466:
25:
175:
461:{\displaystyle {\frac {d}{dt}}\int _{a\left(t\right)}^{b\left(t\right)}f\left(t,x\right)dx=\int _{a\left(t\right)}^{b\left(t\right)}{\frac {\partial f\left(t,x\right)}{\partial t}}dx+f\left(t,b\left(t\right)\right)b^{\prime }\left(t\right)-f\left(t,a\left(t\right)\right)a^{\prime }\left(t\right)}
1702:
1416:
688:
913:
1200:
1551:
575:
785:
1280:
1405:
1348:
565:
1458:
1235:
1043:
999:
951:
1540:
774:
1107:
1049:
while the second corrects for expanding or shrinking area. The fact that mean curvature represents the rate of change in area follows from applying the above equation to
1073:
1484:
1697:{\displaystyle {\frac {d}{dt}}\int _{S}F\,dS=\int _{S}{\frac {\delta F}{\delta t}}\,dS-\int _{S}CB_{\alpha }^{\alpha }F\,dS+\int _{\gamma }c\,d\gamma }
1115:
162:
35:
93:
65:
1419:
Illustration for the law for surface integrals with a moving contour. Change in area comes from two sources: expansion by curvature
484:
683:{\displaystyle {\frac {d}{dt}}\int _{\Omega }F\,d\Omega =\int _{\Omega }{\frac {\partial F}{\partial t}}\,d\Omega +\int _{S}CF\,dS}
72:
718:
698:
166:
1758:
79:
50:
908:{\displaystyle {\frac {d}{dt}}\int _{S}F\,dS=\int _{S}{\frac {\delta F}{\delta t}}\,dS-\int _{S}CB_{\alpha }^{\alpha }F\,dS}
61:
962:
706:
472:
1763:
1247:
1494:
is a moving surface with a moving contour γ. Suppose that the velocity of the contour γ with respect to
1009:
need not be expression with respect to the exterior normal, as long as the choice of the normal is consistent for
1361:
1304:
533:
1422:
524:
1208:
1016:
972:
958:
921:
139:
86:
115:
1726:
Grinfeld, P. (2010). "Hamiltonian
Dynamic Equations for Fluid Films". Studies in Applied Mathematics.
1707:
The last term captures the change in area due to annexation, as the figure on the right illustrates.
508:
1508:
742:
1286:
1078:
488:
512:
1739:
42:
1052:
1735:
1415:
1727:
1290:
966:
734:
127:
480:
158:
123:
1463:
714:
1752:
1731:
142:
of a particular parameter. In physical applications, that parameter is frequently
1195:{\displaystyle {\frac {d}{dt}}\int _{S}\,dS=-\int _{S}CB_{\alpha }^{\alpha }\,dS}
487:, curved surfaces, including integrals over curved surfaces with moving contour
24:
954:
730:
119:
1002:
504:
135:
528:
476:
131:
527:
field defined in the interior of Ω. Then the rate of change of the
1045:. The first term in the above equation captures the rate of change in
1351:
1294:
1414:
157:
The rate of change of one-dimensional integrals with sufficiently
143:
18:
442:
385:
1502:. Then the rate of change of the time dependent integral:
717:. This law can be considered as the generalization of the
46:
1554:
1511:
1466:
1425:
1364:
1307:
1250:
1211:
1118:
1081:
1055:
1019:
975:
924:
788:
745:
578:
536:
178:
118:, in many applications, one needs to calculate the
1696:
1534:
1478:
1452:
1399:
1342:
1274:
1229:
1194:
1101:
1067:
1037:
993:
945:
907:
768:
682:
559:
460:
1411:Surface integrals with moving contour boundaries
713:must be expressed with respect to the exterior
1275:{\displaystyle C\equiv B_{\alpha }^{\alpha }}
1205:The above equation shows that mean curvature
8:
51:introducing citations to additional sources
1722:
1720:
1400:{\displaystyle B_{\alpha }^{\alpha }=-1/R}
1343:{\displaystyle B_{\alpha }^{\alpha }=-2/R}
560:{\displaystyle \int _{\Omega }F\,d\Omega }
16:Change of time of the value of an integral
1687:
1678:
1664:
1655:
1650:
1637:
1623:
1603:
1597:
1583:
1574:
1555:
1553:
1525:
1516:
1510:
1465:
1438:
1433:
1424:
1389:
1374:
1369:
1363:
1332:
1317:
1312:
1306:
1266:
1261:
1249:
1221:
1216:
1210:
1185:
1179:
1174:
1161:
1144:
1138:
1119:
1117:
1092:
1086:
1080:
1054:
1029:
1024:
1018:
985:
980:
974:
935:
930:
925:
923:
898:
889:
884:
871:
857:
837:
831:
817:
808:
789:
787:
759:
750:
744:
673:
661:
647:
627:
621:
607:
598:
579:
577:
550:
541:
535:
441:
384:
299:
282:
266:
214:
198:
179:
177:
1453:{\displaystyle CB_{\alpha }^{\alpha }dt}
41:Relevant discussion may be found on the
1716:
1293:with respect to area. Note that for a
1407:with respect to the exterior normal.
