25:
6874:
4703:
4142:
6542:
6178:
4391:
3901:
3875:
5840:
5927:
6869:{\displaystyle {\begin{aligned}\int _{L}f(z)\,dz&=\int _{L}(u+iv)(dx+i\,dy)\\&=\int _{L}(u,-v)\cdot (dx,dy)+i\int _{L}(u,-v)\cdot (dy,-dx)\\&=\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} +i\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }.\end{aligned}}}
2889:
2474:
1620:
3473:
3609:
4698:{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }=\int _{a}^{b}{\begin{bmatrix}P{\big (}x(t),y(t){\big )}\\Q{\big (}x(t),y(t){\big )}\end{bmatrix}}\cdot {\begin{bmatrix}y'(t)\\-x'(t)\end{bmatrix}}~dt=\int _{a}^{b}\left(-Q~dx+P~dy\right).}
2611:
3339:
5315:
2326:
5666:
3722:
4137:{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).}
2710:
4287:
2768:
2337:
5444:
3096:. The dot product (green line) of its tangent vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral. (Click on image for a detailed description.)
6537:
1491:
1296:
along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by
5844:
The line integrals of complex functions can be evaluated using a number of techniques. The most direct is to split into real and imaginary parts, reducing the problem to evaluating two real-valued line integrals. The
4801:
3350:
1248:
6173:{\displaystyle {\begin{aligned}\oint _{L}{\frac {1}{z}}\,dz&=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }e^{-it}e^{it}\,dt\\&=i\int _{0}^{2\pi }dt=i(2\pi -0)=2\pi i.\end{aligned}}}
3480:
2999:
4884:
2481:
3231:
7103:
1670:
220:
3100:
The line integral of a vector field can be derived in a manner very similar to the case of a scalar field, but this time with the inclusion of a dot product. Again using the above definitions of
3081:
1341:
6547:
5932:
1185:
6963:
5601:
5140:
6341:
5508:
To establish a complete analogy with the line integral of a vector field, one must go back to the definition of differentiability in multivariable calculus. The gradient is defined from
1450:
1415:
1486:
3688:
2214:
6274:
5558:
5098:
7133:
a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating
5661:
5499:
6407:
1966:
1904:
1874:
1850:
1826:
1792:
1747:
1701:
5626:
5015:
2616:
5530:
7008:
6909:
6229:
5337:
7346:
7341:
7041:
1281:
8308:
4234:
1130:. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly
8296:
5835:{\displaystyle \int _{L}f(z){\overline {dz}}:={\overline {\int _{L}{\overline {f(z)}}\,dz}}=\int _{a}^{b}f(\gamma (t)){\overline {\gamma '(t)}}\,dt.}
3870:{\displaystyle {\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)}
6412:
5872:
computes the integral in terms of the singularities. This also implies the path independence of complex line integral for analytic functions.
8418:
8303:
2884:{\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt}
2469:{\displaystyle \Delta s_{i}=\left|\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\right|\approx \left|\mathbf {r} '(t_{i})\Delta t\right|}
1193:
316:
8286:
8281:
8291:
8276:
7390:
1615:{\displaystyle \int _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)\left|\mathbf {r} '(t)\right|\,dt,}
7578:
8271:
2916:
4184:, and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called
3468:{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\approx \mathbf {r} '(t_{i})\,\Delta t.}
7046:
4714:
3192:
th point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their
7888:
7642:
7218:
7210:
Mathematical
Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms
3659:
3604:{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \mathbf {r} '(t_{i})\,\Delta t,}
89:
5331:
1625:
582:
557:
61:
7331:
7440:
5057:
1292:
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given
1058:
621:
1154:
139:
8386:
8245:
6922:
6231:
has real and complex parts equal to the line integral and the flux integral of the vector field corresponding to the
5853:
to the same integral over a more convenient curve. It also implies that over a closed curve enclosing a region where
2606:{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\left|\mathbf {r} '(t_{i})\right|\Delta t}
577:
295:
108:
68:
4807:
3334:{\displaystyle I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \Delta \mathbf {r} _{i}}
7800:
7716:
7106:
562:
8381:
8313:
7938:
7793:
7761:
7520:
6279:
2127:
for more details.) We then label the distance of the line segment between adjacent sample points on the curve as
1420:
898:
572:
547:
229:
7308:
8014:
7991:
7706:
7161:
5563:
1386:
75:
46:
42:
7336:
8104:
8042:
7837:
7711:
7383:
7318:
1461:
680:
627:
508:
3053:. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.
7590:
7568:
7208:
6912:
6879:
6183:
5509:
1139:
334:
306:
8413:
3060:, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the
3034:
A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always
417:
57:
8398:
8164:
7778:
7600:
7313:
6238:
931:
539:
377:
349:
7263:
4959:
7783:
7553:
5535:
1142:
vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on
802:
766:
543:
422:
311:
301:
5634:
8444:
8202:
8149:
7118:
6966:
6350:
5455:
5310:{\displaystyle \sum _{k=1}^{n}f(\gamma (t_{k}))\,=\sum _{k=1}^{n}f(\gamma _{k})\,\Delta \gamma _{k}.}
3653:
3214:
at all the points on the curve and taking the dot product with each displacement vector gives us the
566:
7610:
7238:
1947:
1885:
1855:
1831:
1807:
1773:
1728:
1682:
402:
8449:
8318:
8089:
7637:
7376:
5865:
1941:
701:
261:
8084:
7756:
7363:
5846:
3069:
1328:
would be the area of the "curtain" created—when the points of the surface that are directly over
1015:
807:
696:
35:
4994:
8212:
8094:
7915:
7863:
7669:
7647:
7515:
7126:
5101:
3126:
2321:{\displaystyle I=\lim _{\Delta s_{i}\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\,\Delta s_{i}.}
2003:
1143:
1084:
1051:
980:
941:
825:
761:
685:
7288:
5868:, the value of the integral is simply zero, or in case the region includes singularities, the
5606:
4955:
8338:
8197:
8109:
7766:
7701:
7674:
7664:
7585:
7573:
7558:
7530:
7134:
5515:
3193:
3057:
1025:
691:
462:
407:
368:
274:
8154:
7773:
7620:
7166:
3699:
1030:
1010:
936:
605:
524:
498:
412:
82:
6984:
6885:
6205:
5452:
is a closed curve (initial and final points coincide), the line integral is often denoted
3691:
8:
8174:
8099:
7986:
7943:
7694:
7679:
7510:
7498:
7485:
7445:
7425:
6974:
4349:
3084:
The trajectory of a particle (in red) along a curve inside a vector field. Starting from
3061:
2910:
2204:
1676:
1005:
975:
965:
852:
706:
503:
359:
242:
237:
8263:
8238:
8069:
8022:
7963:
7928:
7923:
7903:
7898:
7893:
7858:
7805:
7788:
7689:
7563:
7548:
7493:
7460:
7151:
7026:
7020:
6970:
6916:
5921:
5512:, and inner products in complex analysis involve conjugacy (the gradient of a function
3344:
2331:
1760:
1266:
970:
873:
857:
797:
792:
787:
751:
632:
551:
457:
452:
256:
251:
6193:
5334:, the line integral can be evaluated as an integral of a function of a real variable:
8403:
8227:
8159:
7981:
7958:
7832:
7825:
7728:
7543:
7435:
7214:
7122:
6232:
5850:
3050:
2050:
1044:
878:
656:
534:
487:
344:
339:
8361:
8144:
8057:
8037:
7968:
7878:
7820:
7812:
7746:
7659:
7420:
7415:
7176:
7156:
6199:
4951:
4346:
3622:
2746:
1453:
1111:
888:
782:
756:
617:
529:
493:
4282:{\displaystyle \mathbf {F} \colon U\subseteq \mathbb {R} ^{2}\to \mathbb {R} ^{2}}
8423:
8408:
8192:
8047:
8027:
7996:
7973:
7953:
7847:
7503:
7450:
7186:
6187:
5869:
2705:{\displaystyle I=\int _{a}^{b}f(\mathbf {r} (t))\left|\mathbf {r} '(t)\right|dt.}
2171:
can be associated with the signed area of a rectangle with a height and width of
1253:
1020:
893:
847:
842:
729:
642:
587:
3080:
1980:
For a line integral over a scalar field, the integral can be constructed from a
1340:
8333:
8232:
8079:
8032:
7933:
7736:
7181:
7023:, the area of a region enclosed by a smooth, closed, positively oriented curve
4197:
3046:
2083:
1147:
903:
711:
478:
7751:
3347:, we see that the displacement vector between adjacent points on the curve is
8438:
8207:
8062:
7652:
7627:
4971:
3215:
647:
397:
354:
5439:{\displaystyle \int _{L}f(z)\,dz=\int _{a}^{b}f(\gamma (t))\gamma '(t)\,dt.}
8217:
8187:
8052:
7615:
7325:
7171:
6532:{\displaystyle \mathbf {F} (x,y)={\overline {f(x+iy)}}=(u(x+iy),-v(x+iy)),}
4229:
3638:
2725:
1759:. Here, and in the rest of the article, the absolute value bars denote the
1381:
1293:
1189:
have natural continuous analogues in terms of line integrals, in this case
1127:
1123:
637:
382:
7465:
7407:
5899:
5320:
4967:
3122:
2896:
2086:
by introducing the straight line piece between each of the sample points
1981:
1135:
1072:
1000:
8182:
8114:
7868:
7741:
7605:
7595:
7538:
7357:
7138:
6978:
3695:
3041:
Line integrals of vector fields are independent of the parametrization
3035:
2124:
1801:
1131:
746:
670:
392:
387:
291:
7353:
2211:
of the terms as the length of the partitions approaches zero gives us
8376:
8124:
8119:
7430:
5631:
The line integral with respect to the conjugate complex differential
5603:, and the complex inner product would attribute twice a conjugate to
3130:
2208:
1673:
1243:{\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }
675:
665:
1256:
done on an object moving through an electric or gravitational field
24:
8371:
7873:
7399:
6194:
Relation of complex line integral and line integral of vector field
3634:
3065:
1080:
741:
483:
440:
129:
8222:
7475:
8391:
7455:
3228:. Letting the size of the partitions go to zero gives us a sum
2119:. (The approximation of a curve to a polygonal path is called
7470:
4196:
The line integral has many uses in physics. For example, the
2749:
1456:
1146:. Many simple formulae in physics, such as the definition of
1088:
7368:
7125:
actually refers not to path integrals in this sense but to
5323:
as the lengths of the subdivision intervals approach zero.
