2351:
6034:
265:
274:
3714:
3046:. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve.
2331:
3428:
2453:
6123:
1662:
20:
3800:
3709:{\displaystyle \operatorname {curl} \mathbf {F} =\nabla \times \mathbf {F} =\left({\frac {\partial F_{3}}{\partial y}}-{\frac {\partial F_{2}}{\partial z}}\right)\mathbf {e} _{1}-\left({\frac {\partial F_{3}}{\partial x}}-{\frac {\partial F_{1}}{\partial z}}\right)\mathbf {e} _{2}+\left({\frac {\partial F_{2}}{\partial x}}-{\frac {\partial F_{1}}{\partial y}}\right)\mathbf {e} _{3}.}
2673:
3382:
4360:
2861:
1161:
3842:
In recent decades many phenomenological formulations of irreversible dynamics and evolution equations in physics, from the mechanics of complex fluids and solids to chemical kinetics and quantum thermodynamics, have converged towards the geometric idea of "steepest entropy ascent" or "gradient flow"
3857:
Consider the flow of a fluid through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector
3018:
Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one
3223:
2510:
3252:
4208:
2718:
2442:
generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body
3734:
The index of a vector field is an integer that helps describe its behaviour around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value −1 at a saddle singularity but +1 at a source or sink singularity.
3747: − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S. This defines a continuous map from S to S. The index of the vector field at the point is the
3742:
the dimension of the manifold on which the vector field is defined. Take a closed surface (homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimension
866:
953:
457:
636:
2367:
A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length
3099:
1610:
3386:
with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the
91:
can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout
234:, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a
3843:
as a consistent universal modeling framework that guarantees compatibility with the second law of thermodynamics and extends well-known near-equilibrium results such as
Onsager reciprocity to the far-nonequilibrium realm.
3775:
For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the
2982:
3420:
is an operation which takes a vector field and produces another vector field. The curl is defined only in three dimensions, but some properties of the curl can be captured in higher dimensions with the
773:
2668:{\displaystyle V=\nabla f=\left({\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}
2184:—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold
4213:
2015:
4050:
3754:
The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to +1 around a source, and more generally equal to (−1) around a saddle that has
3772:
as a whole is defined when it has just finitely many zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.
1462:
4479:
5092:
3377:{\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial F_{1}}{\partial x}}+{\frac {\partial F_{2}}{\partial y}}+{\frac {\partial F_{3}}{\partial z}},}
4845:
6018:
of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of
2376:
map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
5749:
5474:
2155:
1330:
945:
685:
4355:{\displaystyle {\begin{aligned}\gamma _{x}(0)&=x\\\gamma '_{x}(t)&=V(\gamma _{x}(t))\qquad \forall t\in (-\varepsilon ,+\varepsilon )\subset \mathbb {R} .\end{aligned}}}
528:
5323:
5188:
4203:
2279:
2107:
1257:
2856:{\displaystyle \oint _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\oint _{\gamma }\nabla f(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =f(\gamma (1))-f(\gamma (0)).}
895:
203:
85:
2316:
5574:
1505:
209:-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (
4983:
4894:
5254:
5125:
4574:
4389:
4137:
2234:
5025:
4936:
3928:
1156:{\displaystyle {\bigg (}-x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}{\bigg )}(x_{1}^{2}+x_{2}^{2})=-x_{2}(2x_{1})+x_{1}(2x_{2})=0.}
5644:
5609:
784:
5521:
1837:
5221:
5160:
392:
5858:
5826:
5779:
4110:
6048:
5283:
536:
1880:
1811:
5924:
5799:
5704:
5684:
5664:
5379:
5343:
4872:
4805:
4777:
4757:
4729:
4674:
4654:
4634:
4614:
4594:
4547:
4527:
4503:
4437:
4417:
4177:
4157:
4076:
3988:
3968:
3948:
3899:
3879:
2202:
2178:
2035:
1951:
1931:
1900:
1857:
1785:
1761:
1741:
1717:
1693:
705:
497:
477:
6095:
5970:
5904:
2890:
1352:
is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The
139:. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the
5361:
of two vector fields, which is again a vector field. The Lie bracket has a simple definition in terms of the action of vector fields on smooth functions
3722:
of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by
2405:
streamlines (or fieldlines): the path of a particle influenced by the instantaneous field (i.e., the path of a particle if the field is held fixed).
7403:
6019:
3218:{\displaystyle \int _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\int _{a}^{b}V(\gamma (t))\cdot {\dot {\gamma }}(t)\,\mathrm {d} t.}
6594:
7398:
3751:
of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself.
6685:
1653:
to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.
6709:
3748:
1390:
6904:
1629:
211:
3993:
4680:
2339:
2393:
1353:
714:
6774:
6319:
6274:
6242:
2319:
382:(differentiable any number of times). A vector field can be visualized as assigning a vector to individual points within an
7000:
6187:
1632:. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of
1261:
7461:
7053:
6581:
6159:
3249:
of a vector field on
Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by
1956:
7337:
1197:
116:
6486:
6460:
6441:
6405:
6206:
2038:
6166:
7102:
4446:
48:
7456:
7085:
6694:
3835:
In addition to the magnetic field, other phenomena that were modeled by
Faraday include the electrical field and
3039:
227:
136:
5030:
4813:
7446:
6144:
6173:
4055:
7297:
6704:
6518:
6140:
6015:
7282:
7005:
6779:
5709:
5387:
5358:
4696:
2112:
7327:
6548:
6513:
6155:
6090:
900:
644:
6291:
502:
7332:
7302:
7010:
6966:
6947:
6714:
6658:
5165:
4182:
2239:
2044:
5288:
871:
179:
61:
6869:
6734:
3228:
2399:
streaklines: the line produced by particles passing through a specific fixed point over various times
5534:
4505:
in a finite time. In two or three dimensions one can visualize the vector field as giving rise to a
2288:
7451:
7254:
7119:
6811:
6653:
6508:
4877:
3784:
3053:
and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.
2456:
A vector field that has circulation about a point cannot be written as the gradient of a function.
861:{\displaystyle -x_{2}{\frac {\partial }{\partial x_{1}}}+x_{1}{\frac {\partial }{\partial x_{2}}}}
6951:
6921:
6845:
6835:
6791:
6621:
6574:
6133:
5226:
5097:
4850:
On a compact manifold without boundary, every smooth vector field is complete. An example of an
4552:
4367:
4115:
3001:
2369:
2207:
1673:
1356:
distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a
231:
93:
4988:
4899:
3904:
7292:
6911:
6806:
6719:
6626:
6335:
Beretta, Gian Paolo (2020-05-01). "The fourth law of thermodynamics: steepest entropy ascent".
5614:
5579:
4692:
3829:
2420:
2282:
1903:
452:{\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}}
323:
235:
156:
132:
6262:
5494:
1816:
6941:
6936:
6553:
6429:
6309:
6232:
5193:
4941:
2423:
allow us to use a given set of initial and boundary conditions to deduce, for every point in
2350:
459:
for the unit vectors in the coordinate directions. In these terms, every smooth vector field
340:
5130:
631:{\displaystyle \sum _{i=1}^{n}V_{i}(x_{1},\ldots ,x_{n}){\frac {\partial }{\partial x_{i}}}}
7272:
7210:
7058:
6762:
6752:
6724:
6699:
6609:
6481:. Pure and Applied Mathematics, volume 120 (second ed.). Orlando, FL: Academic Press.
6354:
6074:
6063:
5831:
5804:
5757:
5354:
4088:
4079:
3788:
1333:
160:, whose domain's dimension has no relation to the dimension of its range; for example, the
6180:
5259:
8:
7410:
7092:
6970:
6955:
6884:
6643:
4780:
3422:
3395:
3065:
2439:
2373:
52:
7383:
6358:
2431:
experienced by a charged test particle at that point; the resulting vector field is the
1862:
1793:
7352:
7307:
7204:
7075:
6879:
6567:
6534:
6475:
6470:
6378:
6344:
6039:
5909:
5784:
5689:
5669:
5649:
5364:
5328:
4857:
4790:
4762:
4742:
4714:
4659:
4639:
4619:
4599:
4579:
4532:
4512:
4506:
4488:
4422:
4402:
4162:
4142:
4061:
3973:
3953:
3933:
3884:
3864:
3777:
3723:
3417:
3411:
3388:
2678:
2495:
2187:
2163:
2020:
1936:
1916:
1885:
1842:
1770:
1746:
1726:
1702:
1678:
1605:{\displaystyle V_{i,y}=\sum _{j=1}^{n}{\frac {\partial y_{i}}{\partial x_{j}}}V_{j,x}.}
1337:
690:
482:
462:
300:. The arrows depict the field at discrete points, however, the field exists everywhere.
223:
148:
6889:
5929:
5863:
7287:
7267:
7262:
7169:
7080:
6894:
6874:
6729:
6668:
6482:
6456:
6437:
6401:
6382:
6370:
6315:
6270:
6238:
6033:
6000:
4440:
3399:
3005:
2396:
are 3 types of lines that can be made from (time-dependent) vector fields. They are:
2359:
2355:
2181:
1764:
1383:
is a choice of
Cartesian coordinates, in terms of which the components of the vector
169:
3828:
should be an object of study, which it has become throughout physics in the form of
7425:
7219:
7174:
7097:
7068:
6926:
6859:
6854:
6849:
6839:
6631:
6614:
6362:
4482:
3719:
3050:
2997:
2690:
1349:
88:
2330:
7368:
7277:
7107:
7063:
6829:
6425:
5488:
4736:
3814:
3807:
2424:
379:
161:
56:
36:
6525:
7234:
7159:
7129:
7027:
7020:
6960:
6931:
6801:
6796:
6757:
6543:
6080:
6058:
5528:
4784:
3852:
3819:
2432:
2410:
2358:. Here: abstract composition of curves following a vector field generated with
2343:
1907:
1788:
239:
128:
105:
2402:
pathlines: showing the path that a given particle (of zero mass) would follow.
2372:) of the arrow will be an indication of the wind speed. A "high" on the usual
7440:
7420:
7244:
7239:
7224:
7214:
7164:
7141:
7015:
6975:
6916:
6864:
6663:
6374:
6100:
4688:
3043:
3033:
2697:
1720:
1340:
of smooth functions, where multiplication of functions is defined pointwise.
264:
124:
273:
7347:
7342:
7184:
7151:
7124:
6434:
Vector calculus, linear algebra, and differential forms. A unified approach
6366:
6085:
6008:
2701:
2467:
1646:
1493:
defining a different coordinate system. Then the components of the vector
243:
7190:
7179:
7136:
7037:
6638:
6105:
3836:
3093:
2977:{\displaystyle V(T(p))=T(V(p))\qquad (T\in \mathrm {O} (n,\mathbb {R} ))}
165:
3783:
For a vector field on a compact manifold with finitely many zeroes, the
226:, where they associate an arrow tangent to the surface at each point (a
7415:
7373:
7199:
7112:
6744:
6648:
6559:
6538:
6053:
5996:
5524:
4684:
3246:
3240:
1636:
functions in each coordinate system subject to the transformation law (
708:
140:
6502:
1497:
in the new coordinates are required to satisfy the transformation law
7229:
7194:
6899:
6786:
6529:
4700:
3803:
3038:
A common technique in physics is to integrate a vector field along a
168:
is defined only for smaller subset of the ambient space. Likewise, n
6122:
2452:
7393:
7388:
7378:
6769:
6590:
6477:
An introduction to differentiable manifolds and
Riemannian geometry
6349:
5981:
2471:
2461:
2387:
2379:
1357:
707:. The reason for this notation is that a vector field determines a
219:
131:
done by a force moving along a path, and under this interpretation
119:
extend naturally to vector fields. When a vector field represents
16:
Assignment of a vector to each point in a subset of
Euclidean space
205:
can be represented as a vector-valued function that associates an
4636:(i.e., the vector field is equal to the zero vector at the point
2285:); the collection of all smooth vector fields is also denoted by
1661:
775:, given by differentiating in the direction of the vector field.
