73:
2076:
1111:
1628:(whose axis of rotation is perpendicular to the surface), then the vortex causes a depression in the pressure field. The surface of the liquid inside the vortex is pulled downwards as are any surfaces of equal pressure, which still remain parallel to the liquids surface. The effect is strongest inside the vortex and decreases rapidly with the distance from the vortex axis.
2891:
3014:
1894:
1916:
2530:
884:
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1170:
is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map,
2705:
1620:
Since buoyant force points upwards, in the direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the surface,
239:
1747:
1103:
1554:: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.
2410:
1016:
1355:
736:
2405:
125:
375:
2071:{\displaystyle \mathbf {E} =-\mathbf {\nabla } \Phi =-{\frac {1}{4\pi }}\mathbf {\nabla } \int _{\mathbb {R} ^{3}}{\frac {\operatorname {div} \mathbf {E} (\mathbf {r} ')}{\left\|\mathbf {r} -\mathbf {r} '\right\|}}\,dV(\mathbf {r} ')}
2685:
2182:
1536:
2266:
1566:, a fluid in equilibrium, but in the presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This
2886:{\displaystyle \Phi (\mathbf {r} )=-{\frac {1}{n\omega _{n}}}\int _{\mathbb {R} ^{n}}{\frac {\mathbf {E} (\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{\|\mathbf {r} -\mathbf {r} '\|^{n}}}\,dV(\mathbf {r} ').}
2329:
2579:
1693:
1444:
1238:
459:
1889:{\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\operatorname {div} \mathbf {E} (\mathbf {r} ')}{\left\|\mathbf {r} -\mathbf {r} '\right\|}}\,dV(\mathbf {r} ')}
912:
503:
1256:
1185:
of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:
1616:
2346:
1631:
The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating the negative pressure gradient along the surface of the object:
2595:
1731:
30:
This article is about a general description of a function used in mathematics and physics to describe conservative fields. For the scalar potential of electromagnetism, see
2118:
2525:{\displaystyle \nabla \operatorname {div} (\mathbf {E} *\Gamma )=\nabla ^{2}(\mathbf {E} *\Gamma )=\mathbf {E} *\nabla ^{2}\Gamma =-\mathbf {E} *\delta =-\mathbf {E} }
2935:
valid for
Cartesian coordinates, other coordinate systems such as cylindrical or spherical coordinates will have more complicated representations, derived from the
1449:
1156:
292:
53:
of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a
879:{\displaystyle V(\mathbf {r} )=-\int _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =-\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt,}
2207:
234:{\displaystyle \mathbf {F} =-\nabla P=-\left({\frac {\partial P}{\partial x}},{\frac {\partial P}{\partial y}},{\frac {\partial P}{\partial z}}\right),}
2285:
2538:
1634:
1390:
1188:
1071:. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If
2906:
2969:
3044:
529:
so that it can be expressed as the gradient of a scalar function. The third condition re-expresses the second condition in terms of the
263:. In some cases, mathematicians may use a positive sign in front of the gradient to define the potential. Because of this definition of
1577:
419:
667:
72:
470:
257:
3018:
654:
in physics. Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a
891:
1736:
685:
646:
550:
286:
to be described in terms of a scalar potential only, any of the following equivalent statements have to be true:
3039:
1097:
1011:{\displaystyle \mathbf {r} (t),a\leq t\leq b,\mathbf {r} (a)=\mathbf {r_{0}} ,\mathbf {r} (b)=\mathbf {r} .}
659:
513:
1129:
1114:
Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The
2334:
516:
1350:{\displaystyle \mathbf {F} _{\mathrm {P} }=-mg\ \sin \theta \ \cos \theta =-{1 \over 2}mg\sin 2\theta .}
630:
556:
662:
theorem however, all vector fields can be describable in terms of a scalar potential and corresponding
1707:
2201:
2189:
626:
582:
66:
62:
42:
3034:
2400:{\displaystyle \nabla ^{2}\mathbf {G} =\mathbf {\nabla } (\mathbf {\nabla } \cdot {}\mathbf {G} ).}
2185:
2085:
1123:
704:
634:
598:
707:
694:
666:. In electrodynamics, the electromagnetic scalar and vector potentials are known together as the
651:
605:. The electric potential is in this case the electrostatic potential energy per unit charge. In
590:
530:
31:
540:
2698:-ball. The proof is identical. Alternatively, integration by parts (or, more rigorously, the
2965:
638:
574:
370:{\displaystyle -\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} =P(\mathbf {b} )-P(\mathbf {a} ),}
2936:
2901:
2193:
1115:
1044:
663:
622:
509:
101:
50:
2987:
for an example where the potential is defined without a negative. Other references such as
2985:
2680:{\displaystyle \Gamma (\mathbf {r} )={\frac {1}{n(n-2)\omega _{n}\|\mathbf {r} \|^{n-2}}}}
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1102:
614:
602:
88:
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606:
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Scalar potentials play a prominent role in many areas of physics and engineering. The
1166:
is the height above the surface. This means that gravitational potential energy on a
3028:
2911:
2177:{\displaystyle \Gamma (\mathbf {r} )={\frac {1}{4\pi }}{\frac {1}{\|\mathbf {r} \|}}}
730:
618:
1182:
578:
462:
110:
106:
54:
1380:
be the uniform interval of altitude between contours on the contour map, and let
1181:. However, a ball rolling down a hill cannot move directly downwards due to the
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2269:
1167:
256:
and the second part of the equation is minus the gradient for a function of the
58:
17:
1110:
2096:
1531:{\displaystyle F_{P}=-mg{\Delta x\,\Delta h \over \Delta x^{2}+\Delta h^{2}}.}
655:
378:
577:
is the scalar potential associated with the gravity per unit mass, i.e., the
83:
35:
2088:
and vanishes asymptotically to zero towards infinity, decaying faster than
277:
at that point, its magnitude is the rate of that decrease per unit length.
3013:
581:
due to the field, as a function of position. The gravity potential is the
2261:{\displaystyle \nabla ^{2}\Gamma (\mathbf {r} )+\delta (\mathbf {r} )=0.}
1571:
1567:
249:
2333:
Indeed, convolution of an irrotational vector field with a rotationally
644:
Not every vector field has a scalar potential. Those that do are called
461:
where the integral is over any simple closed path, otherwise known as a
92:
1625:
1538:
However, on a contour map, the gradient is inversely proportional to
1101:
555:
2324:{\displaystyle \Phi =\operatorname {div} (\mathbf {E} *\Gamma ).}
2574:{\displaystyle \Phi =\operatorname {div} (\mathbf {E} *\Gamma )}
637:. Further, the scalar potential is the fundamental quantity in
1688:{\displaystyle F_{B}=-\oint _{S}\nabla p\cdot \,d\mathbf {S} .}
76:
Vector field (right) and corresponding scalar potential (left).
1439:{\displaystyle \theta =\tan ^{-1}{\frac {\Delta h}{\Delta x}}}
1175:
always points straight downwards in the direction of gravity;
613:
have a scalar potential only in the special case when it is a
99:
is frequently omitted if there is no danger of confusion with
1233:{\displaystyle \mathbf {F} _{\mathrm {S} }=-mg\ \sin \theta }
27:
When potential energy difference depends only on displacement
65:) that depends only on its location. A familiar example is
1075:
is defined in terms of the line integral, the ambiguity of
1621:
which can be characterized as the plane of zero pressure.
1079:
reflects the freedom in the choice of the reference point
273:
at any point is the direction of the steepest decrease of
2105:
likewise vanishes towards infinity, decaying faster than
560:
Gravitational potential well of an increasing mass where
512:
and is true for any vector field that is a gradient of a
2337:
is also irrotational. For an irrotational vector field
454:{\displaystyle \oint \mathbf {F} \cdot d\mathbf {l} =0,}
1065:
is a scalar potential of the conservative vector field
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near the Earth's surface. It has a potential energy
915:
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422:
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128:
1118:
of the cross-section are at the surface of the body.
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uniform gravitational field near the Earth's surface
1020:
The fact that the line integral depends on the path
49:
describes the situation where the difference in the
2268:Then the scalar potential is the divergence of the
1907:is an infinitesimal volume element with respect to
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2679:
2573:
2524:
2399:
2323:
2260:
2176:
2070:
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1687:
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1530:
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1349:
1244:is the angle of inclination, and the component of
1232:
1150:
1010:
878:
497:
453:
369:
233:
498:{\displaystyle {\nabla }\times {\mathbf {F} }=0.}
2999:when solving for a function from its gradient.
