482:
979: = 3, this reduces to Newton's theorem that the potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center.
971:
306:
784:
261:
1396:
1126:
107:
605:
872:
1331:
821:
650:
1961:
281:
88:
698:
157:
1299:
534:
301:
149:
1388:
1511:
115:
1617:
2021:
1936:
1913:
477:{\displaystyle \Gamma (x)={\begin{cases}{\frac {1}{2\pi }}\log {|x|},&d=2,\\{\frac {1}{d(2-d)\omega _{d}}}|x|^{2-d},&d\neq 2.\end{cases}}}
1669:
692:: one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data.
1521:
1259:
1647:
1548:
1358:
1538:
1560:
652:
which is to say that the operation of taking the
Newtonian potential of a function is a partial inverse to the Laplace operator. Then
1350:
1174:
1151:
1946:
1771:
1966:
1868:
1766:
1587:
1851:
1231:
1213:
1195:
1761:
2031:
2011:
1956:
1815:
1612:
1226:
1208:
1190:
1674:
1577:
54:, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in
1756:
1751:
1543:
1506:
1365:
1252:
824:
797:
59:
1637:
2026:
1721:
1291:
623:
1592:
1533:
1446:
1516:
539:
67:
1501:
1706:
1570:
1323:
1203:
1040:
119:
111:
684:
will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the
1941:
1931:
1696:
1659:
1632:
1526:
1111:
43:
2016:
1988:
1895:
1781:
1776:
1565:
1478:
1245:
1453:
1221:
1185:
991:
1902:
1622:
1379:
832:
668:. It was an open question whether continuity alone is also sufficient. This was shown to be wrong by
661:
91:
680:
is not twice differentiable. The solution is not unique, since addition of any harmonic function to
330:
1711:
1691:
1607:
1458:
1116:
847:
129:
1555:
1283:
1926:
1642:
966:{\displaystyle f*\Gamma (x)=\lambda \Gamma (x),\quad \lambda =\int _{\mathbb {R} ^{d}}f(y)\,dy.}
1885:
1857:
1739:
1716:
1664:
1582:
1468:
1170:
1147:
1139:
1079:
1067:
688:
for the
Poisson equation in suitably regular domains, and for suitably well-behaved functions
685:
103:
266:
73:
1845:
1821:
1797:
1627:
1373:
1028:
828:
618:
95:
55:
1833:
1602:
1496:
1121:
1099:
126:
47:
995:
513:
1876:
1701:
1684:
1654:
1441:
1162:
1036:
669:
665:
286:
134:
2005:
1921:
1803:
1745:
791:
1809:
1679:
1488:
1463:
1426:
1268:
118:. In modern potential theory, the Newtonian potential is instead thought of as an
99:
1827:
1436:
1315:
152:
63:
31:
1951:
1839:
779:{\displaystyle \Gamma *\mu (x)=\int _{\mathbb {R} ^{d}}\Gamma (x-y)\,d\mu (y)}
256:{\displaystyle u(x)=\Gamma *f(x)=\int _{\mathbb {R} ^{d}}\Gamma (x-y)f(y)\,dy}
17:
51:
986:
is associated to a mass distribution on a sufficiently smooth hypersurface
1473:
1414:
851:
496:
1431:
1419:
1307:
1098:. This makes simple layers particularly suited to the study of the
1237:
695:
The
Newtonian potential is defined more broadly as the convolution
1090:) when crossing the layer. Furthermore, the normal derivative of
1241:
656:
will be a classical solution, that is twice differentiable, if
1043:
