2302:, to take the derivative of a vector-valued function requires the choice of a reference frame (at least when a fixed Cartesian coordinate system is not implied as such). Once a reference frame has been chosen, the derivative of a vector-valued function can be computed using techniques similar to those for computing derivatives of scalar-valued functions. A different choice of reference frame will, in general, produce a different derivative function. The derivative functions in different reference frames have a specific
70:
4615:
2952:
3506:
3380:
3097:
2582:
2753:
2252:
2793:
3236:
3391:
3265:
2987:
1812:
2400:
2636:
2110:
1549:
1320:
4324:
4063:
3136:
2039:
4561:
3791:
4686:
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
2947:{\displaystyle {\frac {{}^{\mathrm {N} }d}{dt}}(\mathbf {r} ^{\mathrm {R} })={\frac {{}^{\mathrm {E} }d}{dt}}(\mathbf {r} ^{\mathrm {R} })+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {r} ^{\mathrm {R} }.}
1432:
249:
2372:. However, many complex problems involve the derivative of a vector function in multiple moving reference frames, which means that the basis vectors will not necessarily be constant. In such a case where the basis vectors
4829:, 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
4391:
1123:
1715:
940:
3665:
4563:
However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a
Hilbert space does not guarantee convergence with respect to the actual
570:
3501:{\displaystyle {\frac {\partial }{\partial q}}(\mathbf {a} \times \mathbf {b} )={\frac {\partial \mathbf {a} }{\partial q}}\times \mathbf {b} +\mathbf {a} \times {\frac {\partial \mathbf {b} }{\partial q}}.}
394:
3375:{\displaystyle {\frac {\partial }{\partial q}}(\mathbf {a} \cdot \mathbf {b} )={\frac {\partial \mathbf {a} }{\partial q}}\cdot \mathbf {b} +\mathbf {a} \cdot {\frac {\partial \mathbf {b} }{\partial q}}.}
1689:
1627:
637:
4220:
782:
466:
1138:
3092:{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {R} }={}^{\mathrm {E} }\mathbf {v} ^{\mathrm {R} }+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {r} ^{\mathrm {R} }}
878:
688:
1437:
1939:
2577:{\displaystyle {\frac {{}^{\mathrm {N} }d\mathbf {a} }{dt}}=\sum _{i=1}^{3}{\frac {da_{i}}{dt}}\mathbf {e} _{i}+\sum _{i=1}^{3}a_{i}{\frac {{}^{\mathrm {N} }d\mathbf {e} _{i}}{dt}}}
4102:
3964:
3833:
4796:
4680:
3556:
2748:{\displaystyle {\frac {{}^{\mathrm {N} }d\mathbf {a} }{dt}}={\frac {{}^{\mathrm {E} }d\mathbf {a} }{dt}}+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {a} }
4802:-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 (
4226:
2247:{\displaystyle {\frac {d\mathbf {a} }{dt}}=\sum _{r=1}^{n}{\frac {\partial \mathbf {a} }{\partial q_{r}}}{\frac {dq_{r}}{dt}}+{\frac {\partial \mathbf {a} }{\partial t}}.}
4435:
838:
3867:
1346:
163:
4128:
4440:
3670:
4734:. 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
995:
893:
3231:{\displaystyle {\frac {\partial }{\partial q}}(p\mathbf {a} )={\frac {\partial p}{\partial q}}\mathbf {a} +p{\frac {\partial \mathbf {a} }{\partial q}}.}
504:
317:
1635:
1573:
4331:
575:
4152:
743:
3561:
44:
5070:
4804:
657:
2584:
where the superscript N to the left of the derivative operator indicates the reference frame in which the derivative is taken.
1807:{\displaystyle {\frac {\partial \mathbf {a} }{\partial q}}=\sum _{i=1}^{n}{\frac {\partial a_{i}}{\partial q}}\mathbf {e} _{i}}
4990:
4963:
2621:
are constant, reference frame E. It also can be shown that the second term on the right hand side is equal to the relative
3890:
413:
976:
is a 2-dimensional set of points embedded in (most commonly) 3-dimensional space. One way to represent a surface is with
5041:
4711:
2633:
itself. Thus, after substitution, the formula relating the derivative of a vector function in two reference frames is
2272:. The total derivative differs from the partial time derivative in that the total derivative accounts for changes in
55:
could be 1 or greater than 1); the dimension of the function's domain has no relation to the dimension of its range.
4643:
854:
40:
4820:
4731:
5007:
4597:
need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.
1315:{\displaystyle (x_{1},x_{2},\dots ,x_{n})=(f_{1}(s,t),f_{2}(s,t),\dots ,f_{n}(s,t))\equiv \mathbf {F} (s,t).}
4072:
3803:
5075:
2770:
One common example where this formula is used is to find the velocity of a space-borne object, such as a
1847:
4876:
4594:
4772:
4656:
4319:{\displaystyle \mathbf {f} =f_{1}\mathbf {e} _{1}+f_{2}\mathbf {e} _{2}+f_{3}\mathbf {e} _{3}+\cdots }
3532:
3105:
is the velocity vector of the rocket as measured from a reference frame E that is fixed to the Earth.
293:
5065:
4577:
3902:
2775:
1544:{\displaystyle {\frac {d\mathbf {r} }{dt}}=f'(t)\mathbf {i} +g'(t)\mathbf {j} +h'(t)\mathbf {k} .}
5060:
4826:
4688:
2034:{\displaystyle {\frac {d\mathbf {a} }{dt}}=\sum _{i=1}^{n}{\frac {da_{i}}{dt}}\mathbf {e} _{i}.}
814:
4830:
4727:
3936:
3846:
2393:
are fixed in reference frame E, but not in reference frame N, the more general formula for the
2315:
1336:
737:
32:
28:
4058:{\displaystyle \mathbf {f} '(t)=\lim _{h\to 0}{\frac {\mathbf {f} (t+h)-\mathbf {f} (t)}{h}}.}
2966:
of the Earth relative to the inertial frame N. Since velocity is the derivative of position,
4953:
4590:
989:
973:
4107:
4065:
Most results of the finite-dimensional case also hold in the infinite-dimensional case too,
2364:
each has a derivative of identically zero. This often holds true for problems dealing with
4861:
4753:, whose domain's dimension has no relation to the dimension of its range; for example, the
4406:
2314:
The above formulas for the derivative of a vector function rely on the assumption that the
651:
52:
5036:
8:
4647:
4146:
977:
801:
36:
4937:
Kane, Thomas R.; Levinson, David A. (1996). "1–9 Differentiation of Vector
Functions".
4866:
4816:
4743:
2299:
1903:
1699:
1343:
by simply differentiating the components in the
Cartesian coordinate system. Thus, if
841:
5031:
407:
has its tail at the origin and its head at the coordinates evaluated by the function.