625:. The potential play a prominent role in the
549:that satisfies these conditions is said to be
2592:) with the Newtonian potential given then by
1387:be the distance between two contours. Then
508:The first of these conditions represents the
8:
2845:
2823:
2659:
2650:
2168:
2160:
593:is the scalar potential associated with the
523:. The second condition is a requirement of
105:). The scalar potential is an example of a
1611:{\displaystyle \mathbf {f_{B}} =-\nabla p.}
1366:, parallel to the ground, is greatest when
1171:the three-dimensional negative gradient of
267:in terms of the gradient, the direction of
1162:is the gravitational potential energy and
1092:Altitude as gravitational potential energy
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2857:
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1451:
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1404:
1392:
1316:
1267:
1266:
1261:
1258:
1199:
1198:
1193:
1190:
1131:
1000:
983:
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968:
951:
916:
914:
866:
848:
827:
819:
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808:
793:
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778:
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764:
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738:
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129:
127:
1109:
71:
2952:
2924:
2907:Fundamental theorem of vector analysis
1045:fundamental theorem of line integrals
7:
2931:The second part of this equation is
1043:of a conservative vector field. The
2991:The Calculus with Analytic Geometry
2937:fundamental theorem of the gradient
1698:Scalar potential in Euclidean space
1122:An example is the (nearly) uniform
510:fundamental theorem of the gradient
2709:
2599:
2565:
2542:
2491:
2482:
2464:
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2434:
2414:
2377:
2369:
2351:
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2289:
2221:
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1961:
1936:
1932:
1751:
1664:
1599:
1509:
1493:
1485:
1478:
1427:
1419:
1268:
1200:
720:with respect to a reference point
475:
214:
206:
191:
183:
168:
160:
140:
25:
2914:(isopotential) lines and surfaces
1702:In 3-dimensional Euclidean space
1024:only through its terminal points
650:, corresponding to the notion of
3012:
2869:
2836:
2827:
2810:
2801:
2783:
2774:
2716:
2654:
2606:
2558:
2518:
2501:
2474:
2457:
2427:
2387:
2361:
2305:
2245:
2228:
2200:is equal to the negative of the
2196:, meaning that the Laplacian of
2164:
2129:
2057:
2030:
2021:
2003:
1994:
1921:
1875:
1848:
1839:
1821:
1812:
1758:
1726:{\displaystyle \mathbb {R} ^{3}}
1678:
1587:
1583:
1545:, which is not similar to force
1262:
1194:
1001:
984:
974:
970:
952:
917:
849:
828:
820:
794:
779:
771:
747:
484:
438:
427:
377:where the integration is over a
357:
340:
326:
315:
130:
541:fundamental theorem of the curl
67:potential energy due to gravity
2877:
2864:
2818:
2797:
2791:
2778:
2720:
2712:
2637:
2625:
2610:
2602:
2585:-dimensional Euclidean space (
2568:
2554:
2467:
2453:
2437:
2423:
2391:
2373:
2315:
2301:
2249:
2241:
2232:
2224:
2133:
2125:
2065:
2052:
2039:
2016:
2011:
1998:
1883:
1870:
1857:
1834:
1829:
1816:
1762:
1754:
994:
988:
962:
956:
927:
921:
863:
857:
841:
838:
832:
824:
783:
775:
751:
743:
668:electromagnetic four-potential
583:gravitational potential energy
361:
353:
344:
336:
1:
1735:, the scalar potential of an
1624:If the liquid has a vertical
1558:Pressure as buoyant potential
1051:is defined in this way, then
2964:(2 ed.). pp. 3–4.