222:
of
Euclidean space, but also make sense on other subsets such as
109:
40:
6985:
6556:
An interactive application to show the effects of vector fields
4735:
if each of its flow curves exists for all time. In particular,
1666:
1194:, the operations of scalar multiplication and vector addition,
144:
6230:
5357:
with each other. Their failure to commute is described by the
254:
19:
3799:
2428:
2383:
120:
101:
2414:
868:
describes a counterclockwise rotation around the origin in
97:
4807:
exists for all time; it is described by a smooth mapping
2475:
2499:
if there exists a real-valued function (a scalar field)
768:{\displaystyle V\colon C^{\infty }(S)\to C^{\infty }(S)}
3787:
states that the vector field’s index is the manifold’s
2334:
The flow field around an airplane is a vector field in
374:
is a continuous vector field. It is common to focus on
6453:
Foundations of differentiable manifolds and Lie groups
6096:
Vector fields in cylindrical and spherical coordinates
5291:
5033:
4944:
2354:
Vector fields are commonly used to create patterns in
2291:
215:) in passing from one coordinate system to the other.
112:
force, as it changes from one point to another point.
6014:
Algebraically, vector fields can be characterized as
5932:
5912:
5866:
5834:
5807:
5787:
5760:
5712:
5692:
5672:
5652:
5617:
5582:
5537:
5497:
5390:
5367:
5331:
5262:
5229:
5196:
5168:
5133:
5100:
4991:
4902:
4880:
4860:
4816:
4793:
4765:
4745:
4717:
4662:
4642:
4622:
4602:
4582:
4555:
4535:
4515:
4491:
4449:
4425:
4405:
4370:
4211:
4185:
4165:
4145:
4118:
4091:
4064:
3996:
3976:
3956:
3936:
3907:
3887:
3867:
3431:
3255:
3102:
2893:
2721:
2513:
2447:
2242:
2210:
2190:
2166:
2115:
2047:
2023:
1959:
1939:
1919:
1888:
1865:
1845:
1819:
1796:
1773:
1749:
1729:
1705:
1681:
1508:
1393:
1264:
1200:
956:
903:
874:
787:
717:
693:
647:
539:
505:
485:
465:
395:
182:
64:
6029:
3049:
The line integral is constructed analogously to the
2865:
6147:. Unsourced material may be challenged and removed.
5353:The flows associated to two vector fields need not
4443:. It is not always possible to extend the interval
4399:(or less commonly, flow lines) of the vector field
6474:
5964:
5918:
5898:
5852:
5820:
5793:
5773:
5743:
5698:
5678:
5658:
5638:
5603:
5568:
5515:
5468:
5373:
5337:
5317:
5277:
5248:
5215:
5182:
5154:
5119:
5086:
5019:
4977:
4930:
4888:
4866:
4839:
4799:
4771:
4751:
4723:
4668:
4648:
4628:
4608:
4588:
4568:
4541:
4529:. If we drop a particle into this flow at a point
4521:
4497:
4473:
4431:
4411:
4383:
4354:
4197:
4171:
4151:
4131:
4104:
4070:
4044:
3982:
3962:
3942:
3922:
3893:
3873:
3708:
3376:
3217:
2976:
2855:
2667:
2310:
2273:
2228:
2196:
2172:
2149:
2101:
2029:
2010:{\displaystyle X:C^{\infty }(M)\to C^{\infty }(M)}
2009:
1945:
1925:
1894:
1874:
1851:
1831:
1805:
1779:
1755:
1735:
1711:
1687:
1604:
1456:
1324:
1251:
1155:
939:
889:
860:
767:
699:
679:
630:
522:
491:
471:
451:
197:
79:
6337:Philosophical Transactions of the Royal Society A
6226:
6224:
1913:An alternative definition: A smooth vector field
1036:
959:
242:to the manifold). Vector fields are one kind of
7438:
6049:Eisenbud–Levine–Khimshiashvili signature formula
3019:semiaxis and integrating gives an antigradient.
2390:vector is associated to each point in the fluid.
1343:
378:vector fields, meaning that each component is a
230:). More generally, vector fields are defined on
6020:differential calculus over commutative algebras
4739:vector fields on a manifold are complete. If
6424:
6256:
6254:
6221:
3022:
2281:(especially when thinking of vector fields as
1457:{\displaystyle V_{x}=(V_{1,x},\dots ,V_{n,x})}
711:from the space of smooth functions to itself,
282:Two representations of the same vector field:
6575:
6260:
4485:. The flow may for example reach the edge of
4474:{\displaystyle (-\varepsilon ,+\varepsilon )}
2413:. The fieldlines can be revealed using small
2338:, here visualized by bubbles that follow the
1656:
1642:) relating the different coordinate systems.
6307:
4045:{\displaystyle \gamma '(t)=V(\gamma (t))\,.}
6251:
5087:{\textstyle x(t)={\frac {x_{0}}{1-tx_{0}}}}
4576:in the flow depending on the initial point
3405:
255:Vector fields on subsets of Euclidean space
151:(which represents the rotation of a flow).
6582:
6568:
6292:"An Introduction to Differential Geometry"
4840:{\displaystyle \mathbf {R} \times M\to M.}
3729:
3227:To show vector field topology one can use
6436:. Upper Saddle River, NJ: Prentice Hall.
6348:
6231:Galbis, Antonio; Maestre, Manuel (2012).
6207:Learn how and when to remove this message
5176:
4882:
4706:
4341:
4038:
3425:. In three dimensions, it is defined by
3203:
2964:
185:
67:
6589:
3798:
3394:The divergence can also be defined on a
2466:Vector fields can be constructed out of
2451:
2349:
2329:
1660:
212:covariance and contravariance of vectors
143:(which represents the rate of change of
100:, or the strength and direction of some
23:A portion of the vector field (sin
18:
6469:
6334:
5325:so cannot be defined for all values of
1645:Vector fields are thus contrasted with
7439:
6450:
6395:
6289:
6234:Vector Analysis Versus Vector Calculus
2715:(1)) in a conservative field is zero:
2394:Streamlines, streaklines and pathlines
1902:. In other words, a vector field is a
154:A vector field is a special case of a
135:is exhibited as a special case of the
6563:
6007:, and combining these yields general
5991:th exterior power of vectors) yields
5744:{\displaystyle W\circ f=f_{*}\circ V}
5469:{\displaystyle (f):=X(Y(f))-Y(X(f)).}
3794:
3718:The curl measures the density of the
3402:that measures the length of vectors.
2150:{\displaystyle f,g\in C^{\infty }(M)}
1332:make the smooth vector fields into a
218:Vector fields are often discussed on
6455:. New York-Berlin: Springer-Verlag.
6145:adding citations to reliable sources
6116:
4656:), then the particle will remain at
2427:, a magnitude and direction for the
1628:Such a transformation law is called
1499:
1354:transformation properties of vectors
1325:{\displaystyle (V+W)(p):=V(p)+W(p),}
947:is rotationally invariant, compute:
3096:), the line integral is defined as
2294:
940:{\displaystyle x_{1}^{2}+x_{2}^{2}}
680:{\displaystyle V_{1},\ldots ,V_{n}}
13:
6290:Lerman, Eugene (August 19, 2011).
5975:
5348:
4938:. For, the differential equation
4307:
3677:
3662:
3647:
3632:
3595:
3580:
3565:
3550:
3513:
3498:
3483:
3468:
3446:
3362:
3347:
3332:
3317:
3302:
3287:
3270:
3205:
3131:
2950:
2793:
2772:
2750:
2641:
2633:
2605:
2597:
2575:
2567:
2545:
2537:
2520:
2448:Gradient field in Euclidean spaces
2248:
2211:
2133:
1993:
1971:
1567:
1552:
1018:
1014:
983:
979:
842:
838:
807:
803:
751:
729:
612:
608:
523:{\displaystyle {\mathbf {R} }^{n}}
433:
429:
402:
398:
389:One standard notation is to write
117:differential and integral calculus
14:
7473:
6496:
5318:{\textstyle t={\frac {1}{x_{0}}}}
5183:{\displaystyle t\in \mathbb {R} }
4711:By definition, a vector field on
4198:{\displaystyle \varepsilon >0}
2866:Central field in euclidean spaces
2274:{\displaystyle C^{\infty }(M,TM)}
2102:{\displaystyle X(fg)=fX(g)+X(f)g}
1743:. More precisely, a vector field
1252:{\displaystyle (fV)(p):=f(p)V(p)}
127:of a vector field represents the
6121:
6032:
5479:
4818:
3693:
3611:
3529:
3453:
3439:
3277:
3263:
3136:
3120:
3027:
2798:
2782:
2755:
2739:
890:{\displaystyle \mathbf {R} ^{2}}
877:
509:
272:
263:
198:{\displaystyle \mathbb {R} ^{n}}
172:, a vector field on a domain in
80:{\displaystyle \mathbb {R} ^{n}}
6418:
6132:needs additional citations for
4306:
2939:
2689:, and is used in the method of
2311:{\textstyle {\mathfrak {X}}(M)}
137:fundamental theorem of calculus
6622:Differentiable/Smooth manifold
6389:
6328:
6301:
6283:
5959:
5933:
5893:
5867:
5627:
5592:
5569:{\displaystyle f_{*}:TM\to TN}
5557:
5507:
5460:
5457:
5451:
5445:
5436:
5433:
5427:
5421:
5412:
5406:
5403:
5391:
5272:
5266:
5143:
5137:
5043:
5037:
5001:
4995:
4959:
4953:
4912:
4906:
4828:
4759:is a complete vector field on
4468:
4450:
4334:
4316:
4303:
4300:
4294:
4281:
4268:
4262:
4232:
4226:
4035:
4032:
4026:
4020:
4011:
4005:
3917:
3911:
3846:
3200:
3194:
3176:
3173:
3167:
3161:
3124:
3116:
3042:, also called determining its
2971:
2968:
2954:
2940:
2936:
2933:
2927:
2921:
2912:
2909:
2903:
2897:
2847:
2844:
2838:
2832:
2823:
2820:
2814:
2808:
2786:
2778:
2743:
2735:
2305:
2299:
2268:
2253:
2223:
2214:
2144:
2138:
2093:
2087:
2078:
2072:
2060:
2051:
2004:
1998:
1985:
1982:
1976:
1839:is the identity mapping where
1649:, which associate a number or
1451:
1407:
1316:
1310:
1301:
1295:
1286:
1280:
1277:
1265:
1246:
1240:
1234:
1228:
1219:
1213:
1210:
1201:
1144:
1128:
1112:
1096:
1077:
1041:
762:
756:
743:
740:
734:
603:
571:
1:
6549:Vector fields and field lines
6112:
5027:, has as its unique solution
4549:it will move along the curve
3770:The index of the vector field
3398:, that is, a manifold with a
3234:
1344:Coordinate transformation law
249:
176:-dimensional Euclidean space
6311:Vectors and Vector Operators
6267:An Introduction to Manifolds
4889:{\displaystyle \mathbb {R} }
4787:generated by the flow along
3806:field lines of an iron bar (
3000:. We say central fields are
1859:denotes the projection from
897:. To show that the function
7:
7328:Classification of manifolds
6514:Encyclopedia of Mathematics
6091:Time-dependent vector field
6025:
5999:and exterior powers yields
5995:-vector fields; taking the
5249:{\displaystyle x_{0}\neq 0}
5120:{\displaystyle x_{0}\neq 0}
4569:{\displaystyle \gamma _{p}}
4384:{\displaystyle \gamma _{x}}
4132:{\displaystyle \gamma _{x}}
3758:contracting dimensions and
3023:Operations on vector fields
2325:
2229:{\displaystyle \Gamma (TM)}
1638:
1618:
10:
7480:
7462:Vector physical quantities
6503:Online Vector Field Editor
5020:{\displaystyle x(0)=x_{0}}
4931:{\displaystyle V(x)=x^{2}}
3923:{\displaystyle \gamma (t)}
3850:
3824:emphasized that the field
3409:
3238:
3031:
3011:The point 0 is called the
3006:orthogonal transformations
2459:
1657:Vector fields on manifolds
641:for some smooth functions
7404:over commutative algebras
7361:
7320:
7253:
7150:
7046:
6993:
6984:
6820:
6743:
6682:
6602:
6314:. CRC Press. p. 29.
