:
1505:
1470:
1449:
1394:which describes the stress on the
1351:
1323:
1305:
1250:
1237:
1220:
1211:
1138:
1120:
1105:
1078:
1072:
1057:
1044:
1027:
1018:
641:
594:
567:
542:
483:
471:
455:
415:
403:
384:
317:
305:
286:
260:
14:
2160:Low-priority mathematics articles
115:Knowledge:WikiProject Mathematics
2155:Start-Class mathematics articles
2064:
546:
535:
527:Furthermore, it is unclear what
493:
462:
436:
428:
394:
374:
363:
338:
330:
296:
276:
231:
223:
212:
118:Template:WikiProject Mathematics
82:
72:
51:
20:
973:the essential new part skipped.
967:I agree, the derivation is bad.
135:This article has been rated as
497:
489:
466:
458:
446:
421:
398:
387:
342:
323:
300:
289:
1:
1777:{\displaystyle \mathrm {d} t}
586:mean. My interpretations are
355:that can be rearranged to be
109:and see a list of open tasks.
1387:{\displaystyle \sigma _{ij}}
989:16:11, 8 November 2014 (UTC)
962:00:40, 3 December 2009 (UTC)
908:16:11, 8 November 2014 (UTC)
703:09:12, 4 February 2009 (UTC)
689:16:14, 8 November 2014 (UTC)
520:03:21, 21 October 2016 (UTC)
194:10:06, 5 December 2023 (UTC)
2176:
2141:11:57, 28 July 2023 (UTC)
2046:19:53, 24 June 2011 (UTC)
2018:19:30, 10 June 2010 (UTC)
881:12:52, 30 July 2009 (UTC)
718:10:09, 30 July 2009 (UTC)
173:02:04, 28 June 2020 (UTC)
134:
67:
46:
679:Correct. I put this in.
141:project's priority scale
1640:which is a function of
1357:{\displaystyle \Sigma }
98:WikiProject Mathematics
2125:
2089:
1999:
1985:is just a function of
1979:
1959:
1888:
1778:
1750:
1660:
1634:
1614:
1563:
1540:
1481:
1428:
1408:
1388:
1358:
1334:
1284:
1261:
1186:
1185:{\displaystyle H_{2}0}
1149:
946:
855:
775:
669:
627:
580:
554:
504:
349:
247:
28:This article is rated
2126:
2090:
2000:
1980:
1960:
1889:
1779:
1751:
1661:
1635:
1615:
1564:
1541:
1482:
1429:
1409:
1389:
1359:
1335:
1285:
1262:
1187:
1150:
947:
945:{\displaystyle F_{i}}
856:
776:
670:
628:
581:
555:
505:
350:
248:
2099:
2060:
1989:
1969:
1901:
1791:
1763:
1673:
1644:
1624:
1579:
1553:
1497:
1441:
1418:
1398:
1368:
1348:
1297:
1274:
1200:
1166:
1007:
929:
789:
743:
637:
590:
564:
531:
359:
257:
208:
121:mathematics articles
1659:{\displaystyle r,t}
1158:should to tell me:
1101:
995:Physical Background
850:
830:
806:
664:
2121:
2120:
2085:
2084:
1995:
1975:
1955:
1953:
1924:
1884:
1774:
1746:
1656:
1630:
1610:
1609:
1606:
1559:
1536:
1527:
1509:
1477:
1468:
1453:
1424:
1414:th subunit in the
1404:
1384:
1354:
1330:
1321:
1309:
1280:
1257:
1248:
1215:
1182:
1145:
1136:
1124:
1103:
1087:
1076:
1055:
1022:
942:
851:
834:
816:
792:
771:
665:
650:
623:
576:
550:
500:
345:
243:
90:Mathematics portal
34:content assessment
2052:Lamb form section
2036:comment added by
1998:{\displaystyle t}
1978:{\displaystyle r}
1952:
1923:
1882:
1862:
1839:
1816:
1759:Division through
1736:
1705:
1633:{\displaystyle v}
1605:
1562:{\displaystyle i}
1500:
1444:
1427:{\displaystyle j}
1407:{\displaystyle i}
1300:
1283:{\displaystyle i}
1245:
1206:
1115:
1067:
1052:
1013:
478:
410:
312:
196:
184:comment added by
155:
154:
151:
150:
147:
146:
2167:
2133:Navin123explorer
2130:
2128:
2127:
2122:
2106:
2094:
2092:
2091:
2086:
2067:
2048:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1964:
1962:
1961:
1956:
1954:
1951:
1947:
1941:
1937:
1931:
1925:
1922:
1914:
1906:
1893:
1891:
1890:
1885:
1883:
1881:
1873:
1865:
1863:
1861:
1853:
1845:
1840:
1838:
1830:
1822:
1817:
1815:
1811:
1805:
1801:
1795:
1783:
1781:
1780:
1775:
1770:
1755:
1753:
1752:
1747:
1742:
1737:
1735:
1727:
1719:
1711:
1706:
1704:
1696:
1688:
1680:
1665:
1663:
1662:
1657:
1639:
1637:
1636:
1631:
1619:
1617:
1616:
1611:
1607:
1604:
1600:
1594:
1590:
1584:
1568:
1566:
1565:
1560:
1545:
1543:
1542:
1537:
1532:
1525:
1524:
1508:
1486:
1484:
1483:
1478:
1473:
1466:
1465:
1452:
1433:
1431:
1430:
1425:
1413:
1411:
1410:
1405:
1393:
1391:
1390:
1385:
1383:
1382:
1363:
1361:
1360:
1355:
1339:
1337:
1336:
1331:
1326:
1319:
1318:
1308:
1289:
1287:
1286:
1281:
1266:
1264:
1263:
1258:
1253:
1246:
1244:
1240:
1234:
1233:
1232:
1223:
1217:
1214:
1191:
1189:
1188:
1183:
1178:
1177:
1154:
1152:
1151:
1146:
1141:
1134:
1133:
1123:
1108:
1100:
1095:
1086:
1085:
1075:
1060:
1053:
1051:
1047:
1041:
1040:
1039:
1030:
1024:
1021:
951:
949:
948:
943:
941:
940:
860:
858:
857:
852:
849:
844:
839:
829:
824:
805:
800:
780:
778:
777:
772:
770:
762:
675:, respectively.
674:
672:
671:
666:
663:
658:
649:
648:
632:
630:
629:
624:
622:
621:
612:
611:
602:
601:
585:
583:
582:
577:
559:
557:
556:
551:
549:
538:
509:
507:
506:
501:
496:
479:
477:
469:
465:
453:
439:
431:
411:
409:
401:
397:
382:
377:
366:
354:
352:
351:
346:
341:
333:
313:
311:
303:
299:
284:
279:
252:
250:
249:
244:
242:
234:
226:
215:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
2175:
2174:
2170:
2169:
2168:
2166:
2165:
2164:
2145:
2144:
2097:
2096:
2058:
2057:
2054:
2031:
2027:
2025:Misleading link
1987:
1986:
1967:
1966:
1942:
1932:
1915:
1907:
1899:
1898:
1874:
1866:
1854:
1846:
1831:
1823:
1806:
1796:
1789:
1788:
1761:
1760:
1728:
1720:
1697:
1689:
1671:
1670:
1642:
1641:
1622:
1621:
1595:
1585:
1577:
1576:
1551:
1550:
1513:
1495:
1494:
1454:
1439:
1438:
1416:
1415:
1396:
1395:
1371:
1366:
1365:
1346:
1345:
1310:
1295:
1294:
1272:
1271:
1235:
1224:
1218:
1198:
1197:
1169:
1164:
1163:
1125:
1077:
1042:
1031:
1025:
1005:
1004:
997:
932:
927:
926:
861:
787:
786:
781:
741:
740:
640:
635:
634:
613:
603:
593:
588:
587:
562:
561:
529:
528:
470:
454:
402:
383:
357:
356:
304:
285:
255:
254:
206:
205:
202:
200:Sloppy Notation
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
2173:
2171:
2163:
2162:
2157:
2147:
2146:
2118:
2115:
2112:
2109:
2105:
2082:
2079:
2076:
2073:
2070:
2066:
2053:
2050:
2026:
2023:
2022:
2021:
1994:
1974:
1950:
1946:
1940:
1936:
1928:
1921:
1918:
1913:
1910:
1895:
1894:
1880:
1877:
1872:
1869:
1860:
1857:
1852:
1849:
1843:
1837:
1834:
1829:
1826:
1820:
1814:
1810:
1804:
1800:
1773:
1769:
1757:
1756:
1745:
1741:
1734:
1731:
1726:
1723:
1717:
1714:
1710:
1703:
1700:
1695:
1692:
1686:
1683:
1679:
1655:
1652:
1649:
1629:
1603:
1599:
1593:
1589:
1558:
1547:
1546:
1535:
1531:
1523:
1520:
1516:
1512:
1507:
1503:
1488:
1487:
1476:
1472:
1464:
1461:
1457:
1451:
1447:
1423:
1403:
1381:
1378:
1374:
1353:
1341:
1340:
1329:
1325:
1317:
1313:
1307:
1303:
1279:
1268:
1267:
1256:
1252:
1243:
1239:
1231:
1227:
1222:
1213:
1209:
1205:
1181:
1176:
1172:
1156:
1155:
1144:
1140:
1132:
1128:
1122:
1118:
1114:
1111:
1107:
1099:
1094:
1090:
1084:
1080:
1074:
1070:
1066:
1063:
1059:
1050:
1046:
1038:
1034:
1029:
1020:
1016:
1012:
996:
993:
992:
991:
975:
974:
969:
968:
939:
935:
911:
910:
886:
885:
884:
883:
865:
864:
863:
862:
848:
843:
838:
833:
828:
823:
819:
815:
812:
809:
804:
799:
795:
785:
783:
782:
769:
765:
761:
757:
754:
751:
748:
739:
734:
733:
732:
731:
728:mathematically
721:
720:
692:
691:
662:
657:
653:
647:
643:
620:
616:
610:
606:
600:
596:
575:
572:
569:
548:
544:
541:
537:
499:
495:
491:
488:
485:
482:
476:
473:
468:
464:
460:
457:
451:
448:
445:
442:
438:
434:
430:
426:
423:
420:
417:
414:
408:
405:
400:
396:
392:
389:
386:
380:
376:
372:
369:
365:
344:
340:
336:
332:
328:
325:
322:
319:
316:
310:
307:
302:
298:
294:
291:
288:
282:
278:
274:
271:
268:
265:
262:
241:
237:
233:
229:
225:
221:
218:
214:
201:
198:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2172:
2161:
2158:
2156:
2153:
2152:
2150:
2143:
2142:
2138:
2134:
2116:
2113:
2110:
2107:
2080:
2077:
2074:
2068:
2051:
2049:
2047:
2043:
2039:
2035:
2024:
2019:
2015:
2011:
2007:
2006:
2005:
1992:
1972:
1948:
1938:
1926:
1919:
1911:
1878:
1870:
1858:
1850:
1841:
1835:
1827:
1818:
1812:
1802:
1787:
1786:
1785:
1771:
1743:
1732:
1724:
1715:
1712:
1701:
1693:
1684:
1681:
1669:
1668:
1667:
1653:
1650:
1647:
1627:
1601:
1591:
1573:
1570:
1556:
1533:
1521:
1518:
1514:
1501:
1493:
1492:
1491:
1474:
1462:
1459:
1455:
1445:
1437:
1436:
1435:
1421:
1401:
1379:
1376:
1372:
1327:
1315:
1311:
1301:
1293:
1292:
1291:
1277:
1254:
1241:
1229:
1225:
1207:
1203:
1196:
1195:
1194:
1179:
1174:
1170:
1159:
1142:
1130:
1126:
1116:
1112:
1109:
1097:
1092:
1088:
1082:
1068:
1064:
1061:
1048:
1036:
1032:
1014:
1010:
1003:
1002:
1001:
994:
990:
986:
982:
977:
976:
971:
970:
966:
965:
964:
963:
959:
955:
937:
933:
923:
921:
915:
909:
905:
901:
896:
895:
894:
890:
882:
878:
874:
873:Kallikanzarid
869:
868:
867:
866:
846:
841:
831:
826:
821:
817:
813:
810:
807:
802:
797:
793:
784:
763:
755:
752:
749:
746:
738:
737:
736:
735:
729:
725:
724:
723:
722:
719:
715:
711:
710:Kallikanzarid
707:
706:
705:
704:
700:
696:
690:
686:
682:
678:
677:
676:
660:
655:
651:
645:
618:
614:
608:
604:
598:
573:
570:
539:
525:
522:
521:
517:
513:
486:
480:
474:
449:
443:
440:
432:
424:
418:
412:
406:
390:
378:
370:
367:
334:
326:
320:
314:
308:
292:
280:
272:
269:
266:
263:
239:
235:
227:
219:
216:
199:
197:
195:
191:
187:
183:
175:
174:
170:
166:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2055:
2038:66.207.95.54
2032:β Preceding
2028:
1896:
1758:
1574:
1571:
1548:
1489:
1342:
1269:
1160:
1157:
998:
924:
919:
916:
912:
891:
887:
727:
693:
526:
523:
512:AndrewWinter
203:
180:βΒ Preceding
176:
161:
158:Lead Section
137:Low-priority
136:
96:
62:Lowβpriority
40:WikiProjects
981:89.217.22.3
900:89.217.22.3
681:89.217.22.3
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2149:Categories
165:Altair2013
2034:unsigned
1965:because
1784:yields
954:Cacadril
182:unsigned
1569:then.
