2033:
2890:
36:
2579:
221:
starting point while ascending more than one descends or vice versa, resulting in nonzero work done by gravity. On a real staircase, the height above the ground is a scalar potential field: one has to go upward exactly as much as one goes downward in order to return to the same place, in which case the work by gravity totals to zero. This suggests path-independence of work done on the staircase; equivalently, the force field experienced is conservative (see the later section:
3629:
5952:
2885:{\displaystyle {\frac {\partial \varphi }{\partial x}}={\frac {\partial }{\partial x}}\int _{a,b}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }={\frac {\partial }{\partial x}}\int _{a,b}^{x_{1},y}\mathbf {v} \cdot d{\mathbf {r} }+{\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }=0+{\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }}
1259:
199:
3389:
3606:
1061:
2512:
194:
that do not have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff; one decides to scale the cliff by going vertically up it, and the second decides to walk along a winding path that is longer in length than the height of the cliff, but
3179:
220:
illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground (gravitational potential) as one moves along the staircase. The force field experienced by the one moving on the staircase is non-conservative in that one can return to the
168:
In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure.
4862:
195:
at only a small angle to the horizontal. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy. This is because a gravitational field is conservative.
3472:
169:
Therefore, in general, the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements
5425:
2342:
1410:
116:
is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also
711:
1254:{\displaystyle \int _{P_{c}}\mathbf {v} \cdot d\mathbf {r} =\int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} +\int _{P_{2}}\mathbf {v} \cdot d\mathbf {r} =\int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} -\int _{-P_{2}}\mathbf {v} \cdot d\mathbf {r} =0}
5911:
3741:
1835:
1766:
3037:
5175:
2162:
1853:). Since the gradient theorem is applicable for a differentiable path, the path independence of a conservative vector field over piecewise-differential curves is also proved by the proof per differentiable curve component.
4666:
3820:
6253:
4713:
6471:
3384:{\displaystyle {\frac {\partial }{\partial x}}\varphi (x,y)={\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d\mathbf {r} ={\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}P(t,y)dt=P(x,y)}
4279:
6901:
5333:
3467:
5942:
imply anything about the global behavior of a fluid. It is possible for a fluid that travels in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.
4149:
848:
5235:
4942:
2032:
3174:
6658:
5105:
6402:
3962:
1955:
277:
6909:
of a particle moving under the influence of conservative forces is conserved, in the sense that a loss of potential energy is converted to the equal quantity of kinetic energy, or vice versa.
630:
4386:
6060:
The most prominent examples of conservative forces are gravitational force (associated with a gravitational field) and electric force (associated with an electrostatic field). According to
6513:
436:
7035:
1474:
1321:
3601:{\displaystyle \mathbf {v} =P(x,y)\mathbf {i} +Q(x,y)\mathbf {j} ={\frac {\partial \varphi }{\partial x}}\mathbf {i} +{\frac {\partial \varphi }{\partial y}}\mathbf {j} =\nabla \varphi }
5853:
6580:
5858:
6310:
2507:{\displaystyle \int _{a,b}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }=\int _{a,b}^{x_{1},y}\mathbf {v} \cdot d{\mathbf {r} }+\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }.}
6542:
6094:
4567:
4042:
3849:
2004:
1574:
326:
7086:
461:
2036:
Line integral paths used to prove the following statement: if the line integral of a vector field is path-independent, then the vector field is a conservative vector field.
6051:
5324:
5302:
5277:
5199:
5010:
4984:
4886:
4688:
4090:
4064:
3081:
3059:
2959:
2574:
2059:
1900:
1876:
1596:
1439:
625:
573:
527:
5731:
1693:
6607:
7126:
7106:
7059:
4213:
2552:
2079:
1525:
547:
505:
485:
380:
5765:
5702:
5110:
2084:
5594:
1289:
5614:
5502:
5472:
5056:
4470:
4310:
4189:
3989:
3635:
2917:
2290:
2258:
2226:
2194:
1501:
1316:
1016:
949:
922:
875:
767:
740:
353:
6687:
192:
6181:
3919:
5033:
6842:
6822:
6802:
6782:
6762:
6742:
6722:
6354:
6334:
6275:
6174:
6154:
6134:
6114:
5810:
5788:
5681:
5658:
5634:
5566:
5539:
5445:
5255:
4962:
4906:
4708:
4587:
4538:
4518:
4490:
4443:
4417:
4350:
4330:
4233:
4110:
4013:
3889:
3869:
3121:
3101:
2937:
2532:
2337:
2317:
2024:
1975:
1680:
1660:
1640:
1616:
1545:
1056:
1036:
989:
969:
895:
787:
400:
297:
6409:
1785:
4238:
2964:
6948:
6849:
6061:
4115:
797:
4592:
3746:
4153:
For this reason, such vector fields are sometimes referred to as curl-free vector fields or curl-less vector fields. They are also referred to as
7154:
57:
5637:
5545:
2026:
is path-independent, meaning that the line integral depends on only both path endpoints regardless of which path between them is chosen.
5505:
vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vice versa.
3401:
7195:
79:
5204:
4911:
627:
is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:
407:
4857:{\displaystyle \mathbf {v} (x,y,z)~{\stackrel {\text{def}}{=}}~\left(-{\frac {y}{x^{2}+y^{2}}},{\frac {x}{x^{2}+y^{2}}},0\right).}
3612:. This proof method can be straightforwardly expanded to a higher dimensional orthogonal coordinate system (e.g., a 3-dimensional
3126:
582:
states that, under some regularity conditions, any vector field can be expressed as the sum of a conservative vector field and a
5916:
3392:
1779:
1441:
is that its integral along a path depends on only the endpoints of that path, not the particular route taken. In other words,
7224:
6476:
4393:
6612:
156:
done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define
6933:
5420:{\displaystyle \oint _{P_{c}}\mathbf {v} \cdot d\mathbf {r} =\iint _{A}(\nabla \times \mathbf {v} )\cdot d\mathbf {a} =0}
5065:
6359:
5924:
3924:
1917:
239:
202:
Depiction of two possible paths to integrate. In green is the simplest possible path; blue shows a more convoluted curve
5928:
7108:
to be continuously differentiable. There must be a reason for the definition of conservative vector fields to require
4389:
3613:
3609:
2296:
1850:
1842:
1405:{\textstyle \displaystyle \int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} =\int _{-P_{2}}\mathbf {v} \cdot d\mathbf {r} .}
4356:
50:
44:
6482:
4161:
7009:
1448:
61:
7243:
216:
5836:
6953:
6938:
6551:
4154:
1619:
579:
7061:
is not necessarily continuously differentiable, the condition of being differentiable is enough, since the
6284:
4423:(roughly speaking, a single piece open space without a hole within it), the converse of this is also true:
6943:
6906:
6278:
4420:
149:
109:
706:{\displaystyle \int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} =\int _{P_{2}}\mathbf {v} \cdot d\mathbf {r} }
6918:
4987:
1905:
6518:
6070:
4543:
4018:
3825:
1980:
1550:
302:
7068:
443:
6928:
6065:
5769:
any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed
5571:
6034:
5307:
5285:
5260:
5182:
4993:
4967:
4869:
4671:
4073:
4047:
3064:
3042:
2942:
2557:
2061:
is a continuous vector field which line integral is path-independent. Then, let's make a function
2042:
1883:
1859:
1579:
1422:
608:
556:
510:
6923:
6693:
6054:
4067:
1838:
122:
5707:
5327:
3398:
A similar approach for the line integral path shown in the right of the right figure results in
6585:
5906:{\displaystyle {\boldsymbol {\omega }}~{\stackrel {\text{def}}{=}}~\nabla \times \mathbf {v} .}
7220:
7191:
7150:
7111:
7091:
7044:
5519:
5515:
5059:
4198:
3891:
and is thus irrotational. However, it is not conservative and does not have path independence.
3736:{\displaystyle \mathbf {v} =\left(-{\frac {y}{x^{2}+y^{2}}},{\frac {x}{x^{2}+y^{2}}},0\right)}
2537:
2064:
1510:
532:
490:
470:
365:
5737:
5687:
897:
where the two endpoints are coincident. Two expressions are equivalent since any closed path
7179:
7062:
5813:
5576:
5201:
does not have the path-independence property discussed above so is not conservative even if
3617:
1683:
1264:
595:
583:
550:
157:
126:
5599:
5480:
5450:
5038:
4448:
4288:
4167:
3967:
3628:
2895:
2263:
2231:
2199:
2167:
1479:
1294:
994:
927:
900:
853:
745:
718:
331:
6973:
6663:
172:
141:
93:
3898:
17:
5015:
7248:
7169:
Need to verify if exact differentials also exist for non-orthogonal coordinate systems.
6827:
6807:
6787:
6767:
6747:
6727:
6707:
6339:
6319:
6260:
6159:
6139:
6119:
6099:
5795:
5773:
5666:
5643:
5619:
5551:
5524:
5430:
5240:
4947:
4891:
4693:
4572:
4523:
4503:
4475:
4428:
4402:
4335:
4315:
4218:
4095:
3998:
3874:
3854:
3106:
3086:
2922:
2517:
2322:
2302:
2009:
1960:
1665:
1645:
1625:
1601:
1530:
1041:
1021:
974:
954:
880:
772:
385:
282:
153:
125:. An irrotational vector field is necessarily conservative provided that the domain is
7237:
7183:
5920:
1771:
113:
1830:{\displaystyle \mathbf {v} \cdot d\mathbf {r} =\nabla {\varphi }\cdot d\mathbf {r} }
5951:
1912:
1761:{\displaystyle \int _{P}\mathbf {v} \cdot d{\mathbf {r} }=\varphi (B)-\varphi (A).}
360:
234:
211:
101:
3032:{\displaystyle {\displaystyle \mathbf {v} }=P(x,y)\mathbf {i} +Q(x,y)\mathbf {j} }
2295:
Let's choose the path shown in the left of the right figure where a 2-dimensional
1443:
if it is a conservative vector field, then its line integral is path-independent.
5170:{\displaystyle \oint _{C}\mathbf {v} \cdot \mathbf {e} _{\phi }~d{\phi }=2\pi .}
2157:{\displaystyle \varphi (x,y)=\int _{a,b}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }}
1846:
7129:
4192:
3992:
1775:
1504:
356:
4661:{\displaystyle U=\mathbb {R} ^{3}\setminus \{(0,0,z)\mid z\in \mathbb {R} \}}
3815:{\displaystyle U=\mathbb {R} ^{3}\setminus \{(0,0,z)\mid z\in \mathbb {R} \}}
6701:
6248:{\displaystyle \mathbf {F} _{G}=-{\frac {GmM}{r^{2}}}{\hat {\mathbf {r} }},}
5831:
5825:
198:
133:
5282:
Say again, in a simply connected open region, an irrotational vector field
1902:
is (line integral) path-independent, then it is a conservative vector field
7145:
Stewart, James (2015). "16.3 The
Fundamental Theorem of Line Integrals"".
