5568:
5017:
3733:
5563:{\displaystyle {\begin{aligned}{\frac {\partial P_{v}}{\partial u}}-{\frac {\partial P_{u}}{\partial v}}&={\frac {\partial (\mathbf {F} \circ {\boldsymbol {\psi }})}{\partial u}}\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}-{\frac {\partial (\mathbf {F} \circ {\boldsymbol {\psi }})}{\partial v}}\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}\\&={\frac {\partial {\boldsymbol {\psi }}}{\partial v}}\cdot (J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} ){\frac {\partial {\boldsymbol {\psi }}}{\partial u}}-{\frac {\partial {\boldsymbol {\psi }}}{\partial u}}\cdot (J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} ){\frac {\partial {\boldsymbol {\psi }}}{\partial v}}&&{\text{(chain rule)}}\\&={\frac {\partial {\boldsymbol {\psi }}}{\partial v}}\cdot \left(J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} -{(J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} )}^{\mathsf {T}}\right){\frac {\partial {\boldsymbol {\psi }}}{\partial u}}\end{aligned}}}
3239:
3728:{\displaystyle {\begin{aligned}\oint _{\partial \Sigma }{\mathbf {F} (\mathbf {x} )\cdot \,\mathrm {d} \mathbf {\Gamma } }&=\oint _{\gamma }{\mathbf {F} ({\boldsymbol {\psi }}(\mathbf {y} ))J_{\mathbf {y} }({\boldsymbol {\psi }})\mathbf {e} _{u}(\mathbf {e} _{u}\cdot \,\mathrm {d} \mathbf {y} )+\mathbf {F} ({\boldsymbol {\psi }}(\mathbf {y} ))J_{\mathbf {y} }({\boldsymbol {\psi }})\mathbf {e} _{v}(\mathbf {e} _{v}\cdot \,\mathrm {d} \mathbf {y} )}\\&=\oint _{\gamma }{\left(\left(\mathbf {F} ({\boldsymbol {\psi }}(\mathbf {y} ))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}(\mathbf {y} )\right)\mathbf {e} _{u}+\left(\mathbf {F} ({\boldsymbol {\psi }}(\mathbf {y} ))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}(\mathbf {y} )\right)\mathbf {e} _{v}\right)\cdot \,\mathrm {d} \mathbf {y} }\end{aligned}}}
5006:
4670:
1917:
1510:
5001:{\displaystyle {\begin{aligned}{\frac {\partial P_{u}}{\partial v}}&={\frac {\partial (\mathbf {F} \circ {\boldsymbol {\psi }})}{\partial v}}\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}+(\mathbf {F} \circ {\boldsymbol {\psi }})\cdot {\frac {\partial ^{2}{\boldsymbol {\psi }}}{\partial v\,\partial u}}\\{\frac {\partial P_{v}}{\partial u}}&={\frac {\partial (\mathbf {F} \circ {\boldsymbol {\psi }})}{\partial u}}\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}+(\mathbf {F} \circ {\boldsymbol {\psi }})\cdot {\frac {\partial ^{2}{\boldsymbol {\psi }}}{\partial u\,\partial v}}\end{aligned}}}
6344:
6914:
1912:{\displaystyle {\begin{aligned}&\iint _{\Sigma }\left(\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\,\mathrm {d} y\,\mathrm {d} z+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\,\mathrm {d} z\,\mathrm {d} x+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\,\mathrm {d} x\,\mathrm {d} y\right)\\&=\oint _{\partial \Sigma }{\Bigl (}F_{x}\,\mathrm {d} x+F_{y}\,\mathrm {d} y+F_{z}\,\mathrm {d} z{\Bigr )}.\end{aligned}}}
5956:
3137:
6693:
8643:
7144:
8160:
2936:
6339:{\displaystyle {\begin{aligned}\left(A-A^{\mathsf {T}}\right)\mathbf {e} _{1}&={\begin{bmatrix}0\\a_{3}\\-a_{2}\end{bmatrix}}=\mathbf {a} \times \mathbf {e} _{1}\\\left(A-A^{\mathsf {T}}\right)\mathbf {e} _{2}&={\begin{bmatrix}-a_{3}\\0\\a_{1}\end{bmatrix}}=\mathbf {a} \times \mathbf {e} _{2}\\\left(A-A^{\mathsf {T}}\right)\mathbf {e} _{3}&={\begin{bmatrix}a_{2}\\-a_{1}\\0\end{bmatrix}}=\mathbf {a} \times \mathbf {e} _{3}\end{aligned}}}
970:
3969:
6925:
4647:
7976:
6909:{\displaystyle {\begin{aligned}{\frac {\partial P_{v}}{\partial u}}-{\frac {\partial P_{u}}{\partial v}}&={\frac {\partial {\boldsymbol {\psi }}}{\partial v}}\cdot (\nabla \times \mathbf {F} )\times {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}=(\nabla \times \mathbf {F} )\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}\times {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}\end{aligned}}}
6682:
9096:
7613:
2902:, and proved using more sophisticated machinery. While powerful, these techniques require substantial background, so the proof below avoids them, and does not presuppose any knowledge beyond a familiarity with basic vector calculus and linear algebra. At the end of this section, a short alternative proof of Stokes' theorem is given, as a corollary of the generalized Stokes' theorem.