2535:More generally, the formula
1570:is the negative gradient of
1253:perpendicular to gravity is
87:is a fundamental concept in
729:is defined in terms of the
703:), and its components have
3061:
3045:Scalar physical quantities
2993:(5 ed.), p. 1199
2694:is the volume of the unit
1095:
1041:path independence property
617:. Certain aspects of the
61:: a directionless value (
34:. For all other uses, see
29:
2700:properties of convolution
2115:Written another way, let
1737:irrotational vector field
686:conservative vector field
2343:, it can be shown that
674:Integrability conditions
1098:Gravitational potential
660:Helmholtz decomposition
658:velocity field. By the
122:is defined such that:
118:, the scalar potential
2887:
2681:
2575:
2526:
2401:
2325:
2262:
2178:
2072:
1890:
1727:
1689:
1612:
1532:
1440:
1351:
1234:
1152:
1119:
1107:
1012:
880:
621:can be described by a
570:
499:
455:
408:evaluated at location
381:passing from location
371:
235:
77:
2995:avoid using the term
2888:
2682:
2576:
2527:
2402:
2326:
2263:
2179:
2073:
1891:
1728:
1690:
1613:
1533:
1441:
1352:
1235:
1153:
1151:{\displaystyle U=mgh}
1113:
1105:
1013:
881:
559:
500:
456:
372:
258:Cartesian coordinates
236:
75:
3021:at Wikimedia Commons
2960:Goldstein, Herbert.
2706:
2596:
2539:
2411:
2347:
2286:
2208:
2202:Dirac delta function
2190:fundamental solution
2119:
2078:This holds provided
1917:
1748:
1708:
1635:
1578:
1450:
1391:
1257:
1189:
1130:
1039:is, in essence, the
913:
737:
471:
420:
293:
126:
43:mathematical physics
2962:Classical Mechanics
2335:invariant potential
2186:Newtonian potential
1124:gravitational field
818:
708:partial derivatives
635:classical mechanics
599:electrostatic force
585:per unit mass. In
313:
2883:
2677:
2571:
2522:
2397:
2321:
2258:
2174:
2068:
1886:
1723:
1685:
1608:
1528:
1436:
1347:
1230:
1148:
1120:
1108:
1008:
876:
804:
652:conservative force
591:electric potential
571:
543:. A vector field
495:
451:
367:
299:
231:
78:
51:potential energies
32:electric potential
3017:Media related to
2971:978-0-201-02918-5
2855:
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2675:
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2152:
2043:
1958:
1861:
1781:
1523:
1434:
1324:
1300:
1288:
1220:
1116:inflection points
639:quantum mechanics
597:, i.e., with the
575:gravity potential
221:
198:
175:
16:(Redirected from
3052:
3019:Scalar potential
3016:
3000:
2994:
2989:Louis Leithold,
2982:
2976:
2975:
2957:
2940:
2929:
2902:Gradient theorem
2892:
2890:
2889:
2884:
2876:
2872:
2856:
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2852:
2843:
2839:
2830:
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2786:
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2406:
2404:
2403:
2398:
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2330:
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2327:
2322:
2308:
2281:
2277:
2267:
2265:
2264:
2259:
2248:
2231:
2220:
2219:
2199:
2194:Laplace equation
2183:
2181:
2180:
2175:
2173:
2171:
2167:
2155:
2153:
2151:
2140:
2132:
2111:
2104:
2094:
2083:
2077:
2075:
2074:
2069:
2064:
2060:
2044:
2042:
2038:
2037:
2033:
2024:
2014:
2010:
2006:
1997:
1985:
1983:
1982:
1981:
1980:
1975:
1964:
1959:
1957:
1946:
1935:
1924:
1912:
1906:
1895:
1893:
1892:
1887:
1882:
1878:
1862:
1860:
1856:
1855:
1851:
1842:
1832:
1828:
1824:
1815:
1803:
1801:
1800:
1799:
1798:
1793:
1782:
1780:
1769:
1761:
1743:
1734:
1732:
1730:
1729:
1724:
1722:
1721:
1716:
1694:
1692:
1691:
1686:
1681:
1663:
1662:
1647:
1646:
1617:
1615:
1614:
1609:
1592:
1591:
1590:
1553:
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1537:
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1534:
1529:
1524:
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1505:
1504:
1491:
1476:
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1437:
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1433:
1425:
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1386:
1379:
1369:
1365:
1356:
1354:
1353:
1348:
1325:
1317:
1298:
1286:
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1272:
1271:
1265:
1252:
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1239:
1237:
1236:
1231:
1218:
1205:
1204:
1203:
1197:
1180:
1174:
1165:
1161:
1157:
1155:
1154:
1149:
1087:
1078:
1074:
1070:
1064:
1060:
1050:
1047:implies that if
1038:
1032:
1023:
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1015:
1014:
1009:
1004:
987:
979:
978:
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831:
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797:
782:
774:
769:
768:
750:
728:
719:
683:
664:vector potential
656:solenoidal field
633:formulations of
623:Yukawa potential
569:
553:(conservative).