6269:. Springer. p. 149.
5639:{\displaystyle W:N\to TN}
5604:{\displaystyle V:M\to TM}
4985:, with initial condition
4679:Typical applications are
4616:is a stationary point of
3229:line integral convolution
2474:operator (denoted by the
7120:Riemann curvature tensor
6544:3D Magnetic field viewer
6237:. Springer. p. 12.
5516:{\displaystyle f:M\to N}
4978:{\textstyle x'(t)=x^{2}}
3406:Curl in three dimensions
1832:{\displaystyle p\circ F}
232:differentiable manifolds
6554:Vector field simulation
6451:Warner, Frank (1983) .
5860:, then the Lie bracket
5216:{\displaystyle x_{0}=0}
4693:one-parameter subgroups
4056:Picard–Lindelöf theorem
3858:field as its velocity.
3730:Index of a vector field
2485:defined on an open set
1674:differentiable manifold
362:. If each component of
94:three dimensional space
87:. A vector field on a
7457:Functions and mappings
6912:Manifold with boundary
6627:Differential structure
6367:10.1098/rsta.2019.0168
6261:Tu, Loring W. (2010).
5966:
5920:
5900:
5854:
5822:
5795:
5775:
5745:
5700:
5680:
5660:
5640:
5605:
5576:. Given vector fields
5570:
5517:
5470:
5375:
5339:
5319:
5279:
5250:
5217:
5184:
5156:
5155:{\displaystyle x(t)=0}
5121:
5088:
5021:
4979:
4932:
4890:
4868:
4841:
4801:
4773:
4753:
4725:
4707:Complete vector fields
4670:
4650:
4630:
4610:
4590:
4570:
4543:
4523:
4499:
4475:
4433:
4413:
4385:
4356:
4199:
4173:
4153:
4133:
4106:
4072:
4046:
3984:
3964:
3944:
3924:
3895:
3875:
3811:
3766:expanding dimensions.
3710:
3378:
3219:
2978:
2857:
2669:
2457:
2363:
2347:
2312:
2275:
2230:
2198:
2174:
2151:
2103:
2031:
2011:
1947:
1927:
1896:
1876:
1853:
1833:
1807:
1781:
1757:
1737:
1719:is an assignment of a
1713:
1689:
1669:
1606:
1548:
1458:
1326:
1253:
1184:and a smooth function
1157:
941:
891:
862:
769:
701:
681:
632:
560:
524:
493:
473:
453:
324:vector-valued function
199:
157:vector-valued function
133:conservation of energy
81:
47:is an assignment of a
32:
7447:Differential topology
6398:Differential geometry
6308:Dawber, P.G. (1987).
5980:Replacing vectors by
5967:
5921:
5901:
5855:
5853:{\displaystyle i=1,2}
5823:
5821:{\displaystyle W_{i}}
5796:
5776:
5774:{\displaystyle V_{i}}
5746:
5701:
5681:
5661:
5641:
5606:
5571:
5527:is an induced map on
5518:
5471:
5376:
5340:
5320:
5280:
5251:
5218:
5185:
5157:
5122:
5089:
5022:
4980:
4933:
4891:
4869:
4842:
4802:
4774:
4754:
4726:
4671:
4651:
4631:
4611:
4591:
4571:
4544:
4524:
4500:
4476:
4434:
4414:
4386:
4357:
4200:
4174:
4154:
4134:
4107:
4105:{\displaystyle C^{1}}
4073:
4047:
3985:
3965:
3945:
3925:
3901:, one defines curves
3896:
3876:
3861:Given a vector field
3802:
3785:Poincaré-Hopf theorem
3711:
3379:
3220:
3056:Given a vector field
2979:
2858:
2670:
2460:Further information:
2455:
2353:
2333:
2313:
2276:
2231:
2199:
2175:
2152:
2104:
2032:
2012:
1948:
1928:
1897:
1877:
1854:
1834:
1808:
1782:
1758:
1738:
1714:
1690:
1664:
1607:
1528:
1459:
1327:
1254:
1158:
942:
892:
863:
770:
702:
682:
633:
540:
525:
494:
474:
454:
341:Cartesian coordinates
200:
82:
22:
7059:Covariant derivative
6610:Topological manifold
6141:improve this article
6075:atmospheric dynamics
5930:
5910:
5864:
5832:
5805:
5785:
5758:
5710:
5690:
5670:
5650:
5615:
5580:
5535:
5495:
5388:
5365:
5329:
5289:
5278:{\displaystyle x(t)}
5260:
5227:
5194:
5166:
5131:
5098:
5031:
4989:
4942:
4900:
4878:
4858:
4814:
4791:
4763:
4743:
4715:
4660:
4640:
4620:
4600:
4580:
4553:
4533:
4513:
4489:
4447:
4423:
4403:
4368:
4209:
4183:
4163:
4143:
4116:
4089:
4080:Lipschitz continuous
4062:
3994:
3974:
3954:
3934:
3905:
3885:
3865:
3817:, in his concept of
3789:Euler characteristic
3429:
3253:
3100:
2891:
2719:
2511:
2289:
2240:
2208:
2204:is often denoted by
2188:
2164:
2113:
2045:
2021:
1957:
1937:
1917:
1886:
1863:
1843:
1817:
1794:
1771:
1747:
1727:
1703:
1679:
1665:A vector field on a
1506:
1391:
1262:
1198:
1166:Given vector fields
954:
901:
872:
785:
715:
691:
645:
537:
503:
483:
463:
393:
386:-dimensional space.
368:is continuous, then
322:is represented by a
180:
62:
7093:Exterior derivative
6695:Atiyah–Singer index
6644:Riemannian manifold
6400:. Springer-Verlag.
6396:Sharpe, R. (1997).
6359:2020RSPTA.37890168B
5491:between manifolds,
4781:one-parameter group
4737:compactly supported
4441:equivalence classes
4261:
3950:such that for each
3423:exterior derivative
3412:Curl (mathematics)
3396:Riemannian manifold
3157:
2876:-vector field over
2440:gravitational field
2421:Maxwell's equations
2386:. In this case, a
2374:barometric pressure
1363:Thus, suppose that
1076:
1058:
936:
918:
781:: The vector field
51:to each point in a
7399:Secondary calculus
7353:Singularity theory
7308:Parallel transport
7076:De Rham cohomology
6715:Generalized Stokes
6343:(2170): 20190168.
6297:. Definition 3.23.