3745:
3132:{\displaystyle \oint _{\partial \Sigma }{\mathbf {F} (\mathbf {x} )\cdot \,\mathrm {d} \mathbf {\Gamma } }=\oint _{\gamma }{\mathbf {F} ({\boldsymbol {\psi }}(\mathbf {\gamma } ))\cdot \,\mathrm {d} {\boldsymbol {\psi }}(\mathbf {\gamma } )}=\oint _{\gamma }{\mathbf {F} ({\boldsymbol {\psi }}(\mathbf {y} ))\cdot J_{\mathbf {y} }({\boldsymbol {\psi }})\,\mathrm {d} \gamma }}
4429:
7302:
7139:{\displaystyle {\begin{aligned}\iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \,d\mathbf {\Sigma } &=\iint _{D}{(\nabla \times \mathbf {F} )({\boldsymbol {\psi }}(u,v))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}(u,v)\times {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}(u,v)\,\mathrm {d} u\,\mathrm {d} v}\end{aligned}}}
9424:
6505:
8744:
8155:{\displaystyle \oint _{\partial \Sigma }{\mathbf {F} \cdot \,\mathrm {d} \mathbf {\gamma } }=\oint _{\partial \Sigma }{\omega _{\mathbf {F} }}=\int _{\Sigma }{\mathrm {d} \omega _{\mathbf {F} }}=\int _{\Sigma }{\star \omega _{\nabla \times \mathbf {F} }}=\iint _{\Sigma }{\nabla \times \mathbf {F} \cdot \,\mathrm {d} \mathbf {\Sigma } }}
7390:
11768:
2601:
10741:
10627:
1505:
11226:
7151:
3964:{\displaystyle \mathbf {P} (u,v)=\left(\mathbf {F} ({\boldsymbol {\psi }}(u,v))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}(u,v)\right)\mathbf {e} _{u}+\left(\mathbf {F} ({\boldsymbol {\psi }}(u,v))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}(u,v)\right)\mathbf {e} _{v}}
7857:
5935:
9209:
2807:
4642:{\displaystyle \oint _{\partial \Sigma }{\mathbf {F} (\mathbf {x} )\cdot \,\mathrm {d} \mathbf {l} }=\oint _{\gamma }{\mathbf {P} (\mathbf {y} )\cdot \,\mathrm {d} \mathbf {l} }=\oint _{\gamma }{({P_{u}}(u,v)\mathbf {e} _{u}+{P_{v}}(u,v)\mathbf {e} _{v})\cdot \,\mathrm {d} \mathbf {l} }}
4280:
4153:
11331:
9620:
9824:
11603:
2512:
11054:
9512:
5669:
10656:
10542:
1416:
11103:
6677:{\displaystyle \left({(J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} )}-{(J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} )}^{\mathsf {T}}\right)\mathbf {x} =(\nabla \times \mathbf {F} )\times \mathbf {x} ,\quad {\text{for all}}\,\mathbf {x} \in \mathbb {R} ^{3}}
9091:{\displaystyle {\begin{aligned}\gamma _{1}:\to D;\quad &\gamma _{1}(t)=(t,0)\\\gamma _{2}:\to D;\quad &\gamma _{2}(s)=(1,s)\\\gamma _{3}:\to D;\quad &\gamma _{3}(t)=(1-t,1)\\\gamma _{4}:\to D;\quad &\gamma _{4}(s)=(0,1-s)\end{aligned}}}
7608:{\displaystyle \oint _{\gamma }{({P_{u}}(u,v)\mathbf {e} _{u}+{P_{v}}(u,v)\mathbf {e} _{v})\cdot \,\mathrm {d} \mathbf {l} }=\iint _{D}\left({\frac {\partial P_{v}}{\partial u}}-{\frac {\partial P_{u}}{\partial v}}\right)\,\mathrm {d} u\,\mathrm {d} v}
9892:
Above
Helmholtz's theorem gives an explanation as to why the work done by a conservative force in changing an object's position is path independent. First, we introduce the Lemma 2-2, which is a corollary of and a special case of Helmholtz's theorem.