6545:
464:
105:
6744:
is independent of the moving path chosen (dependent on only the points
6466:{\displaystyle \Phi _{G}~{\stackrel {\text{def}}{=}}-{\frac {GmM}{r}}}
2006:, it is conservative if and only if its line integral along a path in
145:
5934:
For a two-dimensional field, the vorticity acts as a measure of the
222:
4274:{\displaystyle \nabla \times (\nabla \varphi )\equiv \mathbf {0} .}
5950:
3627:
2031:
197:
137:
4425:
Every irrotational vector field in a simply connected open space
2514:
By the path independence, its partial derivative with respect to
6896:{\displaystyle W=\oint _{C}\mathbf {F} \cdot d{\mathbf {r} }=0.}
4396:(also called Clairaut's theorem on equality of mixed partials).
3462:{\textstyle {\frac {\partial }{\partial y}}\varphi (x,y)=Q(x,y)}
5923:
will remain irrotational. This result can be derived from the
4144:{\displaystyle \nabla \times \mathbf {v} \equiv \mathbf {0} .}
3616:) so the converse statement is proved. Another proof is found
843:{\displaystyle \int _{P_{c}}\mathbf {v} \cdot d\mathbf {r} =0}
29:
6980:(Fifth ed.). W.H.Freedman and Company. pp. 550–561.
5230:{\displaystyle \nabla \times \mathbf {v} \equiv \mathbf {0} }
4937:{\displaystyle \nabla \times \mathbf {v} \equiv \mathbf {0} }
1618:
is a differentiable path (i.e., it can be parameterized by a
2299:
is used. The second segment of this path is parallel to the
5915:
The vorticity of an irrotational field is zero everywhere.
3169:{\displaystyle d\mathbf {r} =dx\mathbf {i} +dy\mathbf {j} }
1856:
So far it has been proven that a conservative vector field
3871:-axis (so not a simply connected space), has zero curl in
2292:
regardless of which path between these points is chosen.
1318:
and the last equality holds due to the path independence
792:
The path independence is also equivalently expressed as
121:; in three dimensions, this means that it has vanishing
112:. A conservative vector field has the property that its
5326:
as conservative). This can be proved directly by using
2164:
over an arbitrary path between a chosen starting point
2029:
The proof of this converse statement is the following.
1598:
is a conservative vector field that is continuous) and
7149:(8th ed.). Cengage Learning. pp. 1127–1134.
6653:{\displaystyle \mathbf {F} =F(r){\hat {\mathbf {r} }}}
3404:
1324:
225:). The situation depicted in the print is impossible.
7114:
7094:
7071:
7047:
7012:
6852:
6830:
6810:
6790:
6770:
6750:
6730:
6710:
6666:
6615:
6588:
6554:
6521:
6485:
6412:
6362:
6342:
6322:
6287:
6263:
6184:
6162:
6142:
6122:
6102:
6073:
6037:
5955:
Examples of potential and gradient fields in physics:
5861:
5839:
5798:
5776:
5740:
5710:
5690:
5669:
5646:
5622:
5602:
5579:
5554:
5527:
5483:
5453:
5433:
5336:
5310:
5288:
5263:
5243:
5207:
5185:
5113:
5068:
5041:
5018:
4996:
4970:
4950:
4914:
4894:
4872:
4716:
4696:
4674:
4595:
4575:
4546:
4526:
4506:
4478:
4451:
4431:
4405:
4359:
4338:
4318:
4291:
4241:
4221:
4201:
4170:
4118:
4098:
4076:
4050:
4021:
4001:
3970:
3927:
3901:
3877:
3857:
3828:
3749:
3638:
3475:
3182:
3129:
3109:
3089:
3067:
3045:
2969:
2967:
2945:
2925:
2898:
2582:
2560:
2540:
2520:
2345:
2325:
2305:
2266:
2234:
2202:
2170:
2087:
2067:
2045:
2012:
1983:
1963:
1920:
1886:
1862:
1788:
1696:
1668:
1648:
1628:
1604:
1582:
1553:
1533:
1513:
1482:
1451:
1425:
1325:
1297:
1267:
1064:
1044:
1024:
997:
977:
957:
930:
903:
883:
856:
800:
775:
748:
721:
633:
611:
559:
535:
513:
493:
473:
446:
410:
388:
368:
334:
305:
285:
242:
175:
6609:. It can be shown that any vector field of the form
4193:
continuously differentiable up to the 2nd derivative
6582:associated with the gravitational potential energy
5981:, (gravitational or electrostatic) potential energy
5636:. The irrotational vector fields correspond to the
5544:. The conservative vector fields correspond to the
5100:{\displaystyle \mathbf {v} =\mathbf {e} _{\phi }/r}
2228:. Since it is path-independent, it depends on only
7120:
7100:
7080:
7053:
7029:
6895:
6836:
6816:
6796:
6776:
6756:
6736:
6716:
6681:
6652:
6601:
6574:
6536:
6507:
6465:
6397:{\displaystyle \mathbf {F} _{G}=-\nabla \Phi _{G}}
6396:
6348:
6328:
6304:
6269:
6247:
6168:
6148:
6128:
6108:
6088:
6045:
5905:
5847:
5804:
5782:
5759:
5725:
5696:
5675:
5652:
5628:
5608:
5588:
5560:
5533:
5496:
5466:
5439:
5419:
5318:
5296:
5279:is defined is not a simply connected open space.
5271:
5249:
5229:
5193:
5169:
5099:
5050:
5027:
5004:
4978:
4956:
4936:
4900:
4880:
4856:
4702:
4682:
4660:
4581:
4561:
4532:
4512:
4484:
4464:
4437:
4411:
4380:
4344:
4324:
4304:
4273:
4227:
4207:
4183:
4143:
4104:
4084:
4058:
4036:
4007:
3983:
3957:{\displaystyle \mathbf {v} :U\to \mathbb {R} ^{3}}
3956:
3913:
3883:
3863:
3843:
3814:
3735:
3600:
3461:
3383:
3168:
3115:
3095:
3075:
3053:
3031:
2953:
2931:
2911:
2884:
2568:
2546:
2526:
2506:
2331:
2311:
2284:
2252:
2220:
2188:
2156:
2073:
2053:
2018:
1998:
1969:
1950:{\displaystyle \mathbf {v} :U\to \mathbb {R} ^{n}}
1949:
1894:
1870:
1829:
1760:
1688:fundamental theorem of calculus for line integrals
1674:
1654:
1634:
1610:
1590:
1568:
1539:
1519:
1495:
1468:
1433:
1404:
1310:
1283:
1253:
1050:
1030:
1010:
983:
963:
943:
916:
889:
869:
842:
781:
761:
734:
705:
619:
567:
541:
521:
499:
479:
455:
430:
394:
374:
347:
320:
291:
272:{\displaystyle \mathbf {v} :U\to \mathbb {R} ^{n}}
271:
186:
27:Vector field that is the gradient of some function
4353:. This result can be easily proved by expressing
6053:is conservative, then the force is said to be a
328:, is said to be conservative if there exists a
6356:. The force of gravity is conservative because
5938:rotation of fluid elements. The vorticity does
5919:states that a fluid that is irrotational in an
4589:-axis (so not a simply connected space), i.e.,
4381:{\displaystyle \nabla \times (\nabla \varphi )}
1878:is line integral path-independent. Conversely,
590:Path independence and conservative vector field
223:Path independence and conservative vector field
132:Conservative vector fields appear naturally in
1419:A key property of a conservative vector field
529:is continuous. When the equation above holds,
160:that is independent of the actual path taken.
6994:, 6th edition, Elsevier Academic Press (2005)
6508:{\displaystyle {\frac {\mathbf {F} _{G}}{m}}}
2939:are independent to each other. Let's express
431:{\displaystyle \mathbf {v} =\nabla \varphi .}
8:
7030:{\displaystyle \mathbf {v} =\nabla \varphi }
4655:
4617:
3809:
3771:
1469:{\displaystyle \mathbf {v} =\nabla \varphi }
2339:axis. The line integral along this path is
1841:for an orthogonal coordinate system (e.g.,
6804:done in going around a simple closed loop
6031:If the vector field associated to a force
5107:, so the integral over the unit circle is
769:between a given pair of path endpoints in
7113:
7093:
7070:
7046:
7013:
7011:
6949:Longitudinal and transverse vector fields
6881:
6880:
6869:
6863:
6851:
6829:
6809:
6789:
6769:
6749:
6729:
6709:
6665:
6639:
6637:
6636:
6616:
6614:
6593:
6587:
6561:
6555:
6553:
6528:
6523:
6520:
6494:
6489:
6486:
6484:
6445:
6434:
6429:
6427:
6426:
6417:
6411:
6388:
6369:
6364:
6361:
6341:
6321:
6291:
6289:
6288:
6286:
6262:
6231:
6229:
6228:
6220:
6203:
6191:
6186:
6183:
6161:
6141:
6121:
6101:
6080:
6075:
6072:
6038:
6036:
5895:
5878:
5873:
5871:
5870:
5862:
5860:
5840:
5838:
5797:
5775:
5745:
5739:
5709:
5689:
5668:
5645:
5621:
5601:
5578:
5553:
5526:
5488:
5482:
5458:
5452:
5432:
5406:
5392:
5377:
5365:
5354:
5346:
5341:
5335:
5311:
5309:
5289:
5287:
5264:
5262:
5242:
5222:
5214:
5206:
5186:
5184:
5150:
5138:
5133:
5124:
5118:
5112:
5089:
5083:
5078:
5069:
5067:
5040:
5017:
4997:
4995:
4971:
4969:
4949:
4929:
4921:
4913:
4893:
4873:
4871:
4831:
4818:
4808:
4796:
4783:
4773:
4754:
4749:
4747:
4746:
4717:
4715:
4695:
4675:
4673:
4651:
4650:
4608:
4604:
4603:
4594:
4574:
4553:
4549:
4548:
4545:
4525:
4505:
4477:
4456:
4450:
4430:
4404:
4358:
4337:
4317:
4296:
4290:
4263:
4240:
4220:
4200:
4175:
4169:
4133:
4125:
4117:
4097:
4077:
4075:
4051:
4049:
4028:
4024:
4023:
4020:
4000:
3975:
3969:
3948:
3944:
3943:
3928:
3926:
3900:
3876:
3856:
3835:
3831:
3830:
3827:
3805:
3804:
3762:
3758:
3757:
3748:
3713:
3700:
3690:
3678:
3665:
3655:
3639:
3637:
3620:as the converse of the gradient theorem.