695:Etoombs
139:on the
2010:MArras
1897:since
36:scale.
2137:talk
2042:talk
2014:talk
985:talk
958:talk
904:talk
877:talk
714:talk
699:talk
685:talk
633:and
560:and
516:talk
190:talk
169:talk
1666:is
131:Low
2151::
2139:)
2108:β
2104:β
2075:Ο
2072:β
2069:β
2044:)
2016:)
1917:β
1909:β
1876:β
1868:β
1856:β
1848:β
1833:β
1825:β
1730:β
1722:β
1699:β
1691:β
1515:Ο
1511:β
1506:Ξ©
1502:β«
1456:Ο
1450:Ξ£
1446:β«
1373:Ο
1352:Ξ£
1306:Ξ©
1302:β«
1290::
1212:Ξ©
1208:β«
1204:Ο
1121:Ξ©
1117:β«
1089:Ο
1079:β
1073:Ξ©
1069:β«
1019:Ξ©
1015:β«
1011:Ο
987:)
960:)
906:)
879:)
818:Ξ΄
811:β
794:Ο
753:β
747:Ο
716:)
701:)
687:)
652:Ο
642:β
595:β
574:Ο
571:β
568:β
543:β
540:β
518:)
510:.
487:β
484:β
472:β
456:β
444:Ο
441:β
433:β
425:Ο
419:β
416:β
404:β
391:Ο
385:β
371:Ο
335:β
327:Ο
321:β
318:β
306:β
293:Ο
287:β
273:Ο
267:Ο
264:β
261:β
240:Ο
236:β
228:β
220:Ο
192:)
171:)
2135:(
2117:0
2114:=
2111:u
2081:0
2078:=
2065:u
2040:(
2020:)
2012:(
2008:(
1993:t
1973:r
1949:t
1945:d
1939:r
1935:d
1927:=
1920:t
1912:r
1879:t
1871:r
1859:r
1851:v
1842:+
1836:t
1828:v
1819:=
1813:t
1809:d
1803:v
1799:d
1772:t
1768:d
1744:r
1740:d
1733:r
1725:v
1716:+
1713:t
1709:d
1702:t
1694:v
1685:=
1682:v
1678:d
1654:t
1651:,
1648:r
1628:v
1602:t
1598:d
1592:v
1588:d
1557:i
1534:V
1530:d
1522:j
1519:i
1475:S
1471:d
1463:j
1460:i
1422:j
1402:i
1380:j
1377:i
1328:V
1324:d
1316:i
1312:f
1278:i
1255:V
1251:d
1242:t
1238:d
1230:i
1226:u
1221:d
1180:0
1175:2
1171:H
1143:V
1139:d
1131:i
1127:f
1113:+
1110:V
1106:d
1098:j
1093:i
1083:j
1065:=
1062:V
1058:d
1049:t
1045:d
1037:i
1033:u
1028:d
983:(
956:(
938:i
934:F
920:f
902:(
875:(
847:j
842:i
837:T
832:+
827:j
822:i
814:p
808:=
803:j
798:i
768:T
764:+
760:I
756:p
750:=
712:(
697:(
683:(
661:i
656:j
646:j
619:j
615:v
609:i
605:v
599:j
547:v
536:v
514:(
498:)
494:F
490:(
481:+
475:t
467:)
463:j
459:(
450:=
447:)
437:u
429:u
422:(
413:+
407:t
399:)
395:u
388:(
379:=
375:g
368:=
364:s
343:)
339:u
331:u
324:(
315:+
309:t
301:)
297:u
290:(
281:=
277:g
270:+
232:u
224:u
217:=
213:F
188:(
167:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