2894:, so what is of concern is how to boil down the three-dimensional complicated problem (Stokes' theorem) to a two-dimensional rudimentary problem (Green's theorem). When proving this theorem, mathematicians normally deduce it as a special case of a
8576:
10359:
The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately if the M is simply connected. However, recall that simple-connection only guarantees the existence of a
11589:
7704:
5765:
7937:
9171:
4404:
11520:
2669:
4157:
4030:
7297:{\displaystyle \iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \,\mathrm {d} \mathbf {\Sigma } =\iint _{D}\left({\frac {\partial P_{v}}{\partial u}}-{\frac {\partial P_{u}}{\partial v}}\right)\,\mathrm {d} u\,\mathrm {d} v}
11231:
9523:
11783:
of conservative forces. However, both uses of homotopy appear sufficiently frequently that some sort of terminology is necessary to disambiguate, and the term "tubular homotopy" adopted here serves well enough for that
1387:
9741:
10371:. In other words, the possibility of finding a continuous homotopy, but not being able to integrate over it, is actually eliminated with the benefit of higher mathematics. We thus obtain the following theorem.
5022:
11419:
10954:
10490:
10147:
6496:
9419:{\displaystyle {\begin{aligned}\Gamma _{i}(t)&=H(\gamma _{i}(t))&&i=1,2,3,4\\\Gamma (t)&=H(\gamma (t))=(\Gamma _{1}\oplus \Gamma _{2}\oplus \Gamma _{3}\oplus \Gamma _{4})(t)\end{aligned}}}
9443:
4675:
118:
7670:
9214:
8749:
6930:
6698:
5961:
3244:
1515:
8485:
9712:
9669:
1236:
10368:
5575:
2390:
2028:
10410:
10206:
9932:
8328:
8226:
2250:
9882:
8285:
In this section, we will introduce a theorem that is derived from Stokes' theorem and characterizes vortex-free vector fields. In classical mechanics and fluid dynamics it is called
11763:{\displaystyle \mathbf {e} _{1}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\mathbf {e} _{2}={\begin{bmatrix}0\\1\\0\end{bmatrix}},\mathbf {e} _{3}={\begin{bmatrix}0\\0\\1\end{bmatrix}}.}
7881:
7383:
2596:{\displaystyle \oint _{\partial \Sigma }\mathbf {F} \,\cdot \,\mathrm {d} {\mathbf {\Gamma } }=\iint _{\Sigma }\nabla \times \mathbf {F} \,\cdot \,\mathrm {d} \mathbf {\Sigma } .}
9101:
4285:
10793:
7699:
6404:
5697:
2858:
2656:
2507:
2453:
2313:
2182:
2089:
1972:
1207:
1142:
1071:
5756:
10736:{\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{\Sigma }\mathbf {\nabla } \times \mathbf {B} \cdot \mathrm {d} \mathbf {S} .}
10622:{\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{\Sigma }\mathbf {\nabla } \times \mathbf {E} \cdot \mathrm {d} \mathbf {S} .}
10651:
10537:
7967:
2880:
2829:
2478:
1500:{\displaystyle \iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \mathrm {d} \mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {\Gamma } .}
11451:
11357:
11221:{\displaystyle \mathbf {t} _{u}={\frac {1}{h_{u}}}{\frac {\partial \varphi }{\partial u}}\,,\mathbf {t} _{v}={\frac {1}{h_{v}}}{\frac {\partial \varphi }{\partial v}}.}
11525:
8711:
10928:
10888:
10864:
8687:
2430:
2410:
2333:
2206:
2153:
2060:
1411:
1178:
2623:
10908:
10837:
4025:
3998:
11459:
10817:
2278:
2133:
2113:
7617:
We can substitute the conclusion of STEP2 into the left-hand side of Green's theorem above, and substitute the conclusion of STEP3 into the right-hand side.