3584:
3564:
3556:
3536:
3528:
3502:
3476:
3474:
3405:
3403:
3324:
3311:
3306:
3287:
3279:
3268:
3256:
3243:
3238:
3219:
3183:
3181:
3161:
3147:
3133:
3128:
3108:
3088:
3068:
3066:
3046:
3044:
3024:
2998:
2970:
2968:
2966:
2946:
2944:
2924:
2903:
2897:
2876:
2875:
2864:
2852:
2839:
2834:
2815:
2800:
2799:
2788:
2776:
2763:
2758:
2739:
2730:
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2718:
2704:
2699:
2688:
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2659:
2648:
2636:
2625:
2606:
2583:
2581:
2561:
2559:
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2519:
2495:
2494:
2483:
2471:
2458:
2453:
2440:
2439:
2428:
2414:
2409:
2398:
2385:
2384:
2373:
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2350:
2344:
2324:
2304:
2265:
2233:
2201:
2169:
2148:
2147:
2136:
2124:
2113:
2086:
2066:
2046:
2044:
2011:
1990:
1986:
1985:
1982:
1962:
1941:
1937:
1936:
1921:
1919:
1887:
1885:
1863:
1861:
1822:
1811:
1800:
1789:
1787:
1719:
1718:
1707:
1701:
1695:
1667:
1647:
1627:
1603:
1583:
1581:
1560:
1556:
1555:
1552:
1532:
1512:
1487:
1481:
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1393:
1382:
1374:
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1354:
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1296:
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1177:
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1118:
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1105:
1093:
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1074:
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1063:
1043:
1023:
1002:
996:
976:
956:
935:
929:
908:
902:
882:
861:
855:
829:
818:
810:
805:
799:
774:
753:
747:
726:
720:
698:
687:
679:
674:
662:
651:
643:
638:
632:
612:
610:
560:
558:
534:
514:
512:
492:
472:
445:
411:
409:
387:
367:
339:
333:
312:
308:
307:
304:
284:
263:
259:
258:
243:
241:
179:
174:
80:Learn how and when to remove this message
6515:associated with the gravitational force
6479:. In other words, the gravitation field
6015:, (gravitational or electrostatic) force
4332:is also an irrotational vector field in
43:This article includes a list of general
7065:, that proves the path independence of
6965:
5863:
5848:{\displaystyle {\boldsymbol {\omega }}}
5841:
5447:which boundary is a simple closed path
5304:has the path-independence property (so
4614:
3768:
7038:
6575:{\displaystyle {\frac {\Phi _{G}}{m}}}
5570:, that is, to the forms which are the
5514:More abstractly, in the presence of a
5476:In a simply connected open region, any
3393:second fundamental theorem of calculus
1780:second fundamental theorem of calculus
580:fundamental theorem of vector calculus
136:: They are vector fields representing
6305:{\displaystyle {\hat {\mathbf {r} }}}
5927:, obtained by taking the curl of the
5855:of a vector field can be defined by:
4569:with removing all coordinates on the
3851:with removing all coordinates on the
2319:axis so there is no change along the
850:for any piecewise smooth closed path
7:
7140:
7138:
7002:
7000:
6990:George B. Arfken and Hans J. Weber,
6700:can be interpreted to mean that the
3995:) vector field, with an open subset
3391:where the last equality is from the
6992:Mathematical Methods for Physicists
1770:This holds as a consequence of the
7072:
7021:
6590:
6558:
6414:
6385:
6381:
5889:
5386:
5208:
4915:
4369:
4360:
4251:
4242:
4119:
3592:
3575:
3567:
3547:
3539:
3411:
3407:
3293:
3289:
3225:
3221:
3189:
3185:
2821:
2817:
2745:
2741:
2675:
2671:
2612:
2608:
2594:
2586:
1808:
1460:
605:A line integral of a vector field
447:
419:
49:it lacks sufficient corresponding
25:
5962:Scalar fields, scalar potentials:
152:. For a conservative system, the
7014:
6882:
6870:
6640:
6617:
6537:{\displaystyle \mathbf {F} _{G}}
6524:
6490:
6365:
6292:
6232:
6187:
6089:{\displaystyle \mathbf {F} _{G}}
6076:
6039:
5896:
5427:for any smooth oriented surface
5407:
5393:
5366:
5355:
5312:
5290:
5265:
5223:
5215:
5187:
5134:
5125:
5079:
5070:
4998:
4972:
4930:
4922:
4874:
4718:
4676:
4562:{\displaystyle \mathbb {R} ^{3}}
4264:
4134:
4126:
4078:
4052:
4037:{\displaystyle \mathbb {R} ^{n}}
3929:
3844:{\displaystyle \mathbb {R} ^{3}}
3640:
3608:is proved for the 2-dimensional
3585:
3557:
3529:
3503:
3477:
3280:
3269:
3162:
3148:
3134:
3069:
3047:
3025:
2999:
2971:
2947:
2877:
2865:
2801:
2789:
2731:
2719:
2661:
2649:
2562:
2496:
2484:
2441:
2429:
2386:
2374:
2149:
2137:
2047:
1999:{\displaystyle \mathbb {R} ^{n}}
1922:
1888:
1864:
1823:
1801:
1790:
1720:
1708:
1584:
1569:{\displaystyle \mathbb {R} ^{n}}
1453:
1427:
1394:
1383:
1355:
1344:
1241:
1230:
1202:
1191:
1166:
1155:
1130:
1119:
1094:
1083:
830:
819:
699:
688:
663:
652:
613:
561:
515:
507:is continuously differentiable,
412:
321:{\displaystyle \mathbb {R} ^{n}}
244:
34:
7081:{\displaystyle \nabla \varphi }
6660:is conservative, provided that
5999:Vector fields, gradient fields:
4066:is called irrotational if its
3921:(3-dimensional space), and let
3123:axes respectively, then, since
715:for any pair of integral paths
456:{\displaystyle \nabla \varphi }
7190:, Courier Dover Publications,
6676:
6670:
6644:
6633:
6627:
6477:gravitational potential energy
6296:
6236:
5518:, vector fields correspond to
5397:
5383:
5012:around the unit circle in the
4986:is irrotational. However, the
4740:
4722:
4638:
4620:
4375:
4366:
4257:
4248:
3939:
3792:
3774:
3525:
3513:
3499:
3487:
3456:
3444:
3435:
3423:
3378:
3366:
3351:
3339:
3213:
3201:
3021:
3009:
2995:
2983:
2279:
2267:
2247:
2235:
2215:
2203:
2183:
2171:
2103:
2091:
1932:
1752:
1746:
1737:
1731:
254:
1:
6934:Complex lamellar vector field
6548:of the gravitation potential
5596:of a function (scalar field)
4668:. Now, define a vector field
4520:is not simply connected. Let
4472:conservative vector field in
4312:conservative vector field in
2554:to have partial derivatives,
1880:if a continuous vector field
1772:definition of a line integral
6046:{\displaystyle \mathbf {F} }
6009:, gravitational acceleration
5925:vorticity transport equation
5917:Kelvin's circulation theorem
5319:{\displaystyle \mathbf {v} }
5297:{\displaystyle \mathbf {v} }
5272:{\displaystyle \mathbf {v} }
5194:{\displaystyle \mathbf {v} }
5005:{\displaystyle \mathbf {v} }
4979:{\displaystyle \mathbf {v} }
4888:has zero curl everywhere in
4881:{\displaystyle \mathbf {v} }
4683:{\displaystyle \mathbf {v} }
4085:{\displaystyle \mathbf {0} }
4059:{\displaystyle \mathbf {v} }
3076:{\displaystyle \mathbf {j} }
3054:{\displaystyle \mathbf {i} }
2954:{\displaystyle \mathbf {v} }
2576:needs to be continuous.) is
2569:{\displaystyle \mathbf {v} }
2054:{\displaystyle \mathbf {v} }
1895:{\displaystyle \mathbf {v} }
1871:{\displaystyle \mathbf {v} }
1591:{\displaystyle \mathbf {v} }
1434:{\displaystyle \mathbf {v} }
620:{\displaystyle \mathbf {v} }
568:{\displaystyle \mathbf {v} }
522:{\displaystyle \mathbf {v} }
7219:. Oxford University Press.
7130:continuously differentiable
6062:Newton's law of gravitation
5474:. So, it is concluded that
4421:simply connected open space
4390:Cartesian coordinate system
4162:identity of vector calculus
3993:continuously differentiable
3614:spherical coordinate system
3610:Cartesian coordinate system
3083:are unit vectors along the
2297:Cartesian coordinate system
1837:in the line integral is an
1505:continuously differentiable
357:continuously differentiable
7265:
6976:; Tromba, Anthony (2003).
5823:
5726:{\displaystyle d\omega =0}
4155:longitudinal vector fields
3624:Irrotational vector fields
593:
7217:Elementary Fluid Dynamics
6602:{\displaystyle \Phi _{G}}
6021:, electric field strength
5972:, gravitational potential
1415:Conservative vector field
924:can be made by two path;
98:conservative vector field
18:Irrotational vector field
7188:Elements of Gas Dynamics
7121:{\displaystyle \varphi }
7101:{\displaystyle \varphi }
7054:{\displaystyle \varphi }
4208:{\displaystyle \varphi }
2547:{\displaystyle \varphi }
2074:{\displaystyle \varphi }
1520:{\displaystyle \varphi }
542:{\displaystyle \varphi }
500:{\displaystyle \varphi }
480:{\displaystyle \varphi }
375:{\displaystyle \varphi }
217:Ascending and Descending
7215:Acheson, D. J. (1990).
6954:Solenoidal vector field
6939:Helmholtz decomposition
5929:Navier–Stokes equations
5760:{\displaystyle d^{2}=0}
5697:{\displaystyle \omega }
4496:The above statement is
3632:The above vector field
2196:and an arbitrary point
1620:differentiable function
64:more precise citations.