5014:
4986:
4959:
4851:
4762:
4565:
4394:
1899:
1563:
5017:
47:. The input of a vector-valued function could be a scalar or a vector (that is, the
4683:
4066:
2963:
2764:
2622:
2104:
160:, these specific types of vector-valued functions are given by expressions such as
119:
64:
2778:
using measurements of the rocket's velocity relative to the ground. The velocity
1427:{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }
244:{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }
4980:
4871:
4754:
4651:
4631:
3871:
154:
5008:
Vector-valued functions and their properties (from Lake Tahoe
Community College)
4145:
is a
Hilbert space, then one can easily show that any derivative (and any other
4834:
4723:
4700:
4069:. Differentiation can also be defined to functions of several variables (e.g.,
2339:
are constant, that is, fixed in the reference frame in which the derivative of
5054:
4719:
3914:
410:
The vector shown in the graph to the right is the evaluation of the function
106:
A common example of a vector-valued function is one that depends on a single
5037:
3 Dimensional vector-valued functions (from East
Tennessee State University)
5044:
4896:
4838:
4639:
4607:
4586:
3905:
3114:
2365:
1920:
is regarded as a vector function of a single scalar variable, such as time
1630:
69:
4386:{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3},\ldots }
4758:
4556:{\displaystyle \mathbf {f} '(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).}
3786:{\displaystyle \mathbf {f} '(t)=(f_{1}'(t),f_{2}'(t),\ldots ,f_{n}'(t)).}
3113:
The derivative of a product of vector functions behaves similarly to the
1868:
133:
107:
4819:, where they associate an arrow tangent to the surface at each point (a
2303:
4735:
1925:
1551:
The vector derivative admits the following physical interpretation: if
1340:
1118:{\displaystyle (x,y,z)=(f(s,t),g(s,t),h(s,t))\equiv \mathbf {F} (s,t).}
647:
2588:, the first term on the right hand side is equal to the derivative of
2309:
5022:
4761:
is defined only for smaller subset of the ambient space. Likewise, n
935:{\displaystyle {\hat {\mathbf {y} }}=X{\hat {\boldsymbol {\beta }}},}
733:
48:
3660:{\displaystyle \mathbf {f} (t)=(f_{1}(t),f_{2}(t),\ldots ,f_{n}(t))}
2298:
Whereas for scalar-valued functions there is only a single possible
1324:
4812:
4726:
done by a force moving along a path, and under this interpretation
4714:
extend naturally to vector fields. When a vector field represents
1567:
1330:
96:
indicating a range of solutions and the vector when evaluated near
4798:
can be represented as a vector-valued function that associates an
16:
Function valued in a vector space; typically a real or complex one
4815:
of
Euclidean space, but also make sense on other subsets such as
4704:
4635:
3510:
2369:
565:{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} }
501:
In 2D, We can analogously speak about vector-valued functions as
2784:
in inertial reference frame N of a rocket R located at position
4739:
2771:
4951:
4614:
2984:
in reference frames N and E, respectively. By substitution,
389:{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }
4856:
4715:
4696:
484:
2767:
of the reference frame E relative to the reference frame N.
1684:{\displaystyle {\frac {d\mathbf {v} }{dt}}=\mathbf {a} (t).}
1622:{\displaystyle \mathbf {v} (t)={\frac {d\mathbf {r} }{dt}}.}
4945:
4692:
4571:
3930:
115:
1135:-dimensional space, one similarly has the representation
967:
632:{\displaystyle \mathbf {r} (t)=\langle f(t),g(t)\rangle }
5012:
4585:
too. However, not as many classical results hold in the
4215:{\displaystyle \mathbf {f} =(f_{1},f_{2},f_{3},\ldots )}
3884:
2368:
in a fixed coordinate system, or for simple problems in
777:{\displaystyle \mathbf {y} =A\mathbf {x} +\mathbf {b} ,}
3108:
1131:
is a vector-valued function. For a surface embedded in
292:, and the domain of this vector-valued function is the
4808:) in passing from one coordinate system to the other.
4707:
force, as it changes from one point to another point.
314:. It can also be referred to in a different notation:
4941:. Sunnyvale, California: McGraw-Hill. pp. 29–37.
4775:
4659:
4443:
4409:
4334:
4229:
4155:
4110:
4075:
3967:
3849:
3806:
3673:
3564:
3535:
3394:
3268:
3139:
2990:
2796:
2639:
2403:
2113:
1942:
1718:
1638:
1576:
1440:
1349:
1141:
998:
896:
857:
817:
746:
736:. Closely related is the affine case (linear up to a
660:
578:
507:
461:{\displaystyle \langle 2\cos t,\,4\sin t,\,t\rangle }
416:
320:
166:
3117:of scalar functions. Specifically, in the case of
2310:
Derivative of a vector function with nonfixed bases
4895:In fact, these relations are derived applying the
4790:
4674:
4555:
4429:
4385:
4318:
4214:
4122:
4096:
4057:
3961:can be defined as in the finite-dimensional case:
3861:
3837:then the partial derivatives of the components of
3827:
3785:
3659:
3550:
3500:
3374:
3230:
3091:
2946:
2747:
2576:
2246:
2033:
1806:
1683:
1621:
1543:
1426:
1314:
1117:
934:
872:
832:
776:
682:
631:
564:
460:
388:
243:
4837:to the manifold). Vector fields are one kind of
2262:to indicate the total derivative operator, as in
1325:Derivative of a three-dimensional vector function
5052:
3991:
1629:Likewise, the derivative of the velocity is the
1924:, then the equation above reduces to the first
650:case the function can be expressed in terms of
487:is the path traced by the tip of the vector as
4982:Vector-Valued Functions and their Applications
4825:More generally, vector fields are defined on
964:matrix of fixed (empirically based) numbers.
873:{\displaystyle {\hat {\boldsymbol {\beta }}}}
800:The linear case arises often, for example in
483:; i.e., somewhat more than 3 rotations). The
4936:
626:
596:
455:
417:
383:
338:
4746:(which represents the rotation of a flow).
3951:is a Hilbert space, then the derivative of
3799:is a function of several variables, say of
890:) of estimated values of model parameters:
683:{\displaystyle \mathbf {y} =A\mathbf {x} ,}
4136:is an infinite-dimensional vector space).
3118:
2278:due to the time variance of the variables
1566:of a particle, then the derivative is the
4985:. Springer Science & Business Media.
4979:Hu, Chuang-Gan; Yang, Chung-Chun (2013).
4952:Galbis, Antonio; Maestre, Manuel (2012).
4932:
4930:
4928:
4926:
4924:
4922:
4920:
4918:
4916:
4778:
4662:
4084:
3815:
3538:
2103:can be expressed, in a form known as the
451:
435:
132:as the result. In terms of the standard
4805:covariance and contravariance of vectors
4738:(which represents the rate of change of
4695:, or the strength and direction of some
4618:A portion of the vector field (sin
4613:
4572:Other infinite-dimensional vector spaces
3931:Functions with values in a Hilbert space
2585:
2394:
1906:in which the derivative is being taken.