7315:
completes the proof. Green's theorem asserts the following: for any region D bounded by the
Jordans closed curve γ and two scalar-valued smooth functions
7852:{\displaystyle F_{x}\mathbf {e} _{1}+F_{y}\mathbf {e} _{2}+F_{z}\mathbf {e} _{3}\mapsto F_{x}\,\mathrm {d} x+F_{y}\,\mathrm {d} y+F_{z}\,\mathrm {d} z.}
5930:{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}={\begin{bmatrix}A_{32}-A_{23}\\A_{13}-A_{31}\\A_{21}-A_{12}\end{bmatrix}}}
10434:
10091:
1945:. In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of
2802:{\displaystyle \oint _{\partial \Sigma }(\mathbf {F} \,\cdot \,\mathrm {d} {\mathbf {\Gamma } })\,\mathbf {g} =\iint _{\Sigma }\left\mathbf {g} ,}
1241:
4275:{\displaystyle {P_{v}}(u,v)=\left(\mathbf {F} ({\boldsymbol {\psi }}(u,v))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial v}}(u,v)\right)}
4148:{\displaystyle {P_{u}}(u,v)=\left(\mathbf {F} ({\boldsymbol {\psi }}(u,v))\cdot {\frac {\partial {\boldsymbol {\psi }}}{\partial u}}(u,v)\right)}
12074:
L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, American
Mathematical Society Translations, Ser. 2, Vol. 11,
11998:
Pérez-Garrido, Antonio (2024). "Recovering seldom-used theorems of vector calculus and their application to problems of electromagnetism".
8630:
of "homotopic" or "homotopy"; the latter omit condition . So from now on we refer to homotopy (homotope) in the sense of theorem 2-1 as a
11326:{\displaystyle h_{u}=\left\|{\frac {\partial \varphi }{\partial u}}\right\|,h_{v}=\left\|{\frac {\partial \varphi }{\partial v}}\right\|,}
214:
9615:{\displaystyle 0=\oint _{\Gamma }\mathbf {F} \,\mathrm {d} \Gamma =\sum _{i=1}^{4}\oint _{\Gamma _{i}}\mathbf {F} \,\mathrm {d} \Gamma }
12114:
11779:
There do exist textbooks that use the terms "homotopy" and "homotopic" in the sense of
Theorem 2-1. Indeed, this is very convenient
11375:
11056:
Note that, in some textbooks on vector analysis, these are assigned to different things. For example, in some text book's notation,
6444:
8286:
11891:
11841:
11814:
9819:{\displaystyle 0=\oint _{\Gamma _{1}}\mathbf {F} \,\mathrm {d} \Gamma +\oint _{\Gamma _{3}}\mathbf {F} \,\mathrm {d} \Gamma }
1921:
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Surfaces such as the
480:
455:
7631:
11049:{\displaystyle \mathbf {e} _{u}={\begin{bmatrix}1\\0\end{bmatrix}},\mathbf {e} _{v}={\begin{bmatrix}0\\1\end{bmatrix}}.}
11867:
1089:
of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence:
956:
519:
37:
12059:
9674:
9631:
9507:{\displaystyle \iint _{S}\nabla \times \mathbf {F} \,\mathrm {d} S=\oint _{\Gamma }\mathbf {F} \,\mathrm {d} \Gamma }
5664:{\displaystyle J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} -(J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} )^{\mathsf {T}}}
475:
193:
10500:
4651:
We have successfully reduced one side of Stokes' theorem to a 2-dimensional formula; we now turn to the other side.