7122:
7102:
7082:
7055:
7031:
6944:Laplacian vector field
6897:
6838:
6818:
6798:
6778:
6758:
6738:
6718:
6704:in going from a point
6683:
6654:
6603:
6576:
6538:
6509:
6467:
6398:
6350:
6330:
6306:
6279:gravitational constant
6271:
6249:
6170:
6150:
6136:located at a distance
6130:
6110:
6090:
6047:
6028:
5907:
5849:
5806:
5784:
5761:
5727:
5698:
5677:
5654:
5630:
5610:
5590:
5589:{\displaystyle d\phi }
5562:
5535:
5498:
5468:
5441:
5421:
5320:
5298:
5273:
5251:
5231:
5195:
5171:
5101:
5052:
5029:
5006:
4980:
4958:
4938:
4902:
4882:
4858:
4704:
4684:
4662:
4583:
4563:
4534:
4514:
4486:
4466:
4439:
4413:
4382:
4346:
4326:
4306:
4275:
4229:
4209:
4185:
4145:
4106:
4086:
4060:
4038:
4009:
3985:
3958:
3915:
3892:
3885:
3865:
3845:
3816:
3737:
3602:
3463:
3385:
3170:
3117:
3097:
3077:
3055:
3033:
2955:
2933:
2913:
2886:
2570:
2548:
2528:
2508:
2333:
2313:
2286:
2254:
2222:
2190:
2158:
2075:
2055:
2037:
2020:
2000:
1971:
1951:
1896:
1872:
1831:
1762:
1676:
1656:
1642:with an initial point
1636:
1612:
1592:
1570:
1541:
1521:
1497:
1470:
1435:
1406:
1312:
1285:
1284:{\displaystyle -P_{2}}
1255:
1052:
1032:
1012:
985:
965:
945:
918:
891:
871:
844:
783:
763:
736:
707:
621:
569:
543:
523:
501:
481:
457:
432:
396:
376:
349:
322:
293:
273:
203:
188:
7123:
7103:
7083:
7056:
7032:
6919:Beltrami vector field
6898:
6839:
6819:
6799:
6784:), and that the work
6779:
6759:
6739:
6719:
6684:
6655:
6604:
6577:
6539:
6510:
6468:
6399:
6351:
6331:
6316:vector pointing from
6307:
6272:
6250:
6176:, obeys the equation
6171:
6151:
6131:
6111:
6091:
6048:
5954:
5908:
5850:
5807:
5785:
5762:
5728:
5699:
5678:
5655:
5631:
5611:
5609:{\displaystyle \phi }
5591:
5563:
5536:
5499:
5497:{\displaystyle C^{1}}
5469:
5467:{\displaystyle P_{c}}
5442:
5422:
5321:
5299:
5274:
5252:
5232:
5196:
5172:
5102:
5053:
5051:{\displaystyle 2\pi }
5030:
5007:
4981:
4959:
4939:
4903:
4883:
4859:
4705:
4685:
4663:
4584:
4564:
4535:
4515:
4487:
4467:
4465:{\displaystyle C^{1}}
4440:
4414:
4383:
4347:
4327:
4307:
4305:{\displaystyle C^{1}}
4276:
4230:
4210:
4186:
4184:{\displaystyle C^{2}}
4146:
4107:
4087:
4061:
4039:
4010:
3986:
3984:{\displaystyle C^{1}}
3959:
3916:
3886:
3866:
3846:
3817:
3738:
3631:
3603:
3464:
3386:
3171:
3118:
3098:
3078:
3056:
3034:
2956:
2934:
2914:
2912:{\displaystyle x_{1}}
2887:
2571:
2549:
2529:
2509:
2334:
2314:
2287:
2285:{\displaystyle (x,y)}
2255:
2253:{\displaystyle (a,b)}
2223:
2221:{\displaystyle (x,y)}
2191:
2189:{\displaystyle (a,b)}
2159:
2076:
2056:
2035:
2021:
2001:
1977:is an open subset of
1972:
1952:
1897:
1873:
1851:spherical coordinates
1832:
1763:
1677:
1662:and a terminal point
1657:
1637:
1613:
1593:
1571:
1547:as an open subset of
1542:
1522:
1498:
1496:{\displaystyle C^{1}}
1471:
1436:
1407:
1313:
1311:{\displaystyle P_{2}}
1286:
1256:
1053:
1033:
1013:
1011:{\displaystyle P_{2}}
986:
966:
946:
944:{\displaystyle P_{1}}
919:
917:{\displaystyle P_{c}}
892:
872:
870:{\displaystyle P_{c}}
845:
784:
764:
762:{\displaystyle P_{2}}
737:
735:{\displaystyle P_{1}}
708:
622:
570:
544:
524:
502:
482:
458:
433:
397:
377:
350:
348:{\displaystyle C^{1}}
323:
299:is an open subset of
294:
274:
207:Intuitive explanation
201:
189:
7112:
7092:
7069:
7045:
7010:
6850:
6828:
6808:
6788:
6768:
6748:
6728:
6708:
6682:{\displaystyle F(r)}
6664:
6613:
6586:
6552:
6519:
6483:
6410:
6360:
6340:
6320:
6285:
6261:
6182:
6160:
6140:
6120:
6100:
6071:
6035:
5859:
5837:
5796:
5774:
5738:
5708:
5688:
5667:
5644:
5620:
5600:
5577:
5552:
5525:
5481:
5451:
5431:
5334:
5308:
5286:
5261:
5241:
5205:
5183:
5111:
5066:
5039:
5016:
4994:
4968:
4948:
4912:
4892:
4870:
4714:
4694:
4672:
4593:
4573:
4544:
4524:
4504:
4476:
4449:
4429:
4403:
4357:
4336:
4316:
4289:
4239:
4219:
4199:
4168:
4116:
4096:
4074:
4048:
4019:
3999:
3968:
3925:
3899:
3875:
3855:
3826:
3747:
3636:
3473:
3402:
3180:
3127:
3107:
3087:
3065:
3043:
2965:
2943:
2923:
2896:
2580:
2558:
2538:
2518:
2343:
2323:
2303:
2264:
2232:
2200:
2168:
2085:
2065:
2043:
2010:
1981:
1961:
1918:
1884:
1860:
1786:
1694:
1666:
1646:
1626:
1602:
1580:
1551:
1531:
1511:
1480:
1449:
1423:
1322:
1295:
1265:
1062:
1042:
1022:
995:
975:
971:to another endpoint
955:
928:
901:
881:
854:
798:
773:
746:
719:
631:
609:
557:
533:
511:
491:
471:
444:
408:
386:
366:
332:
303:
283:
240:
187:{\displaystyle d{R}}
173:
7088:, does not require
6929:Conservative system
6694:conservative forces
6066:gravitational force
5990:, Coulomb potential
5947:Conservative forces
5572:exterior derivative
4500:true in general if
3914:{\displaystyle n=3}
3335:
3267:
2863:
2787:
2717:
2647:
2482:
2427:
2372:
2135:
1904:, so the following
7118:
7098:
7078:
7051:
7027:
6924:Conservative force
6893:
6834:
6814:
6794:
6774:
6754:
6734:
6714:
6679:
6650:
6599:
6572:
6534:
6505:
6463:
6394:
6346:
6326:
6302:
6267:
6245:
6166:
6146:
6126:
6106:
6086:
6055:conservative force
6043:
6029:
5903:
5845:
5802:
5780:
5757:
5723:
5694:
5673:
5662:, that is, to the
5650:
5626:
5606:
5586:
5558:
5531:
5494:
5464:
5437:
5417:
5316:
5294:
5269:
5247:
5227:
5191:
5167:
5097:
5048:
5028:{\displaystyle xy}
5025:
5002:
4976:
4954:
4934:
4898:
4878:
4854:
4700:
4680:
4658:
4579:
4559:
4530:
4510:
4482:
4462:
4435:
4409:
4378:
4342:
4322:
4302:
4271:
4225:
4205:
4181:
4141:
4102:
4082:
4056:
4034:
4005:
3981:
3954:
3911:
3893:
3881:
3861:
3841:
3812:
3733:
3598:
3459:
3381:
3302:
3234:
3166:
3113:
3093:
3073:
3051:
3029:
2975:
2951:
2929:
2909:
2882:
2830:
2754:
2684:
2621:
2566:
2544:
2524:
2504:
2449:
2394:
2346:
2329:
2309:
2282:
2250:
2218:
2186:
2154:
2109:
2071:
2051:
2038:
2016:
1996:
1967:
1947:
1892:
1868:
1839:exact differential
1827:
1758:
1672:
1652:
1632:
1608:
1588:
1566:
1537:
1517:
1493:
1466:
1431:
1402:
1401:
1308:
1291:is the reverse of
1281:
1251:
1048:
1028:
1008:
981:
961:
941:
914:
887:
867:
840:
779:
759:
732:
703:
617:
565:
539:
519:
497:
477:
453:
428:
392:
372:
345:
318:
289:
269:
204:
184:
164:Informal treatment
7156:978-1-285-74062-1
6837:{\displaystyle 0}
6817:{\displaystyle C}
6797:{\displaystyle W}
6777:{\displaystyle B}
6757:{\displaystyle A}
6737:{\displaystyle B}
6717:{\displaystyle A}
6698:path independence
6647:
6570:
6503:
6461:
6439:
6437:
6425:
6349:{\displaystyle m}
6329:{\displaystyle M}
6299:
6270:{\displaystyle G}
6239:
6226:
6169:{\displaystyle m}
6149:{\displaystyle r}
6129:{\displaystyle M}
6109:{\displaystyle m}
6096:acting on a mass
5888:
5883:
5881:
5869:
5805:{\displaystyle U}
5783:{\displaystyle 1}
5676:{\displaystyle 1}
5653:{\displaystyle 1}
5629:{\displaystyle U}
5561:{\displaystyle 1}
5534:{\displaystyle 1}
5516:Riemannian metric
5440:{\displaystyle A}
5250:{\displaystyle U}
5146:
5060:polar coordinates
4957:{\displaystyle U}
4944:at everywhere in
4901:{\displaystyle U}
4838:
4803:
4764:
4759:
4757:
4745:
4703:{\displaystyle U}
4582:{\displaystyle z}
4533:{\displaystyle U}
4513:{\displaystyle U}
4485:{\displaystyle U}
4438:{\displaystyle U}
4412:{\displaystyle U}
4394:Schwarz's theorem
4345:{\displaystyle U}
4325:{\displaystyle U}
4228:{\displaystyle U}
4105:{\displaystyle U}
4008:{\displaystyle U}
3884:{\displaystyle U}
3864:{\displaystyle z}
3720:
3685:
3582:
3554:
3418:
3300:
3232:
3196:
3116:{\displaystyle y}
3096:{\displaystyle x}
2932:{\displaystyle x}
2828:
2752:
2682:
2619:
2601:
2527:{\displaystyle x}
2332:{\displaystyle y}
2312:{\displaystyle x}
2019:{\displaystyle U}
1970:{\displaystyle U}
1911:For a continuous
1908:statement holds:
1675:{\displaystyle B}
1655:{\displaystyle A}
1635:{\displaystyle U}
1611:{\displaystyle P}
1540:{\displaystyle U}
1051:{\displaystyle A}
1031:{\displaystyle B}
984:{\displaystyle B}
964:{\displaystyle A}
951:from an endpoint
890:{\displaystyle U}
782:{\displaystyle U}
