2968:
3924:
268:
178:
1265:
1517:
2864:
2466:
2695:
For example, a hollow sphere does not produce any net gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the resultant field is that of all masses not including the sphere, which can be inside and outside the sphere).
1388:
615:
1136:
2540:
1979:
1405:
878:
131:) is a vector field â a vector at each point of space (and time). It is defined so that the gravitational force experienced by a particle is equal to the mass of the particle multiplied by the gravitational field at that point.
2725:
1585:
779:
2357:
2182:
1031:
2950:
1747:
1827:
2699:
Although this follows in one or two lines of algebra from Gauss's law for gravity, it took Isaac Newton several pages of cumbersome calculus to derive it directly using his law of gravity; see the article
255:
931:
480:
1306:
2055:
543:
2299:
2567:
Gauss's law can be used to easily derive the gravitational field in certain cases where a direct application of Newton's law would be more difficult (but not impossible). See the article
1663:
2247:
2471:
2110:
294:
204:
695:
326:
2348:
506:
784:
3164:
1884:
1522:
700:
963:
2873:
3834:
3466:
1761:. It's reasonable to expect the gravitational field from a point mass to be spherically symmetric. (We omit the proof for simplicity.) By making this assumption,
1685:
2351:
1260:{\displaystyle \mathbf {g} (\mathbf {r} )=-G\int \rho (\mathbf {s} ){\frac {(\mathbf {r} -\mathbf {s} )}{|\mathbf {r} -\mathbf {s} |^{3}}}d^{3}\mathbf {s} .}
1892:
3272:
1768:
2116:
3194:
953:
60:
3770:
3443:
3698:
2623:
In particular, a parallel combination of two parallel infinite plates of equal mass per unit area produces no gravitational field between them.
3749:
3257:
1512:{\displaystyle \nabla \cdot \mathbf {g} (\mathbf {r} )=-4\pi G\int \rho (\mathbf {s} )\ \delta (\mathbf {r} -\mathbf {s} )\ d^{3}\mathbf {s} .}
2967:
3775:
3427:
3400:
3242:
3157:
2859:{\displaystyle {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\phi (\mathbf {x} ,t)-{1 \over 8\pi G}(\nabla \phi (\mathbf {x} ,t))^{2}}
887:
410:
of the gravitational field. Note that according to the law it is always negative (or zero), and never positive. This can be contrasted with
3520:
3451:
3574:
2461:{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}\,{\frac {\partial \phi }{\partial r}}\right)=4\pi G\rho (r)}
3827:
3755:
3303:
3286:
3227:
3408:
3237:
3074:
2256:
3150:
2215:
2593:) of any finite thickness, the gravitational field outside the plate is perpendicular to the plate, towards it, with magnitude 2
3209:
3039:
209:
3515:
3437:
436:
3708:
2651:
times the total mass per unit length at a smaller distance (from the axis), regardless of any masses at a larger distance.
3960:
3928:
3820:
3232:
655:
98:. Gauss's law for gravity has the same mathematical relation to Newton's law that Gauss's law for electrostatics bears to
32:
This article is about Gauss's law concerning the gravitational field. For analogous laws concerning different fields, see
3950:
3909:
2998:
2301:
This provides an alternate means of calculating the gravitational potential and gravitational field. Although computing
37:
3493:
1299:, each of which is integrated from ââ to +â.) If we take the divergence of both sides of this equation with respect to
3387:
3377:
3252:
1985:
642:
3335:
939:
Although the two forms are equivalent, one or the other might be more convenient to use in a particular computation.
1672:
are also necessary to prove Newton's law, such as the assumption that the field is zero infinitely far from a mass.
3965:
3693:
128:
1634:
3487:
3482:
2953:
2713:
1621:
3703:
3413:
3955:
3590:
3584:
1383:{\displaystyle \nabla \cdot \left({\frac {\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )}
3551:
2061:
274:
184:
3618:
2867:
1613:
1114:
3419:
3329:
3535:
517:
389:
95:
610:{\displaystyle \oint _{\partial V}\mathbf {g} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {g} \,dV}
301:
3854:
3667:
3646:
3639:
3579:
2309:
directly from Gauss's law, one or the other approach may be an easier computation in a given situation.
1075:
A proof using vector calculus is shown in the box below. It is mathematically identical to the proof of
936:
It is possible to derive the integral form from the differential form using the reverse of this method.
3204:
2312:
In radially symmetric systems, the gravitational potential is a function of only one variable (namely,
2250:
2315:
3843:
3734:
3653:
3632:
3625:
3500:
3355:
3313:
3115:
2982:
1399:
1068:
64:
697:
we can apply the divergence theorem to the integral form of Gauss's law for gravity, which becomes:
488:
379:
121:
115:
76:
3879:
3796:
3262:
3247:
3219:
2987:
2719:
2535:{\displaystyle \mathbf {g} (\mathbf {r} )=-\mathbf {e_{r}} {\frac {\partial \phi }{\partial r}}.}
2197:
1625:
1605:
537:
353:
103:
41:
2544:
When solving the equation it should be taken into account that in the case of finite densities â
1856:
3066:
3059:
2668:
In the case of a spherically symmetric mass distribution we can conclude (by using a spherical
2604:
More generally, for a mass distribution with the density depending on one
Cartesian coordinate
3874:
3869:
3780:
3688:
3432:
3360:
3131:
3089:
3070:
414:
for electricity, where the flux can be either positive or negative. The difference is because
3889:
3762:
3456:
3350:
3123:
3018:
2669:
2636:
2598:
2586:
2571:
for more details on how these derivations are done. Three such applications are as follows:
2568:
2209:
2205:
1084:
369:
138:
99:
83:
enclosed. Gauss's law for gravity is often more convenient to work from than Newton's law.
72:
17:
3884:
3461:
3382:
3342:
3308:
3003:
634:
3119:
3106:
Moody, M. V.; Paik, H. J. (1 March 1993). "Gauss's law test of gravity at short range".