68:
920:
861:
5053:
4978:
4955:Vector Analysis Versus Vector Calculus
4913:
4749:A vector field is a special case of a
4730:is exhibited as a special case of the
1909:
968:Parametric representation of a surface
73:A graph of the vector-valued function
5013:
4811:Vector fields are often discussed on
4097:{\displaystyle t\in \mathbb {R} ^{n}}
3885:Infinite-dimensional vector functions
3828:{\displaystyle t\in \mathbb {R} ^{m}}
1693:
4149:) can be computed componentwise: if
3925:infinite-dimensional vector function
3891:Infinite-dimensional-vector function
3109:Derivative and vector multiplication
2397:of a vector in reference frame N is
954:in the previous generic form) is an
844:is expressed linearly in terms of a
740:) where the function takes the form
2293:
2256:Some authors prefer to use capital
2043:
1335:Many vector-valued functions, like
13:
5042:"Position Vector Valued Functions"
4712:differential and integral calculus
3486:
3476:
3445:
3435:
3401:
3397:
3385:
3360:
3350:
3319:
3309:
3275:
3271:
3216:
3206:
3183:
3175:
3146:
3142:
3083:
3066:
3052:
3038:
3024:
3010:
2996:
2935:
2918:
2904:
2887:
2856:
2836:
2805:
2731:
2717:
2684:
2648:
2626:
2542:
2412:
2343:is being taken, and therefore the
2232:
2222:
2173:
2163:
2093:, then the ordinary derivative of
1783:
1768:
1732:
1722:
1708:with respect to a scalar variable
1434:is a vector-valued function, then
14:
5087:
5071:Vectors (mathematics and physics)
5001:
4939:Dynamics: Theory and Applications
4722:of a vector field represents the
4576:Most of the above hold for other
3384:Similarly, the derivative of the
3127:is a scalar variable function of
58:
4791:{\displaystyle \mathbb {R} ^{n}}
4765:, a vector field on a domain in
4675:{\displaystyle \mathbb {R} ^{n}}
4606:This section is an excerpt from
4446:
4367:
4352:
4337:
4300:
4275:
4250:
4231:
4157:
4033:
4010:
3970:
3676:
3566:
3551:{\displaystyle \mathbb {R} ^{n}}
3480:
3466:
3458:
3439:
3422:
3414:
3354:
3340:
3332:
3313:
3296:
3288:
3241:
3210:
3193:
3162:
3077:
3032:
3004:
2929:
2881:
2830:
2741:
2694:
2658:
2553:
2490:
2422:
2226:
2167:
2121:
2018:
1950:
1794:
1726:
1665:
1646:
1601:
1578:
1534:
1509:
1484:
1448:
1420:
1400:
1380:
1351:
1290:
1093:
901:
767:
759:
748:
673:
662:
580:
558:
538:
509:
322:
296:of the domains of the functions
237:
217:
197:
168:
4732:fundamental theorem of calculus
4600:
2790:can be found using the formula
4889:
4547:
4538:
4532:
4513:
4507:
4488:
4482:
4466:
4460:
4454:
4424:
4418:
4209:
4164:
4043:
4037:
4026:
4014:
3998:
3984:
3978:
3777:
3774:
3768:
3743:
3737:
3718:
3712:
3696:
3690:
3684:
3654:
3651:
3645:
3623:
3617:
3601:
3595:
3582:
3576:
3570:
3426:
3410:
3300:
3284:
3166:
3155:
2893:
2876:
2842:
2825:
1675:
1669:
1588:
1582:
1530:
1524:
1505:
1499:
1480:
1474:
1416:
1410:
1396:
1390:
1376:
1370:
1361:
1355:
1306:
1294:
1283:
1280:
1268:
1246:
1234:
1218:
1206:
1193:
1187:
1142:
1109:
1097:
1086:
1083:
1071:
1062:
1050:
1041:
1029:
1023:
1017:
999:
923:
905:
864:
824:
641:
623:
617:
608:
602:
590:
584:
554:
548:
534:
528:
519:
513:
380:
374:
365:
359:
350:
344:
332:
326:
233:
227:
213:
207:
193:
187:
178:
172:
82:) = ⟨2 cos
1:
4906:
4769:-dimensional Euclidean space
2594:in the reference frame where
992:of any point on the surface:
39:is a set of multidimensional
3895:If the values of a function
3515:-dimensional vector function
2625:of the two reference frames
491:increases from zero through
45:infinite-dimensional vectors
7:
4845:
3388:of two vector functions is
3256:that are both functions of
2089:is only a function of time
10:
5092:
4605:
4593:function with values in a
3888:
2054:is a function of a number
1328:
980:, in which two parameters
833:{\displaystyle {\hat {y}}}
62:
4578:topological vector spaces
3862:{\displaystyle n\times m}
3529:with values in the space
1846:. It is also called the
840:of predicted values of a
804:, where for instance the
4958:. Springer. p. 12.
4882:
4827:differentiable manifolds
3667:. Its derivative equals
2776:inertial reference frame
2395:ordinary time derivative
2304:kinematical relationship
1926:ordinary time derivative
23:, also referred to as a
4689:three dimensional space
4682:. A vector field on a
3115:derivative of a product
2978:are the derivatives of
1337:scalar-valued functions
4792:
4751:vector-valued function
4728:conservation of energy
4676:
4642:is an assignment of a
4627:
4568:of the Hilbert space.
4557:
4431:
4387:
4320:
4216:
4124:
4123:{\displaystyle t\in Y}
4098:
4059:
3863:
3829:
3787:
3661:
3552:
3502:
3376:
3232:
3093:
2948:
2749:
2578:
2523:
2460:
2248:
2159:
2035:
1988:
1808:
1764:
1685:
1623:
1545:
1428:
1316:
1119:
936:
874:
834:
797:vector of parameters.
778:
716:vector of inputs, and
684:
633:
566:
462:
390:
245:
103:
21:vector-valued function
4793:
4677:
4617:
4595:suitable Banach space
4591:absolutely continuous
4558:
4432:
4430:{\displaystyle f'(t)}
4388:
4321:
4217:
4125:
4099:
4060:
3945:is a real number and
3864:
3830:
3788:
3662:
3553:
3503:
3377:
3233:
3119:scalar multiplication
3094:
2949:
2750:
2579:
2503:
2440:
2249:
2139:
2036:
1968:
1809:
1744:
1702:of a vector function
1686:
1624:
1546:
1429:
1317:
1120:
990:Cartesian coordinates
948:(playing the role of
937:
875:
835:
779:
685:
634:
567:
463:
391:
246:
114:, often representing
72:
63:Further information:
29:mathematical function
4862:Multivalued function
4773:
4657:
4441:
4407:
4332:
4227:
4153:
4108:
4073:
3965:
3903:infinite-dimensional
3847:
3804:
3671:
3562:
3533:
3392:
3266:
3137:
2988:
2794:
2637:
2401:
2111:
2058:of scalar variables
1940:
1835:in the direction of
1716:
1636:
1574:
1438:
1347:
1139:
996:
988:determine the three