6040:Mathematics portal
5962:
5916:
5896:
5850:
5818:
5791:
5771:
5741:
5696:
5676:
5656:
5636:
5601:
5566:
5513:
5466:
5371:
5335:
5315:
5275:
5246:
5213:
5180:
5152:
5117:
5084:
5017:
4975:
4928:
4886:
4864:
4837:
4797:
4769:
4749:
4721:
4666:
4646:
4626:
4606:
4586:
4566:
4539:
4519:
4495:
4471:
4429:
4409:
4381:
4352:
4350:
4249:
4195:
4179:so that, for some
4169:
4149:
4129:
4102:
4068:
4042:
3980:
3960:
3940:
3920:
3891:
3871:
3812:
3795:Physical intuition
3778:hairy ball theorem
3706:
3389:divergence theorem
3374:
3215:
3143:
2974:
2853:
2665:
2496:conservative field
2458:
2382:field of a moving
2364:
2348:
2308:
2271:
2226:
2194:
2170:
2147:
2099:
2027:
2007:
1943:
1923:
1892:
1875:{\displaystyle TM}
1872:
1849:
1829:
1806:{\displaystyle TM}
1803:
1777:
1753:
1733:
1709:
1685:
1670:
1602:
1464:and suppose that (
1454:
1322:
1249:
1153:
1062:
1044:
937:
922:
904:
887:
858:
765:
697:
677:
628:
530:can be written as
520:
489:
479:on an open subset
469:
449:
195:
77:
33:
7434:
7433:
7316:
7315:
7081:Differential form
6735:Whitney embedding
6669:Differential form
6321:978-0-85274-585-4
6276:978-1-4419-7399-3
6244:978-1-4614-2199-3
6217:
6216:
6209:
6191:
5919:{\displaystyle f}
5794:{\displaystyle f}
5699:{\displaystyle V}
5679:{\displaystyle f}
5659:{\displaystyle W}
5374:{\displaystyle f}
5338:{\displaystyle t}
5313:
5082:
4874:on the real line
4867:{\displaystyle V}
4800:{\displaystyle X}
4772:{\displaystyle M}
4752:{\displaystyle X}
4724:{\displaystyle M}
4669:{\displaystyle p}
4649:{\displaystyle p}
4629:{\displaystyle V}
4609:{\displaystyle p}
4589:{\displaystyle p}
4542:{\displaystyle p}
4522:{\displaystyle S}
4498:{\displaystyle S}
4432:{\displaystyle S}
4412:{\displaystyle V}
4172:{\displaystyle S}
4152:{\displaystyle x}
4071:{\displaystyle V}
3983:{\displaystyle I}
3963:{\displaystyle t}
3943:{\displaystyle S}
3894:{\displaystyle S}
3874:{\displaystyle V}
3684:
3654:
3602:
3572:
3520:
3490:
3400:Riemannian metric
3369:
3339:
3309:
3191:
2655:
2619:
2589:
2559:
2360:OpenSimplex noise
2356:computer graphics
2197:{\displaystyle M}
2173:{\displaystyle M}
2030:{\displaystyle X}
1946:{\displaystyle M}
1926:{\displaystyle X}
1895:{\displaystyle M}
1852:{\displaystyle p}
1780:{\displaystyle M}
1756:{\displaystyle F}
1736:{\displaystyle M}
1723:to each point in
1712:{\displaystyle M}
1688:{\displaystyle M}
1626:
1625:
1581:
1484:functions of the
1032:
997:
856:
821:
700:{\displaystyle S}
626:
492:{\displaystyle S}
472:{\displaystyle V}
447:
416:
7469:
7426:Stratified space
7384:Fréchet manifold
7098:Interior product
6991:
6990:
6688:
6584:
6577:
6570:
6561:
6560:
6522:
6492:
6480:
6471:Boothby, William
6466:
6447:
6412:
6411:
6393:
6387:
6386:
6352:
6332:
6326:
6325:
6305:
6299:
6298:
6296:
6287:
6281:
6280:
6258:
6249:
6248:
6228:
6212:
6205:
6201:
6198:
6192:
6190:
6149:
6125:
6117:
6042:
6037:
6036:
5971:
5969:
5968:
5965:{\displaystyle }
5963:
5958:
5957:
5945:
5944:
5925:
5923:
5922:
5917:
5905:
5903:
5902:
5899:{\displaystyle }
5897:
5892:
5891:
5879:
5878:
5859:
5857:
5856:
5851:
5827:
5825:
5824:
5819:
5817:
5816:
5800:
5798:
5797:
5792:
5780:
5778:
5777:
5772:
5770:
5769:
5750:
5748:
5747:
5742:
5734:
5733:
5706:if the equation
5705:
5703:
5702:
5697:
5685:
5683:
5682:
5677:
5665:
5663:
5662:
5657:
5645:
5643:
5642:
5637:
5610:
5608:
5607:
5602:
5575:
5573:
5572:
5567:
5547:
5546:
5522:
5520:
5519:
5514:
5475:
5473:
5472:
5467:
5380:
5378:
5377:
5372:
5344:
5342:
5341:
5336:
5324:
5322:
5321:
5316:
5314:
5312:
5311:
5299:
5285:is undefined at
5284:
5282:
5281:
5276:
5255:
5253:
5252:
5247:
5239:
5238:
5222:
5220:
5219:
5214:
5206:
5205:
5189:
5187:
5186:
5181:
5179:
5161:
5159:
5158:
5153:
5126:
5124:
5123:
5118:
5110:
5109:
5093:
5091:
5090:
5085:
5083:
5081:
5080:
5079:
5060:
5059:
5050:
5026:
5024:
5023:
5018:
5016:
5015:
4984:
4982:
4981:
4976:
4974:
4973:
4952:
4937:
4935:
4934:
4929:
4927:
4926:
4895:
4893:
4892:
4887:
4885:
4873:
4871:
4870:
4865:
4846:
4844:
4843:
4838:
4821:
4806:
4804:
4803:
4798:
4778:
4776:
4775:
4770:
4758:
4756:
4755:
4750:
4730:
4728:
4727:
4722:
4675:
4673:
4672:
4667:
4655:
4653:
4652:
4647:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4595:
4593:
4592:
4587:
4575:
4573:
4572:
4567:
4565:
4564:
4548:
4546:
4545:
4540:
4528:
4526:
4525:
4520:
4504:
4502:
4501:
4496:
4483:real number line
4480:
4478:
4477:
4472:
4438:
4436:
4435:
4430:
4418:
4416:
4415:
4410:
4390:
4388:
4387:
4382:
4380:
4379:
4361:
4359:
4358:
4353:
4351:
4344:
4293:
4292:
4257:
4225:
4224:
4204:
4202:
4201:
4196:
4178:
4176:
4175:
4170:
4158:
4156:
4155:
4150:
4138:
4136:
4135:
4130:
4128:
4127:
4111:
4109:
4108:
4103:
4101:
4100:
4077:
4075:
4074:
4069:
4051:
4049:
4048:
4043:
4004:
3989:
3987:
3986:
3981:
3969:
3967:
3966:
3961:
3949:
3947:
3946:
3941:
3929:
3927:
3926:
3921:
3900:
3898:
3897:
3892:
3880:
3878:
3877:
3872:
3720:angular momentum
3715:
3713:
3712:
3707:
3702:
3701:
3696:
3690:
3686:
3685:
3683:
3675:
3674:
3673:
3660:
3655:
3653:
3645:
3644:
3643:
3630:
3620:
3619:
3614:
3608:
3604:
3603:
3601:
3593:
3592:
3591:
3578:
3573:
3571:
3563:
3562:
3561:
3548:
3538:
3537:
3532:
3526:
3522:
3521:
3519:
3511:
3510:
3509:
3496:
3491:
3489:
3481:
3480:
3479:
3466:
3456:
3442:
3383:
3381:
3380:
3375:
3370:
3368:
3360:
3359:
3358:
3345:
3340:
3338:
3330:
3329:
3328:
3315:
3310:
3308:
3300:
3299:
3298:
3285:
3280:
3266:
3224:
3222:
3221:
3216:
3208:
3193:
3192:
3184:
3156:
3151:
3139:
3134:
3123:
3112:
3111:
3091:
3087:
3083:
3071:
3063:
3059:
3051:Riemann integral
2998:orthogonal group
2995:
2983:
2981:
2980:
2975:
2967:
2953:
2882:
2875:
2862:
2860:
2859:
2854:
2801:
2796:
2785:
2771:
2770:
2758:
2753:
2742:
2731:
2730:
2691:gradient descent
2687:
2686:
2674:
2672:
2671:
2666:
2661:
2657:
2656:
2654:
2653:
2652:
2639:
2631:
2620:
2618:
2617:
2616:
2603:
2595:
2590:
2588:
2587:
2586:
2573:
2565:
2560:
2558:
2557:
2556:
2543:
2535:
2317:
2315:
2314:
2309:
2298:
2297:
2280:
2278:
2277:
2272:
2252:
2251:
2235:
2233:
2232:
2227:
2203:
2201:
2200:
2195:
2179:
2177:
2176:
2171:
2160:If the manifold
2156:
2154:
2153:
2148:
2137:
2136:
2108:
2106:
2105:
2100:
2036:
2034:
2033:
2028:
2016:
2014:
2013:
2008:
1997:
1996:
1975:
1974:
1953:is a linear map
1952:
1950:
1949:
1944:
1932:
1930:
1929:
1924:
1901:
1899:
1898:
1893:
1881:
1879:
1878:
1873:
1858:
1856:
1855:
1850:
1838:
1836:
1835:
1830:
1812:
1810:
1809:
1804:
1786:
1784:
1783:
1778:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1718:
1716:
1715:
1710:
1694:
1692:
1691:
1686:
1620:
1611:
1609:
1608:
1603:
1598:
1597:
1582:
1580:
1579:
1578:
1565:
1564:
1563:
1550:
1547:
1542:
1524:
1523:
1500:
1463:
1461:
1460:
1455:
1450:
1449:
1425:
1424:
1403:
1402:
1386:
1382:
1331:
1329:
1328:
1323:
1258:
1256:
1255:
1250:
1193:
1187:
1183:
1177:
1171:
1162:
1160:
1159:
1154:
1143:
1142:
1127:
1126:
1111:
1110:
1095:
1094:
1075:
1070:
1057:
1052:
1040:
1039:
1033:
1031:
1030:
1029:
1013:
1011:
1010:
998:
996:
995:
994:
978:
976:
975:
963:
962:
946:
944:
943:
938:
935:
930:
917:
912:
896:
894:
893:
888:
886:
885:
880:
867:
865:
864:
859:
857:
855:
854:
853:
837:
835:
834:
822:
820:
819:
818:
802:
800:
799:
774:
772:
771:
766:
755:
754:
733:
732:
706:
704:
703:
698:
686:
684:
683:
678:
676:
675:
657:
656:
637:
635:
634:
629:
627:
625:
624:
623:
607:
602:
601:
583:
582:
570:
569:
559:
554:
529:
527:
526:
521:
519:
518:
513:
512:
498:
496:
495:
490:
478:
476:
475:
470:
458:
456:
455:
450:
448:
446:
445:
444:
428:
417:
415:
414:
413:
397:
373:
367:
361:
338:
317:
311:
299:
276:
267:
204:
202:
201:
196:
194:
193:
188:
115:The elements of
86:
84:
83:
78:
76:
75:
70:
55:, most commonly
27:, sin
7479:
7478:
7472:
7471:
7470:
7468:
7467:
7466:
7452:Vector calculus
7437:
7436:
7435:
7430:
7369:Banach manifold
7362:Generalizations
7357:
7312:
7249:
7146:
7108:Ricci curvature
7064:Cotangent space
7042:
6980:
6822:
6816:
6775:Exponential map
6739:
6684:
6678:
6598:
6588:
6507:
6499:
6489:
6463:
6444:
6421:
6416:
6415:
6408:
6394:
6390:
6333:
6329:
6322:
6306:
6302:
6294:
6288:
6284:
6277:
6263:"Vector fields"
6259:
6252:
6245:
6229:
6222:
6213:
6202:
6196:
6193:
6150:
6148:
6138:
6126:
6115:
6110:
6038:
6031:
6028:
5978:
5976:Generalizations
5953:
5949:
5940:
5936:
5931:
5928:
5927:
5911:
5908:
5907:
5887:
5883:
5874:
5870:
5865:
5862:
5861:
5833:
5830:
5829:
5812:
5808:
5806:
5803:
5802:
5786:
5783:
5782:
5765:
5761:
5759:
5756:
5755:
5729:
5725:
5711:
5708:
5707:
5691:
5688:
5687:
5671:
5668:
5667:
5651:
5648:
5647:
5616:
5613:
5612:
5581:
5578:
5577:
5542:
5538:
5536:
5533:
5532:
5529:tangent bundles
5496:
5493:
5492:
5489:smooth function
5485:
5389:
5386:
5385:
5366:
5363:
5362:
5351:
5349:The Lie bracket
5330:
5327:
5326:
5307:
5303:
5298:
5290:
5287:
5286:
5261:
5258:
5257:
5234:
5230:
5228:
5225:
5224:
5201:
5197:
5195:
5192:
5191:
5175:
5167:
5164:
5163:
5132:
5129:
5128:
5105:
5101:
5099:
5096:
5095:
5075:
5071:
5061:
5055:
5051:
5049:
5032:
5029:
5028:
5011:
5007:
4990:
4987:
4986:
4969:
4965:
4945:
4943:
4940:
4939:
4922:
4918:
4901:
4898:
4897:
4881:
4879:
4876:
4875:
4859:
4856:
4855:
4817:
4815:
4812:
4811:
4792:
4789:
4788:
4785:diffeomorphisms
4764:
4761:
4760:
4744:
4741:
4740:
4716:
4713:
4712:
4709:
4697:exponential map
4661:
4658:
4657:
4641:
4638:
4637:
4621:
4618:
4617:
4601:
4598:
4597:
4581:
4578:
4577:
4560:
4556:
4554:
4551:
4550:
4534:
4531:
4530:
4514:
4511:
4510:
4490:
4487:
4486:
4448:
4445:
4444:
4424:
4421:
4420:
4404:
4401:
4400:
4393:integral curves
4375:
4371:
4369:
4366:
4365:
4349:
4348:
4340:
4288:
4284:
4271:
4253:
4246:
4245:
4235:
4220:
4216:
4212:
4210:
4207:
4206:
4184:
4181:
4180:
4164:
4161:
4160:
4144:
4141:
4140:
4139:for each point
4123:
4119:
4117:
4114:
4113:
4096:
4092:
4090:
4087:
4086:
4063:
4060:
4059:
3997:
3995:
3992:
3991:
3975:
3972:
3971:
3970:in an interval
3955:
3952:
3951:
3935:
3932:
3931:
3906:
3903:
3902:
3886:
3883:
3882:
3866:
3863:
3862:
3855:
3849:
3815:Michael Faraday
3808:magnetic dipole
3797:
3732:
3724:Stokes' theorem
3697:
3692:
3691:
3676:
3669:
3665:
3661:
3659:
3646:
3639:
3635:
3631:
3629:
3628:
3624:
3615:
3610:
3609:
3594:
3587:
3583:
3579:
3577:
3564:
3557:
3553:
3549:
3547:
3546:
3542:
3533:
3528:
3527:
3512:
3505:
3501:
3497:
3495:
3482:
3475:
3471:
3467:
3465:
3464:
3460:
3452:
3438:
3430:
3427:
3426:
3414:
3408:
3361:
3354:
3350:
3346:
3344:
3331:
3324:
3320:
3316:
3314:
3301:
3294:
3290:
3286:
3284:
3276:
3262:
3254:
3251:
3250:
3243:
3237:
3204:
3183:
3182:
3152:
3147:
3135:
3130:
3119:
3107:
3103:
3101:
3098:
3097:
3089:
3085:
3073:
3069:
3061:
3057:
3036:
3030:
3025:
2985:
2963:
2949:
2892:
2889:
2888:
2877:
2871:
2868:
2797:
2792:
2781:
2766:
2762:
2754:
2749:
2738:
2726:
2722:
2720:
2717:
2716:
2684:
2683:
2677:The associated
2648:
2644:
2640:
2632:
2630:
2612:
2608:
2604:
2596:
2594:
2582:
2578:
2574:
2566:
2564:
2552:
2548:
2544:
2536:
2534:
2533:
2529:
2512:
2509:
2508:
2481:A vector field
2464:
2450:
2425:Euclidean space
2411:Magnetic fields
2328:
2293:
2292:
2290:
2287:
2286:
2247:
2243:
2241:
2238:
2237:
2209:
2206:
2205:
2189:
2186:
2185:
2165:
2162:
2161:
2132:
2128:
2114:
2111:
2110:
2046:
2043:
2042:
2022:
2019:
2018:
1992:
1988:
1970:
1966:
1958:
1955:
1954:
1938:
1935:
1934:
1918:
1915:
1914:
1887:
1884:
1883:
1864:
1861:
1860:
1844:
1841:
1840:
1818:
1815:
1814:
1795:
1792:
1791:
1772:
1769:
1768:
1748:
1745:
1744:
1728:
1725:
1724:
1704:
1701:
1700:
1680:
1677:
1676:
1659:
1587:
1583:
1574:
1570:
1566:
1559:
1555:
1551:
1549:
1543:
1532:
1513:
1509:
1507:
1504:
1503:
1492:
1479:
1470:
1439:
1435:
1414:
1410:
1398:
1394:
1392:
1389:
1388:
1384:
1380:
1371:
1364:
1346:
1263:
1260:
1259:
1199:
1196:
1195:
1189:
1185:
1179:
1173:
1167:
1138:
1134:
1122:
1118:
1106:
1102:
1090:
1086:
1071:
1066:
1053:
1048:
1035:
1034:
1025:
1021:
1017:
1012:
1006:
1002:
990:
986:
982:
977:
971:
967:
958:
957:
955:
952:
951:
931:
926:
913:
908:
902:
899:
898:
881:
876:
875:
873:
870:
869:
849:
845:
841:
836:
830:
826:
814:
810:
806:
801:
795:
791:
786:
783:
782:
750:
746:
728:
724:
716:
713:
712:
692:
689:
688:
671:
667:
652:
648:
646:
643:
642:
619:
615:
611:
606:
597:
593:
578:
574:
565:
561:
555:
544:
538:
535:
534:
514:
508:
507:
506:
504:
501:
500:
484:
481:
480:
464:
461:
460:
440:
436:
432:
427:
409:
405:
401:
396:
394:
391:
390:
380:smooth function
369:
363:
359:
350:
343:
326:
313:
307:
306:Given a subset
304:
303:
302:
301:
283:
279:
278:
277:
269:
268:
257:
252:
189:
184:
183:
181:
178:
177:
162:position vector
147:of a flow) and
71:
66:
65:
63:
60:
59:
57:Euclidean space
37:vector calculus
17:
12:
11:
5:
7477:
7476:
7465:
7464:
7459:
7454:
7449:
7432:
7431:
7429:
7428:
7423:
7418:
7413:
7408:
7407:
7406:
7396:
7391:
7386:
7381:
7376:
7371:
7365:
7363:
7359:
7358:
7356:
7355:
7350:
7345:
7340:
7335:
7330:
7324:
7322:
7318:
7317:
7314:
7313:
7311:
7310:
7305:
7300:
7295:
7290:
7285:
7280:
7275:
7270:
7265:
7259:
7257:
7251:
7250:
7248:
7247:
7242:
7237:
7232:
7227:
7222:
7217:
7207:
7202:
7197:
7187:
7182:
7177:
7172:
7167:
7162:
7156:
7154:
7148:
7147:
7145:
7144:
7139:
7134:
7133:
7132:
7122:
7117:
7116:
7115:
7105:
7100:
7095:
7090:
7089:
7088:
7078:
7073:
7072:
7071:
7061:
7056:
7050:
7048:
7044:
7043:
7041:
7040:
7035:
7030:
7025:
7024:
7023:
7013:
7008:
7003:
6997:
6995:
6988:
6982:
6981:
6979:
6978:
6973:
6963:
6958:
6944:
6939:
6934:
6929:
6924:
6922:Parallelizable
6919:
6914:
6909:
6908:
6907:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6862:
6857:
6852:
6842:
6832:
6826:
6824:
6818:
6817:
6815:
6814:
6809:
6804:
6802:Lie derivative
6799:
6797:Integral curve
6794:
6789:
6784:
6783:
6782:
6772:
6767:
6766:
6765:
6758:Diffeomorphism
6755:
6749:
6747:
6741:
6740:
6738:
6737:
6732:
6727:
6722:
6717:
6712:
6707:
6702:
6697:
6691:
6689:
6680:
6679:
6677:
6676:
6671:
6666:
6661:
6656:
6651:
6646:
6641:
6636:
6635:
6634:
6629:
6619:
6618:
6617:
6606:
6604:
6603:Basic concepts
6600:
6599:
6587:
6586:
6579:
6572:
6564:
6558:
6557:
6551:
6546:
6541:
6532:
6523:
6509:"Vector field"
6505:
6498:
6497:External links
6495:
6494:
6493:
6487:
6467:
6461:
6448:
6442:
6430:Hubbard, B. B.
6426:Hubbard, J. H.
6420:
6417:
6414:
6413:
6406:
6388:
6327:
6320:
6300:
6282:
6275:
6250:
6243:
6219:
6218:
6215:
6214:
6156:"Vector field"
6129:
6127:
6120:
6114:
6111:
6109:
6108:
6103:
6098:
6093:
6088:
6083:
6081:Lie derivative
6078:
6061:
6059:Field strength
6056:
6051:
6045:
6044:
6043:
6027:
6024:
5977:
5974:
5961:
5956:
5952:
5948:
5943:
5939:
5935:
5915:
5895:
5890:
5886:
5882:
5877:
5873:
5869:
5849:
5846:
5843:
5840:
5837:
5815:
5811:
5790:
5768:
5764:
5740:
5737:
5732:
5728:
5724:
5721:
5718:
5715:
5695:
5675:
5655:
5646:, we say that
5635:
5632:
5629:
5626:
5623:
5620:
5600:
5597:
5594:
5591:
5588:
5585:
5565:
5562:
5559:
5556:
5553:
5550:
5545:
5541:
5512:
5509:
5506:
5503:
5500:
5484:
5478:
5477:
5476:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5370:
5350:
5347:
5334:
5310:
5306:
5302:
5297:
5294:
5274:
5271:
5268:
5265:
5245:
5242:
5237:
5233:
5223:). Hence for
5212:
5209:
5204:
5200:
5178:
5174:
5171:
5151:
5148:
5145:
5142:
5139:
5136:
5116:
5113:
5108:
5104:
5078:
5074:
5070:
5067:
5064:
5058:
5054:
5048:
5045:
5042:
5039:
5036:
5014:
5010:
5006:
5003:
5000:
4997:
4994:
4972:
4968:
4964:
4961:
4958:
4955:
4951:
4948:
4925:
4921:
4917:
4914:
4911:
4908:
4905:
4884:
4863:
4848:
4847:
4836:
4833:
4830:
4827:
4824:
4820:
4796:
4768:
4748:
4720:
4708:
4705:
4665:
4645:
4625:
4605:
4585:
4563:
4559:
4538:
4518:
4494:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4428:
4419:and partition
4408:
4378:
4374:
4347:
4343:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4305:
4302:
4299:
4296:
4291:
4287:
4283:
4280:
4277:
4274:
4272:
4270:
4267:
4264:
4260:
4256:
4252:
4248:
4247:
4244:
4241:
4238:
4236:
4234:
4231:
4228:
4223:
4219:
4215:
4214:
4194:
4191:
4188:
4168:
4148:
4126:
4122:
4099:
4095:
4067:
4041:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4003:
4000:
3979:
3959:
3939:
3919:
3916:
3913:
3910:
3890:
3870:
3853:Integral curve
3851:Main article:
3848:
3845:
3820:lines of force
3796:
3793:
3731:
3728:
3705:
3700:
3695:
3689:
3682:
3679:
3672:
3668:
3664:
3658:
3652:
3649:
3642:
3638:
3634:
3627:
3623:
3618:
3613:
3607:
3600:
3597:
3590:
3586:
3582:
3576:
3570:
3567:
3560:
3556:
3552:
3545:
3541:
3536:
3531:
3525:
3518:
3515:
3508:
3504:
3500:
3494:
3488:
3485:
3478:
3474:
3470:
3463:
3459:
3455:
3451:
3448:
3445:
3441:
3437:
3434:
3410:Main article:
3407:
3404:
3373:
3367:
3364:
3357:
3353:
3349:
3343:
3337:
3334:
3327:
3323:
3319:
3313:
3307:
3304:
3297:
3293:
3289:
3283:
3279:
3275:
3272:
3269:
3265:
3261:
3258:
3239:Main article:
3236:
3233:
3214:
3211:
3207:
3202:
3199:
3196:
3190:
3187:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3155:
3150:
3146:
3142:
3138:
3133:
3129:
3126:
3122:
3118:
3115:
3110:
3106:
3032:Main article:
3029:
3026:
3024:
3021:
3015:of the field.