1212:
4409:
460:
2342:
1980:
10380:
10176:
9902:
8298:
8196:
3150:
2214:
1925:, for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in
796:
470:
445:
127:
10517:
and the integral form of these equations. For
Faraday's law, Stokes' theorem is applied to the electric field,
9844:
1025:
12124:
12075:
578:
525:
406:
10510:
7318:
232:
204:
10514:
315:
8571:{\displaystyle \int _{c_{0}}\mathbf {F} \,\mathrm {d} c_{0}=\int _{c_{1}}\mathbf {F} \,\mathrm {d} c_{1}}
2257:
829:
437:
275:
247:
10155:
satisfying to is crucial;the question is whether such a homotopy can be taken for arbitrary loops. If
5011:
2895:
2185:
1112:
700:
664:
441:
320:
209:
199:
10763:
7675:
6380:
5674:
2834:
2632:
2483:
2435:
2283:
2158:
2065:
1948:
1183:
1118:
1047:
12104:
8274:
5709:
464:
10209:
300:
12119:
1938:
599:
159:
10634:
10520:
7950:
2863:
2812:
2461:
11584:{\displaystyle {\textbf {a}}\cdot A{\textbf {b}}={\textbf {b}}\cdot A^{\mathsf {T}}{\textbf {a}}}
11424:
10509:, Stokes' theorem provides the justification for the equivalence of the differential form of the
913:
705:
594:
11336:
2155:. It now suffices to transfer this notion of boundary along a continuous map to our surface in
10364:
homotopy satisfying ; we seek a piecewise smooth homotopy satisfying those conditions instead.
10164:
949:
878:
839:
723:
659:
583:
11987:
11964:
11947:
11921:
8696:
7932:{\displaystyle \star \omega _{\nabla \times \mathbf {F} }=\mathrm {d} \omega _{\mathbf {F} },}
11844:
10913:
10873:
10849:
8672:
8626:". However, "homotopic" or "homotopy" in above-mentioned sense are different (stronger than)
8233:
8175:
2415:
2395:
2318:
2191:
2138:
2045:
1396:
1163:
923:
589:
360:
305:
266:
172:
9166:{\displaystyle \partial D=\gamma _{1}\oplus \gamma _{2}\oplus \gamma _{3}\oplus \gamma _{4}}
4399:{\displaystyle \mathbf {P} (u,v)={P_{u}}(u,v)\mathbf {e} _{u}+{P_{v}}(u,v)\mathbf {e} _{v}.}
2608:
12082:
11515:{\displaystyle {\textbf {a}}\cdot A{\textbf {b}}={\textbf {a}}^{\mathsf {T}}A{\textbf {b}}}
10893:
10822:
8690:
5953:
is linear, so it is determined by its action on basis elements. But by direct calculation
4664:
4003:
3976:
2039:
1926:
928:
908:
834:
503:
422:
396:
310:
8:
10796:
7970:
1390:
1149:
903:
873:
863:
750:
604:
401:
257:
140:
135:
11770:
Note that, in some textbooks on vector analysis, these are assigned to different things.
12109:
12007:
11983:
11859:
11833:
11360:
10802:
7312:
4660:
2891:
2263:
2118:
2098:
1930:
1103:
1082:
868:
771:
755:
695:
685:
649:
530:
449:
355:
350:
154:
149:
12055:
11887:
11863:
11837:
11829:
11810:
8666:
7306:
2899:
1145:
942:
776:
554:
432:
385:
242:
237:
2921:, we reduce the dimension by using the natural parametrization of the surface. Let
12086:
12017:
11100:
respectively. In this article, however, these are two completely different things.