associated to a charge distribution on a closed surface. If
470:
1127:
Green's function for the three-variable
Laplace equation
1027:. Simple layer potentials are continuous and solve the
1167:
Elliptic
Partial Differential Equations of Second Order
875:
800:
701:
626:
542:
516:
309:
289:
269:
160:
137:
76:
1975:
1912:
1867:
1790:
1732:
1487:
1407:
1342:
1275:
102:, who first discovered it and proved that it was a
965:
815:
778:
644:
599:
528:
476:
295:
275:
255:
143:
82:
1962:Statal Institute of Higher Education Isaac Newton
507:
850:(or, more generally, a finite measure) that is
502:(sometimes sign conventions may vary; compare (
1253:
1146:, Providence: American Mathematical Society,
8:
1058:is the product of a continuous function on
1260:
1246:
1238:
1094:is a well-defined continuous function on
1035:. They appear naturally in the study of
953:
933:
929:
928:
926:
874:
799:
760:
734:
730:
729:
727:
700:
625:
600:{\displaystyle \Gamma (x)=-1/(4\pi |x|).}
586:
578:
564:
541:
515:
441:
436:
427:
418:
390:
363:
355:
354:
333:
325:
308:
288:
268:
246:
208:
204:
203:
201:
159:
136:
75:
50:that acts as the inverse to the negative
1512:Newton's law of universal gravitation
794:. It satisfies the Poisson equation
503:
116:Newton's law of universal gravitation
110:, where it served as the fundamental
7:
1670:Newton's theorem of revolving orbits
672:who gave an example of a continuous
70:at the origin, the Newtonian kernel
1618:Leibniz–Newton calculus controversy
1359:standing on the shoulders of giants
1985:
1019:, then the Newtonian potential of
900:
882:
801:
742:
702:
627:
543:
310:
270:
216:
176:
77:
25:
58:. In its general nature, it is a
1947:Isaac Newton Group of Telescopes
827:. Moreover, when the measure is
1967:Newton International Fellowship
1648:generalized Gauss–Newton method
1561:Newton's method in optimization
1082:undergoes a jump discontinuity
915:
108:special case of three variables
2022:Partial differential equations
1144:Partial Differential Equations
950:
944:
909:
903:
891:
885:
773:
767:
757:
745:
717:
711:
591:
587:
579:
569:
552:
546:
437:
428:
411:
399:
364:
356:
319:
313:
243:
237:
231:
219:
191:
185:
170:
164:
27:Green's function for Laplacian
1:
831:, the Newtonian potential is
816:{\displaystyle \Delta w=\mu }
125:The Newtonian potential of a
1588:Newton's theorem about ovals
1066: − 1)-dimensional
508:Gilbarg & Trudinger 1983
1957:Sir Isaac Newton Sixth Form
1613:Corpuscular theory of light
1539:Schrödinger–Newton equation
1227:Encyclopedia of Mathematics
1220:Solomentsev, E.D. (2001) ,
1209:Encyclopedia of Mathematics
1202:Solomentsev, E.D. (2001) ,
1191:Encyclopedia of Mathematics
1184:Solomentsev, E.D. (2001) ,
645:{\displaystyle \Delta w=f,}
263:where the Newtonian kernel
2048:
1366:Notes on the Jewish Temple
1102:for the Laplace equation.