2973:
2970:
2966:
2962:
2959:
2956:
2952:
2948:
2945:
2942:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2867:
2864:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2800:
2795:
2791:
2788:
2784:
2780:
2777:
2774:
2769:
2765:
2761:
2757:
2752:
2748:
2745:
2741:
2737:
2734:
2729:
2725:
2681:is called the
2664:
2660:
2651:
2647:
2643:
2638:
2635:
2629:
2626:
2623:
2615:
2611:
2607:
2602:
2599:
2593:
2585:
2581:
2577:
2572:
2569:
2563:
2555:
2551:
2547:
2542:
2539:
2532:
2528:
2525:
2522:
2519:
2516:
2491:gradient field
2449:
2446:
2445:
2444:
2436:
2433:electric field
2418:
2408:
2407:
2406:
2403:
2400:
2391:
2377:
2344:wingtip vortex
2327:
2324:
2307:
2304:
2301:
2296:
2270:
2267:
2264:
2261:
2258:
2255:
2250:
2246:
2225:
2222:
2219:
2216:
2213:
2193:
2169:
2146:
2143:
2140:
2135:
2131:
2127:
2124:
2121:
2118:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2026:
2006:
2003:
2000:
1995:
1991:
1987:
1984:
1981:
1978:
1973:
1969:
1965:
1962:
1942:
1933:on a manifold
1922:
1908:tangent bundle
1891:
1871:
1868:
1848:
1828:
1825:
1822:
1802:
1799:
1789:tangent bundle
1776:
1752:
1732:
1721:tangent vector
1708:
1684:
1658:
1655:
1624:
1623:
1614:
1612:
1601:
1596:
1593:
1590:
1586:
1577:
1573:
1569:
1562:
1558:
1554:
1546:
1541:
1538:
1535:
1531:
1527:
1522:
1519:
1516:
1512:
1488:
1475:
1468:
1453:
1448:
1445:
1442:
1438:
1434:
1431:
1428:
1423:
1420:
1417:
1413:
1409:
1406:
1401:
1397:
1376:
1369:
1348:In physics, a
1345:
1342:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1164:
1163:
1152:
1149:
1146:
1141:
1137:
1133:
1130:
1125:
1121:
1117:
1114:
1109:
1105:
1101:
1098:
1093:
1089:
1085:
1082:
1079:
1074:
1069:
1065:
1061:
1056:
1051:
1047:
1043:
1038:
1028:
1024:
1020:
1016:
1009:
1005:
1001:
993:
989:
985:
981:
974:
970:
966:
961:
934:
929:
925:
921:
916:
911:
907:
884:
879:
852:
848:
844:
840:
833:
829:
825:
817:
813:
809:
805:
798:
794:
790:
764:
761:
758:
753:
749:
745:
742:
739:
736:
731:
727:
723:
720:
696:
674:
670:
666:
663:
660:
655:
651:
639:
638:
622:
618:
614:
610:
605:
600:
596:
592:
589:
586:
581:
577:
573:
568:
564:
558:
553:
550:
547:
543:
517:
511:
488:
468:
443:
439:
435:
431:
426:
423:
420:
412:
408:
404:
400:
355:
348:
281:
280:
271:
270:
262:
261:
260:
259:
258:
256:
253:
251:
248:
240:tangent bundle
228:tangent vector
192:
187:
104:, such as the
96:, such as the
74:
69:
15:
9:
6:
4:
3:
2:
7475:
7474:
7463:
7460:
7458:
7455:
7453:
7450:
7448:
7445:
7444:
7442:
7427:
7424:
7422:
7421:Supermanifold
7419:
7417:
7414:
7412:
7409:
7405:
7402:
7401:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7367:
7366:
7364:
7360:
7354:
7351:
7349:
7346:
7344:
7341:
7339:
7336:
7334:
7331:
7329:
7326:
7325:
7323:
7319:
7309:
7306:
7304:
7301:
7299:
7296:
7294:
7291:
7289:
7286:
7284:
7281:
7279:
7276:
7274:
7271:
7269:
7266:
7264:
7261:
7260:
7258:
7256:
7252:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7221:
7218:
7216:
7212:
7208:
7206:
7203:
7201:
7198:
7196:
7192:
7188:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7157:
7155:
7153:
7149:
7143:
7142:Wedge product
7140:
7138:
7135:
7131:
7128:
7127:
7126:
7123:
7121:
7118:
7114:
7111:
7110:
7109:
7106:
7104:
7101:
7099:
7096:
7094:
7091:
7087:
7086:Vector-valued
7084:
7083:
7082:
7079:
7077:
7074:
7070:
7067:
7066:
7065:
7062:
7060:
7057:
7055:
7052:
7051:
7049:
7045:
7039:
7036:
7034:
7031:
7029:
7026:
7022:
7019:
7018:
7017:
7016:Tangent space
7014:
7012:
7009:
7007:
7004:
7002:
6999:
6998:
6996:
6992:
6989:
6987:
6983:
6977:
6974:
6972:
6968:
6964:
6962:
6959:
6957:
6953:
6949:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6928:
6925:
6923:
6920:
6918:
6915:
6913:
6910:
6906:
6903:
6902:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6856:
6853:
6851:
6847:
6843:
6841:
6837:
6833:
6831:
6828:
6827:
6825:
6819:
6813:
6810:
6808:
6805:
6803:
6800:
6798:
6795:
6793:
6790:
6788:
6785:
6781:
6780:in Lie theory
6778:
6777:
6776:
6773:
6771:
6768:
6764:
6761:
6760:
6759:
6756:
6754:
6751:
6750:
6748:
6746:
6742:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6711:
6708:
6706:
6703:
6701:
6698:
6696:
6693:
6692:
6690:
6687:
6683:Main results
6681:
6675:
6672:
6670:
6667:
6665:
6664:Tangent space
6662:
6660:
6657:
6655:
6652:
6650:
6647:
6645:
6642:
6640:
6637:
6633:
6630:
6628:
6625:
6624:
6623:
6620:
6616:
6613:
6612:
6611:
6608:
6607:
6605:
6601:
6596:
6592:
6585:
6580:
6578:
6573:
6571:
6566:
6565:
6562:
6555:
6552:
6550:
6547:
6545:
6542:
6540:
6536:
6533:
6531:
6527:
6524:
6520:
6516:
6515:
6510:
6506:
6504:
6501:
6500:
6490:
6488:0-12-116053-X
6484:
6479:
6478:
6472:
6468:
6464:
6462:0-387-90894-3
6458:
6454:
6449:
6445:
6443:0-13-657446-7
6439:
6435:
6431:
6427:
6423:
6422:
6409:
6407:0-387-94732-9
6403:
6399:
6392:
6384:
6380:
6376:
6372:
6368:
6364:
6360:
6356:
6351:
6346:
6342:
6338:
6331:
6323:
6317:
6313:
6312:
6304:
6293:
6286:
6278:
6272:
6268:
6264:
6257:
6255:
6246:
6240:
6236:
6235:
6227:
6225:
6220:
6211:
6208:
6200:
6189:
6186:
6182:
6179:
6175:
6172:
6168:
6165:
6161:
6158: –
6157:
6153:
6152:Find sources:
6146:
6142:
6136:
6135:
6130:This article
6128:
6124:
6119:
6118:
6107:
6104:
6102:
6101:Tensor fields
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6076:
6071:
6070:
6069:balanced flow
6066:
6065:Gradient flow
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6046:
6041:
6035:
6030:
6023:
6021:
6017:
6012:
6010:
6009:tensor fields
6006:
6004:
6001:differential
5998:
5994:
5990:
5986:
5984:
5973:
5954:
5950:
5946:
5941:
5937:
5913:
5888:
5884:
5880:
5875:
5871:
5847:
5844:
5841:
5838:
5835:
5813:
5809:
5788:
5766:
5762:
5752:
5738:
5735:
5730:
5726:
5722:
5719:
5716:
5713:
5693:
5673:
5653:
5633:
5630:
5624:
5621:
5618:
5598:
5595:
5589:
5586:
5583:
5563:
5560:
5554:
5551:
5548:
5543:
5539:
5530:
5526:
5510:
5504:
5501:
5498:
5490:
5482:
5463:
5454:
5448:
5442:
5439:
5430:
5424:
5418:
5415:
5409:
5400:
5397:
5394:
5384:
5383:
5382:
5368:
5360:
5356:
5346:
5332:
5308:
5304:
5300:
5295:
5292:
5269:
5263:
5243:
5240:
5235:
5231:
5210:
5207:
5202:
5198:
5172:
5169:
5149:
5146:
5140:
5134:
5114:
5111:
5106:
5102:
5076:
5072:
5068:
5065:
5062:
5056:
5052:
5046:
5040:
5034:
5012:
5008:
5004:
4998:
4992:
4970:
4966:
4962:
4956:
4949:
4946:
4923:
4919:
4915:
4909:
4903:
4861:
4854:vector field
4853:
4834:
4831:
4825:
4822:
4810:
4809:
4808:
4794:
4786:
4782:
4766:
4746:
4738:
4734:
4718:
4704:
4702:
4698:
4694:
4690:
4689:geodesic flow
4686:
4682:
4677:
4663:
4643:
4623:
4603:
4583:
4561:
4557:
4536:
4516:
4508:
4492:
4484:
4481:to the whole
4465:
4462:
4459:
4456:
4453:
4442:
4426:
4406:
4398:
4394:
4376:
4372:
4362:
4345:
4337:
4331:
4328:
4325:
4322:
4319:
4313:
4310:
4297:
4289:
4285:
4278:
4275:
4273:
4265:
4258:
4254:
4250:
4242:
4239:
4237:
4229:
4221:
4217:
4192:
4189:
4186:
4166:
4146:
4124:
4120:
4097:
4093:
4085:
4081:
4065:
4057:
4052:
4039:
4029:
4023:
4017:
4014:
4008:
4001:
3998:
3977:
3957:
3937:
3914:
3908:
3888:
3868:
3859:
3854:
3844:
3840:
3838:
3833:
3831:
3827:
3823:
3821:
3816:
3809:
3805:
3801:
3792:
3790:
3786:
3781:
3779:
3773:
3771:
3767:
3765:
3761:
3757:
3752:
3750:
3746:
3741:
3736:
3727:
3725:
3721:
3716:
3703:
3698:
3687:
3680:
3670:
3666:
3656:
3650:
3640:
3636:
3625:
3621:
3616:
3605:
3598:
3588:
3584:
3574:
3568:
3558:
3554:
3543:
3539:
3534:
3523:
3516:
3506:
3502:
3492:
3486:
3476:
3472:
3461:
3457:
3449:
3443:
3435:
3432:
3424:
3419:
3413:
3403:
3401:
3397:
3392:
3390:
3384:
3371:
3365:
3355:
3351:
3341:
3335:
3325:
3321:
3311:
3305:
3295:
3291:
3281:
3273:
3267:
3259:
3256:
3248:
3242:
3232:
3230:
3225:
3212:
3209:
3197:
3188:
3185:
3179:
3170:
3164:
3158:
3153:
3148:
3144:
3140:
3127:
3113:
3108:
3104:
3095:
3081:
3077:
3067:
3054:
3052:
3047:
3045:
3044:line integral
3041:
3035:
3034:Line integral
3028:Line integral
3020:
3016:
3014:
3009:
3007:
3003:
2999:
2993:
2989:
2960:
2957:
2946:
2943:
2930:
2924:
2918:
2915:
2906:
2900:
2894:
2886:
2885:central field
2880:
2874:
2863:
2850:
2841:
2835:
2829:
2826:
2817:
2811:
2805:
2802:
2789:
2775:
2767:
2763:
2759:
2746:
2732:
2727:
2723:
2714:
2710:
2706:
2703:
2699:
2698:path integral
2694:
2692:
2688:
2685:gradient flow
2680:
2675:
2662:
2658:
2649:
2645:
2636:
2627:
2624:
2621:
2613:
2609:
2600:
2591:
2583:
2579:
2570:
2561:
2553:
2549:
2540:
2530:
2526:
2523:
2517:
2514:
2506:
2502:
2498:
2497:
2492:
2488:
2484:
2479:
2477:
2473:
2469:
2468:scalar fields
2463:
2454:
2441:
2437:
2434:
2430:
2426:
2422:
2419:
2416:
2412:
2409:
2404:
2401:
2398:
2397:
2395:
2392:
2389:
2385:
2381:
2378:
2375:
2371:
2366:
2365:
2361:
2357:
2352:
2345:
2341:
2337:
2332:
2323:
2321:
2302:
2284:
2265:
2262:
2259:
2256:
2244:
2220:
2217:
2191:
2183:
2180:is smooth or
2167:
2158:
2141:
2129:
2125:
2122:
2119:
2116:
2096:
2090:
2084:
2081:
2075:
2069:
2066:
2063:
2057:
2054:
2048:
2040:
2024:
2001:
1989:
1979:
1967:
1963:
1960:
1940:
1920:
1911:
1909:
1905:
1889:
1869:
1866:
1846:
1826:
1823:
1820:
1800:
1797:
1790:
1774:
1766:
1750:
1730:
1722:
1706:
1698:
1682:
1675:
1668:
1663:
1654:
1652:
1648:
1647:scalar fields
1643:
1641:
1640:
1635:
1631:
1630:contravariant
1622:
1615:
1613:
1599:
1594:
1591:
1588:
1584:
1575:
1571:
1560:
1556:
1544:
1539:
1536:
1533:
1529:
1525:
1520:
1517:
1514:
1510:
1502:
1501:
1498:
1496:
1491:
1487:
1483:
1478:
1474:
1467:
1446:
1443:
1440:
1436:
1432:
1429:
1426:
1421:
1418:
1415:
1411:
1404:
1399:
1395:
1379:
1375:
1368:
1361:
1359:
1355:
1351:
1341:
1339:
1335:
1319:
1313:
1307:
1304:
1298:
1292:
1289:
1283:
1274:
1271:
1268:
1243:
1237:
1231:
1225:
1222:
1216:
1207:
1204:
1192:
1182:
1176:
1170:
1150:
1147:
1139:
1135:
1131:
1123:
1119:
1115:
1107:
1103:
1099:
1091:
1087:
1083:
1080:
1072:
1067:
1063:
1059:
1054:
1049:
1045:
1026:
1022:
1007:
1003:
999:
991:
987:
972:
968:
964:
950:
949:
948:
932:
927:
923:
919:
914:
909:
905:
882:
850:
846:
831:
827:
823:
815:
811:
796:
792:
788:
780:
776:
759:
747:
737:
725:
721:
718:
710:
694:
672:
668:
664:
661:
658:
653:
649:
620:
616:
598:
594:
590:
587:
584:
579:
575:
566:
562:
556:
551:
548:
545:
541:
533:
532:
531:
515:
486:
466:
441:
437:
424:
421:
418:
410:
406:
387:
385:
381:
377:
372:
366:
358:
354:
347:
342:
337:
333:
329:
325:
321:
316:
310:
298:
294:
290:
286:
275:
266:
247:
245:
241:
237:
233:
229:
225:
221:
216:
214:
213:
208:
190:
175:
171:
167:
163:
159:
158:
152:
150:
146:
142:
138:
134:
130:
126:
125:line integral
122:
118:
113:
111:
110:gravitational
107:
103:
99:
95:
90:
72:
58:
54:
50:
46:
42:
38:
30:
26:
21:
7348:Moving frame
7343:Morse theory
7333:Gauge theory
7125:Tensor field
7054:Closed/Exact
7033:Vector field
7032:
7001:Distribution
6942:Hypercomplex
6937:Quaternionic
6674:Vector field
6673:
6632:Smooth atlas
6535:Vector field
6526:Vector field
6512:
6476:
6452:
6433:
6419:Bibliography
6397:
6391:
6340:
6336:
6330:
6310:
6303:
6285:
6266:
6233:
6203:
6194:
6184:
6177:
6170:
6163:
6151:
6139:Please help
6134:verification
6131:
6086:Scalar field
6073:
6068:
6064:
6013:
6002:
5992:
5988:
5982:
5979:
5926:-related to
5801:-related to
5753:
5686:-related to
5486:
5483:-relatedness
5480:
5352:
4896:is given by
4851:
4849:
4732:
4710:
4678:
4397:trajectories
4396:
4392:
4363:
4083:
4053:
3860:
3856:
3841:
3834:
3830:field theory
3825:
3818:
3813:
3782:
3774:
3769:
3768:
3763:
3759:
3755:
3753:
3744:
3739:
3737:
3733:
3717:
3415:
3393:
3385:
3244:
3226:
3094:real numbers
3079:
3075:
3066:parametrized
3060:and a curve
3055:
3048:
3037:
3017:
3012:
3010:
2991:
2987:
2884:
2883:is called a
2878:
2872:
2869:
2712:
2708:
2704:
2702:closed curve
2695:
2682:
2676:
2504:
2500:
2494:
2490:
2489:is called a
2486:
2482:
2480:
2465:
2335:
2159:
1912:
1697:vector field
1696:
1671:
1650:
1644:
1637:
1633:
1627:
1616:
1494:
1489:
1485:
1481:
1476:
1472:
1465:
1377:
1373:
1366:
1362:
1347:
1190:
1180:
1174:
1168:
1165:
778:
777:
640:
388:
383:
375:
370:
364:
356:
352:
345:
339:in standard
335:
331:
327:
320:vector field
319:
314:
308:
305:
296:
292:
288:
284:
244:tensor field
220:open subsets
217:
210:
206:
173:
155:
153:
114:
45:vector field
44:
34:
28:
24:
7293:Levi-Civita
7283:Generalized
7255:Connections
7205:Lie algebra
7137:Volume form
7038:Vector flow
7011:Pushforward
7006:Lie bracket
6905:Lie algebra
6870:G-structure
6659:Pushforward
6639:Submanifold
6106:Slope field
6016:derivations
5359:Lie bracket
4779:, then the
4391:are called
4364:The curves
4082:there is a
3881:defined on
3847:Flow curves
3837:light field
2340:streamlines
1188:defined on
1178:defined on
170:coordinates
166:space curve
7441:Categories
7416:Stratifold
7374:Diffeology
7170:Associated
6971:Symplectic
6956:Riemannian
6885:Hyperbolic
6812:Submersion
6720:Hopf–Rinow
6654:Submersion
6649:Smooth map
6539:PlanetMath
6350:1908.05768
6197:April 2012
6167:newspapers
6113:References
6054:Field line
5997:dual space
5525:derivative
4852:incomplete
4731:is called
4701:Lie groups
3247:divergence
3241:Divergence
3235:Divergence
3008:around 0.
2700:along any
2507:such that
2470:using the
2443:increases.
2342:showing a
2039:derivation
2017:such that
709:linear map
250:Definition
141:divergence
7298:Principal
7273:Ehresmann
7230:Subbundle
7220:Principal
7195:Fibration
7175:Cotangent
7047:Covectors
6900:Lie group
6880:Hermitian
6823:manifolds
6792:Immersion
6787:Foliation
6725:Noether's
6710:Frobenius
6705:De Rham's
6700:Darboux's
6591:Manifolds
6530:Mathworld
6519:EMS Press
6383:201058607
6375:1471-2962
5736:∘
5731:∗
5717:∘
5628:→
5593:→
5558:→
5544:∗
5508:→
5440:−
5241:≠
5173:∈
5112:≠
5066:−
4829:→
4823:×
4558:γ
4466:ε
4457:ε
4454:−
4373:γ
4338:⊂
4332:ε
4323:ε
4320:−
4314:∈
4308:∀
4286:γ
4251:γ
4218:γ
4187:ε
4121:γ
4024:γ
3999:γ
3909:γ
3678:∂
3663:∂
3657:−
3648:∂
3633:∂
3596:∂
3581:∂
3575:−
3566:∂
3551:∂
3540:−
3514:∂
3499:∂
3493:−
3484:∂
3469:∂
3450:×
3447:∇
3436:
3363:∂
3348:∂
3333:∂
3318:∂
3303:∂
3288:∂
3274:⋅
3271:∇
3260:
3189:˙
3186:γ
3180:⋅
3165:γ
3145:∫
3128:⋅
3109:γ
3105:∫
3002:invariant
2947:∈
2836:γ
2827:−
2812:γ
2790:⋅
2773:∇
2768:γ
2764:∮
2747:⋅
2728:γ
2724:∮
2642:∂
2634:∂
2625:…
2606:∂
2598:∂
2576:∂
2568:∂
2546:∂
2538:∂
2521:∇
2370:magnitude
2249:∞
2212:Γ
2134:∞
2126:∈
1994:∞
1986:→
1972:∞
1824:∘
1787:into the
1568:∂
1553:∂
1530:∑
1430:…
1336:over the
1084:−
1019:∂
1015:∂
984:∂
980:∂
965:−
843:∂
839:∂
808:∂
804:∂
789:−
752:∞
744:→
730:∞
722::
662:…
613:∂
609:∂
588:…
542:∑
434:∂
430:∂
422:…
403:∂
399:∂
7394:Orbifold
7389:K-theory
7379:Diffiety
7103:Pullback
6917:Oriented
6895:Kenmotsu
6875:Hadamard
6821:Types of
6770:Geodesic
6595:Glossary
6473:(1986).
6432:(1999).
6026:See also
5985:-vectors
5487:Given a
5162:for all
4950:′
4733:complete
4695:and the
4681:pathline
4259:′
4002:′
3804:Magnetic
2472:gradient
2462:Gradient
2417:filings.
2388:velocity
2380:Velocity
2326:Examples
2283:sections
2182:analytic
2109:for all
1813:so that
1672:Given a
1358:covector
224:surfaces
106:magnetic
7338:History
7321:Related
7235:Tangent
7213:)
7193:)
7160:Adjoint
7152:Bundles
7130:density
7028:Torsion
6994:Vectors
6986:Tensors
6969:)
6954:)
6950:,
6948:Pseudo−
6927:Poisson
6860:Finsler
6855:Fibered
6850:Contact
6848:)
6840:Complex
6838:)
6807:Section
6521:, 2001
6355:Bibcode
6181:scholar
5751:holds.
5355:commute
4112:-curve
4054:By the
3084:(where
2996:is the
2320:fraktur
1906:of the
1904:section
1765:mapping
1372:, ...,
779:Example
238:of the
236:section
41:physics
7303:Vector
7288:Koszul
7268:Cartan
7263:Affine
7245:Vector
7240:Tensor
7225:Spinor
7215:Normal
7211:Stable
7165:Affine
7069:bundle
7021:bundle
6967:Almost
6890:Kähler
6846:Almost
6836:Almost
6830:Closed
6730:Sard's
6686:(list)
6485:
6459:
6440:
6404:
6381:
6373:
6318:
6273:
6241:
6183:
6176:
6169:
6162:
6154:
6005:-forms
5523:, the
4691:, and
4084:unique
3826:itself
3749:degree
3013:center
3004:under
2984:where
2711:(0) =
2478:: ∇).