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522:
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607:fluid dynamics
595:electric field
587:electrostatics
514:differentiable
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731:line integral
724:
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688:(also called
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619:nuclear force
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280:In order for
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2339:
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2108:
2101:
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1898:
1744:is given by
1740:
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1376:
1372:
1359:
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1183:normal force
1177:
1121:
1081:
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1040:
1035:
1026:
1019:
905:
896:
892:parametrized
722:
716:
711:
700:
693:
690:irrotational
689:
680:
677:
647:conservative
645:
643:
579:acceleration
572:
566:
562:
551:irrotational
545:
535:
525:
507:
463:Jordan curve
410:
399:
395:
389:
387:to location
383:
282:
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269:
244:
114:
111:vector field
107:scalar field
100:
96:
81:
79:
55:scalar field
46:
40:
2270:convolution
2095:and if the
1357:This force
1168:contour map
631:Hamiltonian
59:three-space
3035:Potentials
3029:Categories
2947:References
2097:divergence
2086:continuous
1061:, so that
894:path from
705:continuous
627:Lagrangian
539:using the
379:Jordan arc
109:. Given a
2997:potential
2846:‖
2832:−
2824:‖
2806:−
2795:⋅
2753:∫
2740:ω
2727:−
2710:Φ
2667:−
2660:‖
2651:‖
2642:ω
2632:−
2600:Γ
2581:holds in
2566:Γ
2563:∗
2552:
2543:Φ
2515:−
2509:δ
2506:∗
2498:−
2492:Γ
2483:∇
2479:∗
2465:Γ
2462:∗
2445:∇
2435:Γ
2432:∗
2421:
2415:∇
2382:⋅
2378:∇
2370:∇
2352:∇
2313:Γ
2310:∗
2299:
2290:Φ
2239:δ
2222:Γ
2213:∇
2169:‖
2161:‖
2149:π
2123:Γ
2026:−
1991:
1967:∫
1962:∇
1955:π
1943:−
1937:Φ
1933:∇
1929:−
1844:−
1809:
1785:∫
1778:π
1752:Φ
1671:⋅
1665:∇
1656:∮
1652:−
1600:∇
1597:−
1510:Δ
1494:Δ
1486:Δ
1479:Δ
1467:−
1428:Δ
1420:Δ
1414:
1406:−
1395:θ
1342:θ
1336:
1314:−
1308:θ
1305:
1296:θ
1293:
1278:−
1228:θ
1225:
1210:−
943:≤
937:≤
845:⋅
806:∫
802:−
787:⋅
762:∫
758:−
712:potential
701:potential
601:per unit
480:×
476:∇
432:⋅
424:∮
348:−
320:⋅
301:∫
297:−
215:∂
207:∂
192:∂
184:∂
169:∂
161:∂
150:−
141:∇
138:−
84:potential
36:potential
2896:See also
2874:′
2841:′
2815:′
2788:′
2702:) gives
2062:′
2040:‖
2035:′
2017:‖
2008:′
1913:. Then
1880:′
1858:‖
1853:′
1835:‖
1826:′
1572:pressure
1446:so that
854:′
250:gradient
2192:of the
2184:be the
1733:
1704:
261:x, y, z
248:is the
93:physics
82:scalar
2968:
2687:where
2590:> 2
2407:Hence
2280:Γ
1896:where
1626:vortex
1299:
1287:
1240:where
1219:
1158:where
886:where
710:, the
603:charge
241:where
97:scalar
63:scalar
2919:Notes
2278:with
890:is a
699:, or
697:-free
684:is a
2984:See
2966:ISBN
2933:only
1373:Let
1056:= –∇
1033:and
695:curl
629:and
589:the
565:= –∇
531:curl
393:and
91:and
2549:div
2418:div
2296:div
2272:of
2099:of
2084:is
1988:div
1806:div
1562:In
1402:tan
1333:sin
1302:cos
1290:sin
1222:sin
903:to
714:of
678:If
533:of
404:is
252:of
57:in
41:In
3031::
2282::
2256:0.
2204::
2112:.
2107:1/
2090:1/
1910:r'
1903:r'
1899:dV
1574::
1088:.