854:, then the convolution of
495:is the volume of the unit
60:singular integral operator
846:is a compactly supported
790:is a compactly supported
66:with a function having a
1517:post-Newtonian expansion
1397:Corruptions of Scripture
1389:Ancient Kingdoms Amended
1204:"Simple-layer potential"
609:The Newtonian potential
68:mathematical singularity
1707:Absolute space and time
1571:truncated Newton method
1544:Newton's laws of motion
1507:Newton's law of cooling
1041:electrostatic potential
866:outside the support of
660:is bounded and locally
276:{\displaystyle \Gamma }
120:electrostatic potential
112:gravitational potential
83:{\displaystyle \Gamma }
1942:Isaac Newton Telescope
1932:Isaac Newton Institute
1702:Newton–Puiseux theorem
1697:Parallelogram of force
1685:kissing number problem
1675:Newton–Euler equations
1578:Gauss–Newton algorithm
1527:gravitational constant
1169:, New York: Springer,
1112:Double layer potential
1039:in the context of the
1025:simple layer potential
967:
852:rotationally invariant
817:
780:
646:
601:
530:
478:
297:
277:
257:
145:
84:
1896:Isaac Newton Gargoyle
1806: (nephew-in-law)
1782:Copernican Revolution
1777:Scientific Revolution
1638:Newton–Cotes formulas
1502:Newton's inequalities
1479:Structural coloration
968:
818:
781:
647:
617:is a solution of the
602:
531:
510:)). For example, for
479:
298:
278:
258:
146:
85:
1903:Astronomers Monument
1593:Newton–Pepys problem
1566:Apollonius's problem
1534:Newton–Cartan theory
1447:Newton–Okounkov body
1380:hypotheses non fingo
1369: (c. 1680)
1023:is referred to as a
873:
798:
699:
624:
540:
514:
307:
287:
267:
158:
135:
92:fundamental solution
74:
1712:Luminiferous aether
1660:Newton's identities
1633:Newton's cannonball
1608:Classical mechanics
1598:Newtonian potential
1459:Newtonian telescope
1222:"Surface potential"
848:continuous function
529:{\displaystyle d=3}
130:integrable function
127:compactly supported
98:. It is named for
36:Newtonian potential
2032:Singular integrals
2012:Harmonic functions
1937:Isaac Newton Medal
1742: (birthplace)
1556:Newtonian dynamics
1454:Newton's reflector
1186:"Newton potential"
1070:, then at a point
963:
813:
776:
642:
597:
526:
474:
469:
293:
273:
253:
151:is defined as the
141:
80:
1999:
1998:
1891: (sculpture)
1858:Abraham de Moivre
1812: (professor)
1740:Woolsthorpe Manor
1692:Newton's quotient
1665:Newton polynomial
1623:Newton's notation
1354: (1661–1665)
1080:normal derivative
1068:Hausdorff measure
1005:into two regions
982:When the measure
686:Dirichlet problem
662:Hölder continuous
425:
346:
296:{\displaystyle d}
144:{\displaystyle f}
104:harmonic function
16:(Redirected from
2039:
2027:Potential theory
1987:
1882: (monotype)
1846:William Stukeley
1842: (disciple)
1822:Benjamin Pulleyn
1798:Catherine Barton
1717:Newtonian series
1628:Rotating spheres
1374:General Scholium
1269:Sir Isaac Newton
1262:
1255:
1248:
1239:
1234:
1216:
1198:
1179:
1156:
1117:Green's function
1057:
1029:Laplace equation
992:Lyapunov surface
972:
970:
969:
964:
940:
939:
938:
937:
932:
861:
823:in the sense of
822:
820:
819:
814:
785:
783:
782:
777:
741:
740:
739:
738:
733:
651:
649:
648:
643:
619:Poisson equation
606:
604:
603:
598:
590:
582:
568:
535:
533:
532:
527:
483:
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475:
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451:
440:
431:
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391:
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359:
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274:
262:
260:
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215:
214:
213:
212:
207:
150:
148:
147:
142:
96:Laplace equation
89:
87:
86:
81:
56:potential theory
40:Newton potential
21:
2047:
2046:
2042:
2041:
2040:
2038:
2037:
2036:
2002:
2001:
2000:
1995:
1994:
1993:
1992:
1991:
1984:
1971:
1927:Newton's cradle
1908:
1863:
1836: (student)
1834:William Whiston
1830: (student)
1786:
1767:Religious views
1728:
1643:Newton's method
1603:Newtonian fluid
1497:Bucket argument
1483:
1403:
1338:
1271:
1266:
1219:
1201:
1183:
1177:
1163:Trudinger, Neil
1160:
1154:
1138:
1135:
1122:Riesz potential
1108:
1100:Neumann problem
1044:
1018:
1011:
1001:) that divides
927:
922:
871:
870:
859:
796:
795:
728:
723:
697:
696:
622:
621:
538:
537:
512:
511:
494:
468:
467:
456:
435:
414:
395:
387:
386:
372:
338:
326:
305:
304:
285:
284:
265:
264:
202:
197:
156:
155:
133:
132:
72:
71:
48:vector calculus
28:
23:
22:
15:
12:
11:
5:
2045:
2043:
2035:
2034:
2029:
2024:
2019:
2014:
2004:
2003:
1997:
1996:
1983:
1982:
1981:
1980:
1979:
1977:
1973:
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1964:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1918:
1916:
1910:
1909:
1907:
1906:
1899:
1892:
1883:
1873:
1871:
1865:
1864:
1862:
1861:
1860: (friend)
1855:
1854: (friend)
1849:
1848: (friend)
1843:
1837:
1831:
1825:
1819:
1818: (mentor)
1816:William Clarke
1813:
1807:
1801:
1794:
1792:
1788:
1787:
1785:
1784:
1779:
1774:
1772:Occult studies
1769:
1764:
1759:
1754:
1749:
1743:
1736:
1734:
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1714:
1709:
1704:
1699:
1694:
1689:
1688:
1687:
1677:
1672:
1667:
1662:
1657:
1655:Newton fractal
1652:
1651:
1650:
1640:
1635:
1630:
1625:
1620:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1583:Newton's rings
1580:
1575:
1574:
1573:
1568:
1558:
1553:
1552:
1551:
1541:
1536:
1531:
1530:
1529:
1524:
1519:
1509:
1504:
1499:
1493:
1491:
1485:
1484:
1482:
1481:
1476:
1471:
1469:Newton's metal
1466:
1461:
1456:
1451:
1450:
1449:
1442:Newton polygon
1439:
1434:
1429:
1424:
1423:
1422:
1411:
1409:
1405:
1404:
1402:
1401:
1393:
1385:
1376:" (1713;
1370:
1362:
1355:
1346:
1344:
1343:Other writings
1340:
1339:
1337:
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1328:
1320:
1312:
1304:
1296:
1288:
1279:
1277:
1273:
1272:
1267:
1265:
1264:
1257:
1250:
1242:
1236:
1235:
1217:
1199:
1181:
1175:
1158:
1152:
1134:
1131:
1130:
1129:
1124:
1119:
1114:
1107:
1104:
1037:electrostatics
1016:
1009:
962:
959:
956:
952:
949:
946:
943:
936:
931:
925:
921:
918:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
862:satisfies for
812:
809:
806:
803:
775:
772:
769:
766:
763:
759:
756:
753:
750:
747:
744:
737:
732:
726:
722:
719:
716:
713:
710:
707:
704:
670:Henrik Petrini
641:
638:
635:
632:
629:
596:
593:
589:
585:
581:
577:
574:
571:
567:
563:
560:
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551:
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545:
525:
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519:
490:
471:
466:
463:
460:
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455:
450:
447:
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439:
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421:
417:
413:
410:
407:
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401:
398:
394:
389:
388:
385:
382:
379:
376:
373:
371:
366:
362:
358:
353:
350:
344:
341:
337:
332:
331:
329:
324:
321:
318:
315:
312:
303:is defined by
292:
272:
252:
249:
245:
242:
239:
236:
233:
230:
227:
224:
221:
218:
211:
206:
200:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
140:
79:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2044:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2009:
2007:
1990:
1986:
1978:
1974:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1922:Newton (unit)
1920:
1919:
1917:
1915:
1911:
1905:
1904:
1900:
1898:
1897:
1893:
1890:
1888:
1884:
1881:
1879:
1875:
1874:
1872:
1870:
1866:
1859:
1856:
1853:
1852:William Jones
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1824: (tutor)
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1804:John Conduitt
1802:
1800: (niece)
1799:
1796:
1795:
1793:
1789:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1747:
1746:Cranbury Park
1744:
1741:
1738:
1737:
1735:
1733:Personal life
1731:
1723:
1720:
1719:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1686:
1683:
1682:
1681:
1680:Newton number
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1649:
1646:
1645:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1601:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1581:
1579:
1576:
1572:
1569:
1567:
1564:
1563:
1562:
1559:
1557:
1554:
1550:
1549:Kepler's laws
1547:
1546:
1545:
1542:
1540:
1537:
1535:
1532:
1528:
1525:
1523:
1522:parameterized
1520:
1518:
1515:
1514:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1494:
1492:
1490:
1486:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1448:
1445:
1444:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1421:
1418:
1417:
1416:
1413:
1412:
1410:
1408:Contributions
1406:
1399:
1398:
1394:
1391:
1390:
1386:
1383:
1381:
1375:
1371:
1368:
1367:
1363:
1361:" (1675)
1360:
1356:
1353:
1352:
1348:
1347:
1345:
1341:
1334:
1333:
1329:
1326:
1325:
1321:
1318:
1317:
1313:
1310:
1309:
1305:
1302:
1301:
1297:
1294:
1293:
1289:
1286:
1285:
1281:
1280:
1278:
1274:
1270:
1263:
1258:
1256:
1251:
1249:
1244:
1243:
1240:
1233:
1229:
1228:
1223:
1218:
1215:
1211:
1210:
1205:
1200:
1197:
1193:
1192:
1187:
1182:
1178:
1176:3-540-41160-7
1172:
1168:
1164:
1161:Gilbarg, D.;
1159:
1155:
1153:0-8218-0772-2
1149:
1145:
1141:
1137:
1136:
1132:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1109:
1105:
1103:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1056:
1052:
1048:
1042:
1038:
1034:
1030:
1026:
1022:
1015:
1008:
1004:
1000:
997:
993:
989:
985:
980:
978:
975:In dimension
973:
960:
957:
954:
947:
941:
934:
923:
919:
916:
912:
906:
897:
894:
888:
879:
876:
869:
865:
857:
853:
849:
845:
840:
838:
834:
830:
826:
825:distributions
810:
807:
804:
793:
792:Radon measure
789:
770:
764:
761:
754:
751:
748:
735:
724:
720:
714:
708:
705:
693:
691:
687:
683:
679:
675:
671:
667:
663:
659:
655:
639:
636:
633:
630:
620:
616:
612:
607:
594:
583:
575:
572:
565:
561:
558:
555:
549:
523:
520:
517:
509:
505:
501:
499:
493:
489:
484:
464:
461:
458:
453:
448:
445:
442:
432:
419:
415:
408:
405:
402:
396:
392:
383:
380:
377:
374:
369:
360:
351:
348:
342:
339:
335:
327:
322:
316:
290:
283:in dimension
250:
247:
240:
234:
228:
225:
222:
209:
198:
194:
188:
182:
179:
173:
167:
161:
154:
138:
131:
128:
123:
121:
117:
113:
109:
105:
101:
97:
93:
90:which is the
69:
65:
62:, defined by
61:
57:
53:
49:
45:
41:
37:
33:
19:
18:Newton kernel
2017:Isaac Newton
1989:Isaac Newton
1901:
1894:
1886:
1877:
1810:Isaac Barrow
1748: (home)
1597:
1489:Newtonianism
1464:Newton scale
1427:Impact depth
1400: (1754)
1395:
1392: (1728)
1387:
1377:
1364:
1349:
1335: (1711)
1330:
1327: (1707)
1322:
1319: (1704)
1314:
1311: (1704)
1306:
1303: (1687)
1298:
1295: (1684)
1290:
1287: (1671)
1282:
1276:Publications
1225:
1207:
1189:
1166:
1143:
1095:
1091:
1087:
1083:
1075:
1071:
1063:
1059:
1054:
1050:
1046:
1032:
1024:
1020:
1013:
1006:
1002:
998:
996:Hölder class
987:
983:
981:
976:
974:
867:
863:
855:
843:
841:
836:
787:
694:
689:
681:
677:
673:
664:as shown by
657:
653:
614:
610:
608:
497:
491:
487:
485:
124:
100:Isaac Newton
39:
35:
29:
1889:by Paolozzi
1828:Roger Cotes
1437:Newton disc
1351:Quaestiones
1324:Arithmetica
1140:Evans, L.C.