3904:
3894:
3864:
3661:
2992:
2973:
1080:
1076:
411:
365:
154:
91:
87:
33:
1974:{\displaystyle g(r)\oint _{\partial V}(-\mathbf {e_{r}} )\cdot d\mathbf {A} =-4\pi GM}
3944:
3598:
3267:
2701:
2663:
2590:
2580:
873:{\displaystyle \int _{V}(\nabla \cdot \mathbf {g} )\ dV=\int _{V}(-4\pi G\rho )\ dV.}
357:
142:
2635:) cylindrically symmetric mass distribution we can conclude (by using a cylindrical
3556:
3365:
2552:
has to be continuous at boundaries (discontinuities of the density), and zero for
1580:{\displaystyle \nabla \cdot \mathbf {g} (\mathbf {r} )=-4\pi G\rho (\mathbf {r} )}
884:; the only way this can happen is if the integrands are equal. Hence we arrive at
2597:
times the mass per unit area, independent of the distance to the plate (see also
40:. For Gauss's theorem, a mathematical theorem relevant to all of these laws, see
3525:
3127:
1044:
774:{\displaystyle \int _{V}\nabla \cdot \mathbf {g} \ dV=-4\pi G\int _{V}\rho \ dV}
2177:{\displaystyle \mathbf {g} (\mathbf {r} )=-GM{\frac {\mathbf {e_{r}} }{r^{2}}}}
1026:{\displaystyle \mathbf {g} (\mathbf {r} )=-{\frac {GM}{r^{2}}}\mathbf {e_{r}} }
3739:
2963:
1064:
957:
509:
2945:{\displaystyle 4\pi G\rho (\mathbf {x} ,t)=\nabla ^{2}\phi (\mathbf {x} ,t).}
3859:
3744:
3726:
3530:
3507:
2654:
For example, inside an infinite uniform hollow cylinder, the field is zero.
1742:{\displaystyle \oint _{\partial V}\mathbf {g} \cdot d\mathbf {A} =-4\pi GM.}
536:
The two forms of Gauss's law for gravity are mathematically equivalent. The
3135:
3142:
3801:
3611:
3604:
3008:
2201:
2196:
Since the gravitational field has zero curl (equivalently, gravity is a
1133:, where this contribution is calculated by Newton's law. The result is:
3812:
3173:
1596:
It is impossible to mathematically prove Newton's law from Gauss's law
1587:
which is the differential form of Gauss's law for gravity, as desired.
1519:
Using the "sifting property" of the Dirac delta function, we arrive at
525:
48:
141:
of the gravitational field over a closed surface, analogous to how
3393:
3104:
For usage of the term "Gauss's law for gravity" see, for example,
2616:
times the difference in mass per unit area on either side of this
1822:{\displaystyle \mathbf {g} (\mathbf {r} )=-g(r)\,\mathbf {e_{r}} }
2305:
via
Poisson's equation is mathematically equivalent to computing
86:
The form of Gauss's law for gravity is mathematically similar to
3013:
1675:
The proof of Newton's law from these assumptions is as follows:
621:
is a closed region bounded by a simple closed oriented surface â
407:
158:
102:. This is because both Newton's law and Coulomb's law describe
80:
68:
3816:
3146:
2249:
Then the differential form of Gauss's law for gravity becomes
2731:
2870:
to this
Lagrangian, the result is Gauss's law for gravity:
1616:). In addition to Gauss's law, the assumption is used that
933:
which is the differential form of Gauss's law for gravity.
1592:
Deriving Newton's law from Gauss's law and irrotationality
880:
This has to hold simultaneously for every possible volume
2684:
times only the total mass within a smaller distance than
3090:
The mechanics problem solver, by Fogiel, pp 535â536
1591:
430:
The differential form of Gauss's law for gravity states
250:{\displaystyle \mathbf {g} \cdot d\mathbf {A} =-4\pi GM}
926:{\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho ,}
475:{\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho ,}
1063:
is the mass of the particle, which is assumed to be a
278:
188:
2876:
2728:
2474:
2360:
2318:
2259:
2218:
2119:
2064:
1988:
1895:
1859:
1771:
1688:
1637:
1525:
1408:
1309:
1139:
1105:, can be calculated by adding up the contribution to
966:
956:, which states that the gravitational field due to a
890:
787:
703:
658:
546:
491:
439:
304:
277:
212:
187:
171:
The integral form of Gauss's law for gravity states:
2191:
29:
Restatement of Newton's law of universal gravitation
3789:
3721:
3679:
3565:
3544:
3475:
3322:
3296:
3285:
3218:
3187:
3180:
1604:but does not contain any information regarding the
328:) denotes a surface integral over a closed surface,
3058:
2944:
2858:
2534:
2460:
2342:
2293:
2241:
2176:
2104:
2049:
1973:
1878:
1821:
1741:
1668:Even these are not enough: Boundary conditions on
1657:
1600:, because Gauss's law specifies the divergence of
1579:
1511:
1382:
1259:
1025:
925:
872:
773:
689:
609:
500:
474:
406:The left-hand side of this equation is called the
320:
288:
249:
198:
1841:depends only on the magnitude, not direction, of
1749:Apply this law to the situation where the volume
2050:{\displaystyle g(r)\cdot (-4\pi r^{2})=-4\pi GM}
1113:) due to every bit of mass in the universe (see
2643:from the center is inward with a magnitude of 2
2200:) as mentioned above, it can be written as the
398:is the total mass enclosed within the surface â
79:over any closed surface is proportional to the
3467:Degenerate Higher-Order Scalar-Tensor theories
2676:from the center is inward with a magnitude of
2192:Poisson's equation and gravitational potential
1845:). Plugging this in, and using the fact that â
364:, and whose direction is the outward-pointing
3828:
3158:
2294:{\displaystyle \nabla ^{2}\phi =4\pi G\rho .}
1682:Start with the integral form of Gauss's law:
1117:). To do this, we integrate over every point
145:is a surface integral of the magnetic field.
8:
2352:Del in cylindrical and spherical coordinates
1658:{\displaystyle \nabla \times \mathbf {g} =0}
952:Gauss's law for gravity can be derived from
59:, is a law of physics that is equivalent to
3051:
3049:
2242:{\displaystyle \mathbf {g} =-\nabla \phi .}
637:for more details). The gravitational field
3835:
3821:
3813:
3293:
3184:
3165:
3151:
3143:
2688:. All the mass at a greater distance than
645:vector field defined on a neighborhood of
418:can be either positive or negative, while
2925:
2913:
2892:
2875:
2850:
2832:
2802:
2785:
2765:
2739:
2730:
2729:
2727:
2692:from the center has no resultant effect.