10506:
10217:
8264:
6919:
2910:
1934:
1098:
1078:
786:
680:
654:
515:
427:
391:
12079:
11870:
10931:
5671:. We claim this matrix in fact describes a cross product. Here the superscript "
1041:
998:
918:
791:
745:
740:
627:
540:
485:
6687:
1942:
1922:
801:
609:
376:
3737:
12098:
11453:
10939:
10935:
2092:
1094:
1086:
781:
545:
295:
252:
8642:
1382:{\displaystyle \mathbf {F} (x,y,z)=(F_{x}(x,y,z),F_{y}(x,y,z),F_{z}(x,y,z))}
10867:
2666:
Stokes' theorem can be viewed as a special case of the following identity:
2035:
1074:
535:
280:
10151:
Above Lemma 2-2 follows from theorem 2–1. In Lemma 2-2, the existence of
8247:
This concept is very fundamental in mechanics; as we'll prove later, if
2626:
2336:
1977:
A more detailed statement will be given for subsequent discussions. Let
1021:
898:
7944:
4654:
644:
568:
290:
285:
189:
12021:
2882:
is a uniform scalar field, the standard Stokes' theorem is recovered.
9193:
are piecewise smooth homotopic, there is a piecewise smooth homotopy
5700:
2253:
2031:
573:
563:
10631:
For Ampère's law, Stokes' theorem is applied to the magnetic field,
969:
12012:
10413:
9935:
8627:
8331:
8191:
639:
381:
338:
27:
7307:
Fourth step of the elementary proof (reduction to Green's theorem)
11858:
Atsuo
Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
1037:
6377:
represents an orthonormal basis in the coordinate directions of
2911:
First step of the elementary proof (parametrization of integral)
985:. The direction of positive circulation of the bounding contour
11414:{\displaystyle {\textbf {a}},{\textbf {b}}\in \mathbb {R} ^{n}}
9938:
8334:
7618:
10485:{\displaystyle \int _{c_{0}}\mathbf {F} \,\mathrm {d} c_{0}=0}
10142:{\displaystyle \int _{c_{0}}\mathbf {F} \,\mathrm {d} c_{0}=0}
8580:
Some textbooks such as
Lawrence call the relationship between
6491:{\displaystyle {(J_{{\boldsymbol {\psi }}(u,v)}\mathbf {F} )}}
5010:
Conveniently, the second term vanishes in the difference, by
11809:(7th ed.). Brooks/Cole Cengage Learning. p. 1122.
5572:
But now consider the matrix in that quadratic form—that is,
2933:
be as in that section, and note that by change of variables
9884:, so that the desired equality follows almost immediately.
3738:
Second step in the elementary proof (defining the pullback)
11952:. Graduate Texts in Mathematics. Vol. 218. Springer.
8174:
In this section, we will discuss the irrotational field (
2661:
7311:
Combining the second and third steps, and then applying
3236:, we can expand the previous equation in coordinates as
3197:
be an orthonormal basis in the coordinate directions of
2890:
The proof of the theorem consists of 4 steps. We assume
10416:
and simply connected with an irrotational vector field
7861:
Write the differential 1-form associated to a function
11729:
11678:
11627:
11022:
10978:
10942:
Jordan curves, so that the integrals are well-defined.
10367:
Fortunately, the gap in regularity is resolved by the
8598:
stated in theorem 2-1 as "homotopic" and the function
6264:
6140:
6016:
5839:
5782:
4659:
First, calculate the partial derivatives appearing in
11606:
11528:
11462:
11427:
11378:
11339:
11234:
11106:
10957:
10916:
10896:
10876:
10852:
10825:
10805:
10766:
10659:
10637:
10545:
10523:
10437:
10383:
10179:
10094:
9905:
9847:
9744:
9677:
9634:
9526:
9446:
9212:
9104:
8747:
8699:
8675:
8488:
8301:
8199:
7979:
7953:
7884:
7707:
7678:
7634:
7393:
7321:
7154:
6928:
6696:
6686:
We can now recognize the difference of partials as a
6508:
6447:
6383:
5959:
5768:
5712:
5677:
5578:
5020:
4673:
4432:
4288:
4160:
4033:
4006:
3979:
3748:
3242:
2939:
2866:
2837:
2815:
2672:
2635:
2611:
2515:
2486:
2464:
2438:
2418:
2398:
2345:
2321:
2286:
2266:
2217:
2194:
2161:
2141:
2121:
2101:
2068:
2048:
1983:
1951:
1513:
1419:
1399:
1244:
1215:
1186:
1166:
1121:
1050:
40:
8293:
Theorem 2-1 (Helmholtz's theorem in fluid dynamics).