601:Path independence
395:{\displaystyle U}
292:{\displaystyle U}
214:lithograph print
90:
89:
82:
16:(Redirected from
7256:
7230:
7202:
7200:
7176:
7170:
7167:
7161:
7160:
7142:
7133:
7127:
7125:
7124:
7119:
7107:
7105:
7104:
7099:
7087:
7085:
7084:
7079:
7063:Gradient theorem
7060:
7058:
7057:
7052:
7039:path-independent
7036:
7034:
7033:
7028:
7017:
7004:
6995:
6988:
6982:
6981:
6974:Marsden, Jerrold
6970:
6902:
6900:
6899:
6894:
6886:
6885:
6873:
6868:
6867:
6843:
6841:
6840:
6835:
6823:
6821:
6820:
6815:
6803:
6801:
6800:
6795:
6783:
6781:
6780:
6775:
6763:
6761:
6760:
6755:
6743:
6741:
6740:
6735:
6723:
6721:
6720:
6715:
6688:
6686:
6685:
6680:
6659:
6657:
6656:
6651:
6649:
6648:
6643:
6638:
6620:
6608:
6606:
6605:
6600:
6598:
6597:
6581:
6579:
6578:
6573:
6571:
6566:
6565:
6556:
6543:
6541:
6540:
6535:
6533:
6532:
6527:
6514:
6512:
6511:
6506:
6504:
6499:
6498:
6493:
6487:
6472:
6470:
6469:
6464:
6462:
6457:
6446:
6441:
6440:
6438:
6435:
6433:
6428:
6423:
6422:
6421:
6403:
6401:
6400:
6395:
6393:
6392:
6374:
6373:
6368:
6355:
6353:
6352:
6347:
6335:
6333:
6332:
6327:
6311:
6309:
6308:
6303:
6301:
6300:
6295:
6290:
6276:
6274:
6273:
6268:
6254:
6252:
6251:
6246:
6241:
6240:
6235:
6230:
6227:
6225:
6224:
6215:
6204:
6196:
6195:
6190:
6175:
6173:
6172:
6167:
6155:
6153:
6152:
6147:
6135:
6133:
6132:
6127:
6115:
6113:
6112:
6107:
6095:
6093:
6092:
6087:
6085:
6084:
6079:
6052:
6050:
6049:
6044:
6042:
5998:
5961:
5912:
5910:
5909:
5904:
5899:
5886:
5885:
5884:
5882:
5879:
5877:
5872:
5867:
5866:
5854:
5852:
5851:
5846:
5844:
5814:simply connected
5811:
5809:
5808:
5803:
5791:
5789:
5787:
5786:
5781:
5768:
5766:
5764:
5763:
5758:
5750:
5749:
5732:
5730:
5729:
5724:
5703:
5701:
5700:
5695:
5684:
5682:
5680:
5679:
5674:
5661:
5659:
5657:
5656:
5651:
5635:
5633:
5632:
5627:
5615:
5613:
5612:
5607:
5595:
5593:
5592:
5587:
5569:
5567:
5565:
5564:
5559:
5542:
5540:
5538:
5537:
5532:
5503:
5501:
5500:
5495:
5493:
5492:
5473:
5471:
5470:
5465:
5463:
5462:
5446:
5444:
5443:
5438:
5426:
5424:
5423:
5418:
5410:
5396:
5382:
5381:
5369:
5358:
5353:
5352:
5351:
5350:
5325:
5323:
5322:
5317:
5315:
5303:
5301:
5300:
5295:
5293:
5278:
5276:
5275:
5270:
5268:
5256:
5254:
5253:
5248:
5236:
5234:
5233:
5228:
5226:
5218:
5200:
5198:
5197:
5192:
5190:
5176:
5174:
5173:
5168:
5154:
5144:
5143:
5142:
5137:
5128:
5123:
5122:
5106:
5104:
5103:
5098:
5093:
5088:
5087:
5082:
5073:
5057:
5055:
5054:
5049:
5034:
5032:
5031:
5026:
5011:
5009:
5008:
5003:
5001:
4985:
4983:
4982:
4977:
4975:
4963:
4961:
4960:
4955:
4943:
4941:
4940:
4935:
4933:
4925:
4907:
4905:
4904:
4899:
4887:
4885:
4884:
4879:
4877:
4863:
4861:
4860:
4855:
4850:
4846:
4839:
4837:
4836:
4835:
4823:
4822:
4809:
4804:
4802:
4801:
4800:
4788:
4787:
4774:
4762:
4761:
4760:
4758:
4755:
4753:
4748:
4743:
4721:
4709:
4707:
4706:
4701:
4689:
4687:
4686:
4681:
4679:
4667:
4665:
4664:
4659:
4654:
4613:
4612:
4607:
4588:
4586:
4585:
4580:
4568:
4566:
4565:
4560:
4558:
4557:
4552:
4539:
4537:
4536:
4531:
4519:
4517:
4516:
4511:
4491:
4489:
4488:
4483:
4471:
4469:
4468:
4463:
4461:
4460:
4444:
4442:
4441:
4436:
4418:
4416:
4415:
4410:
4387:
4385:
4384:
4379:
4351:
4349:
4348:
4343:
4331:
4329:
4328:
4323:
4311:
4309:
4308:
4303:
4301:
4300:
4280:
4278:
4277:
4272:
4267:
4234:
4232:
4231:
4226:
4214:
4212:
4211:
4206:
4190:
4188:
4187:
4182:
4180:
4179:
4150:
4148:
4147:
4142:
4137:
4129:
4111:
4109:
4108:
4103:
4091:
4089:
4088:
4083:
4081:
4065:
4063:
4062:
4057:
4055:
4043:
4041:
4040:
4035:
4033:
4032:
4027:
4014:
4012:
4011:
4006:
3990:
3988:
3987:
3982:
3980:
3979:
3963:
3961:
3960:
3955:
3953:
3952:
3947:
3932:
3920:
3918:
3917:
3912:
3890:
3888:
3887:
3882:
3870:
3868:
3867:
3862:
3850:
3848:
3847:
3842:
3840:
3839:
3834:
3821:
3819:
3818:
3813:
3808:
3767:
3766:
3761:
3742:
3740:
3739:
3734:
3732:
3728:
3721:
3719:
3718:
3717:
3705:
3704:
3691:
3686:
3684:
3683:
3682:
3670:
3669:
3656:
3643:
3607:
3605:
3604:
3599:
3588:
3583:
3581:
3573:
3565:
3560:
3555:
3553:
3545:
3537:
3532:
3506:
3480:
3468:
3466:
3465:
3460:
3419:
3417:
3406:
3390:
3388:
3387:
3382:
3334:
3323:
3316:
3315:
3301:
3299:
3288:
3283:
3272:
3266:
3255:
3248:
3247:
3233:
3231:
3220:
3197:
3195:
3184:
3175:
3173:
3172:
3167:
3165:
3151:
3137:
3122:
3120:
3119:
3114:
3102:
3100:
3099:
3094:
3082:
3080:
3079:
3074:
3072:
3060:
3058:
3057:
3052:
3050:
3038:
3036:
3035:
3030:
3028:
3002:
2976:
2974:
2960:
2958:
2957:
2952:
2950:
2938:
2936:
2935:
2930:
2918:
2916:
2915:
2910:
2908:
2907:
2891:
2889:
2888:
2883:
2881:
2880:
2868:
2862:
2851:
2844:
2843:
2829:
2827:
2816:
2805:
2804:
2792:
2786:
2775:
2768:
2767:
2753:
2751:
2740:
2735:
2734:
2722:
2716:
2709:
2708:
2698:
2683:
2681:
2670:
2665:
2664:
2652:
2646:
2635:
2620:
2618:
2607:
2602:
2600:
2592:
2584:
2575:
2573:
2572:
2567:
2565:
2553:
2551:
2550:
2545:
2533:
2531:
2530:
2525:
2513:
2511:
2510:
2505:
2500:
2499:
2487:
2481:
2470:
2463:
2462:
2445:
2444:
2432:
2426:
2419:
2418:
2408:
2390:
2389:
2377:
2371:
2360:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2291:
2289:
2288:
2283:
2259:
2257:
2256:
2251:
2227:
2225:
2224:
2219:
2195:
2193:
2192:
2187:
2163:
2161:
2160:
2155:
2153:
2152:
2140:
2134:
2123:
2080:
2078:
2077:
2072:
2060:
2058:
2057:
2052:
2050:
2025:
2023:
2022:
2017:
2005:
2003:
2002:
1997:
1995:
1994:
1989:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1946:
1945:
1940:
1925:
1901:
1899:
1898:
1893:
1891:
1877:
1875:
1874:
1869:
1867:
1836:
1834:
1833:
1828:
1826:
1815:
1804:
1793:
1767:
1765:
1764:
1759:
1724:
1723:
1711:
1706:
1705:
1684:gradient theorem
1681:
1679:
1678:
1673:
1661:
1659:
1658:
1653:
1641:
1639:
1638:
1633:
1617:
1615:
1614:
1609:
1597:
1595:
1594:
1589:
1587:
1575:
1573:
1572:
1567:
1565:
1564:
1559:
1546:
1544:
1543:
1538:
1526:
1524:
1523:
1518:
1502:
1500:
1499:
1494:
1492:
1491:
1475:
1473:
1472:
1467:
1456:
1440:
1438:
1437:
1432:
1430:
1411:
1409:
1408:
1403:
1397:
1386:
1381:
1380:
1379:
1378:
1358:
1347:
1342:
1341:
1340:
1339:
1317:
1315:
1314:
1309:
1307:
1306:
1290:
1288:
1287:
1282:
1280:
1279:
1260:
1258:
1257:
1252:
1244:
1233:
1228:
1227:
1226:
1225:
1205:
1194:
1189:
1188:
1187:
1186:
1169:
1158:
1153:
1152:
1151:
1150:
1133:
1122:
1117:
1116:
1115:
1114:
1097:
1086:
1081:
1080:
1079:
1078:
1057:
1055:
1054:
1049:
1037:
1035:
1034:
1029:
1017:
1015:
1014:
1009:
1007:
1006:
990:
988:
987:
982:
970:
968:
967:
962:
950:
948:
947:
942:
940:
939:
923:
921:
920:
915:
913:
912:
896:
894:
893:
888:
876:
874:
873:
868:
866:
865:
849:
847:
846:
841:
833:
822:
817:
816:
815:
814:
788:
786:
785:
780:
768:
766:
765:
760:
758:
757:
741:
739:
738:
733:
731:
730:
712:
710:
709:
704:
702:
691:
686:
685:
684:
683:
666:
655:
650:
649:
648:
647:
626:
624:
623:
618:
616:
596:Gradient theorem
584:solenoidal field
574:
572:
571:
566:
564:
551:scalar potential
548:
546:
545:
540:
528:
526:
525:
520:
518:
506:
504:
503:
498:
486:
484:
483:
478:
462:
460:
459:
454:
437:
435:
434:
429:
415:
401:
399:
398:
393:
381:
379:
378:
373:
354:
352:
351:
346:
344:
343:
327:
325:
324:
319:
317:
316:
311:
298:
296:
295:
290:
278:
276:
275:
270:
268:
267:
262:
247:
193:
191:
190:
185:
183:
158:potential energy
142:physical systems
127:simply connected
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
7264:
7263:
7259:
7258:
7257:
7255:
7254:
7253:
7244:Vector calculus
7234:
7233:
7227:
7214:
7211:
7209:Further reading
7206:
7205:
7198:
7178:
7177:
7173:
7168:
7164:
7157:
7144:
7143:
7136:
7110:
7109:
7090:
7089:
7067:
7066:
7043:
7042:
7008:
7007:
7005:
6998:
6989:
6985:
6978:Vector calculus
6972:
6971:
6967:
6962:
6915:
6859:
6848:
6847:
6826:
6825:
6806:
6805:
6786:
6785:
6766:
6765:
6746:
6745:
6726:
6725:
6706:
6705:
6689:is integrable.