833:subharmonic
666:Otto Hölder
153:convolution
64:convolution
32:mathematics
2006:Categories
1976:Categories
1952:XMM-Newton
1869:Depictions
1840:John Keill
1762:Apple tree
1757:Later life
1752:Early life
1332:De Analysi
1133:References
1062:with the (
1031:except on
676:for which
504:Evans 1998
1791:Relations
1300:Principia
1232:EMS Press
1214:EMS Press
1196:EMS Press
924:∫
917:λ
901:Γ
898:λ
883:Γ
880:∗
811:μ
802:Δ
765:μ
752:−
743:Γ
725:∫
709:μ
706:∗
703:Γ
628:Δ
576:π
559:−
544:Γ
462:≠
446:−
416:ω
406:−
352:
343:π
311:Γ
271:Γ
226:−
217:Γ
199:∫
180:∗
177:Γ
78:Γ
52:Laplacian
1914:Namesake
1880:by Blake
1474:Spectrum
1415:Calculus
1384: )
1284:Fluxions
1165:(1983),
1142:(1998),
1106:See also
829:positive
536:we have
44:operator
1432:Inertia
1420:fluxion
1316:Queries
1308:Opticks
1292:De Motu
1017:−
506:) and (
106:in the
94:of the
1887:Newton
1878:Newton
1173:
1150:
1078:, the
42:is an
34:, the
1722:table
858:with
786:when
500:-ball
486:Here
1171:ISBN
1148:ISBN
1012:and
1074:of
994:of
990:(a
842:If
835:on
613:of
349:log
114:in
46:in
38:or
30:In
2008::
1230:,
1224:,
1212:,
1206:,
1194:,
1188:,
1049:=
839:.
465:2.
122:.
1382:"
1378:"
1372:"
1357:"
1261:e
1254:t
1247:v
1180:.
1157:.
1096:S
1092:w
1088:y
1086:(
1084:f
1076:S
1072:y
1064:d
1060:S
1055:H
1053:d
1051:f
1047:μ
1045:d
1033:S
1021:μ
1014:D
1010:+
1007:D
1003:R
999:C
988:S
984:μ
977:d
961:.
958:y
955:d
951:)
948:y
945:(
942:f
935:d
930:R
920:=
913:,
910:)
907:x
904:(
895:=
892:)
889:x
886:(
877:f
868:f
864:x
860:Γ
856:f
844:f
837:R
808:=
805:w
788:μ
774:)
771:y
768:(
762:d
758:)
755:y
749:x
746:(
736:d
731:R
721:=
718:)
715:x
712:(
690:f
682:w
678:w
674:f
658:f
654:w
640:,
637:f
634:=
631:w
615:f
611:w
595:.
592:)
588:|
584:x
580:|
573:4
570:(
566:/
562:1
556:=
553:)
550:x
547:(
524:3
521:=
518:d
498:d
492:d
488:ω
459:d
454:,
449:d
443:2
438:|
433:x
429:|
420:d
412:)
409:d
403:2
400:(
397:d
393:1
384:,
381:2
378:=
375:d
370:,
365:|
361:x
357:|
340:2
336:1
328:{
323:=
320:)
317:x
314:(
291:d
251:y
248:d
244:)
241:y
238:(
235:f
232:)
229:y
223:x
220:(
210:d
205:R
195:=
192:)
189:x
186:(
183:f
174:=
171:)
168:x
165:(
162:u
139:f
20:)
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