1230:{\displaystyle B_{\alpha }^{\alpha }}
1038:{\displaystyle B_{\alpha }^{\alpha }}
994:{\displaystyle B_{\alpha }^{\alpha }}
7:
946:{\displaystyle {\delta }/{\delta }t}
1241:of area. An evolution governed by
705:is the fundamental concept in the
651:
638:
630:
622:
611:
599:
569:is governed by the following law:
554:
542:
329:
302:
14:
485:differential geometry of surfaces
1732:10.1111/j.1467-9590.2010.00485.x
1237:can be appropriately called the
701:. The velocity of the interface
161:integrands, is governed by this
34:relies largely or entirely on a
23:
719:fundamental theorem of calculus
167:fundamental theorem of calculus
1535:{\displaystyle \int _{S}F\,dS}
769:{\displaystyle \int _{S}F\,dS}
507:and consider a time-dependent
1:
1102:{\displaystyle \int _{S}\,dS}
483:, and surface integrals over
62:"Time evolution of integrals"
1460:and expansion by annexation
963:calculus of moving surfaces
707:calculus of moving surfaces
473:calculus of moving surfaces
1780:
729:A related law governs the
479:for volume integrals over
1068:{\displaystyle F\equiv 1}
965:, originally proposed by
709:. In the above equation,
699:velocity of the interface
1698:
1536:
1487:
1480:
1454:
1401:
1344:
1276:
1231:
1196:
1103:
1069:
1039:
995:
947:
909:
770:
684:
561:
462:
1759:Differential calculus
1699:
1537:
1481:
1455:
1418:
1402:
1345:
1277:
1232:
1197:
1104:
1070:
1040:
1003:mean curvature tensor
996:
948:
910:
771:
685:
562:
511:Ω with a smooth
463:
116:differential calculus
1552:
1509:
1464:
1423:
1362:
1305:
1248:
1209:
1116:
1079:
1053:
1017:
1001:is the trace of the
973:
922:
786:
743:
576:
534:
523:be a time-dependent
176:
47:improve this article
1660:
1479:{\displaystyle cdt}
1443:
1379:
1322:
1287:mean curvature flow
1271:
1226:
1184:
1034:
990:
957:is the fundamental
894:
475:provides analogous
298:
230:
1694:
1646:
1532:
1488:
1476:
1450:
1429:
1397:
1365:
1340:
1308:
1272:
1257:
1227:
1212:
1192:
1170:
1099:
1065:
1035:
1020:
991:
976:
943:
905:
880:
766:
680:
557:
458:
262:
194:
1764:Integral calculus
1621:
1568:
1132:
855:
802:
725:Surface integrals
645:
592:
481:Euclidean domains
336:
192:
134:, as well as the
112:
111:
97:
1771:
1743:
1724:
1703:
1701:
1700:
1695:
1683:
1682:
1659:
1654:
1642:
1641:
1622:
1620:
1612:
1604:
1602:
1601:
1579:
1578:
1569:
1567:
1556:
1541:
1539:
1538:
1533:
1521:
1520:
1485:
1483:
1482:
1477:
1459:
1457:
1456:
1451:
1442:
1437:
1406:
1404:
1403:
1398:
1393:
1378:
1373:
1349:
1347:
1346:
1341:
1336:
1321:
1316:
1291:steepest descent
1281:
1279:
1278:
1273:
1270:
1265:
1236:
1234:
1233:
1228:
1225:
1220:
1201:
1199:
1198:
1193:
1183:
1178:
1166:
1165:
1143:
1142:
1133:
1131:
1120:
1108:
1106:
1105:
1100:
1091:
1090:
1074:
1072:
1071:
1066:
1044:
1042:
1041:
1036:
1033:
1028:
1000:
998:
997:
992:
989:
984:
967:Jacques Hadamard
952:
950:
949:
944:
939:
934:
929:
914:
912:
911:
906:
893:
888:
876:
875:
856:
854:
846:
838:
836:
835:
813:
812:
803:
801:
790:
775:
773:
772:
767:
755:
754:
735:surface integral
689:
687:
686:
681:
666:
665:
646:
644:
636:
628:
626:
625:
603:
602:
593:
591:
580:
566:
564:
563:
558:
546:
545:
495:Volume integrals
467:
465:
464:
459:
457:
446:
445:
436:
432:
431:
400:
389:
388:
379:
375:
374:
337:
335:
327:
326:
322:
300:
297:
296:
281:
280:
252:
248:
229:
228:
213:
212:
193:
191:
180:
130:whose domain of
128:surface integral
107:
104:
98:
96:
55:
27:
19:
1779:
1778:
1774:
1773:
1772:
1770:
1769:
1768:
1749:
1748:
1747:
1746:
1725:
1718:
1713:
1674:
1633:
1613:
1605:
1593:
1570:
1560:
1550:
1549:
1512:
1507:
1506:
1462:
1461:
1421:
1420:
1413:
1360:
1359:
1303:
1302:
1289:and represents
1285:is the popular
1246:
1245:
1207:
1206:
1157:
1134:
1124:
1114:
1113:
1082:
1077:
1076:
1051:
1050:
1015:
1014:
1005:. In this law,
971:
970:
920:
919:
867:
847:
839:
827:
804:
794:
784:
783:
746:
741:
740:
727:
657:
637:
629:
617:
594:
584:
574:
573:
537:
532:
531:
503:be a time-like
497:
447:
437:
421:
411:
407:
390:
380:
364:
354:
350:
328:
312:
308:
301:
286:
270:
238:
234:
218:
202:
184:
174:
173:
155:
108:
102:
99:
56:
54:
40:
28:
17:
12:
11:
5:
1777:
1775:
1767:
1766:
1761:
1751:
1750:
1745:
1744:
1715:
1714:
1712:
1709:
1705:
1704:
1693:
1690:
1686:
1681:
1677:
1673:
1670:
1667:
1663:
1658:
1653:
1649:
1645:
1640:
1636:
1632:
1629:
1626:
1619:
1616:
1611:
1608:
1600:
1596:
1592:
1589:
1586:
1582:
1577:
1573:
1566:
1563:
1559:
1543:
1542:
1531:
1528:
1524:
1519:
1515:
1475:
1472:
1469:
1449:
1446:
1441:
1436:
1432:
1428:
1412:
1409:
1396:
1392:
1388:
1385:
1382:
1377:
1372:
1368:
1339:
1335:
1331:
1328:
1325:
1320:
1315:
1311:
1283:
1282:
1269:
1264:
1260:
1256:
1253:
1239:shape gradient
1224:
1219:
1215:
1203:
1202:
1191:
1188:
1182:
1177:
1173:
1169:
1164:
1160:
1156:
1153:
1150:
1147:
1141:
1137:
1130:
1127:
1123:
1098:
1095:
1089:
1085:
1064:
1061:
1058:
1032:
1027:
1023:
988:
983:
979:
942:
938:
933:
928:
916:
915:
904:
901:
897:
892:
887:
883:
879:
874:
870:
866:
863:
860:
853:
850:
845:
842:
834:
830:
826:
823:
820:
816:
811:
807:
800:
797:
793:
779:The law reads
777:
776:
765:
762:
758:
753:
749:
731:rate of change
726:
723:
691:
690:
679:
676:
672:
669:
664:
660:
656:
653:
650:
643:
640:
635:
632:
624:
620:
616:
613:
610:
606:
601:
597:
590:
587:
583:
556:
553:
549:
544:
540:
496:
493:
469:
468:
456:
453:
450:
444:
440:
435:
430:
427:
424:
420:
417:
414:
410:
406:
403:
399:
396:
393:
387:
383:
378:
373:
370:
367:
363:
360:
357:
353:
349:
346:
343:
340:
334:
331:
325:
321:
318:
315:
311:
307:
304:
295:
292:
289:
285:
279:
276:
273:
269:
265:
261:
258:
255:
251:
247:
244:
241:
237:
233:
227:
224:
221:
217:
211:
208:
205:
201:
197:
190:
187:
183:
154:
151:
120:rate of change
110:
109:
45:. Please help
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
1776:
1765:
1762:
1760:
1757:
1756:
1754:
1741:
1737:
1733:
1729:
1723:
1721:
1717:
1710:
1708:
1691:
1688:
1684:
1679:
1675:
1671:
1668:
1665:
1661:
1656:
1651:
1647:
1643:
1638:
1634:
1630:
1627:
1624:
1617:
1614:
1609:
1606:
1598:
1594:
1590:
1587:
1584:
1580:
1575:
1571:
1564:
1561:
1557:
1548:
1547:
1546:
1529:
1526:
1522:
1517:
1513:
1505:
1504:
1503:
1501:
1497:
1493:
1490:Suppose that
1473:
1470:
1467:
1447:
1444:
1439:
1434:
1430:
1426:
1417:
1410:
1408:
1394:
1390:
1386:
1383:
1380:
1375:
1370:
1366:
1357:
1353:
1337:
1333:
1329:
1326:
1323:
1318:
1313:
1309:
1300:
1296:
1292:
1288:
1267:
1262:
1258:
1254:
1251:
1244:
1243:
1242:
1240:
1222:
1217:
1213:
1189:
1186:
1180:
1175:
1171:
1167:
1162:
1158:
1154:
1151:
1148:
1145:
1139:
1135:
1128:
1125:
1121:
1112:
1111:
1110:
1096:
1093:
1087:
1083:
1062:
1059:
1056:
1048:
1030:
1025:
1021:
1012:
1008:
1004:
986:
981:
977:
968:
964:
960:
956:
940:
936:
931:
926:
902:
899:
895:
890:
885:
881:
877:
872:
868:
864:
861:
858:
851:
848:
843:
840:
832:
828:
824:
821:
818:
814:
809:
805:
798:
795:
791:
782:
781:
780:
763:
760:
756:
751:
747:
739:
738:
737:
736:
732:
724:
722:
720:
716:
712:
708:
704:
700:
696:
677:
674:
670:
667:
662:
658:
654:
648:
641:
633:
618:
614:
608:
604:
595:
588:
585:
581:
572:
571:
570:
567:
551:
547:
538:
530:
526:
522:
518:
514:
510:
506:
502:
494:
492:
490:
486:
482:
478:
474:
454:
451:
448:
438:
433:
428:
425:
422:
418:
415:
412:
408:
404:
401:
397:
394:
391:
381:
376:
371:
368:
365:
361:
358:
355:
351:
347:
344:
341:
338:
332:
323:
319:
316:
313:
309:
305:
293:
290:
287:
283:
277:
274:
271:
267:
263:
259:
256:
253:
249:
245:
242:
239:
235:
231:
225:
222:
219:
215:
209:
206:
203:
199:
195:
188:
185:
181:
172:
171:
170:
168:
164:
160:
152:
150:
148:
145:
141:
137:
133:
129:
125:
121:
117:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
1706:
1544:
1499:
1495:
1491:
1489:
1355:
1350:, and for a
1298:
1284:
1238:
1204:
1046:
1010:
1006:
917:
778:
728:
710:
702:
694:
692:
568:
520:
516:
500:
498:
470:
156:
153:Introduction
146:
113:
100:
90:
83:
76:
69:
57:
33:
132:integration
1753:Categories
1711:References
1354:of radius
1297:of radius
955:derivative
918:where the
489:boundaries
73:newspapers
1740:0022-2526
1692:γ
1680:γ
1676:∫
1657:α
1652:α
1635:∫
1631:−
1615:δ
1607:δ
1595:∫
1572:∫
1514:∫
1440:α
1435:α
1384:−
1376:α
1371:α
1327:−
1319:α
1314:α
1268:α
1263:α
1255:≡
1223:α
1218:α
1181:α
1176:α
1159:∫
1155:−
1136:∫
1109:is area:
1084:∫
1060:≡
1031:α
1026:α
987:α
982:α
937:δ
927:δ
891:α
886:α
869:∫
865:−
849:δ
841:δ
829:∫
806:∫
748:∫
659:∫
652:Ω
639:∂
631:∂
623:Ω
619:∫
612:Ω
600:Ω
596:∫
555:Ω
543:Ω
539:∫
525:invariant
515:boundary
505:parameter
443:′
402:−
386:′
330:∂
303:∂
264:∫
196:∫
163:extension
140:functions
136:integrand
103:July 2012
43:talk page
959:operator
529:integral
477:formulas
961:in the
733:of the
697:is the
513:surface
165:of the
114:Within
87:scholar
1738:
1352:circle
1295:sphere
1075:since
715:normal
693:where
519:. Let
509:domain
159:smooth
138:, are
124:volume
89:
82:
75:
68:
60:
122:of a
94:JSTOR
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