6662:
6661:
6611:
6610:
6589:
6584:
6583:
6557:
6550:
6549:
6522:
6517:
6516:
6488:
6481:
6480:
6447:
6413:
6408:
6407:
6384:
6363:
6358:
6357:
6338:
6337:
6318:
6317:
6283:
6282:
6259:
6258:
6216:
6205:
6185:
6180:
6179:
6158:
6157:
6138:
6137:
6118:
6117:
6098:
6097:
6074:
6069:
6068:
6033:
6032:
6027:
6024:
6007:
5996:
5993:
5988:
5979:
5970:
5959:
5949:
5857:
5856:
5835:
5834:
5828:
5822:
5794:
5793:
5772:
5771:
5770:
5741:
5736:
5735:
5734:
5706:
5705:
5686:
5685:
5665:
5664:
5663:
5642:
5641:
5640:
5618:
5617:
5598:
5597:
5575:
5574:
5550:
5549:
5548:
5523:
5522:
5521:
5512:
5484:
5479:
5478:
5454:
5449:
5448:
5429:
5428:
5373:
5342:
5337:
5332:
5331:
5328:Stokes' theorem
5306:
5305:
5284:
5283:
5259:
5258:
5239:
5238:
5203:
5202:
5181:
5180:
5132:
5114:
5109:
5108:
5077:
5064:
5063:
5037:
5036:
5014:
5013:
4992:
4991:
4966:
4965:
4946:
4945:
4910:
4909:
4890:
4889:
4868:
4867:
4827:
4814:
4813:
4792:
4779:
4778:
4769:
4765:
4712:
4711:
4692:
4691:
4670:
4669:
4602:
4591:
4590:
4571:
4570:
4547:
4542:
4541:
4522:
4521:
4502:
4501:
4474:
4473:
4452:
4447:
4446:
4427:
4426:
4401:
4400:
4355:
4354:
4334:
4333:
4314:
4313:
4292:
4287:
4286:
4237:
4236:
4217:
4216:
4197:
4196:
4195:) scalar field
4171:
4166:
4165:
4114:
4113:
4094:
4093:
4072:
4071:
4046:
4045:
4022:
4017:
4016:
3997:
3996:
3971:
3966:
3965:
3942:
3923:
3922:
3897:
3896:
3873:
3872:
3853:
3852:
3829:
3824:
3823:
3756:
3745:
3744:
3709:
3696:
3695:
3674:
3661:
3660:
3651:
3647:
3634:
3633:
3626:
3574:
3566:
3546:
3538:
3471:
3470:
3410:
3400:
3399:
3307:
3292:
3239:
3224:
3188:
3178:
3177:
3125:
3124:
3105:
3104:
3085:
3084:
3063:
3062:
3041:
3040:
2963:
2962:
2941:
2940:
2921:
2920:
2899:
2894:
2893:
2835:
2820:
2759:
2744:
2700:
2674:
2611:
2593:
2585:
2578:
2577:
2556:
2555:
2536:
2535:
2516:
2515:
2454:
2410:
2341:
2340:
2321:
2320:
2301:
2300:
2262:
2261:
2230:
2229:
2198:
2197:
2166:
2165:
2083:
2082:
2063:
2062:
2041:
2040:
2027:
2008:
2007:
1984:
1979:
1978:
1959:
1958:
1935:
1916:
1915:
1882:
1881:
1858:
1857:
1784:
1783:
1697:
1692:
1691:
1664:
1663:
1644:
1643:
1624:
1623:
1600:
1599:
1578:
1577:
1554:
1549:
1548:
1529:
1528:
1509:
1508:
1507:) scalar field
1483:
1478:
1477:
1447:
1446:
1421:
1420:
1417:
1370:
1362:
1331:
1326:
1320:
1319:
1298:
1293:
1292:
1271:
1263:
1262:
1217:
1209:
1178:
1173:
1142:
1137:
1106:
1101:
1070:
1065:
1060:
1059:
1040:
1039:
1020:
1019:
998:
993:
992:
973:
972:
953:
952:
931:
926:
925:
904:
899:
898:
879:
878:
857:
852:
851:
806:
801:
796:
795:
771:
770:
749:
744:
743:
722:
717:
716:
675:
670:
639:
634:
629:
628:
607:
606:
603:
598:
592:
555:
554:
531:
530:
509:
508:
489:
488:
469:
468:
442:
441:
406:
405:
384:
383:
364:
363:
335:
330:
329:
306:
301:
300:
281:
280:
257:
238:
237:
231:
209:
171:
170:
166:
94:vector calculus
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
7262:
7260:
7252:
7251:
7246:
7236:
7235:
7232:
7231:
7225:
7210:
7207:
7204:
7203:
7201:, pp. 194–196.
7196:
7180:Liepmann, H.W.
7171:
7162:
7155:
7134:
7117:
7097:
7077:
7074:
7050:
7026:
7023:
7020:
7016:
6996:
6983:
6964:
6963:
6961:
6958:
6957:
6956:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6914:
6911:
6892:
6889:
6884:
6879:
6876:
6872:
6866:
6862:
6858:
6855:
6833:
6813:
6793:
6773:
6753:
6733:
6713:
6678:
6675:
6672:
6669:
6646:
6642:
6635:
6632:
6629:
6626:
6623:
6619:
6596:
6592:
6569:
6564:
6560:
6531:
6526:
6502:
6497:
6492:
6460:
6456:
6453:
6450:
6444:
6432:
6420:
6416:
6391:
6387:
6383:
6380:
6377:
6372:
6367:
6345:
6325:
6298:
6294:
6266:
6244:
6238:
6234:
6223:
6219:
6214:
6211:
6208:
6202:
6199:
6194:
6189:
6165:
6145:
6125:
6116:due to a mass
6105:
6083:
6078:
6041:
6026:
6025:
6023:
6022:
6016:
6010:
6005:
6000:
5994:
5992:
5991:
5986:
5982:
5977:
5973:
5968:
5963:
5956:
5948:
5945:
5902:
5898:
5894:
5891:
5876:
5865:
5843:
5824:Main article:
5821:
5818:
5801:
5779:
5756:
5753:
5748:
5744:
5722:
5719:
5716:
5713:
5693:
5672:
5649:
5625:
5605:
5585:
5582:
5557:
5530:
5511:
5508:
5491:
5487:
5461:
5457:
5436:
5416:
5413:
5409:
5405:
5402:
5399:
5395:
5391:
5388:
5385:
5380:
5376:
5372:
5368:
5364:
5361:
5357:
5349:
5345:
5340:
5314:
5292:
5267:
5246:
5225:
5221:
5217:
5213:
5210:
5189:
5166:
5163:
5160:
5157:
5153:
5149:
5141:
5136:
5131:
5127:
5121:
5117:
5096:
5092:
5086:
5081:
5076:
5072:
5047:
5044:
5024:
5021:
5000:
4974:
4953:
4932:
4928:
4924:
4920:
4917:
4897:
4876:
4853:
4849:
4845:
4842:
4834:
4830:
4826:
4821:
4817:
4812:
4807:
4799:
4795:
4791:
4786:
4782:
4777:
4772:
4768:
4752:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4720:
4699:
4678:
4657:
4653:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4611:
4606:
4601:
4598:
4578:
4556:
4551:
4529:
4509:
4481:
4459:
4455:
4434:
4408:
4399:Provided that
4377:
4374:
4371:
4368:
4365:
4362:
4341:
4321:
4299:
4295:
4270:
4266:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4224:
4204:
4178:
4174:
4140:
4136:
4132:
4128:
4124:
4121:
4101:
4092:everywhere in
4080:
4054:
4031:
4026:
4004:
3978:
3974:
3951:
3946:
3941:
3938:
3935:
3931:
3910:
3907:
3904:
3880:
3860:
3838:
3833:
3811:
3807:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3765:
3760:
3755:
3752:
3731:
3727:
3724:
3716:
3712:
3708:
3703:
3699:
3694:
3689:
3681:
3677:
3673:
3668:
3664:
3659:
3654:
3650:
3646:
3642:
3625:
3622:
3597:
3594:
3591:
3587:
3580:
3577:
3572:
3569:
3563:
3559:
3552:
3549:
3544:
3541:
3535:
3531:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3505:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3479:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3416:
3413:
3409:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3333:
3330:
3327:
3322:
3319:
3314:
3310:
3305:
3298:
3295:
3291:
3286:
3282:
3278:
3275:
3271:
3265:
3262:
3259:
3254:
3251:
3246:
3242:
3237:
3230:
3227:
3223:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3194:
3191:
3187:
3164:
3160:
3157:
3154:
3150:
3146:
3143:
3140:
3136:
3132:
3112:
3092:
3071:
3049:
3027:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3001:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2973:
2949:
2928:
2906:
2902:
2879:
2874:
2871:
2867:
2861:
2858:
2855:
2850:
2847:
2842:
2838:
2833:
2826:
2823:
2819:
2814:
2811:
2808:
2803:
2798:
2795:
2791:
2785:
2782:
2779:
2774:
2771:
2766:
2762:
2757:
2750:
2747:
2743:
2738:
2733:
2728:
2725:
2721:
2715:
2712:
2707:
2703:
2697:
2694:
2691:
2687:
2680:
2677:
2673:
2668:
2663:
2658:
2655:
2651:
2645:
2642:
2639:
2634:
2631:
2628:
2624:
2617:
2614:
2610:
2605:
2599:
2596:
2591:
2588:
2564:
2543:
2523:
2503:
2498:
2493:
2490:
2486:
2480:
2477:
2474:
2469:
2466:
2461:
2457:
2452:
2448:
2443:
2438:
2435:
2431:
2425:
2422:
2417:
2413:
2407:
2404:
2401:
2397:
2393:
2388:
2383:
2380:
2376:
2370:
2367:
2364:
2359:
2356:
2353:
2349:
2328:
2308:
2281:
2278:
2275:
2272:
2269:
2249:
2246:
2243:
2240:
2237:
2217:
2214:
2211:
2208:
2205:
2185:
2182:
2179:
2176:
2173:
2151:
2146:
2143:
2139:
2133:
2130:
2127:
2122:
2119:
2116:
2112:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2070:
2049:
2015:
1993:
1988:
1966:
1944:
1939:
1934:
1931:
1928:
1924:
1910:
1890:
1866:
1825:
1821:
1818:
1814:
1810:
1807:
1803:
1799:
1796:
1792:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1722:
1717:
1714:
1710:
1704:
1700:
1690:) states that
1671:
1651:
1631:
1607:
1586:
1563:
1558:
1536:
1516:
1490:
1486:
1465:
1462:
1459:
1455:
1429:
1416:
1413:
1400:
1396:
1392:
1389:
1385:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1350:
1346:
1338:
1334:
1329:
1305:
1301:
1278:
1274:
1270:
1250:
1247:
1243:
1239:
1236:
1232:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1197:
1193:
1185:
1181:
1176:
1172:
1168:
1164:
1161:
1157:
1149:
1145:
1140:
1136:
1132:
1128:
1125:
1121:
1113:
1109:
1104:
1100:
1096:
1092:
1089:
1085:
1077:
1073:
1068:
1047:
1027:
1005:
1001:
980:
960:
938:
934:
911:
907:
886:
864:
860:
839:
836:
832:
828:
825:
821:
813:
809:
804:
778:
756:
752:
729:
725:
701:
697:
694:
690:
682:
678:
673:
669:
665:
661:
658:
654:
646:
642:
637:
615:
602:
599:
594:Main article:
591:
588:
563:
538:
517:
496:
476:
452:
449:
427:
424:
421:
418:
414:
391:
371:
342:
338:
315:
310:
288:
266:
261:
256:
253:
250:
246:
230:
227:
212:M. C. Escher's
208:
205:
182:
178:
165:
162:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7261:
7250:
7247:
7245:
7242:
7241:
7239:
7228:
7222:
7218:
7213:
7212:
7208:
7199:
7197:0-486-41963-0
7193:
7189:
7185:
7181:
7175:
7172:
7166:
7163:
7158:
7152:
7148:
7141:
7139:
7135:
7131:
7115:
7095:
7075:
7064:
7048:
7040:
7024:
7018:
7003:
7001:
6997:
6993:
6987:
6984:
6979:
6975:
6969:
6966:
6959:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6916:
6912:
6910:
6908:
6903:
6890:
6887:
6877:
6874:
6864:
6860:
6856:
6853:
6845:
6831:
6811:
6791:
6771:
6751:
6731:
6711:
6703:
6699:
6695:
6690:
6673:
6667:
6630:
6624:
6621:
6594:
6567:
6562:
6547:
6529:
6500:
6495:
6478:
6473:
6458:
6454:
6451:
6448:
6442:
6430:
6418:
6405:
6389:
6378:
6375:
6370:
6343:
6323:
6315:
6280:
6264:
6255:
6242:
6221:
6217:
6212:
6209:
6206:
6200:
6197:
6192:
6177:
6163:
6143:
6123:
6103:
6081:
6067:
6063:
6058:
6056:
6020:
6017:
6014:
6011:
6008:
6002:
6001:
5995:
5989:
5983:
5980:
5974:
5971:
5965:
5964:
5958:
5957:
5953:
5946:
5944:
5941:
5937:
5932:
5930:
5926:
5922:
5921:inviscid flow
5918:
5913:
5900:
5892:
5874:
5833:
5827:
5819:
5817:
5815:
5799:
5792:are exact if
5777:
5754:
5751:
5746:
5742:
5720:
5717:
5714:
5711:
5691:
5670:
5647:
5639:
5623:
5603:
5583:
5580:
5573:
5555:
5547:
5543:
5528:
5520:differential
5517:
5509:
5507:
5506:
5489:
5485:
5477:
5459:
5455:
5434:
5414:
5411:
5403:
5400:
5389:
5378:
5374:
5370:
5362:
5359:
5347:
5343:
5338:
5329:
5280:
5244:
5219:
5211:
5177:
5164:
5161:
5158:
5155:
5151:
5147:
5139:
5129:
5119:
5115:
5094:
5090:
5084:
5074:
5061:
5045:
5042:
5022:
5019:
4989:
4951:
4926:
4918:
4895:
4864:
4851:
4847:
4843:
4840:
4832:
4828:
4824:
4819:
4815:
4810:
4805:
4797:
4793:
4789:
4784:
4780:
4775:
4770:
4766:
4750:
4737:
4734:
4731:
4728:
4725:
4697:
4647:
4644:
4641:
4635:
4632:
4629:
4626:
4623:
4609:
4599:
4596:
4576:
4554:
4527:
4507:
4499:
4494:
4492:
4479:
4457:
4453:
4432:
4422:
4406:
4397:
4395:
4391:
4372:
4363:
4352:
4339:
4319:
4297:
4293:
4281:
4268:
4260:
4254:
4245:
4222:
4202:
4194:
4176:
4172:
4164:that for any
4163:
4158:
4156:
4151:
4138:
4130:
4122:
4099:
4069:
4029:
4002:
3994:
3976:
3972:
3949:
3936:
3933:
3908:
3905:
3902:
3878:
3858:
3836:
3801:
3798:
3795:
3789:
3786:
3783:
3780:
3777:
3763:
3753:
3750:
3729:
3725:
3722:
3714:
3710:
3706:
3701:
3697:
3692:
3687:
3679:
3675:
3671:
3666:
3662:
3657:
3652:
3648:
3644:
3630:
3623:
3621:
3619:
3615:
3611:
3595:
3589:
3578:
3570:
3561:
3550:
3542:
3533:
3522:
3519:
3516:
3510:
3507:
3496:
3493:
3490:
3484:
3481:
3453:
3450:
3447:
3441:
3438:
3432:
3429:
3426:
3420:
3414:
3396:
3394:
3375:
3372:
3369:
3363:
3360:
3357:
3354:
3348:
3345:
3342:
3336:
3331:
3328:
3325:
3320:
3317:
3312:
3308:
3303:
3296:
3284:
3276:
3273:
3263:
3260:
3257:
3252:
3249:
3244:
3240:
3235:
3228:
3216:
3210:
3207:
3204:
3198:
3192:
3158:
3155:
3152:
3144:
3141:
3138:
3130:
3110:
3090:
3018:
3015:
3012:
3006:
3003:
2992:
2989:
2986:
2980:
2977:
2926:
2904:
2900:
2872:
2869:
2859:
2856:
2853:
2848:
2845:
2840:
2836:
2831:
2824:
2812:
2809:
2806:
2796:
2793:
2783:
2780:
2777:
2772:
2769:
2764:
2760:
2755:
2748:
2736:
2726:
2723:
2713:
2710:
2705:
2701:
2695:
2692:
2689:
2685:
2678:
2666:
2656:
2653:
2643:
2640:
2637:
2632:
2629:
2626:
2622:
2615:
2603:
2597:
2589:
2541:
2521:
2501:
2491:
2488:
2478:
2475:
2472:
2467:
2464:
2459:
2455:
2450:
2446:
2436:
2433:
2423:
2420:
2415:
2411:
2405:
2402:
2399:
2395:
2391:
2381:
2378:
2368:
2365:
2362:
2357:
2354:
2351:
2347:
2326:
2306:
2298:
2293:
2276:
2273:
2270:
2244:
2241:
2238:
2212:
2209:
2206:
2180:
2177:
2174:
2144:
2141:
2131:
2128:
2125:
2120:
2117:
2114:
2110:
2106:
2100:
2097:
2094:
2088:
2068:
2034:
2030:
2013:
1991:
1964:
1942:
1929:
1926:
1914:
1909:
1907:
1906:biconditional
1903:
1854:
1852:
1848:
1844:
1840:
1819:
1816:
1812:
1805:
1797:
1794:
1781:
1777:
1773:
1768:
1755:
1749:
1743:
1740:
1734:
1728:
1725:
1715:
1712:
1702:
1698:
1689:
1686:(also called
1685:
1669:
1649:
1629:
1621:
1605:
1561:
1534:
1514:
1506:
1488:
1484:
1463:
1457:
1445:Suppose that
1444:
1414:
1412:
1398:
1390:
1387:
1375:
1371:
1367:
1363:
1359:
1351:
1348:
1336:
1332:
1327:
1303:
1299:
1276:
1272:
1268:
1248:
1245:
1237:
1234:
1222:
1218:
1214:
1210:
1206:
1198:
1195:
1183:
1179:
1174:
1170:
1162:
1159:
1147:
1143:
1138:
1134:
1126:
1123:
1111:
1107:
1102:
1098:
1090:
1087:
1075:
1071:
1066:
1045:
1025:
1003:
999:
978:
958:
936:
932:
909:
905:
884:
862:
858:
837:
834:
826:
823:
811:
807:
802:
793:
790:
776:
754:
750:
727:
723:
713:
695:
692:
680:
676:
671:
667:
659:
656:
644:
640:
635:
600:
597:
589:
587:
585:
581:
576:
552:
536:
494:
474:
466:
450:
438:
425:
422:
416:
403:
389:
369:
362:
358:
340:
336:
313:
286:
264:
251:
248:
236:
228:
226:
224:
219:
218:
213:
206:
200:
196:
180:
176:
163:
161:
159:
155:
151:
147:
143:
139:
135:
130:
128:
124:
120:
115:
114:line integral
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
7216:
7187:
7174:
7165:
7146:
6991:
6986:
6977:
6968:
6904:
6846:
6697:
6691:
6474:
6406:
6313:
6256:
6178:
6059:
6030:
6018:
6012:
6003:
5984:
5975:
5966:
5939:
5935:
5933:
5914:
5829:
5513:
5504:
5475:
5281:
5178:
4865:
4497:
4495:
4424:
4398:
4284:
4282:
4159:
4152:
3894:
3397:
2294:
2039:
2028:
1913:vector field
1879:
1855:
1769:
1687:
1442:
1418:
794:
791:
714:
604:
577:
549:is called a
463:denotes the
439:
404:
361:scalar field
235:vector field
232:
215:
210:
167:
131:
119:irrotational
118:
104:that is the
102:vector field
97:
91:
76:
67:
48:
6724:to a point
5510:Abstraction
5179:Therefore,
4988:circulation
4283:Therefore,
4112:, i.e., if
3743:defined on
2081:defined as
1847:cylindrical
1682:. Then the
62:introducing
7238:Categories
7226:0198596790
7184:Roshko, A.