4340:
3877:
which happens to be the integrand for the line integral of
2994:{\displaystyle ||\mathbf {r} '(t)||\neq 0\;\;\forall t\in }
7129:, that is, integrals over a space of paths, of a function
7010:
are identical to the vanishing of curl and divergence for
2334:, the distance between subsequent points on the curve, is
16:
Definite integral of a scalar or vector field along a path
7098:{\textstyle {\frac {1}{2i}}\int _{L}{\overline {z}}\,dz.}
3613:
which is the
Riemann sum for the integral defined above.
2045:
denotes some point, call it a sample point, on the curve
1115:
is used as well, although that is typically reserved for
4931:
is on the clockwise side of the forward velocity vector
4796:{\displaystyle \mathbf {r} '(t)^{\perp }=(y'(t),-x'(t))}
7049:
5458:
4803:
is the clockwise perpendicular of the velocity vector
4573:
4462:
1196:
7029:
6987:
6925:
6915:) for any smooth closed curve L. Correspondingly, by
6888:
6545:
6415:
6353:
6282:
6241:
6208:
5930:
5669:
5637:
5609:
5566:
5538:
5518:
5340:
5143:
5060:
4997:
4890:
The flow is computed in an oriented sense: the curve
4810:
4717:
4394:
4237:
3904:
3725:
3662:
3483:
3353:
3234:
2919:
2771:
2619:
2484:
2340:
2217:
1950:
1888:
1858:
1834:
1810:
1776:
1731:
1685:
1665:{\displaystyle \mathbf {r} \colon \to {\mathcal {C}}}
1628:
1494:
1464:
1423:
1389:
1269:
1157:
142:
7105:
This fact is used, for example, in the proof of the
5628:
in the vector field definition of a line integral).
7293:(2nd ed.). New York: McGraw-Hill. p. 103.
7239:"Line integrals are independent of parametrization"
4204:inside a force field represented as a vector field
3064:(which takes the vector field to the corresponding
1940:in space, and the line integral gives the (signed)
49:. Unsourced material may be challenged and removed.
7097:
7035:
7002:
6981:-free). In fact, the Cauchy-Riemann equations for
6957:
6903:
6868:
6531:
6401:
6335:
6268:
6223:
6172:
5834:
5655:
5620:
5595:
5552:
5524:
5493:
5438:
5309:
5092:
5009:
4878:
4795:
4697:
4281:
4136:
3869:
3682:
3603:
3477:Substituting this in the above Riemann sum yields
3467:
3333:
2993:
2883:
2714:
2704:
2605:
2478:Substituting this in the above Riemann sum yields
2468:
2320:
1960:
1898:
1868:
1844:
1820:
1786:
1741:
1695:
1664:
1614:
1480:
1444:
1409:
1335:
1275:
1242:
1179:
214:
6202:, the line integral of a complex-valued function
8436:
3491:
3242:
2492:
2225:
1852:). Line integrals of scalar fields over a curve
1800:may be intuitively interpreted as an elementary
1180:{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }
215:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
6958:{\displaystyle \mathbf {F} ={\overline {f(z)}}}
7206:
5849:may be used to equate the line integral of an
4879:{\displaystyle \mathbf {r} '(t)=(x'(t),y'(t))}
1348:can be thought of as the area under the curve
7384:
7364:Line integral of a vector field – Interactive
5017:is a curve of finite length, parametrized by
4552:
4518:
4504:
4470:
1794:is the domain of integration, and the symbol
1052:
1876:do not depend on the chosen parametrization
6336:{\displaystyle \mathbf {r} (t)=(x(t),y(t))}
4954:, the line integral is defined in terms of
4916:, and the flow is counted as positive when
1445:{\displaystyle U\subseteq \mathbb {R} ^{n}}
7391:
7377:
5501:sometimes referred to in engineering as a
2966:
2965:
2613:which is the Riemann sum for the integral
1116:
1059:
1045:
8419:Regiomontanus' angle maximization problem
7213:. Springer Science & Business Media.
7085:
6919:, the right-hand integrals are zero when
6629:
6572:
6198:Viewing complex numbers as 2-dimensional
6083:
6023:
5955:
5822:
5744:
5596:{\displaystyle {\overline {\gamma '(z)}}}
5546:
5481:
5426:
5363:
5290:
5193:
5083:
4269:
4254:
4072:
4007:
3591:
3455:
3129:(which is the range of the values of the
2874:
2301:
1602:
1521:
1432:
1410:{\displaystyle f\colon U\to \mathbb {R} }
1403:
175:
109:Learn how and when to remove this message
8262:
4945:
4200:done on a particle traveling on a curve
3079:
3068:field), over the curve considered as an
1339:
7767:Differentiating under the integral sign
7286:
7200:
5319:The integral is then the limit of this
4894:has a specified forward direction from
2002:. This can be done by partitioning the
1481:{\displaystyle {\mathcal {C}}\subset U}
1122:The function to be integrated may be a
583:Differentiating under the integral sign
8437:
6882:, the left-hand integral is zero when
4223:
3683:{\displaystyle \mathbf {F} =\nabla G,}
1344:The line integral over a scalar field
1087:to be integrated is evaluated along a
7643:Inverse functions and differentiation
7372:
1909:Geometrically, when the scalar field
7232:
7230:
7112:
3616:
47:adding citations to reliable sources
18:
7332:"Introduction to the Line Integral"
7207:Kwong-Tin Tang (30 November 2006).
6269:{\displaystyle {\overline {f(z)}}.}
3121:, we construct the integral from a
1770:is called the integrand, the curve
1117:line integrals in the complex plane
13:
7441:Free variables and bound variables
7347:"Line Integral Example 2 (part 2)"
7342:"Line Integral Example 2 (part 1)"
7236:
5291:
5100:may be defined by subdividing the
3769:
3671:
3592:
3495:
3456:
3393:
3354:
3316:
3246:
3218:contribution of each partition of
2967:
2597:
2496:
2455:
2383:
2341:
2302:
2229:
1972:. See the animation to the right.
1953:
1891:
1861:
1837:
1813:
1779:
1734:
1688:
1657:
1501:
1467:
1287:
124:Part of a series of articles about
14:
8461:
8246:The Method of Mechanical Theorems
7301:
7227:
5553:{\displaystyle z\in \mathbb {C} }
5093:{\displaystyle \int _{L}f(z)\,dz}
7801:Partial fractions in integration
7717:Stochastic differential equation
6927:
6849:
6834:
6826:
6805:
6791:
6783:
6417:
6409:corresponds to the vector field
6284:
5656:{\displaystyle {\overline {dz}}}
5494:{\textstyle \oint _{L}f(z)\,dz,}
4813:
4720:
4429:
4414:
4406:
4239:
4156:depends solely on the values of
4146:In other words, the integral of
4115:
4089:
4045:
3990:
3969:
3961:
3938:
3924:
3916:
3850:
3829:
3821:
3800:
3779:
3739:
3664:
3567:
3539:
3531:
3431:
3406:
3373:
3359:
3321:
3290:
3282:
3038:to the line of the integration.
2932:
2857:
2836:
2828:
2805:
2791:
2783:
2671:
2648:
2568:
2538:
2431:
2396:
2363:
2278:
1828:(i.e., a differential length of
1630:
1580:
1555:
1514:
1236:
1222:
1214:
1173:
1165:
23:
7939:Jacobian matrix and determinant
7794:Tangent half-angle substitution
7762:Fundamental theorem of calculus
6402:{\displaystyle f(z)=u(z)+iv(z)}
5137:and considering the expression
4191:
3088:, the particle traces the path
3075:
2719:
2715:Line integral of a vector field
1984:using the above definitions of
1336:Line integral of a scalar field
34:needs additional citations for
8015:Arithmetico-geometric sequence
7707:Ordinary differential equation
7280:
7256:
7162:Methods of contour integration
6997:
6991:
6946:
6940:
6898:
6892:
6838:
6830:
6795:
6787:
6759:
6738:
6732:
6717:
6698:
6680:
6674:
6659:
6636:
6614:
6611:
6596:
6569:
6563:
6523:
6520:
6505:
6493:
6478:
6472:
6460:
6445:
6433:
6421:
6396:
6390:
6378:
6372:
6363:
6357:
6330:
6327:
6321:
6312:
6306:
6300:
6294:
6288:
6254:
6248:
6218:
6212:
6148:
6133:
5813:
5807:
5793:
5790:
5784:
5778:
5735:
5729:
5689:
5683:
5584:
5578:
5478:
5472:
5423:
5417:
5406:
5403:
5397:
5391:
5360:
5354:
5287:
5274:
5244:
5241:
5222:
5213:
5200:
5194:
5190:
5187:
5174:
5168:
5080:
5074:
4873:
4870:
4864:
4850:
4844:
4833:
4827:
4821:
4790:
4787:
4781:
4764:
4758:
4747:
4735:
4728:
4614:
4608:
4590:
4584:
4547:
4541:
4532:
4526:
4499:
4493:
4484:
4478:
4418:
4410:
4264:
4128:
4125:
4119:
4111:
4102:
4099:
4093:
4085:
4058:
4055:
4049:
4041:
4004:
3998:
3982:
3979:
3973:
3965:
3928:
3920:
3864:
3858:
3842:
3839:
3833:
3825:
3814:
3808:
3792:
3789:
3783:
3775:
3752:
3749:
3743:
3735:
3588:
3575:
3559:
3556:
3543:
3535:
3501:
3452:
3439:
3423:
3410:
3399:
3377:
3310:
3307:
3294:
3286:
3252:
2988:
2976:
2955:
2950:
2946:
2940:
2926:
2921:
2871:
2865:
2849:
2846:
2840:
2832:
2795:
2787:
2685:
2679:
2661:
2658:
2652:
2644:
2589:
2576:
2558:
2555:
2542:
2534:
2502:
2452:
2439:
2413:
2400:
2389:
2367:
2298:
2295:
2282:
2274:
2242:
1961:{\displaystyle {\mathcal {C}}}
1899:{\displaystyle {\mathcal {C}}}
1869:{\displaystyle {\mathcal {C}}}
1845:{\displaystyle {\mathcal {C}}}
1821:{\displaystyle {\mathcal {C}}}
1787:{\displaystyle {\mathcal {C}}}
1742:{\displaystyle {\mathcal {C}}}
1696:{\displaystyle {\mathcal {C}}}
1652:
1649:
1637:
1594:
1588:
1565:
1559:
1518:
1510:
1399:
1226:
1218:
209:
203:
194:
188:
172:
166:
1:
7838:Integro-differential equation
7712:Partial differential equation
7193:
4962:of complex numbers. Suppose
3188:gives us the position of the
1975:
1375:
509:Integral of inverse functions
7398:
7309:"Integral over trajectories"
7080:
6950:
6911:is analytic (satisfying the
6464:
6258:
6182:This is a typical result of
5817:
5752:
5739:
5702:
5648:
5588:
5510:Riesz representation theorem
4330:line integral across a curve
3049:, but they do depend on its
2745:, the line integral along a
1452:, the line integral along a
1324:plane. The line integral of
1134:or, for a vector field, the
7:
7992:Generalized Stokes' theorem
7779:Integration by substitution
7314:Encyclopedia of Mathematics
7144:
5332:continuously differentiable
4345:, is defined in terms of a
3894:. It follows, given a path
2203:, respectively. Taking the
1138:of the vector field with a
932:Calculus on Euclidean space
350:Logarithmic differentiation
10:
8466:
7521:(ε, δ)-definition of limit
5875:
5010:{\displaystyle L\subset U}
3620:
1944:area bounded by the curve
8414:Proof that 22/7 exceeds π
8351:
8329:
8255:
8203:Gottfried Wilhelm Leibniz
8173:
8150:e (mathematical constant)
8135:
8007:
7914:
7846:
7727:
7529:
7484:
7406:
7337:"Line Integral Example 1"
7119:path integral formulation
7043:is given by the integral
6184:Cauchy's integral formula
5924:. Substituting, we find:
5902:about 0, parametrized by
2121:rectification of a curve,
2078:to approximate the curve
2010:sub-intervals of length
1921:, its graph is a surface
1761:standard (Euclidean) norm
1371:, described by the field.