978:parametric equations
894:
855:
815:
744:
658:
576:
505:
414:
318:
286:coordinate functions
164:
86:, 4 sin
5032:Everything2 article
4646:to each point in a
4531:
4506:
4481:
3767:
3736:
3711:
2586:As shown previously
1910:Ordinary derivative
802:multiple regression
5076:Types of functions
5015:Weisstein, Eric W.
4867:Parametric surface
4788:
4672:
4628:
4589:setting, e.g., an
4553:
4519:
4494:
4469:
4427:
4383:
4316:
4212:
4120:
4094:
4055:
4005:
3869:matrix called the
3859:
3825:
3783:
3755:
3724:
3699:
3657:
3558:can be written as
3548:
3498:
3372:
3244:, for two vectors
3242:dot multiplication
3228:
3089:
2944:
2745:
2574:
2244:
2031:
1804:
1700:partial derivative
1694:Partial derivative
1681:
1619:
1541:
1424:
1312:
1115:
932:
870:
842:dependent variable
830:
784:where in addition
774:
680:
629:
562:
458:
386:
241:
104:
5018:"Vector Function"
4992:978-94-015-8030-4
4965:978-1-4614-2199-3
4852:Coordinate vector
4395:orthonormal basis
4050:
3990:
3923:may be called an
3525:of a real number
3511:Derivative of an
3493:
3452:
3408:
3367:
3326:
3282:
3223:
3190:
3153:
2874:
2823:
2707:
2671:
2572:
2486:
2435:
2239:
2214:
2187:
2134:
2014:
1963:
1900:orthonormal basis
1790:
1739:
1659:
1614:
1461:
926:
908:
867:
827:
288:of the parameter
5083:
5028:
5027:
4996:
4970:
4969:
4949:
4943:
4942:
4934:
4900:
4893:
4797:
4795:
4794:
4789:
4787:
4786:
4781:
4710:The elements of
4681:
4679:
4678:
4673:
4671:
4670:
4665:
4650:, most commonly
4622:, sin
4584:
4562:
4560:
4559:
4554:
4527:
4502:
4477:
4453:
4449:
4436:
4434:
4433:
4428:
4417:
4402:
4392:
4390:
4389:
4384:
4376:
4375:
4370:
4361:
4360:
4355:
4346:
4345:
4340:
4327:
4325:
4323:
4322:
4317:
4309:
4308:
4303:
4297:
4296:
4284:
4283:
4278:
4272:
4271:
4259:
4258:
4253:
4247:
4246:
4234:
4221:
4219:
4218:
4213:
4202:
4201:
4189:
4188:
4176:
4175:
4160:
4144:
4135:
4129:
4127:
4126:
4121:
4103:
4101:
4100:
4095:
4093:
4092:
4087:
4067:mutatis mutandis
4064:
4062:
4061:
4056:
4051:
4046:
4036:
4013:
4007:
4004:
3977:
3973:
3960:
3956:
3950:
3944:
3922:
3912:
3900:
3879:
3868:
3866:
3865:
3860:
3842:
3836:
3834:
3832:
3831:
3826:
3824:
3823:
3818:
3798:
3792:
3790:
3789:
3784:
3763:
3732:
3707:
3683:
3679:
3666:
3664:
3663:
3658:
3644:
3643:
3616:
3615:
3594:
3593:
3569:
3557:
3555:
3554:
3549:
3547:
3546:
3541:
3528:
3524:
3507:
3505:
3504:
3499:
3494:
3492:
3484:
3483:
3474:
3469:
3461:
3453:
3451:
3443:
3442:
3433:
3425:
3417:
3409:
3407:
3396:
3381:
3379:
3378:
3373:
3368:
3366:
3358:
3357:
3348:
3343:
3335:
3327:
3325:
3317:
3316:
3307:
3299:
3291:
3283:
3281:
3270:
3261:
3255:
3249:
3237:
3235:
3234:
3229:
3224:
3222:
3214:
3213:
3204:
3196:
3191:
3189:
3181:
3173:
3165:
3154:
3152:
3141:
3132:
3126:
3121:of a vector, if
3104:
3098:
3096:
3095:
3090:
3088:
3087:
3086:
3080:
3071:
3070:
3069:
3063:
3057:
3056:
3055:
3049:
3043:
3042:
3041:
3035:
3029:
3028:
3027:
3021:
3015:
3014:
3013:
3007:
3001:
3000:
2999:
2993:
2983:
2977:
2971:
2964:angular velocity
2961:
2953:
2951:
2950:
2945:
2940:
2939:
2938:
2932:
2923:
2922:
2921:
2915:
2909:
2908:
2907:
2901:
2892:
2891:
2890:
2884:
2875:
2873:
2865:
2861:
2860:
2859:
2853:
2849:
2841:
2840:
2839:
2833:
2824:
2822:
2814:
2810:
2809:
2808:
2802:
2798:
2789:
2783:
2765:angular velocity
2762:
2754:
2752:
2751:
2746:
2744:
2736:
2735:
2734:
2728:
2722:
2721:
2720:
2714:
2708:
2706:
2698:
2697:
2689:
2688:
2687:
2681:
2677:
2672:
2670:
2662:
2661:
2653:
2652:
2651:
2645:
2641:
2629:with the vector
2627:cross multiplied
2623:angular velocity
2620:
2611:
2602:
2593:
2583:
2581:
2580:
2575:
2573:
2571:
2563:
2562:
2561:
2556:
2547:
2546:
2545:
2539:
2535:
2533:
2532:
2522:
2517:
2499:
2498:
2493:
2487:
2485:
2477:
2476:
2475:
2462:
2459:
2454:
2436:
2434:
2426:
2425:
2417:
2416:
2415:
2409:
2405:
2294:Reference frames
2289:
2277:
2271:
2261:
2253:
2251:
2250:
2245:
2240:
2238:
2230:
2229:
2220:
2215:
2213:
2205:
2204:
2203:
2190:
2188:
2186:
2185:
2184:
2171:
2170:
2161:
2158:
2153:
2135:
2133:
2125:
2124:
2115:
2105:total derivative
2102:
2099:with respect to
2098:
2092:
2088:
2077:
2072:= 1, ...,
2057:
2053:
2044:Total derivative
2040:
2038:
2037:
2032:
2027:
2026:
2021:
2015:
2013:
2005:
2004:
2003:
1990:
1987:
1982:
1964:
1962:
1954:
1953:
1944:
1935:
1932:with respect to
1923:
1919:
1897:
1888:
1879:
1866:
1855:
1848:direction cosine
1845:
1834:
1827:scalar component
1824:
1813:
1811:
1810:
1805:
1803:
1802:
1797:
1791:
1789:
1781:
1780:
1779:
1766:
1763:
1758:
1740:
1738:
1730:
1729:
1720:
1711:
1707:
1690:
1688:
1687:
1682:
1668:
1660:
1658:
1650:
1649:
1640:
1628:
1626:
1625:
1620:
1615:
1613:
1605:
1604:
1595:
1581:
1570:of the particle
1561:
1550:
1548:
1547:
1542:
1537:
1523:
1512:
1498:
1487:
1473:
1462:
1460:
1452:
1451:
1442:
1433:
1431:
1430:
1425:
1423:
1403:
1383:
1354:
1321:
1319:
1318:
1313:
1293:
1267:
1266:
1233:
1232:
1205:
1204:
1186:
1185:
1167:
1166:
1154:
1153:
1134:
1130:
1124:
1122:
1121:
1116:
1096:
987:
983:
963:
953:
947:
941:
939:
938:
933:
928:
927:
919:
910:
909:
904:
899:
889:
879:
877:
876:
871:
869:
868:
860:
850:
839:
837:
836:
831:
829:
828:
820:
810:
796:
789:
783:
781:
780:
775:
770:
762:
751:
731:
721:
715:
708:
702:
695:
689:
687:
686:
681:
676:
665:
638:
636:
635:
630:
583:
571:
569:
568:
563:
561:
541:
512:
497:
490:
482:
478:
474:
467:
465:
464:
459:
406:
395:
393:
392:
387:
325:
313:
307:
301:
291:
283:
272:
261:
250:
248:
247:
242:
240:
220:
200:
171:
158:
152:
146:
140:
131:
113:
102:
95:
65:Parametric curve
5091:
5090:
5086:
5085:
5084:
5082:
5081:
5080:
5066:Vector calculus
5051:
5050:
5004:
4999:
4993:
4974:
4973:
4966:
4950:
4946:
4935:
4914:
4909:
4904:
4903:
4894:
4890:
4885:
4877:Parametrization
4872:Position vector
4848:
4843:
4842:
4782:
4777:
4776:
4774:
4771:
4770:
4755:position vector
4742:of a flow) and
4666:
4661:
4660:
4658:
4655:
4654:
4652:Euclidean space
4632:vector calculus
4611:
4603:
4580:
4574:
4523:
4498:
4473:
4445:
4444:
4442:
4439:
4438:
4410:
4408:
4405:
4404:
4398:
4371:
4366:
4365:
4356:
4351:
4350:
4341:
4336:
4335:
4333:
4330:
4329:
4304:
4299:
4298:
4292:
4288:
4279:
4274:
4273:
4267:
4263:
4254:
4249:
4248:
4242:
4238:
4230:
4228:
4225:
4224:
4223:
4197:
4193:
4184:
4180:
4171:
4167:
4156:
4154:
4151:
4150:
4140:
4131:
4109:
4106:
4105:
4088:
4083:
4082:
4074:
4071:
4070:
4032:
4009:
4008:
4006:
3994:
3969:
3968:
3966:
3963:
3962:
3958:
3952:
3946:
3940:
3933:
3918:
3908:
3896:
3893:
3887:
3875:
3872:Jacobian matrix
3848:
3845:
3844:
3838:
3819:
3814:
3813:
3805:
3802:
3801:
3800:
3794:
3759:
3728:
3703:
3675:
3674:
3672:
3669:
3668:
3639:
3635:
3611:
3607:
3589:
3585:
3565:
3563:
3560:
3559:
3542:
3537:
3536:
3534:
3531:
3530:
3526:
3520:
3517:
3485:
3479:
3475:
3473:
3465:
3457:
3444:
3438:
3434:
3432:
3421:
3413:
3400:
3395:
3393:
3390:
3389:
3359:
3353:
3349:
3347:
3339:
3331:
3318:
3312:
3308:
3306:
3295:
3287:
3274:
3269:
3267:
3264:
3263:
3257:
3251:
3245:
3240:In the case of
3215:
3209:
3205:
3203:
3192:
3182:
3174:
3172:
3161:
3145:
3140:
3138:
3135:
3134:
3128:
3122:
3111:
3100:
3082:
3081:
3076:
3075:
3065:
3064:
3059:
3058:
3051:
3050:
3048:
3047:
3037:
3036:
3031:
3030:
3023:
3022:
3020:
3019:
3009:
3008:
3003:
3002:
2995:
2994:
2992:
2991:
2989:
2986:
2985:
2979:
2973:
2967:
2955:
2934:
2933:
2928:
2927:
2917:
2916:
2911:
2910:
2903:
2902:
2900:
2899:
2886:
2885:
2880:
2879:
2866:
2855:
2854:
2852:
2851:
2850:
2848:
2835:
2834:
2829:
2828:
2815:
2804:
2803:
2801:
2800:
2799:
2797:
2795:
2792:
2791:
2785:
2779:
2756:
2740:
2730:
2729:
2724:
2723:
2716:
2715:
2713:
2712:
2699:
2693:
2683:
2682:
2680:
2679:
2678:
2676:
2663:
2657:
2647:
2646:
2644:
2643:
2642:
2640:
2638:
2635:
2634:
2619:
2613:
2610:
2604:
2601:
2595:
2589:
2564:
2557:
2552:
2551:
2541:
2540:
2538:
2537:
2536:
2534:
2528:
2524:
2518:
2507:
2494:
2489:
2488:
2478:
2471:
2467:
2463:
2461:
2455:
2444:
2427:
2421:
2411:
2410:
2408:
2407:
2406:
2404:
2402:
2399:
2398:
2392:
2385:
2378:
2363:
2356:
2349:
2338:
2331:
2324:
2312:
2300:reference frame
2296:
2288:
2279:
2273:
2263:
2257:
2231:
2225:
2221:
2219:
2206:
2199:
2195:
2191:
2189:
2180:
2176:
2172:
2166:
2162:
2160:
2154:
2143:
2126:
2120:
2116:
2114:
2112:
2109:
2108:
2100:
2094:
2090:
2087:
2079:
2067:
2059:
2055:
2049:
2046:
2022:
2017:
2016:
2006:
1999:
1995:
1991:
1989:
1983:
1972:
1955:
1949:
1945:
1943:
1941:
1938:
1937:
1933:
1921:
1915:
1912:
1904:reference frame
1896:
1890:
1887:
1881:
1878:
1872:
1871:. The vectors
1865:
1857:
1851:
1844:
1836:
1830:
1823:
1815:
1798:
1793:
1792:
1782:
1775:
1771:
1767:
1765:
1759:
1748:
1731:
1725:
1721:
1719:
1717:
1714:
1713:
1712:is defined as
1709:
1703:
1696:
1664:
1651:
1645:
1641:
1639:
1637:
1634:
1633:
1606:
1600:
1596:
1594:
1577:
1575:
1572:
1571:
1562:represents the
1552:
1533:
1516:
1508:
1491:
1483:
1466:
1453:
1447:
1443:
1441:
1439:
1436:
1435:
1419:
1399:
1379:
1350:
1348:
1345:
1344:
1333:
1327:
1289:
1262:
1258:
1228:
1224:
1200:
1196:
1181:
1177:
1162:
1158:
1149:
1145:
1140:
1137:
1136:
1132:
1126:
1092:
997:
994:
993:
985:
981:
970:
955:
949:
943:
918:
917:
900:
898:
897:
895:
892:
891:
881:
859:
858:
856:
853:
852:
845:
819:
818:
816:
813:
812:
805:
791:
785:
766:
758:
747:
745:
742:
741:
723:
717:
710:
704:
703:output vector,
697:
691:
672:
661:
659:
656:
655:
644:
579:
577:
574:
573:
557:
537:
508:
506:
503:
502:
492:
488:
480:
476:
469:
415:
412:
411:
397:
321:
319:
316:
315:
309:
303:
297:
289:
274:
263:
252:
236:
216:
196:
167:
165:
162:
161:
156:
148:
142:
136:
122:
111:
97:
74:
67:
61:
31:of one or more
25:vector function
17:
12:
11:
5:
5089:
5079:
5078:
5073:
5068:
5063:
5061:Linear algebra
5049:
5048:
5039:
5034:
5029:
5010:
5003:
5002:External links
5000:
4998:
4997:
4991:
4975:
4972:
4971:
4964:
4944:
4911:
4910:
4908:
4905:
4902:
4901:
4899:componentwise.
4887:
4886:
4884:
4881:
4880:
4879:
4874:
4869:
4864:
4859:
4854:
4847:
4844:
4835:tangent bundle
4821:tangent vector
4785:
4780:
4699:, such as the
4691:, such as the
4669:
4664:
4612:
4604:
4602:
4599:
4573:
4570:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4530:
4526:
4522:
4518:
4515:
4512:
4509:
4505:
4501:
4497:
4493:
4490:
4487:
4484:
4480:
4476:
4472:
4468:
4465:
4462:
4459:
4456:
4452:
4448:
4426:
4423:
4420:
4416:
4413:
4403: ), and
4382:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4344:
4339:
4315:
4312:
4307:
4302:
4295:
4291:
4287:
4282:
4277:
4270:
4266:
4262:
4257:
4252:
4245:
4241:
4237:
4233:
4211:
4208:
4205:
4200:
4196:
4192:
4187:
4183:
4179:
4174:
4170:
4166:
4163:
4159:
4119:
4116:
4113:
4091:
4086:
4081:
4078:
4054:
4049:
4045:
4042:
4039:
4035:
4031:
4028:
4025:
4022:
4019:
4016:
4012:
4003:
4000:
3997:
3993:
3989:
3986:
3983:
3980:
3976:
3972:
3932:
3929:
3889:Main article:
3886:
3883:
3858:
3855:
3852:
3822:
3817:
3812:
3809:
3782:
3779:
3776:
3773:
3770:
3766:
3762:
3758:
3754:
3751:
3748:
3745:
3742:
3739:
3735:
3731:
3727:
3723:
3720:
3717:
3714:
3710:
3706:
3702:
3698:
3695:
3692:
3689:
3686:
3682:
3678:
3656:
3653:
3650:
3647:
3642:
3638:
3634:
3631:
3628:
3625:
3622:
3619:
3614:
3610:
3606:
3603:
3600:
3597:
3592:
3588:
3584:
3581:
3578:
3575:
3572:
3568:
3545:
3540:
3516:
3509:
3497:
3491:
3488:
3482:
3478:
3472:
3468:
3464:
3460:
3456:
3450:
3447:
3441:
3437:
3431:
3428:
3424:
3420:
3416:
3412:
3406:
3403:
3399:
3371:
3365:
3362:
3356:
3352:
3346:
3342:
3338:
3334:
3330:
3324:
3321:
3315:
3311:
3305:
3302:
3298:
3294:
3290:
3286:
3280:
3277:
3273:
3227:
3221:
3218:
3212:
3208:
3202:
3199:
3195:
3188:
3185:
3180:
3177:
3171:
3168:
3164:
3160:
3157:
3151:
3148:
3144:
3110:
3107:
3085:
3079:
3074:
3068:
3062:
3054:
3046:
3040:
3034:
3026:
3018:
3012:
3006:
2998:
2943:
2937:
2931:
2926:
2920:
2914:
2906:
2898:
2895:
2889:
2883:
2878:
2872:
2869:
2864:
2858:
2847:
2844:
2838:
2832:
2827:
2821:
2818:
2813:
2807:
2743:
2739:
2733:
2727:
2719:
2711:
2705:
2702:
2696:
2692:
2686:
2675:
2669:
2666:
2660:
2656:
2650:
2617:
2608:
2599:
2570:
2567:
2560:
2555:
2550:
2544:
2531:
2527:
2521:
2516:
2513:
2510:
2506:
2502:
2497:
2492:
2484:
2481:
2474:
2470:
2466:
2458:
2453:
2450:
2447:
2443:
2439:
2433:
2430:
2424:
2420:
2414:
2390:
2383:
2376:
2361:
2354:
2347:
2336:
2329:
2322:
2311:
2308:
2295:
2292:
2283:
2243:
2237:
2234:
2228:
2224:
2218:
2212:
2209:
2202:
2198:
2194:
2183:
2179:
2175:
2169:
2165:
2157:
2152:
2149:
2146:
2142:
2138:
2132:
2129:
2123:
2119:
2083:
2063:
2048:If the vector
2045:
2042:
2030:
2025:
2020:
2012:
2009:
2002:
1998:
1994:
1986:
1981:
1978:
1975:
1971:
1967:
1961:
1958:
1952:
1948:
1911:
1908:
1894:
1885:
1876:
1861:
1840:
1819:
1801:
1796:
1788:
1785:
1778:
1774:
1770:
1762:
1757:
1754:
1751:
1747:
1743:
1737:
1734:
1728:
1724:
1695:
1692:
1680:
1677:
1674:
1671:
1667:
1663:
1657:
1654:
1648:
1644:
1618:
1612:
1609:
1603:
1599:
1593:
1590:
1587:
1584:
1580:
1540:
1536:
1532:
1529:
1526:
1522:
1519:
1515:
1511:
1507:
1504:
1501:
1497:
1494:
1490:
1486:
1482:
1479:
1476:
1472:
1469:
1465:
1459:
1456:
1450:
1446:
1422:
1418:
1415:
1412:
1409:
1406:
1402:
1398:
1395:
1392:
1389:
1386:
1382:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1353:
1341:differentiated
1326:
1323:
1311:
1308:
1305:
1302:
1299:
1296:
1292:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1265:
1261:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1231:
1227:
1223:
1220:
1217:
1214:
1211:
1208:
1203:
1199:
1195:
1192:
1189:
1184:
1180:
1176:
1173:
1170:
1165:
1161:
1157:
1152:
1148:
1144:
1114:
1111:
1108:
1105:
1102:
1099:
1095:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
969:
966:
931:
925:
922:
916:
913:
907:
903:
866:
863:
826:
823:
773:
769:
765:
761:
757:
754:
750:
679:
675:
671:
668:
664:
643:
640:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
582:
560:
556:
553:
550:
547:
544:
540:
536:
533:
530:
527:
524:
521:
518:
515:
511:
457:
454:
450:
447:
444:
441:
438:
434:
431:
428:
425:
422:
419:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
324:
239:
235:
232:
229:
226:
223:
219:
215:
212:
209:
206:
203:
199:
195:
192:
189:
186:
183:
180:
177:
174:
170:
118:, producing a
60:
59:Example: Helix
57:
15:
9:
6:
4:
3:
2:
5088:
5077:
5074:
5072:
5069:
5067:
5064:
5062:
5059:
5058:
5056:
5046:
5043:
5040:
5038:
5035:
5033:
5030:
5025:
5024:
5019:
5016:
5011:
5009:
5006:
5005:
4994:
4988:
4984:
4983:
4977:
4976:
4967:
4961:
4957:
4956:
4948:
4940:
4933:
4931:
4929:
4927:
4925:
4923:
4921:
4919:
4917:
4912:
4898:
4892:
4888:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4849:
4840:
4836:
4832:
4828:
4824:
4822:
4818:
4814:
4809:
4807:
4806:
4801:
4783:
4768:
4764:
4760:
4756:
4752:
4747:
4745:
4741:
4737:
4733:
4729:
4725:
4721:
4720:line integral
4717:
4713:
4708:
4706:
4705:gravitational
4702:
4698:
4694:
4690:
4685:
4667:
4653:
4649:
4645:
4641:
4637:
4633:
4625:
4621:
4616:
4609:
4598:
4596:
4592:
4588:
4583:
4579:
4569:
4567:
4550:
4544:
4541:
4535:
4528:
4524:
4520:
4516:
4510:
4503:
4499:
4495:
4491:
4485:
4478:
4474:
4470:
4463:
4457:
4450:
4437:exists, then
4421:
4414:
4411:
4401:
4397:of the space
4396:
4380:
4377:
4372:
4362:
4357:
4347:
4342:
4313:
4310:
4305:
4293:
4289:
4285:
4280:
4268:
4264:
4260:
4255:
4243:
4239:
4235:
4206:
4203:
4198:
4194:
4190:
4185:
4181:
4177:
4172:
4168:
4161:
4148:
4143:
4137:
4134:
4117:
4114:
4111:
4089:
4079:
4076:
4068:
4052:
4047:
4040:
4029:
4023:
4020:
4017:
4001:
3995:
3987:
3981:
3974:
3955:
3949:
3943:
3938:
3928:
3926:
3921:
3916:
3915:Hilbert space
3911:
3907:
3904:
3899:
3892:
3882:
3880:
3878:
3873:
3856:
3853:
3850:
3841:
3820:
3810:
3807:
3797:
3780:
3771:
3764:
3760:
3756:
3752:
3749:
3746:
3740:
3733:
3729:
3725:
3721:
3715:
3708:
3704:
3700:
3693:
3687:
3680:
3648:
3640:
3636:
3632:
3629:
3626:
3620:
3612:
3608:
3604:
3598:
3590:
3586:
3579:
3573:
3543:
3523:
3514:
3508:
3495:
3489:
3470:
3462:
3454:
3448:
3429:
3418:
3404:
3387:
3386:cross product
3382:
3369:
3363:
3344:
3336:
3328:
3322:
3303:
3292:
3278:
3260:
3254:
3248:
3243:
3238:
3225:
3219:
3200:
3197:
3186:
3178:
3169:
3158:
3149:
3131:
3125:
3120:
3116:
3106:
3103:
3072:
3060:
3044:
3016:
2982:
2976:
2970:
2965:
2960:
2959:
2941:
2924:
2912:
2896:
2870:
2867:
2862:
2845:
2819:
2816:
2811:
2788:
2782:
2777:
2773:
2768:
2766:
2761:
2760:
2737:
2725:
2709:
2703:
2700:
2690:
2673:
2667:
2664:
2654:
2632:
2628:
2624:
2616:
2607:
2598:
2592:
2587:
2568:
2565:
2558:
2548:
2529:
2525:
2519:
2514:
2511:
2508:
2504:
2500:
2495:
2482:
2479:
2472:
2468:
2464:
2456:
2451:
2448:
2445:
2441:
2437:
2431:
2428:
2418:
2396:
2389:
2382:
2375:
2371:
2367:
2366:vector fields
2360:
2353:
2346:
2342:
2335:
2328:
2321:
2317:
2307:
2305:
2301:
2291:
2286:
2282:
2276:
2270:
2266:
2260:
2254:
2241:
2235:
2216:
2210:
2207:
2200:
2196:
2192:
2181:
2177:
2155:
2150:
2147:
2144:
2140:
2136:
2130:
2127:
2117:
2106:
2097:
2086:
2082:
2075:
2071:
2066:
2062:
2052:
2041:
2028:
2023:
2010:
2007:
2000:
1996:
1992:
1984:
1979:
1976:
1973:
1969:
1965:
1959:
1956:
1946:
1931:
1927:
1918:
1907:
1905:
1902:fixed in the
1901:
1893:
1884:
1875:
1870:
1864:
1860:
1854:
1849:
1843:
1839:
1833:
1828:
1822:
1818:
1799:
1786:
1776:
1772:
1760:
1755:
1752:
1749:
1745:
1741:
1735:
1706:
1701:
1691:
1678:
1672:
1661:
1655:
1652:
1642:
1632:
1616:
1610:
1607:
1597:
1591:
1585:
1569:
1565:
1559:
1555:
1538:
1527:
1520:
1517:
1513:
1502:
1495:
1492:
1488:
1477:
1470:
1467:
1463:
1457:
1454:
1444:
1413:
1407:
1404:
1393:
1387:
1384:
1373:
1367:
1364:
1358:
1342:
1338:
1332:
1322:
1309:
1303:
1300:
1297:
1286:
1277:
1274:
1271:
1263:
1259:
1255:
1252:
1249:
1243:
1240:
1237:
1229:
1225:
1221:
1215:
1212:
1209:
1201:
1197:
1190:
1182:
1178:
1174:
1171:
1168:
1163:
1159:
1155:
1150:
1146:
1129:
1112:
1106:
1103:
1100:
1089:
1080:
1077:
1074:
1068:
1065:
1059:
1056:
1053:
1047:
1044:
1038:
1035:
1032:
1026:
1020:
1014:
1011:
1008:
1005:
1002:
991:
979:
975:
965:
962:
958:
952:
946:
929:
914:
911:
888:
884:
848:
843:
821:
808:
803:
798:
794:
788:
771:
763:
755:
752:
739:
735:
730:
726:
720:
713:
707:
700:
694:
677:
669:
666:
653:
649:
639:
620:
614:
611:
605:
599:
593:
587:
551:
545:
542:
531:
525:
522:
516:
499:
496:
486:
472:
452:
448:
445:
442:
439:
436:
432:
429:
426:
423:
420:
408:
404:
400:
377:
371:
368:
362:
356:
353:
347:
341:
335:
329:
312:
306:
300:
295:
287:
281:
277:
270:
266:
259:
255:
230:
224:
221:
210:
204:
201:
190:
184:
181:
175:
159:
151:
145:
139:
135:
129:
125:
121:
117:
109:
100:
93:
89:
85:
81:
77:
71:
66:
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
5045:Khan Academy
5021:
4981:
4954:
4947:
4938:
4897:product rule
4891:
4839:tensor field
4813:open subsets
4810:
4803:
4799:
4766:
4750:
4748:
4709:
4640:vector field
4629:
4623:
4619:
4608:Vector field
4601:Vector field
4587:Banach space
4581:
4575:
4399:
4141:
4138:
4132:
3953:
3947:
3941:
3934:
3924:
3919:
3913:, such as a
3909:
3906:vector space
3897:
3894:
3876:
3870:
3839:
3795:
3521:
3518:
3512:
3383:
3258:
3252:
3246:
3239:
3129:
3123:
3112:
3101:
2980:
2974:
2968:
2957:
2956:
2786:
2780:
2769:
2758:
2757:
2630:
2614:
2605:
2596:
2590:
2387:
2380:
2373:
2358:
2351:
2344:
2340:
2333:
2326:
2319:
2313:
2297:
2284:
2280:
2274:
2268:
2264:
2258:
2255:
2095:
2084:
2080:
2073:
2069:
2064:
2060:
2050:
2047:
1929:
1916:
1913:
1891:
1882:
1873:
1862:
1858:
1852:
1841:
1837:
1831:
1826:
1820:
1816:
1704:
1697:
1631:acceleration
1557:
1553:
1334:
1127:
971:
960:
956:
950:
944:
886:
882:
846:
806:
799:
792:
786:
728:
724:
718:
711:
705:
698:
692:
645:
500:
494:
473:= 19.5
470:
409:
402:
398:
310:
304:
298:
294:intersection
285:
279:
275:
268:
264:
257:
253:
149:
143:
137:
134:unit vectors
127:
123:
105:
98:
91:
87:
83:
79:
75:
24:
20:
18:
4763:coordinates
4759:space curve
3957:at a point
3519:A function
2078:, and each
1869:dot product
738:translation
642:Linear case
396:The vector
5055:Categories
4907:References
4736:divergence
3901:lie in an
1329:See also:
734:parameters
732:matrix of
155:Cartesian
110:parameter
5023:MathWorld
4545:…
4381:…
4314:⋯
4207:…
4115:∈
4080:∈
4030:−
3999:→
3854:×
3811:∈
3750:…
3630:…
3487:∂
3477:∂
3471:×
3455:×
3446:∂
3436:∂
3419:×
3402:∂
3398:∂
3361:∂
3351:∂
3345:⋅
3329:⋅
3320:∂
3310:∂
3293:⋅
3276:∂
3272:∂
3217:∂
3207:∂
3184:∂
3176:∂
3147:∂
3143:∂
3073:×
3061:ω
2925:×
2913:ω
2774:, in the
2738:×
2726:ω
2505:∑
2442:∑
2233:∂
2223:∂
2174:∂
2164:∂
2141:∑
1970:∑
1867:or their
1784:∂
1769:∂
1746:∑
1733:∂
1723:∂
1339:, can be
1287:≡
1253:…
1172:…
1090:≡
942:in which
924:^
921:β
906:^
865:^
862:β
849:× 1
825:^
809:× 1
795:× 1
714:× 1
701:× 1
627:⟩
597:⟨
475:(between
456:⟩
443:
427:
418:⟨
384:⟩
339:⟨
49:dimension
33:variables
4846:See also
4817:surfaces
4701:magnetic
4566:topology
4529:′
4504:′
4479:′
4451:′
4415:′
4139:N.B. If
4130:, where
4104:or even
3975:′
3937:argument
3765:′
3734:′
3709:′
3681:′
2318:vectors
1898:form an
1568:velocity
1564:position
1521:′
1496:′
1471:′
1331:Gradient
652:matrices
284:are the
94:⟩
4833:of the
4831:section
4636:physics
4222:(i.e.,
3935:If the
3917:, then
3843:form a
2962:is the
2763:is the
2370:physics
2287:
1825:is the
974:surface
851:vector
811:vector
646:In the
157:3-space
51:of the
41:vectors
27:, is a
5047:module
4989:
4962:
4740:volume
4718:, the
4644:vector
4393:is an
4328:where
3099:where
2954:where
2772:rocket
2755:where
1814:where
790:is an
722:is an
696:is an
690:where
648:linear
308:, and
251:where
120:vector
101:= 19.5
53:domain
35:whose
4883:Notes
4857:Curve
4757:of a
4716:force
4697:force
4684:plane
4648:space
4147:limit
2316:basis
2107:, as
1125:Here
885:<
709:is a
485:helix
468:near
37:range
4987:ISBN
4960:ISBN
4744:curl
4724:work
4693:wind
4638:, a
4634:and
3250:and
2972:and
1856:and
1698:The
984:and
481:6.5π
479:and
273:and
116:time
108:real
4823:).
4703:or
4630:In
3992:lim
3939:of
3874:of
3793:If
1928:of
1914:If
1850:of
1829:of
787:b''
572:or
440:sin
424:cos
153:of
43:or
5057::
5020:.
4915:^
3927:.
3881:.
3262:,
3133:,
2612:,
2603:,
2386:,
2379:,
2357:,
2350:,
2332:,
2325:,
2306:.
2290:.
2269:Dt
1936:,
1889:,
1880:,
972:A
959:×
727:×
654::
498:.
477:6π
302:,
262:,
147:,
141:,
90:,
19:A
5026:.
4995:.
4968:.
4841:.
4800:n
4784:n
4779:R
4767:n
4668:n
4663:R
4626:)
4624:x
4620:y
4610:.
4582:X
4551:.
4548:)
4542:,
4539:)
4536:t
4533:(
4525:3
4521:f
4517:,
4514:)
4511:t
4508:(
4500:2
4496:f
4492:,
4489:)
4486:t
4483:(
4475:1
4471:f
4467:(
4464:=
4461:)
4458:t
4455:(
4447:f
4425:)
4422:t
4419:(
4412:f
4400:X
4378:,
4373:3
4368:e
4363:,
4358:2
4353:e
4348:,
4343:1
4338:e
4326:,
4311:+
4306:3
4301:e
4294:3
4290:f
4286:+
4281:2
4276:e
4269:2
4265:f
4261:+
4256:1
4251:e
4244:1
4240:f
4236:=
4232:f
4210:)
4204:,
4199:3
4195:f
4191:,
4186:2
4182:f
4178:,
4173:1
4169:f
4165:(
4162:=
4158:f
4142:X
4133:Y
4118:Y
4112:t
4090:n
4085:R
4077:t
4053:.