7665:{\displaystyle \mathbb {R} ^{3}\to \mathbb {R} ^{3}}
4655:
Third step of the elementary proof (second equation)
7672:can be identified with the differential 1-forms on
1933:surface. One (advanced) technique is to pass to a
11926:. Modern Birkhauser Classics. Boston: Birkhaeuser.
11762:
11583:
11514:
11445:
11413:
11351:
11325:
11220:
11048:
10922:
10902:
10882:
10858:
10831:
10811:
10787:
10735:
10645:
10621:
10531:
10484:
10404:
10200:
10141:
9926:
9876:
9818:
9706:
9663:
9614:
9506:
9418:
9165:
9090:
8705:
8681:
8570:
8363:be piecewise smooth loops. If there is a function
8322:
8220:
8154:
7961:
7931:
7851:
7693:
7664:
7607:
7377:
7296:
7138:
6922:also includes a triple product—the very same one!
6908:
6676:
6490:
6398:
6338:
5929:
5750:
5691:
5663:
5562:
5000:
4641:
4398:
4274:
4147:
4019:
3992:
3963:
3742:The previous step suggests we define the function
3727:
3131:
2874:
2852:
2823:
2801:
2650:
2617:
2595:
2501:
2472:
2447:
2424:
2404:
2384:
2327:
2307:
2272:
2244:
2200:
2176:
2147:
2127:
2107:
2083:
2054:
2022:
1966:
1911:
1499:
1405:
1381:
1230:
1201:
1172:
1136:
1065:
1001:(i.e., the right hand the fingers circulate along
112:
1897:
1827:
973:An illustration of Stokes' theorem, with surface
12096:
12052:Advanced Calculus: A Differential Forms Approach
12039:(4th ed.). Boston: Pearson. pp. 500–3.
113:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
10501:Maxwell's equations § Circulation and curl
9707:{\displaystyle \Gamma _{2}=\ominus \Gamma _{4}}
9664:{\displaystyle \Gamma _{2}=\ominus \Gamma _{4}}
8637:
7623:
1231:{\displaystyle \partial \Sigma \equiv \Gamma }
1097:of a vector field over a loop is equal to the
1085:of the vector field over some surface, to the
11997:
9520:is lamellar, so the left side vanishes, i.e.
950:
11346:
11340:
2385:{\displaystyle \Gamma (t)=\psi (\gamma (t))}
2023:{\displaystyle \gamma :\to \mathbb {R} ^{2}}
10405:{\displaystyle U\subseteq \mathbb {R} ^{3}}
10230:there exists a continuous tubular homotopy
10201:{\displaystyle M\subseteq \mathbb {R} ^{n}}
9927:{\displaystyle U\subseteq \mathbb {R} ^{3}}
8713:" for reversing the orientation of a path.
8323:{\displaystyle U\subseteq \mathbb {R} ^{3}}
8221:{\displaystyle U\subseteq \mathbb {R} ^{3}}
2245:{\displaystyle \psi :D\to \mathbb {R} ^{3}}
9514:follows immediately from Stokes' theorem.
8632:tubular homotopy (resp. tubular-homotopic)
1389:is defined and has continuous first order
957:
943:
12011:
11966:Essential Calculus: Early Transcendentals
11881:
11854:
11852:
11401:
11159:
10460:
10392:
10188:
10117:
9914:
9877:{\displaystyle c_{3}=\ominus \Gamma _{3}}
9807:
9773:
9603:
9548:
9495:
9468:
8552:
8511:
8310:
8208:
8140:
8002:
7973:. Thus, by generalized Stokes' theorem,
7837:
7815:
7793:
7681:
7652:
7637:
7596:
7587:
7495:
7285:
7276:
7185:
7122:
7113:
6963:
6664:
6653:
6386:
4984:
4824:
4627:
4511:
4466:
3709:
3487:
3390:
3280:
3222:are precisely the partial derivatives of
3119:
3029:
2973:
2840:
2714:
2698:
2694:
2638:
2579:
2575:
2538:
2534:
2489:
2232:
2164:
2095:one and another that is non-compact. Let
2071:
2010:
1954:
1886:
1864:
1842:
1788:
1779:
1700:
1691:
1612:
1603:
1189:
1124:
1111:Stokes' theorem is a special case of the
1053:
73:
10220:if and only if for any continuous loop,
10171:Definition 2-2 (simply connected space).