666:Summand limit (term test)
8165:Stirling's approximation
7638:Implicit differentiation
7586:Rules of differentiation
6913:Cauchy–Riemann equations
5898:be the counterclockwise
5621:{\displaystyle \gamma '}
4711:is the dot product, and
4210:is the line integral of
3692:multivariable chain rule
3208:. As before, evaluating
3110:and its parametrization
1913:is defined over a plane
345:Implicit differentiation
335:Differentiation notation
262:Inverse function theorem
8399:Euler–Maclaurin formula
8304:trigonometric functions
7757:Constant of integration
5847:Cauchy integral theorem
5525:{\displaystyle \gamma }
5326:If the parametrization
3092:along the vector field
808:Helmholtz decomposition
8368:Differential geometry
8213:Infinitesimal calculus
7916:Multivariable calculus
7864:Directional derivative
7670:Second derivative test
7648:Logarithmic derivative
7621:General Leibniz's rule
7516:Order of approximation
7287:Ahlfors, Lars (1966).
7135:probability amplitudes
7099:
7037:
7004:
6959:
6905:
6870:
6533:
6403:
6337:
6270:
6225:
6174:
5894:, and let the contour
5880:Consider the function
5836:
5657:
5622:
5597:
5554:
5526:
5495:
5440:
5311:
5270:
5164:
5094:
5011:
4880:
4797:
4699:
4283:
4138:
3871:
3684:
3605:
3529:
3469:
3335:
3280:
3097:
3056:From the viewpoint of
3027:give the endpoints of
2995:
2885:
2761:, in the direction of
2706:
2607:
2530:
2470:
2322:
2270:
1992:and a parametrization
1962:
1900:
1870:
1846:
1822:
1788:
1743:
1725:give the endpoints of
1697:
1666:
1616:
1482:
1446:
1411:
1372:
1277:
1244:
1181:
942:Limit of distributions
762:Directional derivative
418:Faà di Bruno's formula
216:
8287:logarithmic functions
8282:exponential functions
8198:Generality of algebra
8076:Tests of convergence
7702:Differential equation
7686:Further applications
7675:Extreme value theorem
7665:First derivative test
7559:Differential operator
7531:Differential calculus
7264:"16.2 Line Integrals"
7100:
7038:
7005:
6960:
6906:
6871:
6534:
6404:
6338:
6271:
6226:
6175:
5837:
5658:
5623:
5598:
5555:
5527:
5496:
5441:
5312:
5250:
5144:
5095:
5012:
4946:Complex line integral
4881:
4798:
4700:
4284:
4139:
3872:
3685:
3606:
3509:
3470:
3336:
3260:
3083:
3058:differential geometry
2996:
2886:
2707:
2608:
2510:
2471:
2323:
2250:
1963:
1901:
1871:
1847:
1823:
1789:
1744:
1698:
1667:
1617:
1483:
1447:
1412:
1343:
1278:
1245:
1182:
1026:Mathematical analysis
937:Generalized functions
622:arithmetico-geometric
463:Leibniz integral rule
217:
8352:Miscellaneous topics
8292:hyperbolic functions
8277:irrational functions
8155:Exponential function
8008:Sequences and series
7774:Integration by parts
7127:functional integrals
7047:
7027:
7003:{\displaystyle f(z)}
6985:
6923:
6904:{\displaystyle f(z)}
6886:
6543:
6413:
6351:
6280:
6239:
6224:{\displaystyle f(z)}
6206:
5928:
5864:is analytic without
5667:
5635:
5607:
5564:
5536:
5516:
5456:
5338:
5141:
5058:
5054:. The line integral
4995:
4808:
4715:
4392:
4235:
3902:
3723:
3660:
3481:
3351:
3232:
3172:th point on , then
3140:intervals of length
2917:
2769:
2617:
2482:
2338:
2215:
1948:
1886:
1856:
1832:
1808:
1774:
1729:
1683:
1626:
1492:
1462:
1421:
1387:
1267:
1194:
1155:
1106:curvilinear integral
1031:Nonstandard analysis
499:Lebesgue integration
369:Rules and identities
140:
43:improve this article
8339:List of derivatives
8175:History of calculus
8090:Cauchy condensation
7987:Exterior derivative
7944:Lagrange multiplier
7680:Maximum and minimum
7511:Limit of a sequence
7499:Limit of a function
7446:Graph of a function
7426:Continuous function
6120:
6053:
5986:
5922:complex exponential
5774:
5387:
4991:is a function, and
4651:
4456:
4224:Flow across a curve
4031:
3959:
3125:. We partition the
3062:musical isomorphism
2826:
2640:
1545:
1252:which computes the
702:Cauchy condensation
504:Contour integration
230:Fundamental theorem
157:
8272:rational functions
8239:Method of Fluxions
8085:Alternating series
7982:Differential forms
7964:Partial derivative
7924:Divergence theorem
7806:Quadratic integral
7574:Leibniz's notation
7564:Mean value theorem
7549:Partial derivative
7494:Indeterminate form
7152:Divergence theorem
7095:
7033:
7000:
6955:
6901:
6866:
6864:
6529:
6399:
6333:
6266:
6221:
6170:
6168:
6103:
6036:
5969:
5832:
5760:
5653:
5618:
5593:
5550:
5522:
5491:
5436:
5373:
5307:
5090:
5007:
4876:
4793:
4695:
4637:
4619:
4559:
4442:
4339:, also called the
4279:
4134:
4017:
3945:
3867:
3680:
3627:If a vector field
3601:
3508:
3465:
3345:mean value theorem
3331:
3259:
3098:
2991:
2881:
2812:
2702:
2626:
2603:
2509:
2466:
2332:mean value theorem
2318:
2249:
1958:
1896:
1866:
1842:
1818:
1784:
1739:
1693:
1662:
1612:
1531:
1478:
1442:
1407:
1373:
1273:
1240:
1177:
874:Partial derivative
803:generalized Stokes
697:Alternating series
578:Reduction formulae
567:Heaviside's method
548:tangent half-angle
535:Cylindrical shells
458:Integral transform
453:Lists of integrals
257:Mean value theorem
212:
143:
8432:
8431:
8358:Complex calculus
8347:
8346:
8228:Law of Continuity
8160:Natural logarithm
8145:Bernoulli numbers
8136:Special functions
8095:Direct comparison
7959:Multiple integral
7833:Integral equation
7729:Integral calculus
7660:Stationary points
7634:Other techniques
7579:Newton's notation
7544:Second derivative
7436:Finite difference
7220:978-3-540-30268-1
7167:Nachbin's theorem
7123:quantum mechanics
7113:Quantum mechanics
7083:
7063:
7036:{\displaystyle L}
6953:
6467:
6276:Specifically, if
6261:
6005:
5953:
5851:analytic function
5820:
5755:
5742:
5705:
5663:is defined to be
5651:
5591:
4680:
4665:
4627:
4070:
3764:
3617:Path independence
3490:
3241:
2491:
2224:
2139:. The product of
2053:of sample points
2049:. We can use the
1968:and the graph of
1276:{\displaystyle L}
1069:
1068:
949:
948:
911:
910:
879:Multiple integral
815:
814:
719:
718:
686:Direct comparison
657:Convergence tests
595:
594:
563:Partial fractions
430:
429:
340:Second derivative
119:
118:
111:
93:
8457:
8445:Complex analysis
8362:Contour integral
8260:
8259:
8110:Limit comparison
8019:Types of series
7978:Advanced topics
7969:Surface integral
7813:Trapezoidal rule
7752:Basic properties
7747:Riemann integral
7695:Taylor's theorem
7421:Concave function
7416:Binomial theorem
7393:
7386:
7379:
7370:
7369:
7322:
7295:
7294:
7290:Complex Analysis
7284:
7278:
7277:
7275:
7274:
7260:
7254:
7253:
7251:
7249:
7234:
7225:
7224:
7204:
7177:Surface integral
7157:Gradient theorem
7104:
7102:
7101:
7096:
7084:
7076:
7074:
7073:
7064:
7062:
7051:
7042:
7040:
7039:
7034:
7015:
7009:
7007:
7006:
7001:
6964:
6962:
6961:
6956:
6954:
6949:
6935:
6930:
6910:
6908:
6907:
6902:
6880:Cauchy's theorem
6875:
6873:
6872:
6867:
6865:
6858:
6857:
6852:
6837:
6829:
6824:
6823:
6808:
6794:
6786:
6781:
6780:
6765:
6716:
6715:
6658:
6657:
6642:
6595:
6594:
6559:
6558:
6538:
6536:
6535:
6530:
6468:
6463:
6440:
6420:
6408:
6406:
6405:
6400:
6342:
6340:
6339:
6334:
6287:
6275:
6273:
6272:
6267:
6262:
6257:
6243:
6230:
6228:
6227:
6222:
6179:
6177:
6176:
6171:
6169:
6119:
6111:
6093:
6082:
6081:
6069:
6068:
6052:
6044:
6022:
6021:
6006:
6004:
6003:
5988:
5985:
5977:
5954:
5946:
5944:
5943:
5919:
5915:
5893:
5863:
5841:
5839:
5838:
5833:
5821:
5816:
5806:
5797:
5773:
5768:
5756:
5751:
5743:
5738:
5724:
5722:
5721:
5711:
5706:
5701:
5693:
5679:
5678:
5662:
5660:
5659:
5654:
5652:
5647:
5639:
5627:
5625:
5624:
5619:
5617:
5602:
5600:
5599:
5594:
5592:
5587:
5577:
5568:
5559:
5557:
5556:
5551:
5549:
5531:
5529:
5528:
5523:
5500:
5498:
5497:
5492:
5468:
5467:
5451:
5445:
5443:
5442:
5437:
5416:
5386:
5381:
5350:
5349:
5329:
5316:
5314:
5313:
5308:
5303:
5302:
5286:
5285:
5269:
5264:
5240:
5239:
5212:
5211:
5186:
5185:
5163:
5158:
5136:
5099:
5097:
5096:
5091:
5070:
5069:
5053:
5026:
5016:
5014:
5013:
5008:
4990:
4952:complex analysis
4941:
4930:
4915:
4904:
4893:
4887:
4885:
4883:
4882:
4877:
4863:
4843:
4820:
4816:
4802:
4800:
4799:
4794:
4780:
4757:
4743:
4742:
4727:
4723:
4710:
4704:
4702:
4701:
4696:
4691:
4687:
4678:
4663:
4650:
4645:
4625:
4624:
4623:
4607:
4583:
4564:
4563:
4556:
4555:
4522:
4521:
4508:
4507:
4474:
4473:
4455:
4450:
4438:
4437:
4432:
4417:
4409:
4404:
4403:
4387:
4360:
4347:piecewise smooth
4327:
4288:
4286:
4285:
4280:
4278:
4277:
4272:
4263:
4262:
4257:
4242:
4215:
4209:
4186:path independent
4183:
4172:
4161:
4151:
4143:
4141:
4140:
4135:
4118:
4092:
4071:
4069:
4061:
4048:
4033:
4030:
4025:
3997:
3993:
3972:
3964:
3958:
3953:
3941:
3927:
3919:
3914:
3913:
3893:
3882:
3876:
3874:
3873:
3868:
3857:
3853:
3832:
3824:
3807:
3803:
3782:
3765:
3763:
3755:
3742:
3727:
3718:
3707:
3689:
3687:
3686:
3681:
3667:
3651:
3645:
3632:
3623:Gradient theorem
3610:
3608:
3607:
3602:
3587:
3586:
3574:
3570:
3555:
3554:
3542:
3534:
3528:
3523:
3507:
3474:
3472:
3471:
3466:
3451:
3450:
3438:
3434:
3422:
3421:
3409:
3389:
3388:
3376:
3368:
3367:
3362:
3340:
3338:
3337:
3332:
3330:
3329:
3324:
3306:
3305:
3293:
3285:
3279:
3274:
3258:
3227:
3223:
3213:
3207:
3191:
3187:
3171:
3167:
3158:
3139:
3135:
3120:
3109:
3105:
3026:
3015:
3000:
2998:
2997:
2992:
2958:
2953:
2939:
2935:
2929:
2924:
2908:
2894:
2890:
2888:
2887:
2882:
2864:
2860:
2839:
2831:
2825:
2820:
2808:
2794:
2786:
2781:
2780:
2765:, is defined as
2760:
2747:piecewise smooth
2744:
2711:
2709:
2708:
2703:
2692:
2688:
2678:
2674:
2651:
2639:
2634:
2612:
2610:
2609:
2604:
2596:
2592:
2588:
2587:
2575:
2571:
2554:
2553:
2541:
2529:
2524:
2508:
2475:
2473:
2472:
2467:
2465:
2461:
2451:
2450:
2438:
2434:
2420:
2416:
2412:
2411:
2399:
2379:
2378:
2366:
2353:
2352:
2327:
2325:
2324:
2319:
2314:
2313:
2294:
2293:
2281:
2269:
2264:
2248:
2241:
2240:
2202:
2190:
2170:
2158:
2138:
2118:
2102:
2081:
2077:
2048:
2044:
2028:
2009:
2001:
1997:
1991:
1987:
1971:
1967:
1965:
1964:
1959:
1957:
1956:
1939:
1920:
1912:
1905:
1903:
1902:
1897:
1895:
1894:
1881:
1875:
1873:
1872:
1867:
1865:
1864:
1851:
1849:
1848:
1843:
1841:
1840:
1827:
1825:
1824:
1819:
1817:
1816:
1799:
1793:
1791:
1790:
1785:
1783:
1782:
1769:
1758:
1748:
1746:
1745:
1740:
1738:
1737:
1724:
1713:
1702:
1700:
1699:
1694:
1692:
1691:
1672:is an arbitrary
1671:
1669:
1668:
1663:
1661:
1660:
1633:
1621:
1619:
1618:
1613:
1601:
1597:
1587:
1583:
1572:
1568:
1558:
1544:
1539:
1517:
1506:
1505:
1504:
1487:
1485:
1484:
1479:
1471:
1470:
1454:piecewise smooth
1451:
1449:
1448:
1443:
1441:
1440:
1435:
1416:
1414:
1413:
1408:
1406:
1370:
1352:along a surface
1332:are carved out.
1315:
1284:
1282:
1280:
1279:
1274:
1261:
1251:
1249:
1247:
1246:
1241:
1239:
1225:
1217:
1212:
1211:
1188:
1186:
1184:
1183:
1178:
1176:
1168:
1112:contour integral
1061:
1054:
1047:
995:
960:
926:
925:
922:
889:Surface integral
832:
831:
828:
736:
735:
732:
692:Limit comparison
612:
611:
608:
494:Riemann integral
447:
446:
443:
403:L'Hôpital's rule
360:Taylor's theorem
281:
280:
277:
221:
219:
218:
213:
165:
156:
151:
121:
120:
114:
107:
103:
100:
94:
92:
51:
27:
19:
8465:
8464:
8460:
8459:
8458:
8456:
8455:
8454:
8450:Vector calculus
8435:
8434:
8433:
8428:
8424:Steinmetz solid
8409:Integration Bee
8343:
8325:
8251:
8193:Colin Maclaurin
8169:
8137:
8131:
8003:
7997:Tensor calculus
7974:Volume integral
7910:
7885:Basic theorems
7848:Vector calculus
7842:
7723:
7690:Newton's method
7525:
7504:One-sided limit
7480:
7461:Rolle's theorem
7451:Linear function
7402:
7397:
7307:
7304:
7299:
7298:
7285:
7281:
7272:
7270:
7268:www.whitman.edu
7262:
7261:
7257:
7247:
7245:
7237:Nykamp, Duane.
7235:
7228:
7221:
7205:
7201:
7196:
7191:
7187:Volume integral
7147:
7115:
7075:
7069:
7065:
7055:
7050:
7048:
7045:
7044:
7028:
7025:
7024:
7021:Green's theorem
7011:
6986:
6983:
6982:
6936:
6934:
6926:
6924:
6921:
6920:
6917:Green's theorem
6887:
6884:
6883:
6863:
6862:
6853:
6848:
6847:
6833:
6825:
6819:
6815:
6804:
6790:
6782:
6776:
6772:
6763:
6762:
6711:
6707:
6653:
6649:
6640:
6639:
6590:
6586:
6579:
6554:
6550:
6546:
6544:
6541:
6540:
6441:
6439:
6416:
6414:
6411:
6410:
6352:
6349:
6348:
6283:
6281:
6278:
6277:
6244:
6242:
6240:
6237:
6236:
6207:
6204:
6203:
6196:
6188:residue theorem
6167:
6166:
6112:
6107:
6091:
6090:
6074:
6070:
6058:
6054:
6045:
6040:
6014:
6010:
5996:
5992:
5987:
5978:
5973:
5962:
5945:
5939:
5935:
5931:
5929:
5926:
5925:
5917:
5903:
5881:
5878:
5870:residue theorem
5854:
5799:
5798:
5796:
5769:
5764:
5725:
5723:
5717:
5713:
5712:
5710:
5694:
5692:
5674:
5670:
5668:
5665:
5664:
5640:
5638:
5636:
5633:
5632:
5610:
5608:
5605:
5604:
5570:
5569:
5567:
5565:
5562:
5561:
5545:
5537:
5534:
5533:
5517:
5514:
5513:
5503:cyclic integral
5463:
5459:
5457:
5454:
5453:
5449:
5409:
5382:
5377:
5345:
5341:
5339:
5336:
5335:
5327:
5298:
5294:
5281:
5277:
5265:
5254:
5229:
5225:
5207:
5203:
5181:
5177:
5159:
5148:
5142:
5139:
5138:
5131:
5123:< ... <
5122:
5115:
5105:
5065:
5061:
5059:
5056:
5055:
5028:
5018:
4996:
4993:
4992:
4978:
4948:
4932:
4917:
4906:
4895:
4891:
4856:
4836:
4812:
4811:
4809:
4806:
4805:
4804:
4773:
4750:
4738:
4734:
4719:
4718:
4716:
4713:
4712:
4708:
4656:
4652:
4646:
4641:
4618:
4617:
4600:
4594:
4593:
4576:
4569:
4568:
4558:
4557:
4551:
4550:
4517:
4516:
4510:
4509:
4503:
4502:
4469:
4468:
4458:
4457:
4451:
4446:
4433:
4428:
4427:
4413:
4405:
4399:
4395:
4393:
4390:
4389:
4362:
4352:
4350:parametrization
4290:
4273:
4268:
4267:
4258:
4253:
4252:
4238:
4236:
4233:
4232:
4226:
4211:
4205:
4194:
4174:
4163:
4157:
4147:
4114:
4088:
4062:
4044:
4034:
4032:
4026:
4021:
3989:
3988:
3968:
3960:
3954:
3949:
3937:
3923:
3915:
3909:
3905:
3903:
3900:
3899:
3884:
3878:
3849:
3848:
3828:
3820:
3799:
3798:
3778:
3756:
3738:
3728:
3726:
3724:
3721:
3720:
3709:
3703:
3663:
3661:
3658:
3657:
3647:
3641:
3628:
3625:
3619:
3582:
3578:
3566:
3565:
3550:
3546:
3538:
3530:
3524:
3513:
3494:
3482:
3479:
3478:
3446:
3442:
3430:
3429:
3417:
3413:
3405:
3384:
3380:
3372:
3363:
3358:
3357:
3352:
3349:
3348:
3325:
3320:
3319:
3301:
3297:
3289:
3281:
3275:
3264:
3245:
3233:
3230:
3229:
3225:
3219:
3209:
3206:
3197:
3189:
3185:
3173:
3169:
3165:
3160:
3141:
3137:
3133:
3111:
3107:
3101:
3078:
3017:
3006:
3001:) of the curve
2954:
2949:
2931:
2930:
2925:
2920:
2918:
2915:
2914:
2911:parametrization
2900:
2892:
2856:
2855:
2835:
2827:
2821:
2816:
2804:
2790:
2782:
2776:
2772:
2770:
2767:
2766:
2752:
2728:
2722:
2717:
2670:
2669:
2668:
2664:
2647:
2635:
2630:
2618:
2615:
2614:
2583:
2579:
2567:
2566:
2565:
2561:
2549:
2545:
2537:
2525:
2514:
2495:
2483:
2480:
2479:
2446:
2442:
2430:
2429:
2428:
2424:
2407:
2403:
2395:
2374:
2370:
2362:
2361:
2357:
2348:
2344:
2339:
2336:
2335:
2309:
2305:
2289:
2285:
2277:
2265:
2254:
2236:
2232:
2228:
2216:
2213:
2212:
2201:
2192:
2188:
2172:
2169:
2160:
2156:
2140:
2137:
2128:
2116:
2104:
2100:
2087:
2079:
2067:
2054:
2046:
2042:
2030:
2011:
2007:
1999:
1993:
1989:
1985:
1978:
1969:
1952:
1951:
1949:
1946:
1945:
1942:cross-sectional
1922:
1914:
1910:
1890:
1889:
1887:
1884:
1883:
1877:
1860:
1859:
1857:
1854:
1853:
1836:
1835:
1833:
1830:
1829:
1812:
1811:
1809:
1806:
1805:
1795:
1778:
1777:
1775:
1772:
1771:
1767:
1750:
1733:
1732:
1730:
1727:
1726:
1715:
1704:
1687:
1686:
1684:
1681:
1680:
1677:parametrization
1656:
1655:
1629:
1627:
1624:
1623:
1579:
1578:
1577:
1573:
1554:
1553:
1549:
1540:
1535:
1513:
1500:
1499:
1495:
1493:
1490:
1489:
1466:
1465:
1463:
1460:
1459:
1436:
1431:
1430:
1422:
1419:
1418:
1402:
1388:
1385:
1384:
1378:
1353:
1338:
1298:
1290:
1288:Vector calculus
1268:
1265:
1264:
1263:
1257:
1235:
1221:
1213:
1207:
1203:
1195:
1192:
1191:
1190:
1172:
1164:
1156:
1153:
1152:
1151:
1109:are also used;
1065:
1036:
1035:
1021:Integration Bee
996:
993:
986:
985:
961:
958:
951:
950:
923:
920:
913:
912:
894:Volume integral
829:
824:
817:
816:
733:
728:
721:
720:
690:
609:
604:
597:
596:
588:Risch algorithm
558:Euler's formula
444:
439:
432:
431:
413:General Leibniz
296:generalizations
278:
273:
266:
252:Rolle's theorem
247:
222:
158:
152:
147:
141:
138:
137:
115:
104:
98:
95:
58:"Line integral"
52:
50:
40:
28:
17:
12:
11:
5:
8463:
8453:
8452:
8447:
8430:
8429:
8427:
8426:
8421:
8416:
8411:
8406:
8404:Gabriel's horn
8401:
8396:
8395:
8394:
8389:
8384:
8379:
8374:
8366:
8365:
8364:
8355:
8353:
8349:
8348:
8345:
8344:
8342:
8341:
8336:
8334:List of limits
8330:
8327:
8326:
8324:
8323:
8322:
8321:
8316:
8311:
8301:
8300:
8299:
8289:
8284:
8279:
8274:
8268:
8266:
8257:
8253:
8252:
8250:
8249:
8242:
8235:
8233:Leonhard Euler
8230:
8225:
8220:
8215:
8210:
8205:
8200:
8195:
8190:
8185:
8179:
8177:
8171:
8170:
8168:
8167:
8162:
8157:
8152:
8147:
8141:
8139:
8133:
8132:
8130:
8129:
8128:
8127:
8122:
8117:
8112:
8107:
8102:
8097:
8092:
8087:
8082:
8074:
8073:
8072:
8067:
8066:
8065:
8060:
8050:
8045:
8040:
8035:
8030:
8025:
8017:
8011:
8009:
8005:
8004:
8002:
8001:
8000:
7999:
7994:
7989:
7984:
7976:
7971:
7966:
7961:
7956:
7951:
7946:
7941:
7936:
7934:Hessian matrix
7931:
7926:
7920:
7918:
7912:
7911:
7909:
7908:
7907:
7906:
7901:
7896:
7891:
7889:Line integrals
7883:
7882:
7881:
7876:
7871:
7866:
7861:
7852:
7850:
7844:
7843:
7841:
7840:
7835:
7830:
7829:
7828:
7823:
7815:
7810:
7809:
7808:
7798:
7797:
7796:
7791:
7786:
7776:
7771:
7770:
7769:
7759:
7754:
7749:
7744:
7739:
7737:Antiderivative
7733:
7731:
7725:
7724:
7722:
7721:
7720:
7719:
7714:
7709:
7699:
7698:
7697:
7692:
7684:
7683:
7682:
7677:
7672:
7667:
7657:
7656:
7655:
7650:
7645:
7640:
7632:
7631:
7630:
7625:
7624:
7623:
7613:
7608:
7603:
7598:
7593:
7583:
7582:
7581:
7576:
7566:
7561:
7556:
7551:
7546:
7541:
7535:
7533:
7527:
7526:
7524:
7523:
7518:
7513:
7508:
7507:
7506:
7496:
7490:
7488:
7482:
7481:
7479:
7478:
7473:
7468:
7463:
7458:
7453:
7448:
7443:
7438:
7433:
7428:
7423:
7418:
7412:
7410:
7404:
7403:
7396:
7395:
7388:
7381:
7373:
7367:
7366:
7361:
7351:
7350:
7349:
7344:
7339:
7334:
7323:
7303:
7302:External links
7300:
7297:
7296:
7279:
7255:
7226:
7219:
7198:
7197:
7195:
7192:
7190:
7189:
7184:
7182:Volume element
7179:
7174:
7169:
7164:
7159:
7154:
7148:
7146:
7143:
7114:
7111:
7094:
7091:
7088:
7082:
7079:
7072:
7068:
7061:
7058:
7054:
7032:
6999:
6996:
6993:
6990:
6975:incompressible
6952:
6948:
6945:
6942:
6939:
6933:
6929:
6900:
6897:
6894:
6891:
6861:
6856:
6851:
6846:
6843:
6840:
6836:
6832:
6828:
6822:
6818:
6814:
6811:
6807:
6803:
6800:
6797:
6793:
6789:
6785:
6779:
6775:
6771:
6768:
6766:
6764:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6728:
6725:
6722:
6719:
6714:
6710:
6706:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6661:
6656:
6652:
6648:
6645:
6643:
6641:
6638:
6635:
6632:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6607:
6604:
6601:
6598:
6593:
6589:
6585:
6582:
6580:
6578:
6575:
6571:
6568:
6565:
6562:
6557:
6553:
6549:
6548:
6528:
6525:
6522:
6519:
6516:
6513:
6510:
6507:
6504:
6501:
6498:
6495:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6466:
6462:
6459:
6456:
6453:
6450:
6447:
6444:
6438:
6435:
6432:
6429:
6426:
6423:
6419:
6398:
6395:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6332:
6329:
6326:
6323:
6320:
6317:
6314:
6311:
6308:
6305:
6302:
6299:
6296:
6293:
6290:
6286:
6265:
6260:
6256:
6253:
6250:
6247:
6220:
6217:
6214:
6211:
6195:
6192:
6165:
6162:
6159:
6156:
6153:
6150:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6118:
6115:
6110:
6106:
6102:
6099:
6096:
6094:
6092:
6089:
6086:
6080:
6077:
6073:
6067:
6064:
6061:
6057:
6051:
6048:
6043:
6039:
6035:
6032:
6029:
6026:
6020:
6017:
6013:
6009:
6002:
5999:
5995:
5991:
5984:
5981:
5976:
5972:
5968:
5965:
5963:
5961:
5958:
5952:
5949:
5942:
5938:
5934:
5933:
5920:in using the
5877:
5874:
5831:
5828:
5825:
5819:
5815:
5812:
5809:
5805:
5802:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5772:
5767:
5763:
5759:
5754:
5750:
5747:
5741:
5737:
5734:
5731:
5728:
5720:
5716:
5709:
5704:
5700:
5697:
5691:
5688:
5685:
5682:
5677:
5673:
5650:
5646:
5643:
5616:
5613:
5590:
5586:
5583:
5580:
5576:
5573:
5548:
5544:
5541:
5521:
5490:
5487:
5484:
5480:
5477:
5474:
5471:
5466:
5462:
5435:
5432:
5429:
5425:
5422:
5419:
5415:
5412:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5385:
5380:
5376:
5372:
5369:
5366:
5362:
5359:
5356:
5353:
5348:
5344:
5306:
5301:
5297:
5293:
5289:
5284:
5280:
5276:
5273:
5268:
5263:
5260:
5257:
5253:
5249:
5246:
5243:
5238:
5235:
5232:
5228:
5224:
5221:
5218:
5215:
5210:
5206:
5202:
5199:
5196:
5192:
5189:
5184:
5180:
5176:
5173:
5170:
5167:
5162:
5157:
5154:
5151:
5147:
5127:
5120:
5113:
5089:
5086:
5082:
5079:
5076:
5073:
5068:
5064:
5006:
5003:
5000:
4956:multiplication
4947:
4944:
4875:
4872:
4869:
4866:
4862:
4859:
4855:
4852:
4849:
4846:
4842:
4839:
4835:
4832:
4829:
4826:
4823:
4819:
4815:
4792:
4789:
4786:
4783:
4779:
4776:
4772:
4769:
4766:
4763:
4760:
4756:
4753:
4749:
4746:
4741:
4737:
4733:
4730:
4726:
4722:
4694:
4690:
4686:
4683:
4677:
4674:
4671:
4668:
4662:
4659:
4655:
4649:
4644:
4640:
4636:
4633:
4630:
4622:
4616:
4613:
4610:
4606:
4603:
4599:
4596:
4595:
4592:
4589:
4586:
4582:
4579:
4575:
4574:
4572:
4567:
4562:
4554:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4520:
4515:
4512:
4511:
4506:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4472:
4467:
4464:
4463:
4461:
4454:
4449:
4445:
4441:
4436:
4431:
4426:
4423:
4420:
4416:
4412:
4408:
4402:
4398:
4276:
4271:
4266:
4261:
4256:
4251:
4248:
4245:
4241:
4225:
4222:
4193:
4190:
4162:at the points
4133:
4130:
4127:
4124:
4121:
4117:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4091:
4087:
4084:
4081:
4078:
4075:
4068:
4065:
4060:
4057:
4054:
4051:
4047:
4043:
4040:
4037:
4029:
4024:
4020:
4016:
4013:
4010:
4006:
4003:
4000:
3996:
3992:
3987:
3984:
3981:
3978:
3975:
3971:
3967:
3963:
3957:
3952:
3948:
3944:
3940:
3936:
3933:
3930:
3926:
3922:
3918:
3912:
3908:
3866:
3863:
3860:
3856:
3852:
3847:
3844:
3841:
3838:
3835:
3831:
3827:
3823:
3819:
3816:
3813:
3810:
3806:
3802:
3797:
3794:
3791:
3788:
3785:
3781:
3777:
3774:
3771:
3768:
3762:
3759:
3754:
3751:
3748:
3745:
3741:
3737:
3734:
3731:
3679:
3676:
3673:
3670:
3666:
3621:Main article:
3618:
3615:
3600:
3597:
3594:
3590:
3585:
3581:
3577:
3573:
3569:
3564:
3561:
3558:
3553:
3549:
3545:
3541:
3537:
3533:
3527:
3522:
3519:
3516:
3512:
3506:
3503:
3500:
3497:
3493:
3489:
3486:
3464:
3461:
3458:
3454:
3449:
3445:
3441:
3437:
3433:
3428:
3425:
3420:
3416:
3412:
3408:
3404:
3401:
3398:
3395:
3392:
3387:
3383:
3379:
3375:
3371:
3366:
3361:
3356:
3328:
3323:
3318:
3315:
3312:
3309:
3304:
3300:
3296:
3292:
3288:
3284:
3278:
3273:
3270:
3267:
3263:
3257:
3254:
3251:
3248:
3244:
3240:
3237:
3202:
3181:
3163:
3077:
3074:
3047:absolute value
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2964:
2961:
2957:
2952:
2948:
2945:
2942:
2938:
2934:
2928:
2923:
2909:is a regular
2880:
2877:
2873:
2870:
2867:
2863:
2859:
2854:
2851:
2848:
2845:
2842:
2838:
2834:
2830:
2824:
2819:
2815:
2811:
2807:
2803:
2800:
2797:
2793:
2789:
2785:
2779:
2775:
2721:
2718:
2716:
2713:
2701:
2698:
2695:
2691:
2687:
2684:
2681:
2677:
2673:
2667:
2663:
2660:
2657:
2654:
2650:
2646:
2643:
2638:
2633:
2629:
2625:
2622:
2602:
2599:
2595:
2591:
2586:
2582:
2578:
2574:
2570:
2564:
2560:
2557:
2552:
2548:
2544:
2540:
2536:
2533:
2528:
2523:
2520:
2517:
2513:
2507:
2504:
2501:
2498:
2494:
2490:
2487:
2464:
2460:
2457:
2454:
2449:
2445:
2441:
2437:
2433:
2427:
2423:
2419:
2415:
2410:
2406:
2402:
2398:
2394:
2391:
2388:
2385:
2382:
2377:
2373:
2369:
2365:
2360:
2356:
2351:
2347:
2343:
2317:
2312:
2308:
2304:
2300:
2297:
2292:
2288:
2284:
2280:
2276:
2273:
2268:
2263:
2260:
2257:
2253:
2247:
2244:
2239:
2235:
2231:
2227:
2223:
2220:
2197:
2184:
2165:
2152:
2133:
2112:
2095:
2084:polygonal path
2063:
2038:
1977:
1974:
1955:
1893:
1863:
1839:
1815:
1781:
1736:
1690:
1659:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1632:
1611:
1608:
1605:
1600:
1596:
1593:
1590:
1586:
1582:
1576:
1571:
1567:
1564:
1561:
1557:
1552:
1548:
1543:
1538:
1534:
1530:
1527:
1524:
1520:
1516:
1512:
1509:
1503:
1498:
1488:is defined as
1477:
1474:
1469:
1439:
1434:
1429:
1426:
1405:
1401:
1398:
1395:
1392:
1377:
1374:
1337:
1334:
1289:
1286:
1272:
1238:
1234:
1231:
1228:
1224:
1220:
1216:
1210:
1206:
1202:
1199:
1175:
1171:
1167:
1163:
1160:
1136:scalar product
1100:curve integral
1067:
1066:
1064:
1063:
1056:
1049:
1041:
1038:
1037:
1034:
1033:
1028:
1023:
1018:
1016:List of topics
1013:
1008:
1003:
997:
992:
991:
988:
987:
984:
983:
978:
973:
968:
962:
957:
956:
953:
952:
947:
946:
945:
944:
939:
934:
924:
919:
918:
915:
914:
909:
908:
907:
906:
901:
896:
891:
886:
881:
876:
868:
867:
863:
862:
861:
860:
855:
850:
845:
837:
836:
830:
823:
822:
819:
818:
813:
812:
811:
810:
805:
800:
795:
790:
785:
777:
776:
772:
771:
770:
769:
764:
759:
754:
749:
744:
734:
727:
726:
723:
722:
717:
716:
715:
714:
709:
704:
699:
694:
688:
683:
678:
673:
668:
660:
659:
653:
652:
651:
650:
645:
640:
635:
630:
625:
610:
603:
602:
599:
598:
593:
592:
591:
590:
585:
580:
575:
573:Changing order
570:
560:
555:
537:
532:
527:
519:
518:
517:Integration by
514:
513:
512:
511:
506:
501:
496:
491:
481:
479:Antiderivative
473:
472:
468:
467:
466:
465:
460:
455:
445:
438:
437:
434:
433:
428:
427:
426:
425:
420:
415:
410:
405:
400:
395:
390:
385:
380:
372:
371:
365:
364:
363:
362:
357:
352:
347:
342:
337:
329:
328:
324:
323:
322:
321:
320:
319:
314:
309:
299:
286:
285:
279:
272:
271:
268:
267:
265:
264:
259:
254:
248:
246:
245:
240:
234:
233:
232:
224:
223:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
174:
171:
168:
164:
161:
155:
150:
146:
136:
133:
132:
126:
125:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
8462:
8451:
8448:
8446:
8443:
8442:
8440:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8393:
8390:
8388:
8385:
8383:
8380:
8378:
8375:
8373:
8370:
8369:
8367:
8363:
8360:
8359:
8357:
8356:
8354:
8350:
8340:
8337:
8335:
8332:
8331:
8328:
8320:
8317:
8315:
8312:
8310:
8307:
8306:
8305:
8302:
8298:
8295:
8294:
8293:
8290:
8288:
8285:
8283:
8280:
8278:
8275:
8273:
8270:
8269:
8267:
8265:
8261:
8258:
8254:
8248:
8247:
8243:
8241:
8240:
8236:
8234:
8231:
8229:
8226:
8224:
8221:
8219:
8216:
8214:
8211:
8209:
8208:Infinitesimal
8206:
8204:
8201:
8199:
8196:
8194:
8191:
8189:
8186:
8184:
8181:
8180:
8178:
8176:
8172:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8142:
8140:
8134:
8126:
8123:
8121:
8118:
8116:
8113:
8111:
8108:
8106:
8103:
8101:
8098:
8096:
8093:
8091:
8088:
8086:
8083:
8081:
8078:
8077:
8075:
8071:
8068:
8064:
8061:
8059:
8056:
8055:
8054:
8051:
8049:
8046:
8044:
8041:
8039:
8036:
8034:
8031:
8029:
8026:
8024:
8021:
8020:
8018:
8016:
8013:
8012:
8010:
8006:
7998:
7995:
7993:
7990:
7988:
7985:
7983:
7980:
7979:
7977:
7975:
7972:
7970:
7967:
7965:
7962:
7960:
7957:
7955:
7952:
7950:
7949:Line integral
7947:
7945:
7942:
7940:
7937:
7935:
7932:
7930:
7927:
7925:
7922:
7921:
7919:
7917:
7913:
7905:
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7886:
7884:
7880:
7877:
7875:
7872:
7870:
7867:
7865:
7862:
7860:
7857:
7856:
7854:
7853:
7851:
7849:
7845:
7839:
7836:
7834:
7831:
7827:
7824:
7822:
7821:Washer method
7819:
7818:
7816:
7814:
7811:
7807:
7804:
7803:
7802:
7799:
7795:
7792:
7790:
7787:
7785:
7784:trigonometric
7782:
7781:
7780:
7777:
7775:
7772:
7768:
7765:
7764:
7763:
7760:
7758:
7755:
7753:
7750:
7748:
7745:
7743:
7740:
7738:
7735:
7734:
7732:
7730:
7726:
7718:
7715:
7713:
7710:
7708:
7705:
7704:
7703:
7700:
7696:
7693:
7691:
7688:
7687:
7685:
7681:
7678:
7676:
7673:
7671:
7668:
7666:
7663:
7662:
7661:
7658:
7654:
7653:Related rates
7651:
7649:
7646:
7644:
7641:
7639:
7636:
7635:
7633:
7629:
7626:
7622:
7619:
7618:
7617:
7614:
7612:
7609:
7607:
7604:
7602:
7599:
7597:
7594:
7592:
7589:
7588:
7587:
7584:
7580:
7577:
7575:
7572:
7571:
7570:
7567:
7565:
7562:
7560:
7557:
7555:
7552:
7550:
7547:
7545:
7542:
7540:
7537:
7536:
7534:
7532:
7528:
7522:
7519:
7517:
7514:
7512:
7509:
7505:
7502:
7501:
7500:
7497:
7495:
7492:
7491:
7489:
7487:
7483:
7477:
7474:
7472:
7469:
7467:
7464:
7462:
7459:
7457:
7454:
7452:
7449:
7447:
7444:
7442:
7439:
7437:
7434:
7432:
7429:
7427:
7424:
7422:
7419:
7417:
7414:
7413:
7411:
7409:
7405:
7401:
7394:
7389:
7387:
7382:
7380:
7375:
7374:
7371:
7365:
7362:
7359:
7355:
7354:Path integral
7352:
7348:
7345:
7343:
7340:
7338:
7335:
7333:
7330:
7329:
7327:
7324:
7320:
7316:
7315:
7310:
7306:
7305:
7292:
7291:
7283:
7269:
7265:
7259:
7248:September 18,
7244:
7240:
7233:
7231:
7222:
7216:
7212:
7211:
7203:
7199:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7163:
7160:
7158:
7155:
7153:
7150:
7149:
7142:
7140:
7136:
7132:
7128:
7124:
7120:
7110:
7108:
7092:
7089:
7086:
7077:
7070:
7066:
7059:
7056:
7052:
7030:
7022:
7017:
7014:
6994:
6988:
6980:
6976:
6972:
6968:
6943:
6937:
6931:
6918:
6914:
6895:
6889:
6881:
6876:
6859:
6854:
6844:
6841:
6820:
6816:
6812:
6809:
6801:
6798:
6777:
6773:
6769:
6767:
6756:
6753:
6750:
6747:
6744:
6741:
6735:
6729:
6726:
6723:
6720:
6712:
6708:
6704:
6701:
6695:
6692:
6689:
6686:
6683:
6677:
6671:
6668:
6665:
6662:
6654:
6650:
6646:
6644:
6633:
6630:
6626:
6623:
6620:
6617:
6608:
6605:
6602:
6599:
6591:
6587:
6583:
6581:
6576:
6573:
6566:
6560:
6555:
6551:
6526:
6517:
6514:
6511:
6508:
6502:
6499:
6496:
6490:
6487:
6484:
6481:
6475:
6469:
6457:
6454:
6451:
6448:
6442:
6436:
6430:
6427:
6424:
6393:
6387:
6384:
6381:
6375:
6369:
6366:
6360:
6354:
6346:
6343:parametrizes
6324:
6318:
6315:
6309:
6303:
6297:
6291:
6263:
6251:
6245:
6234:
6215:
6209:
6201:
6191:
6189:
6185:
6180:
6163:
6160:
6157:
6154:
6151:
6145:
6142:
6139:
6136:
6130:
6127:
6124:
6121:
6116:
6113:
6108:
6104:
6100:
6097:
6095:
6087:
6084:
6078:
6075:
6071:
6065:
6062:
6059:
6055:
6049:
6046:
6041:
6037:
6033:
6030:
6027:
6024:
6018:
6015:
6011:
6007:
6000:
5997:
5993:
5989:
5982:
5979:
5974:
5970:
5966:
5964:
5959:
5956:
5950:
5947:
5940:
5936:
5923:
5914:
5910:
5906:
5901:
5897:
5892:
5888:
5884:
5873:
5871:
5867:
5866:singularities
5861:
5857:
5852:
5848:
5842:
5829:
5826:
5823:
5810:
5803:
5800:
5787:
5781:
5775:
5770:
5765:
5761:
5757:
5748:
5745:
5732:
5726:
5718:
5714:
5707:
5698:
5695:
5686:
5680:
5675:
5671:
5644:
5641:
5629:
5614:
5611:
5581:
5574:
5571:
5542:
5539:
5519:
5511:
5506:
5504:
5488:
5485:
5482:
5475:
5469:
5464:
5460:
5446:
5433:
5430:
5427:
5420:
5413:
5410:
5400:
5394:
5388:
5383:
5378:
5374:
5370:
5367:
5364:
5357:
5351:
5346:
5342:
5333:
5324:
5322:
5317:
5304:
5299:
5295:
5282:
5278:
5271:
5266:
5261:
5258:
5255:
5251:
5247:
5236:
5233:
5230:
5226:
5219:
5216:
5208:
5204:
5197:
5182:
5178:
5171:
5165:
5160:
5155:
5152:
5149:
5145:
5135:
5130:
5126:
5119:
5112:
5108:
5103:
5087:
5084:
5077:
5071:
5066:
5062:
5051:
5047:
5043:
5039:
5035:
5031:
5025:
5021:
5004:
5001:
4998:
4989:
4985:
4981:
4976:
4973:
4972:complex plane
4969:
4965:
4961:
4957:
4953:
4943:
4939:
4935:
4928:
4924:
4920:
4913:
4909:
4902:
4898:
4888:
4867:
4860:
4857:
4853:
4847:
4840:
4837:
4830:
4824:
4817:
4784:
4777:
4774:
4770:
4767:
4761:
4754:
4751:
4744:
4739:
4731:
4724:
4705:
4692:
4688:
4684:
4681:
4675:
4672:
4669:
4666:
4660:
4657:
4653:
4647:
4642:
4638:
4634:
4631:
4628:
4620:
4611:
4604:
4601:
4597:
4587:
4580:
4577:
4570:
4565:
4560:
4544:
4538:
4535:
4529:
4523:
4513:
4496:
4490:
4487:
4481:
4475:
4465:
4459:
4452:
4447:
4443:
4439:
4434:
4424:
4421:
4400:
4396:
4385:
4381:
4377:
4373:
4369:
4365:
4359:
4355:
4351:
4348:
4344:
4343:
4342:flux integral
4338:
4334:
4331:
4325:
4321:
4317:
4313:
4309:
4305:
4301:
4297:
4293:
4274:
4259:
4249:
4246:
4243:
4231:
4221:
4219:
4214:
4208:
4203:
4199:
4189:
4187:
4181:
4177:
4170:
4166:
4160:
4155:
4150:
4144:
4131:
4122:
4108:
4105:
4096:
4082:
4079:
4076:
4073:
4066:
4063:
4052:
4038:
4035:
4027:
4022:
4018:
4014:
4011:
4008:
4001:
3994:
3985:
3976:
3955:
3950:
3946:
3942:
3934:
3931:
3910:
3906:
3897:
3891:
3887:
3881:
3861:
3854:
3845:
3836:
3817:
3811:
3804:
3795:
3786:
3772:
3766:
3760:
3757:
3746:
3732:
3729:
3716:
3712:
3706:
3701:
3697:
3693:
3677:
3674:
3668:
3655:
3650:
3644:
3640:
3636:
3631:
3624:
3614:
3611:
3598:
3595:
3583:
3579:
3571:
3562:
3551:
3547:
3525:
3520:
3517:
3514:
3510:
3504:
3498:
3487:
3484:
3475:
3462:
3459:
3447:
3443:
3435:
3426:
3418:
3414:
3402:
3396:
3390:
3385:
3381:
3369:
3364:
3346:
3341:
3326:
3313:
3302:
3298:
3276:
3271:
3268:
3265:
3261:
3255:
3249:
3238:
3235:
3222:
3217:
3216:infinitesimal
3212:
3205:
3201:
3195:
3184:
3180:
3176:
3166:
3157:
3153:
3149:
3145:
3132:
3128:
3124:
3118:
3114:
3104:
3095:
3091:
3087:
3082:
3073:
3071:
3067:
3063:
3059:
3054:
3052:
3048:
3044:
3039:
3037:
3032:
3030:
3024:
3020:
3013:
3009:
3004:
2985:
2982:
2979:
2973:
2970:
2962:
2959:
2943:
2936:
2912:
2907:
2903:
2898:
2878:
2875:
2868:
2861:
2852:
2843:
2822:
2817:
2813:
2809:
2801:
2798:
2777:
2773:
2764:
2759:
2755:
2751:
2748:
2743:
2739:
2735:
2731:
2727:
2712:
2699:
2696:
2693:
2689:
2682:
2675:
2665:
2655:
2641:
2636:
2631:
2627:
2623:
2620:
2600:
2593:
2584:
2580:
2572:
2562:
2550:
2546:
2531:
2526:
2521:
2518:
2515:
2511:
2505:
2499:
2488:
2485:
2476:
2462:
2458:
2447:
2443:
2435:
2425:
2421:
2417:
2408:
2404:
2392:
2386:
2380:
2375:
2371:
2358:
2354:
2349:
2345:
2333:
2328:
2315:
2310:
2306:
2290:
2286:
2271:
2266:
2261:
2258:
2255:
2251:
2245:
2237:
2233:
2221:
2218:
2210:
2206:
2200:
2196:
2187:
2183:
2179:
2175:
2168:
2164:
2155:
2151:
2147:
2143:
2136:
2132:
2126:
2122:
2115:
2111:
2107:
2098:
2094:
2090:
2085:
2075:
2071:
2066:
2062:
2058:
2052:
2041:
2037:
2033:
2027:
2023:
2019:
2015:
2005:
1996:
1983:
1973:
1943:
1937:
1933:
1929:
1925:
1918:
1907:
1880:
1804:of the curve
1803:
1798:
1766:The function
1764:
1763:of a vector.
1762:
1757:
1753:
1722:
1718:
1711:
1707:
1679:of the curve
1678:
1675:
1646:
1643:
1640:
1634:
1609:
1606:
1603:
1598:
1591:
1584:
1574:
1569:
1562:
1550:
1546:
1541:
1536:
1532:
1528:
1525:
1522:
1507:
1496:
1475:
1472:
1458:
1455:
1437:
1427:
1424:
1396:
1393:
1390:
1383:
1368:
1364:
1360:
1356:
1351:
1347:
1342:
1333:
1331:
1327:
1323:
1319:
1313:
1309:
1305:
1301:
1295:
1285:
1270:
1262:along a path
1260:
1255:
1232:
1229:
1208:
1204:
1200:
1197:
1169:
1161:
1158:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1120:
1118:
1114:
1113:
1108:
1107:
1102:
1101:
1096:
1095:
1094:path integral
1091:. The terms
1090:
1086:
1082:
1078:
1077:line integral
1074:
1062:
1057:
1055:
1050:
1048:
1043:
1042:
1040:
1039:
1032:
1029:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
998:
990:
989:
982:
979:
977:
974:
972:
969:
967:
964:
963:
955:
954:
943:
940:
938:
935:
933:
930:
929:
928:
927:
917:
916:
905:
902:
900:
897:
895:
892:
890:
887:
885:
884:Line integral
882:
880:
877:
875:
872:
871:
870:
869:
865:
864:
859:
856:
854:
851:
849:
846:
844:
841:
840:
839:
838:
834:
833:
827:
826:Multivariable
821:
820:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
780:
779:
778:
774:
773:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
739:
738:
737:
731:
725:
724:
713:
710:
708:
705:
703:
700:
698:
695:
693:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
663:
662:
661:
658:
655:
654:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
623:
619:
616:
615:
614:
613:
607:
601:
600:
589:
586:
584:
581:
579:
576:
574:
571:
568:
564:
561:
559:
556:
553:
549:
545:
544:trigonometric
541:
538:
536:
533:
531:
528:
526:
523:
522:
521:
520:
516:
515:
510:
507:
505:
502:
500:
497:
495:
492:
489:
485:
482:
480:
477:
476:
475:
474:
470:
469:
464:
461:
459:
456:
454:
451:
450:
449:
448:
442:
436:
435:
424:
421:
419:
416:
414:
411:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
379:
376:
375:
374:
373:
370:
367:
366:
361:
358:
356:
355:Related rates
353:
351:
348:
346:
343:
341:
338:
336:
333:
332:
331:
330:
326:
325:
318:
315:
313:
312:of a function
310:
308:
307:infinitesimal
305:
304:
303:
300:
297:
293:
290:
289:
288:
287:
283:
282:
276:
270:
269:
263:
260:
258:
255:
253:
250:
249:
244:
241:
239:
236:
235:
231:
228:
227:
226:
225:
206:
200:
197:
191:
185:
182:
179:
176:
169:
162:
159:
153:
148:
144:
135:
134:
131:
128:
127:
123:
122:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
8319:Secant cubed
8244:
8237:
8218:Isaac Newton
8188:Brook Taylor
7948:
7855:Derivatives
7826:Shell method
7554:Differential
7326:Khan Academy
7312:
7289:
7282:
7271:. Retrieved
7267:
7258:
7246:. Retrieved
7243:Math Insight
7242:
7209:
7202:
7172:Line element
7130:
7116:
7107:area theorem
7018:
7012:
6967:irrotational
6877:
6344:
6197:
6181:
5912:
5908:
5904:
5895:
5890:
5886:
5882:
5879:
5859:
5855:
5843:
5630:
5507:
5502:
5447:
5325:
5318:
5133:
5128:
5124:
5117:
5110:
5106:
5049:
5045:
5041:
5037:
5033:
5029:
5023:
5019:
4987:
4983:
4979:
4974:
4963:
4949:
4937:
4933:
4926:
4922:
4918:
4911:
4907:
4900:
4896:
4889:
4706:
4383:
4379:
4375:
4371:
4367:
4363:
4357:
4353:
4341:
4336:
4332:
4329:
4323:
4319:
4315:
4311:
4307:
4303:
4299:
4295:
4291:
4230:vector field
4227:
4217:
4212:
4206:
4201:
4195:
4192:Applications
4185:
4179:
4175:
4168:
4164:
4158:
4153:
4148:
4145:
3895:
3889:
3885:
3879:
3714:
3710:
3704:
3690:then by the
3656:), that is,
3654:conservative
3648:
3642:
3639:scalar field
3629:
3626:
3612:
3476:
3342:
3220:
3210:
3203:
3199:
3194:displacement
3182:
3178:
3174:
3161:
3155:
3151:
3147:
3143:
3116:
3112:
3102:
3099:
3093:
3089:
3085:
3072:1-manifold.