2322:"X").
1667:sphere
1651:scalar
1480:) are
1350:vector
1334:module
376:smooth
145:volume
123:, the
49:vector
7411:Sheaf
7185:Fiber
6961:Rizza
6932:Prime
6763:Local
6753:Curve
6615:Atlas
6379:S2CID
6345:arXiv
6295:(PDF)
6188:JSTOR
6174:books
5127:(and
4685:fluid
4596:. If
4439:into
4058:, if
3738:Let
3082:]
3074:[
3040:curve
2881:\ {0}
2493:or a
2429:force
2384:fluid
2037:is a
1767:from
1763:is a
1471:,...,
351:, …,
295:) = −
164:of a
121:force
102:force
89:plane
53:space
7278:Form
7180:Dual
7113:flow
6976:Tame
6952:Sub−
6865:Flat
6745:Maps
6483:ISBN
6457:ISBN
6438:ISBN
6402:ISBN
6371:ISSN
6316:ISBN
6271:ISBN
6239:ISBN
6160:news
6067:and
5611:and
5190:if
4507:flow
4190:>
3740:n be
3433:curl
3418:curl
3416:The
3245:The
3092:are
3088:and
2696:The
2679:flow
2415:iron
1695:, a
1387:are
1338:ring
318:, a
149:curl
129:work
98:wind
43:, a
39:and
7200:Jet
6363:doi
6341:378
6143:by
6072:in
5906:is
5781:is
5754:If
5666:is
5094:if
4783:of
4699:in
4683:in
4509:on
4395:or
4159:in
4078:is
3930:on
3257:div
3072:in
3068:by
2887:if
2503:on
2476:del
2318:(a
2236:or
1882:to
1699:on
687:on
499:of
312:of
108:or
35:In
7443::
7191:Co
6537:—
6528:—
6517:,
6511:,
6428:;
6377:.
6369:.
6361:.
6353:.
6339:.
6265:.
6253:^
6223:^
6022:.
6011:.
5972:.
5828:,
5531:,
5416::=
5381::
5345:.
5256:,
4703:.
4687:,
4676:.
4205:,
3990:,
3839:.
3832:.
3791:.
3780:.
3726:.
3391:.
3231:.
3078:,
3064:,
2990:,
2986:O(
2870:A
2693:.
2438:A
2157:.
2041::
1910:.
1360:.
1290::=
1223::=
1172:,
1151:0.
334:→
330::
291:,
246:.
7209:(
7189:(
6965:(
6946:(
6844:(
6834:(
6597:)
6593:(
6583:e
6576:t
6569:v
6491:.
6465:.
6446:.
6410:.
6385:.
6365::
6357::
6347::
6324:.
6279:.
6247:.
6210:)
6204:(
6199:)
6195:(
6185:·
6178:·
6171:·
6164:·
6137:.
6003:k
5993:p
5989:p
5987:(
5983:p
5960:]
5955:2
5951:W
5947:,
5942:1
5938:W
5934:[
5914:f
5894:]
5889:2
5885:V
5881:,
5876:1
5872:V
5868:[
5848:2
5845:,
5842:1
5839:=
5836:i
5814:i
5810:W
5789:f
5767:i
5763:V
5739:V
5727:f
5723:=
5720:f
5714:W
5694:V
5674:f
5654:W
5634:N
5631:T
5625:N
5622::
5619:W
5599:M
5596:T
5590:M
5587::
5584:V
5564:N
5561:T
5555:M
5552:T
5549::
5540:f
5511:N
5505:M
5502::
5499:f
5481:f
5464:.
5461:)
5458:)
5455:f
5452:(
5449:X
5446:(
5443:Y
5437:)
5434:)
5431:f
5428:(
5425:Y
5422:(
5419:X
5413:)
5410:f
5407:(
5404:]
5401:Y
5398:,
5395:X
5392:[
5369:f
5333:t
5309:0
5305:x
5301:1
5296:=
5293:t
5273:)
5270:t
5267:(
5264:x
5244:0
5236:0
5232:x
5211:0
5208:=
5203:0
5199:x
5177:R
5170:t
5150:0
5147:=
5144:)
5141:t
5138:(
5135:x
5115:0
5107:0
5103:x
5077:0
5073:x
5069:t
5063:1
5057:0
5053:x
5047:=
5044:)
5041:t
5038:(
5035:x
5013:0
5009:x
5005:=
5002:)
4999:0
4996:(
4993:x
4971:2
4967:x
4963:=
4960:)
4957:t
4954:(
4947:x
4924:2
4920:x
4916:=
4913:)
4910:x
4907:(
4904:V
4883:R
4862:V
4835:.
4832:M
4826:M
4819:R
4795:X
4767:M
4747:X
4719:M
4664:p
4644:p
4624:V
4604:p
4584:p
4562:p
4537:p
4517:S
4493:S
4469:)
4463:+
4460:,
4451:(
4427:S
4407:V
4377:x
4346:.
4342:R
4335:)
4329:+
4326:,
4317:(
4311:t
4304:)
4301:)
4298:t
4295:(
4290:x
4282:(
4279:V
4276:=
4269:)
4266:t
4263:(
4255:x
4243:x
4240:=
4233:)
4230:0
4227:(
4222:x
4193:0
4167:S
4147:x
4125:x
4098:1
4094:C
4066:V
4040:.
4036:)
4033:)
4030:t
4027:(
4021:(
4018:V
4015:=
4012:)
4009:t
4006:(
3978:I
3958:t
3938:S
3918:)
3915:t
3912:(
3889:S
3869:V
3822:,
3810:)
3764:k
3762:−
3760:n
3756:k
3745:n
3704:.
3699:3
3694:e
3688:)
3681:y
3671:1
3667:F
3651:x
3641:2
3637:F
3626:(
3622:+
3617:2
3612:e
3606:)
3599:z
3589:1
3585:F
3569:x
3559:3
3555:F
3544:(
3535:1
3530:e
3524:)
3517:z
3507:2
3503:F
3487:y
3477:3
3473:F
3462:(
3458:=
3454:F
3444:=
3440:F
3372:,
3366:z
3356:3
3352:F
3342:+
3336:y
3326:2
3322:F
3312:+
3306:x
3296:1
3292:F
3282:=
3278:F
3268:=
3264:F
3213:.
3210:t
3206:d
3201:)
3198:t
3195:(
3177:)
3174:)
3171:t
3168:(
3162:(
3159:V
3154:b
3149:a
3141:=
3137:x
3132:d
3125:)
3121:x
3117:(
3114:V
3090:b
3086:a
3080:b
3076:a
3070:t
3062:γ
3058:V
2994:)
2992:R
2988:n
2972:)
2969:)
2965:R
2961:,
2958:n
2955:(
2951:O
2944:T
2941:(
2937:)
2934:)
2931:p
2928:(
2925:V
2922:(
2919:T
2916:=
2913:)
2910:)
2907:p
2904:(
2901:T
2898:(
2895:V
2879:R
2873:C
2851:.
2848:)
2845:)
2842:0
2839:(
2833:(
2830:f
2824:)
2821:)
2818:1
2815:(
2809:(
2806:f
2803:=
2799:x
2794:d
2787:)
2783:x
2779:(
2776:f
2760:=
2756:x
2751:d
2744:)
2740:x
2736:(
2733:V
2713:γ
2709:γ
2707:(
2705:γ
2663:.
2659:)
2650:n
2646:x
2637:f
2628:,
2622:,
2614:3
2610:x
2601:f
2592:,
2584:2
2580:x
2571:f
2562:,
2554:1
2550:x
2541:f
2531:(
2527:=
2524:f
2518:=
2515:V
2505:S
2501:f
2487:S
2483:V
2435:.
2368:(
2362:.
2346:.
2336:R
2306:)
2303:M
2300:(
2295:X
2269:)
2266:M
2263:T
2260:,
2257:M
2254:(
2245:C
2224:)
2221:M
2218:T
2215:(
2192:M
2168:M
2145:)
2142:M
2139:(
2130:C
2123:g
2120:,
2117:f
2097:g
2094:)
2091:f
2088:(
2085:X
2082:+
2079:)
2076:g
2073:(
2070:X
2067:f
2064:=
2061:)
2058:g
2055:f
2052:(
2049:X
2025:X
2005:)
2002:M
1999:(
1990:C
1983:)
1980:M
1977:(
1968:C
1964::
1961:X
1941:M
1921:X
1890:M
1870:M
1867:T
1847:p
1827:F
1821:p
1801:M
1798:T
1775:M
1751:F
1731:M
1707:M
1683:M
1639:1
1634:n
1621:)
1619:1
1617:(
1600:.
1595:x
1592:,
1589:j
1585:V
1576:j
1572:x
1561:i
1557:y
1545:n
1540:1
1537:=
1534:j
1526:=
1521:y
1518:,
1515:i
1511:V
1495:V
1490:i
1486:x
1482:n
1477:n
1473:y
1469:1
1466:y
1452:)
1447:x
1444:,
1441:n
1437:V
1433:,
1427:,
1422:x
1419:,
1416:1
1412:V
1408:(
1405:=
1400:x
1396:V
1385:V
1381:)
1378:n
1374:x
1370:1
1367:x
1365:(
1320:,
1317:)
1314:p
1311:(
1308:W
1305:+
1302:)
1299:p
1296:(
1293:V
1287:)
1284:p
1281:(
1278:)
1275:W
1272:+
1269:V
1266:(
1247:)
1244:p
1241:(
1238:V
1235:)
1232:p
1229:(
1226:f
1220:)
1217:p
1214:(
1211:)
1208:V
1205:f
1202:(
1191:S
1186:f
1181:S
1175:W
1169:V
1148:=
1145:)
1140:2
1136:x
1132:2
1129:(
1124:1
1120:x
1116:+
1113:)
1108:1
1104:x
1100:2
1097:(
1092:2
1088:x
1081:=
1078:)
1073:2
1068:2
1064:x
1060:+
1055:2
1050:1
1046:x
1042:(
1037:)
1027:2
1023:x
1008:1
1004:x
1000:+
992:1
988:x
973:2
969:x
960:(
933:2
928:2
924:x
920:+
915:2
910:1
906:x
883:2
878:R
851:2
847:x
832:1
828:x
824:+
816:1
812:x
797:2
793:x
763:)
760:S
757:(
748:C
741:)
738:S
735:(
726:C
719:V
695:S
673:n
669:V
665:,
659:,
654:1
650:V
621:i
617:x
604:)
599:n
595:x
591:,
585:,
580:1
576:x
572:(
567:i
563:V
557:n
552:1
549:=
546:i
516:n
510:R
487:S
467:V
442:n
438:x
425:,
419:,
411:1
407:x
384:n
371:V
365:V
360:)
357:n
353:x
349:1
346:x
344:(
336:R
332:S
328:V
315:R
309:S
297:r
293:y
289:x
287:(
285:v
207:n
191:n
186:R
174:n
73:n
68:R
31:)
29:x
25:y
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