2627:Cylindrically symmetric mass distribution
2509:
2502:
2497:
2483:
2475:
2473:
2409:
2408:
2402:
2378:
2370:
2361:
2359:
2335:
2330:
2325:
2317:
2264:
2258:
2219:
2217:
2166:
2155:
2150:
2148:
2128:
2120:
2118:
2094:
2080:
2063:
2020:
1987:
1948:
1932:
1927:
1912:
1894:
1870:
1858:
1812:
1807:
1806:
1780:
1772:
1770:
1713:
1702:
1693:
1687:
1644:
1636:
1569:
1540:
1532:
1524:
1501:
1495:
1480:
1472:
1455:
1423:
1415:
1407:
1372:
1344:
1339:
1333:
1328:
1322:
1320:
1308:
1249:
1243:
1230:
1225:
1219:
1211:
1206:
1196:
1188:
1182:
1174:
1148:
1140:
1138:
1016:
1011:
1003:
989:
975:
967:
965:
897:
889:
831:
807:
792:
786:
753:
720:
708:
702:
669:
657:
600:
595:
583:
571:
560:
551:
545:
490:
446:
438:
309:
303:
276:
224:
213:
211:
186:
2672:) that the field strength at a distance
2639:) that the field strength at a distance
2105:{\displaystyle g(r)={\frac {GM}{r^{2}}}}
1121:in space, adding up the contribution to
629:is an infinitesimal piece of the volume
3771:Gravitational interaction of antimatter
3065:(3rd ed.). Prentice Hall. p.
3031:
2658:Spherically symmetric mass distribution
2631:In the case of an infinite uniform (in
2350:), and Poisson's equation becomes (see
1129:) associated with the mass (if any) at
289:{\displaystyle \scriptstyle \partial V}
199:{\displaystyle \scriptstyle \partial V}
948:Deriving Gauss's law from Newton's law
106:interaction in a 3-dimensional space.
3776:Physics in the medieval Islamic world
3699:(2+1)-dimensional topological gravity
3195:Newton's law of universal gravitation
2589:") that for an infinite, flat plate (
1849:is a spherical surface with constant
954:Newton's law of universal gravitation
947:
61:Newton's law of universal gravitation
7:
3521:Asymptotic safety in quantum gravity
1833:is antiparallel to the direction of
690:{\displaystyle M=\int _{V}\rho \ dV}
356:, whose magnitude is the area of an
321:{\displaystyle \oint _{\partial V}}
153:The gravitational flux through any
3273:GibbonsâHawkingâYork boundary term
2910:
2823:
2520:
2512:
2468:while the gravitational field is:
2420:
2412:
2384:
2380:
2261:
2230:
1913:
1694:
1638:
1526:
1409:
1310:
891:
801:
714:
589:
552:
492:
440:
310:
279:
189:
25:
3388:Modified Newtonian dynamics, MOND
3304:Classical theories of gravitation
1624:(has zero curl), as gravity is a
3923:
3922:
2966:
2926:
2893:
2833:
2786:
2766:
2740:
2503:
2499:
2484:
2476:
2343:{\displaystyle r=|\mathbf {r} |}
2331:
2220:
2156:
2152:
2129:
2121:
1949:
1933:
1929:
1813:
1809:
1781:
1773:
1714:
1703:
1645:
1570:
1541:
1533:
1502:
1481:
1473:
1456:
1424:
1416:
1373:
1334:
1323:
1250:
1220:
1212:
1197:
1189:
1175:
1149:
1141:
1017:
1013:
976:
968:
898:
808:
721:
596:
572:
561:
447:
266:
225:
214:
176:
157:is proportional to the enclosed
148:Gauss's law for gravity states:
110:Qualitative statement of the law
57:Gauss's flux theorem for gravity
3210:History of gravitational theory
3061:Introduction to Electrodynamics
3516:Causal dynamical triangulation
3205:Poisson's equation for gravity
2936:
2922:
2903:
2889:
2847:
2843:
2829:
2820:
2796:
2782:
2776:
2762:
2750:
2736:
2488:
2480:
2455:
2449:
2336:
2326:
2133:
2125:
2074:
2068:
2026:
2004:
1998:
1992:
1939:
1921:
1905:
1899:
1803:
1797:
1785:
1777:
1574:
1566:
1545:
1537:
1485:
1469:
1460:
1452:
1428:
1420:
1377:
1369:
1340:
1329:
1226:
1207:
1201:
1185:
1179:
1171:
1153:
1145:
1101:), the gravitational field at
980:
972:
855:
837:
812:
798:
1:
2585:We can conclude (by using a "
532:Relation to the integral form
501:{\displaystyle \nabla \cdot }
3057:Griffiths, David J. (1998).
2704:for this direct derivation.
1303:, and use the known theorem
3378:Infinite derivative gravity
3128:10.1103/PhysRevLett.70.1195
643:continuously differentiable
335:is any closed surface (the
18:Gauss' law for gravity
3982:
3566:Unified-field-theoric and
2711:
2708:Derivation from Lagrangian
2661:
2578:
1879:{\displaystyle 4\pi r^{2}}
1765:takes the following form:
129:gravitational acceleration
113:
31:
3918:
3910:Gauss's law for magnetism
3850:
3709:JackiwâTeitelboim gravity
3488:Canonical quantum gravity
3483:Euclidean quantum gravity
3040:"Gauss's law and gravity"
2999:Gauss's law for magnetism
2954:Lagrangian (field theory)
2722:for Newtonian gravity is
2714:Lagrangian (field theory)
1757:centered on a point-mass
38:Gauss's law for magnetism
3585:Superfluid vacuum theory
3409:Nonsymmetric gravitation
3258:Post-Newtonian formalism
1829:(i.e., the direction of
943:Relation to Newton's law
781:which can be rewritten:
3900:Gauss's law for gravity
3750:Mechanical explanations
3619:Heterotic string theory
3575:Noncommutative geometry
3494:WheelerâDeWitt equation
3220:General relativity (GR)
3200:Gauss's law for gravity
3174:Theories of gravitation
3108:Physical Review Letters
2210:gravitational potential
2187:which is Newton's law.