3055:
3042:
3040:
3033:
3028:
3022:
3018:
3011:
3007:
3002:
2905:
2901:
2762:
2757:
2753:
2741:
2737:
2733:
2729:
2726:vector field
2723:
2477:
2329:
2198:
2194:
2185:
2181:
2177:
2173:
2166:
2162:
2153:
2149:
2145:
2141:
2134:
2130:
2120:
2113:
2109:
2105:
2096:
2092:
2088:
2073:
2069:
2064:
2060:
2056:
2039:
2035:
2031:
2025:
2021:
2017:
2013:
1994:
1979:
1935:
1931:
1927:
1923:
1916:
1908:
1878:
1796:
1765:
1755:
1751:
1720:
1716:
1709:
1705:
1382:scalar field
1379:
1366:
1362:
1358:
1354:
1349:
1345:
1329:
1325:
1321:
1317:
1316:and a curve
1311:
1307:
1303:
1299:
1294:tensor field
1291:
1258:
1140:differential
1128:vector field
1124:scalar field
1121:
1110:
1105:
1104:
1099:
1098:
1093:
1092:
1076:
1070:
883:
540:Substitution
302:Differential
275:Differential
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
8387:of surfaces
8138:and numbers
8100:Dirichlet's
8070:Telescoping
8023:Alternating
7611:L'Hôpital's
7408:Precalculus
7137:in quantum
6973:-free) and
5900:unit circle
5321:Riemann sum
4968:open subset
3700:composition
3123:Riemann sum
3051:orientation
2897:dot product
1982:Riemann sum
1073:mathematics
1001:Precalculus
994:Miscellanea
959:Specialized
866:Definitions
633:Alternating
471:Definitions
284:Definitions
8439:Categories
8183:Adequality
7869:Divergence
7742:Arc length
7539:Derivative
7358:PlanetMath
7273:2020-09-18
7194:References
7139:scattering
6979:divergence
3696:derivative
3159:. Letting
3076:Derivation
3036:tangential
3005:such that
2720:Definition
1976:Derivation
1802:arc length
1703:such that
1376:Definition
1132:arc length
1083:where the
981:Variations
976:Stochastic
966:Fractional
835:Formalisms
798:Divergence
767:Identities
747:Divergence
292:Derivative
243:Continuity
69:newspapers
8382:of curves
8377:Curvature
8264:Integrals
8058:Maclaurin
8038:Geometric
7929:Geometric
7879:Laplacian
7591:linearity
7431:Factorial
7328:modules:
7319:EMS Press
7081:¯
7067:∫
6951:¯
6855:⊥
6842:⋅
6817:∫
6799:⋅
6774:∫
6751:−
6736:⋅
6727:−
6709:∫
6678:⋅
6669:−
6651:∫
6588:∫
6552:∫
6500:−
6465:¯
6259:¯
6235:function
6233:conjugate
6158:π
6143:−
6140:π
6117:π
6105:∫
6060:−
6050:π
6038:∫
5983:π
5971:∫
5937:∮
5818:¯
5801:γ
5782:γ
5762:∫
5753:¯
5740:¯
5715:∫
5703:¯
5672:∫
5649:¯
5612:γ
5589:¯
5572:γ
5560:would be
5543:∈
5520:γ
5461:∮
5411:γ
5395:γ
5375:∫
5343:∫
5296:γ
5292:Δ
5279:γ
5252:∑
5234:−
5220:γ
5217:−
5198:γ
5172:γ
5146:∑
5063:∫
5002:⊂
4771:−
4740:⊥
4658:−
4639:∫
4598:−
4566:⋅
4444:∫
4435:⊥
4422:⋅
4397:∫
4265:→
4250:⊆
4244::
4106:−
4019:∫
3986:⋅
3947:∫
3932:⋅
3907:∫
3846:⋅
3796:⋅
3770:∇
3672:∇
3646:(i.e. if
3593:Δ
3563:⋅
3511:∑
3502:→
3496:Δ
3457:Δ
3427:≈
3403:−
3394:Δ
3355:Δ
3317:Δ
3314:⋅
3262:∑
3253:→
3247:Δ
3196:vectors,
3131:parameter
2974:∈
2968:∀
2960:≠
2853:⋅
2814:∫
2799:⋅
2774:∫
2628:∫
2598:Δ
2512:∑
2503:→
2497:Δ
2456:Δ
2422:≈
2393:−
2384:Δ
2342:Δ
2303:Δ
2252:∑
2243:→
2230:Δ
1674:bijective
1653:→
1635::
1533:∫
1497:∫
1473:⊂
1428:⊆
1400:→
1394::
1380:For some
1230:⋅
1205:∫
1170:⋅
1144:intervals
971:Malliavin
858:Geometric
757:Laplacian
707:Dirichlet
618:Geometric
198:−
145:∫
99:June 2023
8372:Manifold
8105:Integral
8048:Infinite
8043:Harmonic
8028:Binomial
7874:Gradient
7817:Volumes
7628:Quotient
7569:Notation
7400:Calculus
7145:See also
7141:theory.
6186:and the
5804:′
5615:′
5575:′
5532:at some
5414:′
5102:interval
5027:, where
4982: :
4960:addition
4861:′
4841:′
4818:′
4778:′
4755:′
4725:′
4605:′
4581:′
3995:′
3855:′
3805:′
3635:gradient
3572:′
3436:′
3127:interval
3070:immersed
3066:covector
2937:′
2862:′
2676:′
2573:′
2436:′
2004:interval
1585:′
1085:function
1081:integral
1011:Glossary
921:Advanced
899:Jacobian
853:Exterior
783:Gradient
775:Theorems
742:Gradient
681:Integral
643:Binomial
628:Harmonic
488:improper
484:Integral
441:Integral
423:Reynolds
398:Quotient
327:Concepts
163:′
130:Calculus
8309:inverse
8297:inverse
8223:Fluxion
8033:Fourier
7899:Stokes'
7894:Green's
7616:Product
7476:Tangent
7321:, 2001
6200:vectors
5876:Example
4970:of the
3898:, that
3698:of the
3633:is the
3343:By the
3168:be the
3136:) into
2895:is the
2330:By the
2207:of the
2068:): 1 ≤
2029:, then
1320:in the
1006:History
904:Hessian
793:Stokes'
788:Green's
620: (
542: (
486: (
408:Inverse
383:Product
294: (
83:scholar
8392:Tensor
8314:Secant
8080:Abel's
8063:Taylor
7954:Matrix
7904:Gauss'
7486:Limits
7466:Secant
7456:Radian
7217:
6539:then:
6347:, and
5889:) = 1/
4966:is an
4679:
4664:
4626:
4388:, as:
4328:, the
4228:For a
2913:(i.e:
2899:, and
2891:where
2724:For a
1622:where
1417:where
1103:, and
1079:is an
848:Tensor
843:Matrix
730:Vector
648:Taylor
606:Series
238:Limits
85:
78:
71:
64:
56:
8256:Lists
8115:Ratio
8053:Power
7789:Euler
7606:Chain
7596:Power
7471:Slope
5916:with
5448:When
5116:<
5104:into
5022:: →
4707:Here
4370:) = (
4356:: →
4302:) = (
4152:over
3637:of a
2904:: →
2750:curve
2205:limit
2082:as a
2006:into
1754:<
1457:curve
1126:or a
1089:curve
671:Ratio
638:Power
552:Euler
530:Discs
525:Parts
393:Power
388:Chain
317:total
90:JSTOR
76:books
8125:Term
8120:Root
7859:Curl
7250:2020
7215:ISBN
7117:The
6971:curl
5911:) =
5044:) +
5036:) =
4958:and
4198:work
4173:and
3708:and
3694:the
3016:and
2191:and
2159:and
2125:here
2123:see
2103:and
1919:= 2)
1749:and
1714:and
1254:work
1148:work
1075:, a
752:Curl
712:Abel
676:Root
62:news
7601:Sum
7356:at
7121:of
7019:By
6965:is
6878:By
5330:is
4950:In
4905:to
4378:),
4314:),
4216:on
3883:on
3719:is
3702:of
3652:is
3492:lim
3243:lim
3224:on
3146:= (
3045:in
2493:lim
2226:lim
2209:sum
2051:set
2016:= (
1998:of
1882:of
1150:as
1071:In
378:Sum
45:by
8441::
7317:,
7311:,
7266:.
7241:.
7229:^
7131:of
7109:.
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2018:b
2014:t
2012:Δ
2008:n
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1995:r
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1986:f
1970:f
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1936:y
1932:x
1930:(
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565:(
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39:.
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