4048:h
4044:)
4041:t
4038:(
4034:f
4027:)
4024:h
4021:+
4018:t
4015:(
4011:f
4002:0
3996:h
3988:=
3985:)
3982:t
3979:(
3971:f
3959:t
3954:f
3948:X
3942:f
3920:f
3910:X
3898:f
3877:f
3857:m
3851:n
3840:f
3835:,
3821:m
3816:R
3808:t
3796:f
3781:.
3778:)
3775:)
3772:t
3769:(
3761:n
3757:f
3753:,
3747:,
3744:)
3741:t
3738:(
3730:2
3726:f
3722:,
3719:)
3716:t
3713:(
3705:1
3701:f
3697:(
3694:=
3691:)
3688:t
3685:(
3677:f
3655:)
3652:)
3649:t
3646:(
3641:n
3637:f
3633:,
3627:,
3624:)
3621:t
3618:(
3613:2
3609:f
3605:,
3602:)
3599:t
3596:(
3591:1
3587:f
3583:(
3580:=
3577:)
3574:t
3571:(
3567:f
3544:n
3539:R
3527:t
3522:f
3513:n
3496:.
3490:q
3481:b
3467:a
3463:+
3459:b
3449:q
3440:a
3430:=
3427:)
3423:b
3415:a
3411:(
3405:q
3370:.
3364:q
3355:b
3341:a
3337:+
3333:b
3323:q
3314:a
3304:=
3301:)
3297:b
3289:a
3285:(
3279:q
3259:q
3253:b
3247:a
3226:.
3220:q
3211:a
3201:p
3198:+
3194:a
3187:q
3179:p
3170:=
3167:)
3163:a
3159:p
3156:(
3150:q
3130:q
3124:p
3102:v
3084:R
3078:r
3067:E
3053:N
3045:+
3039:R
3033:v
3025:E
3017:=
3011:R
3005:v
2997:N
2981:r
2975:v
2969:v
2958:ω
2942:.
2936:R
2930:r
2919:E
2905:N
2897:+
2894:)
2888:R
2882:r
2877:(
2871:t
2868:d
2863:d
2857:E
2846:=
2843:)
2837:R
2831:r
2826:(
2820:t
2817:d
2812:d
2806:N
2787:r
2781:v
2759:ω
2742:a
2732:E
2718:N
2710:+
2704:t
2701:d
2695:a
2691:d
2685:E
2674:=
2668:t
2665:d
2659:a
2655:d
2649:N
2631:a
2618:3
2615:e
2609:2
2606:e
2600:1
2597:e
2591:a
2569:t
2566:d
2559:i
2554:e
2549:d
2543:N
2530:i
2526:a
2520:3
2515:1
2512:=
2509:i
2501:+
2496:i
2491:e
2483:t
2480:d
2473:i
2469:a
2465:d
2457:3
2452:1
2449:=
2446:i
2438:=
2432:t
2429:d
2423:a
2419:d
2413:N
2391:3
2388:e
2384:2
2381:e
2377:1
2374:e
2362:3
2359:e
2355:2
2352:e
2348:1
2345:e
2341:a
2337:3
2334:e
2330:2
2327:e
2323:1
2320:e
2285:r
2281:q
2275:a
2267:/
2265:D
2259:D
2242:.
2236:t
2227:a
2217:+
2211:t
2208:d
2201:r
2197:q
2193:d
2182:r
2178:q
2168:a
2156:n
2151:1
2148:=
2145:r
2137:=
2131:t
2128:d
2122:a
2118:d
2101:t
2096:a
2091:t
2085:r
2081:q
2076:)
2074:n
2070:r
2068:(
2065:r
2061:q
2056:n
2051:a
2029:.
2024:i
2019:e
2011:t
2008:d
2001:i
1997:a
1993:d
1985:n
1980:1
1977:=
1974:i
1966:=
1960:t
1957:d
1951:a
1947:d
1934:t
1930:a
1922:t
1917:a
1895:3
1892:e
1886:2
1883:e
1877:1
1874:e
1863:i
1859:e
1853:a
1842:i
1838:e
1832:a
1821:i
1817:a
1800:i
1795:e
1787:q
1777:i
1773:a
1761:n
1756:1
1753:=
1750:i
1742:=
1736:q
1727:a
1710:q
1705:a
1679:.
1676:)
1673:t
1670:(
1666:a
1662:=
1656:t
1653:d
1647:v
1643:d
1617:.
1611:t
1608:d
1602:r
1598:d
1592:=
1589:)
1586:t
1583:(
1579:v
1560:)
1558:t
1556:(
1554:r
1539:.
1535:k
1531:)
1528:t
1525:(
1518:h
1514:+
1510:j
1506:)
1503:t
1500:(
1493:g
1489:+
1485:i
1481:)
1478:t
1475:(
1468:f
1464:=
1458:t
1455:d
1449:r
1445:d
1421:k
1417:)
1414:t
1411:(
1408:h
1405:+
1401:j
1397:)
1394:t
1391:(
1388:g
1385:+
1381:i
1377:)
1374:t
1371:(
1368:f
1365:=
1362:)
1359:t
1356:(
1352:r
1310:.
1307:)
1304:t
1301:,
1298:s
1295:(
1291:F
1284:)
1281:)
1278:t
1275:,
1272:s
1269:(
1264:n
1260:f
1256:,
1250:,
1247:)
1244:t
1241:,
1238:s
1235:(
1230:2
1226:f
1222:,
1219:)
1216:t
1213:,
1210:s
1207:(
1202:1
1198:f
1194:(
1191:=
1188:)
1183:n
1179:x
1175:,
1169:,
1164:2
1160:x
1156:,
1151:1
1147:x
1143:(
1133:n
1128:F
1113:.
1110:)
1107:t
1104:,
1101:s
1098:(
1094:F
1087:)
1084:)
1081:t
1078:,
1075:s
1072:(
1069:h
1066:,
1063:)
1060:t
1057:,
1054:s
1051:(
1048:g
1045:,
1042:)
1039:t
1036:,
1033:s
1030:(
1027:f
1024:(
1021:=
1018:)
1015:z
1012:,
1009:y
1006:,
1003:x
1000:(
986:t
982:s
961:k
957:n
951:A
945:X
930:,
915:X
912:=
902:y
887:n
883:k
880:(
847:k
822:y
807:n
793:n
772:,
768:b
764:+
760:x
756:A
753:=
749:y
729:k
725:n
719:A
712:k
706:x
699:n
693:y
678:,
674:x
670:A
667:=
663:y
624:)
621:t
618:(
615:g
612:,
609:)
606:t
603:(
600:f
594:=
591:)
588:t
585:(
581:r
559:j
555:)
552:t
549:(
546:g
543:+
539:i
535:)
532:t
529:(
526:f
523:=
520:)
517:t
514:(
510:r
495:π
493:8
489:t
471:t
453:t
449:,
446:t
437:4
433:,
430:t
421:2
405:)
403:t
401:(
399:r
381:)
378:t
375:(
372:h
369:,
366:)
363:t
360:(
357:g
354:,
351:)
348:t
345:(
342:f
336:=
333:)
330:t
327:(
323:r
311:h
305:g
299:f
290:t
282:)
280:t
278:(
276:h
271:)
269:t
267:(
265:g
260:)
258:t
256:(
254:f
238:k
234:)
231:t
228:(
225:h
222:+
218:j
214:)
211:t
208:(
205:g
202:+
198:i
194:)
191:t
188:(
185:f
182:=
179:)
176:t
173:(
169:r
150:k
144:j
138:i
130:)
128:t
126:(
124:v
112:t
99:z
92:z
88:z
84:z
80:z
78:(
76:r
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