1837:, and the magnitude of
1614:Helmholtz decomposition
1115:superposition principle
339:of an arbitrary volume
53:Gauss's law for gravity
3591:Logarithmic BEC vacuum
3536:Rainbow gravity theory
3414:Scalarâtensor theories
3188:Newtonian gravity (NG)
2946:
2860:
2608:only, gravity for any
2536:
2462:
2344:
2295:
2243:
2178:
2106:
2051:
1975:
1880:
1823:
1753:is a sphere of radius
1743:
1659:
1581:
1513:
1384:
1261:
1027:
927:
874:
775:
691:
611:
518:gravitational constant
502:
476:
422:can only be positive.
390:gravitational constant
360:piece of the surface â
322:
290:
251:
200:
3855:Gauss composition law
3668:Twistor string theory
3647:Type II string theory
3640:Bosonic string theory
3580:Semiclassical gravity
3545:Unified-field-theoric
3330:Poincaré gauge theory
2947:
2861:
2537:
2463:
2345:
2296:
2244:
2179:
2107:
2052:
1976:
1881:
1824:
1744:
1660:
1582:
1514:
1385:
1262:
1028:
928:
875:
776:
692:
612:
503:
477:
323:
291:
252:
201:
67:. It states that the
3961:Carl Friedrich Gauss
3844:Carl Friedrich Gauss
3735:Aristotelian physics
3704:GaussâBonnet gravity
3654:Little string theory
3633:Type 0 string theory
3626:Type I string theory
3501:Loop quantum gravity
3428:Scalarâtensorâvector
3401:Tensorâvectorâscalar
3356:Gauge theory gravity
3314:Theory of everything
2983:Carl Friedrich Gauss
2874:
2868:Hamilton's principle
2726:
2472:
2358:
2316:
2257:
2216:
2117:
2062:
1986:
1893:
1857:
1769:
1686:
1635:
1523:
1406:
1400:Dirac delta function
1307:
1137:
964:
888:
785:
701:
656:
544:
489:
437:
302:
275:
210:
185:
65:Carl Friedrich Gauss
63:. It is named after
3951:Theories of gravity
3552:KaluzaâKlein theory
3120:1993PhRvL..70.1195M
942:
380:gravitational field
122:gravitational field
116:Gravitational field
96:Maxwell's equations
77:gravitational field
3880:Gaussian curvature
3797:Gravitational wave
3680:Generalisations /
3568:quantum-mechanical
3476:Quantum-mechanical
3289:general relativity
3263:Linearized gravity
3055:See, for example,
2988:Divergence theorem
2942:
2856:
2720:Lagrangian density
2532:
2458:
2340:
2291:
2251:Poisson's equation
2239:
2198:conservative force
2174:
2102:
2047:
1971:
1876:
1819:
1739:
1680:
1655:
1626:conservative force
1577:
1509:
1380:
1257:
1092:
1023:
923:
870:
771:
687:
607:
538:divergence theorem
498:
472:
372:for more details),
318:
286:
285:
247:
196:
195:
135:Gravitational flux
42:Divergence theorem
3966:Newtonian gravity
3938:
3937:
3875:Gaussian brackets
3810:
3809:
3781:Theory of impetus
3717:
3716:
3689:Liouville gravity
3433:Conformal gravity
3361:Composite gravity
3351:Bimetric theories
3281:
3280:
2818:
2599:gravity anomalies
2527:
2427:
2391:
2376:
2172:
2100:
1678:
1490:
1465:
1351:
1237:
1090:
1009:
860:
817:
764:
727:
680:
516:is the universal
426:Differential form
388:is the universal
16:(Redirected from
3973:
3926:
3925:
3890:Gaussian surface
3837:
3830:
3823:
3814:
3765:
3763:Entropic gravity
3758:
3682:extensions of GR
3670:
3656:
3649:
3642:
3635:
3628:
3621:
3614:
3607:
3593:
3510:
3503:
3496:
3457:Geometrodynamics
3446:
3422:
3403:
3396:
3345:
3338:
3294:
3185:
3167:
3160:
3153:
3144:
3139:
3114:(9): 1195â1198.
3092:
3087:
3081:
3080:
3064:
3053:
3044:
3043:
3036:
3019:Gaussian surface
2976:
2971:
2970:
2951:
2949:
2948:
2943:
2929:
2918:
2917:
2896:
2865:
2863:
2862:
2857:
2855:
2854:
2836:
2819:
2817:
2803:
2789:
2769:
2743:
2735:
2734:
2670:Gaussian surface
2637:Gaussian surface
2587:Gaussian pillbox
2569:Gaussian surface
2558:
2541:
2539:
2538:
2533:
2528:
2526:
2518:
2510:
2508:
2507:
2506:
2487:
2479:
2467:
2465:
2464:
2459:
2433:
2429:
2428:
2426:
2418:
2410:
2407:
2406:
2392:
2390:
2379:
2377:
2375:
2374:
2362:
2349:
2347:
2346:
2341:
2339:
2334:
2329:
2300:
2298:
2297:
2292:
2269:
2268:
2248:
2246:
2245:
2240:
2223:
2206:scalar potential
2183:
2181:
2180:
2175:
2173:
2171:
2170:
2161:
2160:
2159:
2149:
2132:
2124:
2111:
2109:
2108:
2103:
2101:
2099:
2098:
2089:
2081:
2056:
2054:
2053:
2048:
2025:
2024:
1980:
1978:
1977:
1972:
1952:
1938:
1937:
1936:
1920:
1919:
1885:
1883:
1882:
1877:
1875:
1874:
1828:
1826:
1825:
1820:
1818:
1817:
1816:
1784:
1776:
1748:
1746:
1745:
1740:
1717:
1706:
1701:
1700:
1679:Outline of proof
1664:
1662:
1661:
1656:
1648:
1586:
1584:
1583:
1578:
1573:
1544:
1536:
1518:
1516:
1515:
1510:
1505:
1500:
1499:
1488:
1484:
1476:
1463:
1459:
1427:
1419:
1402:, the result is
1389:
1387:
1386:
1381:
1376:
1356:
1352:
1350:
1349:
1348:
1343:
1337:
1332:
1326:
1321:
1266:
1264:
1263:
1258:
1253:
1248:
1247:
1238:
1236:
1235:
1234:
1229:
1223:
1215:
1210:
1204:
1200:
1192:
1183:
1178:
1152:
1144:
1091:Outline of proof
1083:) starting from
1053:is the radius, |
1032:
1030:
1029:
1024:
1022:
1021:
1020:
1010:
1008:
1007:
998:
990:
979:
971:
932:
930:
929:
924:
901:
879:
877:
876:
871:
858:
836:
835:
815:
811:
797:
796:
780:
778:
777:
772:
762:
758:
757:
725:
724:
713:
712:
696:
694:
693:
688:
678:
674:
673:
652:Given also that
616:
614:
613:
608:
599:
588:
587:
575:
564:
559:
558:
507:
505:
504:
499:
481:
479:
478:
473:
450:
370:surface integral
327:
325:
324:
319:
317:
316:
297:
296:
295:
293:
292:
287:
270:
269:
257:
256:
254:
253:
248:
228:
217:
206:
205:
203:
202:
197:
180:
179:
139:surface integral
73:surface integral
55:, also known as
21:
3981:
3980:
3976:
3975:
3974:
3972:
3971:
3970:
3956:Vector calculus
3941:
3940:
3939:
3934:
3914:
3885:Gaussian period
3846:
3841:
3811:
3806:
3785:
3761:
3754:
3725:
3723:
3713:
3694:Lovelock theory
3681:
3675:
3666:
3652:
3645:
3638:
3631:
3624:
3617:
3610:
3603:
3589:
3567:
3561:
3540:
3506:
3499:
3492:
3471:
3462:Induced gravity
3442:
3438:Scalar theories
3418:
3399:
3392:
3383:Massive gravity
3343:Teleparallelism
3341:
3336:EinsteinâCartan
3334:
3318:
3309:Quantum gravity
3288:
3287:Alternatives to
3277:
3243:Exact solutions
3214:
3176:
3171:
3105:
3101:
3099:Further reading
3096:
3095:
3088:
3084:
3077:
3056:
3054:
3047:
3038:
3037:
3033:
3028:
3023:
3004:Vector calculus
2995:for electricity
2972:
2965:
2962:
2909:
2872:
2871:
2846:
2807:
2724:
2723:
2716:
2710:
2666:
2660:
2629:
2583:
2577:
2565:
2553:
2519:
2511:
2498:
2470:
2469:
2419:
2411:
2398:
2397:
2393:
2383:
2366:
2356:
2355:
2314:
2313:
2260:
2255:
2254:
2214:
2213:
2194:
2189:
2162:
2151:
2115:
2114:
2090:
2082:
2060:
2059:
2016:
1984:
1983:
1928:
1908:
1891:
1890:
1866:
1855:
1854:
1808:
1767:
1766:
1689:
1684:
1683:
1633:
1632:
1594:
1589:
1521:
1520:
1491:
1404:
1403:
1338:
1327:
1316:
1305:
1304:
1298:
1290:
1282:
1239:
1224:
1205:
1184:
1135:
1134:
1067:located at the
1041:
1012:
999:
991:
962:
961:
950:
945:
886:
885:
827:
788:
783:
782:
749:
704:
699:
698:
665:
654:
653:
635:volume integral
579:
547:
542:
541:
534:
528:at each point.
487:
486:
483:
435:
434:
428:
305:
300:
299:
273:
272:
271:
267:
265:
259:
208:
207:
183:
182:
181:
177:
175:
169:
118:
112:
45:
30:
23:
22:
15:
12:
11:
5:
3979:
3977:
3969:
3968:
3963:
3958:
3953:
3943:
3942:
3936:
3935:
3933:
3932:
3919:
3916:
3915:
3913:
3912:
3907:
3902:
3897:
3895:Gaussian units
3892:
3887:
3882:
3877:
3872:
3870:Gauss's method
3867:
3865:Gauss notation
3862:
3857:
3851:
3848:
3847:
3842:
3840:
3839:
3832:
3825:
3817:
3808:
3807:
3805:
3804:
3799:
3793:
3791:
3790:Related topics
3787:
3786:
3784:
3783:
3778:
3773:
3768:
3767:
3766:
3759:
3747:
3742:
3737:
3731:
3729:
3719:
3718:
3715:
3714:
3712:
3711:
3706:
3701:
3696:
3691:
3685:
3683:
3677:
3676:
3674:
3673:
3672:
3671:
3662:Twistor theory
3659:
3658:
3657:
3650:
3643:
3636:
3629:
3622:
3615:
3608:
3596:
3595:
3594:
3582:
3577:
3571:
3569:
3563:
3562:
3560:
3559:
3554:
3548:
3546:
3542:
3541:
3539:
3538:
3533:
3528:
3523:
3518:
3513:
3512:
3511:
3504:
3497:
3485:
3479:
3477:
3473:
3472:
3470:
3469:
3464:
3459:
3454:
3449:
3448:
3447:
3435:
3430:
3425:
3424:
3423:
3411:
3406:
3405:
3404:
3397:
3385:
3380:
3375:
3363:
3358:
3353:
3348:
3347:
3346:
3339:
3326:
3324:
3320:
3319:
3317:
3316:
3311:
3306:
3300:
3298:
3291:
3283:
3282:
3279:
3278:
3276:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3224:
3222:
3216:
3215:
3213:
3212:
3207:
3202:
3197:
3191:
3189:
3182:
3178:
3177:
3172:
3170:
3169:
3162:
3155:
3147:
3141:
3140:
3100:
3097:
3094:
3093:
3082:
3075:
3045:
3030:
3029:
3027:
3024:
3022:
3021:
3016:
3011:
3006:
3001:
2996:
2990:
2985:
2979:
2978:
2977:
2974:Physics portal
2961:
2958:
2941:
2938:
2935:
2932:
2928:
2924:
2921:
2916:
2912:
2908:
2905:
2902:
2899:
2895:
2891:
2888:
2885:
2882:
2879:
2853:
2849:
2845:
2842:
2839:
2835:
2831:
2828:
2825:
2822:
2816:
2813:
2810:
2806:
2801:
2798:
2795:
2792:
2788:
2784:
2781:
2778:
2775:
2772:
2768:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2742:
2738:
2733:
2712:Main article:
2709:
2706:
2662:Main article:
2659:
2656:
2628:
2625:
2579:Main article:
2576:
2573:
2564:
2561:
2531:
2525:
2522:
2517:
2514:
2505:
2501:
2496:
2493:
2490:
2486:
2482:
2478:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2432:
2425:
2422:
2417:
2414:
2405:
2401:
2396:
2389:
2386:
2382:
2373:
2369:
2365:
2338:
2333:
2328:
2324:
2321:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2267:
2263:
2238:
2235:
2232:
2229:
2226:
2222:
2193:
2190:
2185:
2184:
2169:
2165:
2158:
2154:
2147:
2144:
2141:
2138:
2135:
2131:
2127:
2123:
2112:
2097:
2093:
2088:
2085:
2079:
2076:
2073:
2070:
2067:
2057:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2023:
2019:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1981:
1970:
1967:
1964:
1961:
1958:
1955:
1951:
1947:
1944:
1941:
1935:
1931:
1926:
1923:
1918:
1915:
1911:
1907:
1904:
1901:
1898:
1873:
1869:
1865:
1862:
1815:
1811:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1783:
1779:
1775:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1716:
1712:
1709:
1705:
1699:
1696:
1692:
1677:
1666:
1665:
1654:
1651:
1647:
1643:
1640:
1593:
1590:
1576:
1572:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1543:
1539:
1535:
1531:
1528:
1508:
1504:
1498:
1494:
1487:
1483:
1479:
1475:
1471:
1468:
1462:
1458:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1426:
1422:
1418:
1414:
1411:
1379:
1375:
1371:
1368:
1365:
1362:
1359:
1355:
1347:
1342:
1336:
1331:
1325:
1319:
1315:
1312:
1294:
1286:
1278:
1256:
1252:
1246:
1242:
1233:
1228:
1222:
1218:
1214:
1209:
1203:
1199:
1195:
1191:
1187:
1181:
1177:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1151:
1147:
1143:
1089:
1081:electrostatics
1073:
1072:
1058:
1048:
1043:is the radial
1039:
1019:
1015:
1006:
1002:
997:
994:
988:
985:
982:
978:
974:
970:
949:
946:
944:
941:
922:
919:
916:
913:
910:
907:
904:
900:
896:
893:
869:
866:
863:
857:
854:
851:
848:
845:
842:
839:
834:
830:
826:
823:
820:
814:
810:
806:
803:
800:
795:
791:
770:
767:
761:
756:
752:
748:
745:
742:
739:
736:
733:
730:
723:
719:
716:
711:
707:
686:
683:
677:
672:
668:
664:
661:
606:
603:
598:
594:
591:
586:
582:
578:
574:
570:
567:
563:
557:
554:
550:
533:
530:
497:
494:
471:
468:
465:
462:
459:
456:
453:
449:
445:
442:
432:
427:
424:
404:
403:
393:
383:
373:
366:surface normal
344:
329:
315:
312:
308:
298:(also written
284:
281:
246:
243:
240:
237:
234:
231:
227:
223:
220:
216:
194:
191:
173:
168:
165:
164:
163:
155:closed surface
114:Main article:
111:
108:
104:inverse-square
92:electrostatics
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3978:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3948:
3946:
3931:
3930:
3921:
3920:
3917:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3852:
3849:
3845:
3838:
3833:
3831:
3826:
3824:
3819:
3818:
3815:
3803:
3800:
3798:
3795:
3794:
3792:
3788:
3782:
3779:
3777:
3774:
3772:
3769:
3764:
3760:
3757:
3756:FatioâLe Sage
3753:
3752:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3732:
3730:
3728:
3722:Pre-Newtonian
3720:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3686:
3684:
3678:
3669:
3665:
3664:
3663:
3660:
3655:
3651:
3648:
3644:
3641:
3637:
3634:
3630:
3627:
3623:
3620:
3616:
3613:
3609:
3606:
3602:
3601:
3600:
3599:String theory
3597:
3592:
3588:
3587:
3586:
3583:
3581:
3578:
3576:
3573:
3572:
3570:
3564:
3558:
3555:
3553:
3550:
3549:
3547:
3543:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3509:
3505:
3502:
3498:
3495:
3491:
3490:
3489:
3486:
3484:
3481:
3480:
3478:
3474:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3445:
3441:
3440:
3439:
3436:
3434:
3431:
3429:
3426:
3421:
3417:
3416:
3415:
3412:
3410:
3407:
3402:
3398:
3395:
3391:
3390:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3372:
3368:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3344:
3340:
3337:
3333:
3332:
3331:
3328:
3327:
3325:
3321:
3315:
3312:
3310:
3307:
3305:
3302:
3301:
3299:
3295:
3292:
3290:
3284:
3274:
3271:
3269:
3268:ADM formalism
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3225:
3223:
3221:
3217:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3193:
3192:
3190:
3186:
3183:
3179:
3175:
3168:
3163:
3161:
3156:
3154:
3149:
3148:
3145:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3103:
3102:
3098:
3091:
3086:
3083:
3078:
3076:0-13-805326-X
3072:
3068:
3063:
3062:
3052:
3050:
3046:
3041:
3035:
3032:
3025:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2994:
2991:
2989:
2986:
2984:
2981:
2980:
2975:
2969:
2964:
2959:
2957:
2956:for details.