6960:References
6905:The total
5704:such that
5035:-plane is
4235:, we have
1778:, and the
1776:chain rule
402:such that
229:Definition
45:references
7186:(1993) ,
7116:φ
7096:φ
7076:φ
7073:∇
7049:φ
7025:φ
7022:∇
6875:⋅
6861:∮
6702:work done
6645:^
6591:Φ
6559:Φ
6443:−
6415:Φ
6386:Φ
6382:∇
6379:−
6297:^
6237:^
6201:−
5893:×
5890:∇
5864:ω
5842:ω
5832:vorticity
5826:Vorticity
5820:Vorticity
5715:ω
5692:ω
5604:ϕ
5584:ϕ
5401:⋅
5390:×
5387:∇
5375:∬
5360:⋅
5339:∮
5220:≡
5212:×
5209:∇
5162:π
5152:ϕ
5140:ϕ
5130:⋅
5116:∮
5085:ϕ
5046:π
4964:), i.e.,
4927:≡
4919:×
4916:∇
4771:−
4648:∈
4642:∣
4615:∖
4373:φ
4370:∇
4364:×
4361:∇
4261:≡
4255:φ
4252:∇
4246:×
4243:∇
4203:φ
4160:It is an
4131:≡
4123:×
4120:∇
3940:→
3802:∈
3796:∣
3769:∖
3653:−
3596:φ
3593:∇
3576:∂
3571:φ
3568:∂
3548:∂
3543:φ
3540:∂
3421:φ
3412:∂
3408:∂
3304:∫
3294:∂
3290:∂
3274:⋅
3236:∫
3226:∂
3222:∂
3199:φ
3190:∂
3186:∂
2870:⋅
2832:∫
2822:∂
2818:∂
2794:⋅
2756:∫
2746:∂
2742:∂
2724:⋅
2686:∫
2676:∂
2672:∂
2654:⋅
2623:∫
2613:∂
2609:∂
2595:∂
2590:φ
2587:∂
2542:φ
2489:⋅
2451:∫
2434:⋅
2396:∫
2379:⋅
2348:∫
2142:⋅
2111:∫
2089:φ
2069:φ
1933:→
1843:Cartesian
1817:⋅
1813:φ
1809:∇
1795:⋅
1744:φ
1741:−
1729:φ
1713:⋅
1699:∫
1515:φ
1476:for some
1464:φ
1461:∇
1388:⋅
1368:−
1364:∫
1349:⋅
1328:∫
1269:−
1235:⋅
1215:−
1211:∫
1207:−
1196:⋅
1175:∫
1160:⋅
1139:∫
1124:⋅
1103:∫
1088:⋅
1067:∫
824:⋅
803:∫
693:⋅
672:∫
657:⋅
636:∫
537:φ
495:φ
475:φ
451:φ
448:∇
423:φ
420:∇
370:φ
255:→
150:conserved
144:in which
134:mechanics
7147:Calculus
6913:See also
6546:gradient
6404:, where
3822:, i.e.,
1957:, where
487:. Since
465:gradient
279:, where
110:function
108:of some
106:gradient
70:May 2009
6544:is the
6475:is the
6336:toward
6277:is the
4044:. Then
58:improve
7223:
7194:
7153:
7128:to be
7037:to be
6907:energy
6424:
6257:where
5997:
5960:
5887:
5868:
5790:-forms
5683:-forms
5660:-forms
5638:closed
5568:-forms
5541:-forms
5257:where
5237:since
5145:
4763:
4744:
4285:every
3039:where
2892:since
1774:, the
1261:where
991:, and
440:Here,
146:energy
138:forces
47:, but
7249:Force
6312:is a
6156:from
5936:local
5733:. As
5546:exact
5058:; in
4866:Then
4445:is a
4419:is a
4392:with
4388:in a
3964:be a
2534:(for
1849:, or
1622:) in
1527:over
1058:, so
1018:from
100:is a
7221:ISBN
7192:ISBN
7151:ISBN
7006:For
6764:and
6692:For
6314:unit
6281:and
6064:, a
5830:The
4068:curl
3895:Let
3618:here
3103:and
3061:and
2919:and
2260:and
1576:(so
742:and
578:The
553:for
154:work
123:curl
96:, a
6824:is
6436:def
5978:pot
5940:not
5880:def
5812:is
5616:on
4990:of
4756:def
4710:by
4690:on
4540:be
4498:not
4215:on
4070:is
4015:of
3469:so
2961:as
1038:to
877:in
467:of
382:on
148:is
140:of
92:In
7240::
7182:;
7137:^
7041:,
6999:^
6891:0.
6844::
6696:,
6057:.
5931:.
5816:.
5062:,
4493:.
4157:.
3395:.
3176:,
1845:,
1782:.
789:.
586:.
575:.
359:)
233:A
129:.
7229:.
7159:.
7132:.
7019:=
7015:v
6888:=
6883:r
6878:d
6871:F
6865:C
6857:=
6854:W
6832:0
6812:C
6792:W
6772:B
6752:A
6732:B
6712:A
6677:)
6674:r
6671:(
6668:F
6641:r
6634:)
6631:r
6628:(
6625:F
6622:=
6618:F
6595:G
6568:m
6563:G
6530:G
6525:F
6501:m
6496:G
6491:F
6459:r
6455:M
6452:m
6449:G
6431:=
6419:G
6390:G
6376:=
6371:G
6366:F
6344:m
6324:M
6293:r
6265:G
6243:,
6233:r
6222:2
6218:r
6213:M
6210:m
6207:G
6198:=
6193:G
6188:F
6164:m
6144:r
6124:M
6104:m
6082:G
6077:F
6040:F
6019:E
6013:F
6006:G
6004:a
5987:C
5985:V
5976:W
5969:G
5967:V
5901:.
5897:v
5875:=
5800:U
5778:1
5767:,
5755:0
5752:=
5747:2
5743:d
5721:0
5718:=
5712:d
5671:1
5648:1
5624:U
5581:d
5556:1
5529:1
5490:1
5486:C
5460:c
5456:P
5435:A
5415:0
5412:=
5408:a
5404:d
5398:)
5394:v
5384:(
5379:A
5371:=
5367:r
5363:d
5356:v
5348:c
5344:P
5330:,
5313:v
5291:v
5266:v
5245:U
5224:0
5216:v
5188:v
5165:.
5159:2
5156:=
5148:d
5135:e
5126:v
5120:C
5095:r
5091:/
5080:e
5075:=
5071:v
5043:2
5023:y
5020:x
4999:v
4973:v
4952:U
4931:0
4923:v
4908:(
4896:U
4875:v
4852:.
4848:)
4844:0
4841:,
4833:2
4829:y
4825:+
4820:2
4816:x
4811:x
4806:,
4798:2
4794:y
4790:+
4785:2
4781:x
4776:y
4767:(
4751:=
4741:)
4738:z
4735:,
4732:y
4729:,
4726:x
4723:(
4719:v
4698:U
4677:v
4656:}
4652:R
4645:z
4639:)
4636:z
4633:,
4630:0
4627:,
4624:0
4621:(
4618:{
4610:3
4605:R
4600:=
4597:U
4577:z
4555:3
4550:R
4528:U
4508:U
4480:U
4458:1
4454:C
4433:U
4407:U
4376:)
4367:(
4340:U
4320:U
4298:1
4294:C
4269:.
4265:0
4258:)
4249:(
4223:U
4191:(
4177:2
4173:C
4139:.
4135:0
4127:v
4100:U
4079:0
4053:v
4030:n
4025:R
4003:U
3991:(
3977:1
3973:C
3950:3
3945:R
3937:U
3934::
3930:v
3909:3
3906:=
3903:n
3879:U
3859:z
3837:3
3832:R
3810:}
3806:R
3799:z
3793:)
3790:z
3787:,
3784:0
3781:,
3778:0
3775:(
3772:{
3764:3
3759:R
3754:=
3751:U
3730:)
3726:0
3723:,
3715:2
3711:y
3707:+
3702:2
3698:x
3693:x
3688:,
3680:2
3676:y
3672:+
3667:2
3663:x
3658:y
3649:(
3645:=
3641:v
3590:=
3586:j
3579:y
3562:+
3558:i
3551:x
3534:=
3530:j
3526:)
3523:y
3520:,
3517:x
3514:(
3511:Q
3508:+
3504:i
3500:)
3497:y
3494:,
3491:x
3488:(
3485:P
3482:=
3478:v
3457:)
3454:y
3451:,
3448:x
3445:(
3442:Q
3439:=
3436:)
3433:y
3430:,
3427:x
3424:(
3415:y
3379:)
3376:y
3373:,
3370:x
3367:(
3364:P
3361:=
3358:t
3355:d
3352:)
3349:y
3346:,
3343:t
3340:(
3337:P
3332:y
3329:,
3326:x
3321:y
3318:,
3313:1
3309:x
3297:x
3285:=
3281:r
3277:d
3270:v
3264:y
3261:,
3258:x
3253:y
3250:,
3245:1
3241:x
3229:x
3217:=
3214:)
3211:y
3208:,
3205:x
3202:(
3193:x
3163:j
3159:y
3156:d
3153:+
3149:i
3145:x
3142:d
3139:=
3135:r
3131:d
3111:y
3091:x
3070:j
3048:i
3026:j
3022:)
3019:y
3016:,
3013:x
3010:(
3007:Q
3004:+
3000:i
2996:)
2993:y
2990:,
2987:x
2984:(
2981:P
2978:=
2972:v
2948:v
2927:x
2905:1
2901:x
2878:r
2873:d
2866:v
2860:y
2857:,
2854:x
2849:y
2846:,
2841:1
2837:x
2825:x
2813:+
2810:0
2807:=
2802:r
2797:d
2790:v
2784:y
2781:,
2778:x
2773:y
2770:,
2765:1
2761:x
2749:x
2737:+
2732:r
2727:d
2720:v
2714:y
2711:,
2706:1
2702:x
2696:b
2693:,
2690:a
2679:x
2667:=
2662:r
2657:d
2650:v
2644:y
2641:,
2638:x
2633:b
2630:,
2627:a
2616:x
2604:=
2598:x
2563:v
2522:x
2502:.
2497:r
2492:d
2485:v
2479:y
2476:,
2473:x
2468:y
2465:,
2460:1
2456:x
2447:+
2442:r
2437:d
2430:v
2424:y
2421:,
2416:1
2412:x
2406:b
2403:,
2400:a
2392:=
2387:r
2382:d
2375:v
2369:y
2366:,
2363:x
2358:b
2355:,
2352:a
2327:y
2307:x
2280:)
2277:y
2274:,
2271:x
2268:(
2248:)
2245:b
2242:,
2239:a
2236:(
2216:)
2213:y
2210:,
2207:x
2204:(
2184:)
2181:b
2178:,
2175:a
2172:(
2150:r
2145:d
2138:v
2132:y
2129:,
2126:x
2121:b
2118:,
2115:a
2107:=
2104:)
2101:y
2098:,
2095:x
2092:(
2048:v
2014:U
1992:n
1987:R
1965:U
1943:n
1938:R
1930:U
1927::
1923:v
1889:v
1865:v
1824:r
1820:d
1806:=
1802:r
1798:d
1791:v
1756:.
1753:)
1750:A
1747:(
1738:)
1735:B
1732:(
1726:=
1721:r
1716:d
1709:v
1703:P
1670:B
1650:A
1630:U
1606:P
1585:v
1562:n
1557:R
1535:U
1503:(
1489:1
1485:C
1458:=
1454:v
1428:v
1399:.
1395:r
1391:d
1384:v
1376:2
1372:P
1360:=
1356:r
1352:d
1345:v
1337:1
1333:P
1304:2
1300:P
1277:2
1273:P
1249:0
1246:=
1242:r
1238:d
1231:v
1223:2
1219:P
1203:r
1199:d
1192:v
1184:1
1180:P
1171:=
1167:r
1163:d
1156:v
1148:2
1144:P
1135:+
1131:r
1127:d
1120:v
1112:1
1108:P
1099:=
1095:r
1091:d
1084:v
1076:c
1072:P
1046:A
1026:B
1004:2
1000:P
979:B
959:A
937:1
933:P
910:c
906:P
885:U
863:c
859:P
838:0
835:=
831:r
827:d
820:v
812:c
808:P
777:U
755:2
751:P
728:1
724:P
700:r
696:d
689:v
681:2
677:P
668:=
664:r
660:d
653:v
645:1
641:P
614:v
562:v
516:v
426:.
417:=
413:v
390:U
355:(
341:1
337:C
314:n
309:R
287:U
265:n
260:R
252:U
249::
245:v
181:R
177:d
83:)
77:(
72:)
68:(
54:.
20:)
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