909:,
733::
692:,
670:.
641:.
493:0.
80:A
69:.
45:,
2974:.
2939:.
2881:.
2878:)
2870:r
2865:(
2862:V
2859:d
2850:n
2837:r
2828:r
2819:)
2811:r
2802:r
2798:(
2792:)
2784:r
2779:(
2775:E
2764:n
2759:R
2744:n
2736:n
2732:1
2724:=
2721:)
2717:r
2713:(
2696:n
2691:n
2689:ω
2670:2
2664:n
2655:r
2646:n
2638:)
2635:2
2629:n
2626:(
2623:n
2619:1
2614:=
2611:)
2607:r
2603:(
2588:n
2583:n
2569:)
2559:E
2555:(
2546:=
2519:E
2512:=
2502:E
2495:=
2487:2
2475:E
2471:=
2468:)
2458:E
2454:(
2449:2
2441:=
2438:)
2428:E
2424:(
2395:.
2392:)
2388:G
2374:(
2366:=
2362:G
2356:2
2340:G
2319:.
2316:)
2306:E
2302:(
2293:=
2275:E
2253:=
2250:)
2246:r
2242:(
2236:+
2233:)
2229:r
2225:(
2217:2
2198:Γ
2165:r
2157:1
2146:4
2142:1
2137:=
2134:)
2130:r
2126:(
2109:r
2102:E
2092:r
2081:E
2066:)
2058:r
2053:(
2050:V
2047:d
2031:r
2022:r
2012:)
2004:r
1999:(
1995:E
1978:3
1973:R
1952:4
1948:1
1940:=
1926:=
1922:E
1905:)
1901:(
1884:)
1876:r
1871:(
1868:V
1865:d
1849:r
1840:r
1830:)
1822:r
1817:(
1813:E
1796:3
1791:R
1775:4
1771:1
1766:=
1763:)
1759:r
1755:(
1741:E
1719:3
1714:R
1683:.
1679:S
1675:d
1668:p
1660:S
1649:=
1644:B
1640:F
1606:.
1603:p
1594:=
1588:B
1584:f
1551:P
1548:F
1542:x
1540:Δ
1526:.
1518:2
1514:h
1507:+
1502:2
1498:x
1489:h
1482:x
1473:g
1470:m
1464:=
1459:P
1455:F
1431:x
1423:h
1409:1
1398:=
1384:x
1382:Δ
1377:h
1375:Δ
1368:θ
1363:P
1360:F
1345:.
1339:2
1330:g
1327:m
1322:2
1319:1
1311:=
1284:g
1281:m
1275:=
1269:P
1263:F
1250:S
1247:F
1242:θ
1216:g
1213:m
1207:=
1201:S
1195:F
1178:F
1173:U
1164:h
1160:U
1146:h
1143:g
1140:m
1137:=
1134:U
1085:0
1082:r
1077:V
1073:V
1068:F
1063:V
1058:V
1054:F
1049:V
1036:r
1030:0
1027:r
1022:C
1006:.
1002:r
998:=
995:)
992:b
989:(
985:r
981:,
975:0
971:r
966:=
963:)
960:a
957:(
953:r
949:,
946:b
940:t
934:a
931:,
928:)
925:t
922:(
918:r
906:r
900:0
897:r
888:C
874:,
871:t
868:d
864:)
861:t
858:(
850:r
842:)
839:)
836:t
833:(
829:r
825:(
821:F
815:b
810:a
799:=
795:r
791:d
784:)
780:r
776:(
772:F
766:C
755:=
752:)
748:r
744:(
741:V
726:0
723:r
717:F
681:F
567:P
563:F
546:F
536:F
526:F
521:P
490:=
485:F
465:.
449:,
446:0
443:=
439:l
435:d
428:F
414:.
411:b
406:P
402:)
400:b
398:(
396:P
390:b
384:a
365:,
362:)
358:a
354:(
351:P
345:)
341:b
337:(
334:P
331:=
327:l
323:d
316:F
310:b
305:a
283:F
275:P
270:F
265:P
254:P
245:P
243:∇
229:,
225:)
218:z
210:P
201:,
195:y
187:P
178:,
172:x
164:P
154:(
147:=
144:P
135:=
131:F
120:P
115:F
38:.
20:)
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