2955:
2939:
2933:
2930:
2919:
2914:
2906:
2900:
2897:
2886:
2883:
2880:
2877:
2869:
2851:
2840:
2837:
2826:
2814:
2811:
2808:
2804:
2799:
2793:
2790:
2779:
2773:
2770:
2759:
2756:
2753:
2747:
2744:
2721:
2715:
2707:
2705:
2703:
2702:shell theorem
2697:
2693:
2691:
2687:
2683:
2679:
2675:
2671:
2665:
2664:Shell theorem
2657:
2655:
2652:
2650:
2646:
2642:
2638:
2634:
2626:
2624:
2621:
2619:
2615:
2611:
2607:
2602:
2600:
2596:
2592:
2591:Bouguer plate
2588:
2582:
2581:Bouguer plate
2575:Bouguer plate
2574:
2572:
2570:
2562:
2560:
2556:
2551:
2547:
2542:
2529:
2523:
2515:
2494:
2491:
2452:
2446:
2443:
2440:
2437:
2434:
2430:
2423:
2415:
2403:
2399:
2394:
2387:
2371:
2367:
2363:
2353:
2322:
2319:
2310:
2308:
2304:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2265:
2252:
2236:
2233:
2227:
2224:
2211:
2208:, called the
2207:
2203:
2199:
2188:
2167:
2163:
2145:
2142:
2139:
2136:
2113:
2095:
2091:
2086:
2083:
2077:
2071:
2065:
2058:
2044:
2041:
2038:
2035:
2032:
2029:
2021:
2017:
2013:
2010:
2007:
2001:
1995:
1989:
1982:
1968:
1965:
1962:
1959:
1956:
1953:
1945:
1942:
1924:
1916:
1909:
1902:
1896:
1889:
1888:
1887:
1871:
1867:
1863:
1860:
1852:
1848:
1844:
1840:
1836:
1832:
1800:
1794:
1791:
1788:
1764:
1760:
1756:
1752:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1710:
1707:
1697:
1690:
1676:
1673:
1671:
1652:
1649:
1641:
1631:
1630:
1629:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1588:
1563:
1560:
1557:
1554:
1551:
1548:
1529:
1506:
1496:
1492:
1477:
1466:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1412:
1401:
1397:
1393:
1366:
1363:
1360:
1357:
1353:
1345:
1317:
1313:
1302:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1270:
1254:
1244:
1240:
1231:
1216:
1193:
1168:
1165:
1162:
1159:
1156:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1088:
1086:
1085:Coulomb's law
1082:
1078:
1070:
1066:
1062:
1059:
1056:
1052:
1049:
1046:
1042:
1036:
1035:
1034:
1004:
1000:
995:
992:
986:
983:
959:
955:
940:
937:
934:
920:
917:
914:
911:
908:
905:
902:
894:
883:
867:
864:
861:
852:
849:
846:
843:
840:
832:
828:
824:
821:
818:
804:
793:
789:
768:
765:
759:
754:
750:
746:
743:
740:
737:
734:
731:
728:
717:
709:
705:
684:
681:
675:
670:
666:
662:
659:
650:
648:
644:
640:
636:
632:
628:
624:
620:
604:
601:
592:
584:
580:
576:
568:
565:
555:
548:
539:
531:
529:
527:
523:
519:
515:
511:
495:
482:
469:
466:
463:
460:
457:
454:
451:
443:
431:
425:
423:
421:
417:
413:
409:
401:
397:
394:
391:
387:
384:
381:
377:
374:
371:
367:
363:
359:
358:infinitesimal
355:
351:
348:
345:
342:
338:
334:
330:
313:
306:
282:
264:
263:
262:
258:
244:
241:
238:
235:
232:
229:
221:
218:
192:
172:
167:Integral form
166:
162:
160:
156:
151:
150:
149:
146:
144:
143:magnetic flux
140:
136:
132:
130:
127:(also called
126:
123:
117:
109:
107:
105:
101:
100:Coulomb's law
97:
93:
89:
84:
82:
78:
74:
70:
66:
62:
58:
54:
50:
43:
39:
35:
27:
19:
3927:
3899:
3724:theories and
3557:Supergravity
3370:
3366:
3228:Introduction
3199:
3111:
3107:
3085:
3060:
3034:
2717:
2698:
2694:
2689:
2685:
2681:
2677:
2673:
2667:
2653:
2648:
2644:
2640:
2632:
2630:
2622:
2617:
2613:
2609:
2605:
2603:
2594:
2584:
2566:
2563:Applications
2554:
2549:
2545:
2543:
2311:
2306:
2302:
2195:
2186:
1850:
1846:
1842:
1838:
1834:
1830:
1762:
1758:
1754:
1750:
1681:
1674:
1669:
1667:
1622:irrotational
1617:
1609:
1601:
1597:
1595:
1395:
1391:
1300:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1268:
1130:
1126:
1122:
1118:
1110:
1106:
1102:
1098:
1094:
1093:
1074:
1060:
1054:
1050:
1037:
951:
938:
935:
881:
651:
646:
638:
630:
626:
622:
618:
535:
526:mass density
521:
513:
484:
433:
429:
419:
415:
405:
399:
395:
385:
375:
361:
349:
346:
340:
336:
332:
260:
174:
170:
152:
147:
134:
133:
124:
119:
85:
56:
52:
46:
26:
3905:Gauss's law
3526:Causal sets
3420:BransâDicke
3238:Mathematics
2993:Gauss's law
1274:stands for
1077:Gauss's law
1045:unit vector
412:Gauss's law
88:Gauss's law
34:Gauss's law
3945:Categories
3740:CGHS model
3727:toy models
3026:References
1065:point mass
958:point mass
641:must be a
510:divergence
3860:Gauss map
3745:RST model
3531:DGP model
3508:Spin foam
3452:Whitehead
3444:Nordström
3373:) gravity
3323:Classical
3297:Paradigms
3248:Resources
2920:ϕ
2911:∇
2887:ρ
2881:π
2866:Applying
2827:ϕ
2824:∇
2812:π
2800:−
2780:ϕ
2760:ρ
2757:−
2521:∂
2516:ϕ
2513:∂
2495:−
2447:ρ
2441:π
2421:∂
2416:ϕ
2413:∂
2385:∂
2381:∂
2286:ρ
2280:π
2271:ϕ
2262:∇
2234:ϕ
2231:∇
2228:−
2140:−
2039:π
2033:−
2014:π
2008:−
2002:⋅
1963:π
1957:−
1943:⋅
1925:−
1914:∂
1910:∮
1864:π
1853:and area
1792:−
1728:π
1722:−
1708:⋅
1695:∂
1691:∮
1642:×
1639:∇
1564:ρ
1558:π
1552:−
1530:⋅
1527:∇
1478:−
1467:δ
1450:ρ
1447:∫
1441:π
1435:−
1413:⋅
1410:∇
1398:) is the
1367:δ
1364:π
1314:⋅
1311:∇
1217:−
1194:−
1169:ρ
1166:∫
1160:−
987:−
918:ρ
912:π
906:−
895:⋅
892:∇
853:ρ
847:π
841:−
829:∫
805:⋅
802:∇
790:∫
760:ρ
751:∫
744:π
738:−
718:⋅
715:∇
706:∫
676:ρ
667:∫
593:⋅
590:∇
581:∫
566:⋅
553:∂
549:∮
496:⋅
493:∇
467:ρ
461:π
455:−
444:⋅
441:∇
311:∂
307:∮
280:∂
239:π
233:−
219:⋅
190:∂
94:, one of
75:) of the
3929:Category
3802:Graviton
3612:F-theory
3605:M-theory
3181:Standard
3136:10054315
3009:Integral
2960:See also
2202:gradient
540:states:
508:denotes
337:boundary
3233:History
3116:Bibcode
2620:value.
524:is the
378:is the
49:physics
3134:
3073:
1489:
1464:
1390:where
1069:origin
1033:where
859:
816:
763:
726:
679:
617:where
520:, and
485:where
416:charge
354:vector
261:where
3394:AQUAL
3253:Tests
2204:of a
1612:(see
1598:alone
633:(see
392:, and
368:(see
352:is a
137:is a
3132:PMID
3071:ISBN
3014:Flux
2952:See
2718:The
2612:is 2
1606:curl
1079:(in
960:is:
625:and
420:mass
408:flux
159:mass
120:The
90:for
81:mass
69:flux
36:and
3124:doi
2601:).
2557:= 0
2354:):
1620:is
1608:of
47:In
3947::
3130:.
3122:.
3112:70
3110:.
3069:.
3067:50
3048:^
2614:ÏG
2595:ÏG
2559:.
2548:/â
2253::
2212::
1886:,
1628::
1292:ds
1284:ds
1276:ds
1087:.
1057:|.
649:.
627:dV
512:,
343:),
51:,
3836:e
3829:t
3822:v
3371:R
3369:(
3367:f
3166:e
3159:t
3152:v
3138:.
3126::
3118::
3079:.
3042:.
2940:.
2937:)
2934:t
2931:,
2927:x
2923:(
2915:2
2907:=
2904:)
2901:t
2898:,
2894:x
2890:(
2884:G
2878:4
2852:2
2848:)
2844:)
2841:t
2838:,
2834:x
2830:(
2821:(
2815:G
2809:8
2805:1
2797:)
2794:t
2791:,
2787:x
2783:(
2777:)
2774:t
2771:,
2767:x
2763:(
2754:=
2751:)
2748:t
2745:,
2741:x
2737:(
2732:L
2690:r
2686:r
2682:r
2680:/
2678:G
2674:r
2649:r
2647:/
2645:G
2641:r
2633:z
2618:z
2610:z
2606:z
2555:r
2550:r
2546:Ï
2530:.
2524:r
2504:r
2500:e
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2489:)
2485:r
2481:(
2477:g
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2453:r
2450:(
2444:G
2438:4
2435:=
2431:)
2424:r
2404:2
2400:r
2395:(
2388:r
2372:2
2368:r
2364:1
2337:|
2332:r
2327:|
2323:=
2320:r
2307:g
2303:g
2289:.
2283:G
2277:4
2274:=
2266:2
2237:.
2225:=
2221:g
2168:2
2164:r
2157:r
2153:e
2146:M
2143:G
2137:=
2134:)
2130:r
2126:(
2122:g
2096:2
2092:r
2087:M
2084:G
2078:=
2075:)
2072:r
2069:(
2066:g
2045:M
2042:G
2036:4
2030:=
2027:)
2022:2
2018:r
2011:4
2005:(
1999:)
1996:r
1993:(
1990:g
1969:M
1966:G
1960:4
1954:=
1950:A
1946:d
1940:)
1934:r
1930:e
1922:(
1917:V
1906:)
1903:r
1900:(
1897:g
1872:2
1868:r
1861:4
1851:r
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1839:g
1835:r
1831:g
1814:r
1810:e
1804:)
1801:r
1798:(
1795:g
1789:=
1786:)
1782:r
1778:(
1774:g
1763:g
1759:M
1755:r
1751:V
1737:.
1734:M
1731:G
1725:4
1719:=
1715:A
1711:d
1704:g
1698:V
1670:g
1653:0
1650:=
1646:g
1618:g
1610:g
1602:g
1575:)
1571:r
1567:(
1561:G
1555:4
1549:=
1546:)
1542:r
1538:(
1534:g
1507:.
1503:s
1497:3
1493:d
1486:)
1482:s
1474:r
1470:(
1461:)
1457:s
1453:(
1444:G
1438:4
1432:=
1429:)
1425:r
1421:(
1417:g
1396:r
1394:(
1392:ÎŽ
1378:)
1374:r
1370:(
1361:4
1358:=
1354:)
1346:3
1341:|
1335:r
1330:|
1324:r
1318:(
1301:r
1296:z
1288:y
1280:x
1272:s
1269:d
1267:(
1255:.
1251:s
1245:3
1241:d
1232:3
1227:|
1221:s
1213:r
1208:|
1202:)
1198:s
1190:r
1186:(
1180:)
1176:s
1172:(
1163:G
1157:=
1154:)
1150:r
1146:(
1142:g
1131:s
1127:r
1125:(
1123:g
1119:s
1111:r
1109:(
1107:g
1103:r
1099:r
1097:(
1095:g
1071:.
1061:M
1055:r
1051:r
1047:,
1040:r
1038:e
1018:r
1014:e
1005:2
1001:r
996:M
993:G
984:=
981:)
977:r
973:(
969:g
921:,
915:G
909:4
903:=
899:g
882:V
868:.
865:V
862:d
856:)
850:G
844:4
838:(
833:V
825:=
822:V
819:d
813:)
809:g
799:(
794:V
769:V
766:d
755:V
747:G
741:4
735:=
732:V
729:d
722:g
710:V
685:V
682:d
671:V
663:=
660:M
647:V
639:g
631:V
623:V
619:V
605:V
602:d
597:g
585:V
577:=
573:A
569:d
562:g
556:V
522:Ï
514:G
470:,
464:G
458:4
452:=
448:g
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400:V
396:M
386:G
382:,
376:g
362:V
350:A
347:d
341:V
333:V
331:â
314:V
283:V
245:M
242:G
236:4
230:=
226:A
222:d
215:g
193:V
161:.
125:g
71:(
44:.
20:)
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