8641:
2831:is any smooth vector or scalar field in
1507:More explicitly, the equality says that
968:
12049:
11978:
11976:
11962:
11804:
11366:
10687:
10573:
7083:
7043:
7011:
6918:On the other hand, the definition of a
6887:
6862:
6817:
6772:
6568:
6524:
6458:
5623:
5585:
5541:
5491:
5451:
5422:
5382:
5346:
5319:
5294:
5258:
5231:
5199:
5171:
5135:
5107:
4972:
4948:
4918:
4890:
4812:
4788:
4758:
4730:
4237:
4205:
4110:
4078:
3914:
3882:
3818:
3786:
3658:
3630:
3570:
3542:
3450:
3416:
3353:
3319:
3112:
3075:
3036:
3008:
2184:. But we already have such a map: the
481:Differentiating under the integral sign
12097:
12034:
11919:
11915:
11913:
11911:
11909:
11907:
11905:
11903:
11849:
11823:
11568:
11496:
10494:
9887:
8280:
8169:
6600:
6229:
6105:
5981:
5683:
5655:
5523:
2662:Special case of a more general theorem
12078:, Providence, R.I., 1959, pp. 1–114.
11941:
11939:
11937:
11935:
11933:
10841:
7378:{\displaystyle P_{u}(u,v),P_{v}(u,v)}
993:of positive flux through the surface
11973:
11594:
10945:
8689:" for concatenation of paths in the
8182:Definition 2-1 (irrotational field).
12068:
11982:Robert Scheichl, lecture notes for
11956:
11945:
11900:
11773:
11576:
11554:
11544:
11531:
11507:
11489:
11478:
11465:
11391:
11381:
10755:
3973:Now, if the scalar value functions
2905:
1115:. In particular, a vector field on
13:
11930:
11307:
11299:
11263:
11255:
11206:
11198:
11150:
11142:
10917:
10853:
10767:
10721:
10705:
10699:
10682:
10668:
10665:
10607:
10591:
10585:
10568:
10554:
10551:
10462:
10119:
9865:
9813:
9809:
9791:
9779:
9775:
9757:
9695:
9679:
9652:
9636:
9609:
9605:
9587:
9554:
9550:
9538:
9501:
9497:
9485:
9470:
9457:
9391:
9378:
9365:
9352:
9305:
9218:
9105:
8554:
8513:
8142:
8126:
8120:
8098:
8084:
8059:
8052:
8025:
8022:
8004:
7988:
7985:
7955:
7910:
7893:
7839:
7817:
7795:
7598:
7589:
7573:
7558:
7543:
7528:
7497:
7287:
7278:
7262:
7247:
7232:
7217:
7187:
7168:
7160:
7124:
7115:
7089:
7079:
7049:
7039:
6993:
6946:
6938:
6893:
6883:
6868:
6858:
6838:
6823:
6813:
6793:
6778:
6768:
6749:
6734:
6719:
6704:
6622:
5547:
5537:
5428:
5418:
5388:
5378:
5325:
5315:
5300:
5290:
5237:
5227:
5205:
5195:
5180:
5156:
5141:
5131:
5116:
5092:
5073:
5058:
5043:
5028:
4985:
4978:
4962:
4924:
4914:
4899:
4875:
4856:
4841:
4825:
4818:
4802:
4764:
4754:
4739:
4715:
4696:
4681:
4629:
4513:
4468:
4441:
4438:
4426:, and, by the above, it satisfies
4243:
4233:
4116:
4106:
3920:
3910:
3824:
3814:
3711:
3664:
3654:
3576:
3566:
3489:
3392:
3282:
3255:
3252:
3121:
3031:
2975:
2948:
2945:
2778:
2756:
2739:
2728:
2700:
2681:
2678:
2581:
2564:
2559:
2540:
2524:
2521:
2442:
2439:
2419:
2399:
2346:
2322:
2287:
2195:
1888:
1866:
1844:
1820:
1817:
1790:
1781:
1765:
1750:
1735:
1720:
1702:
1693:
1677:
1662:
1647:
1632:
1614:
1605:
1589:
1574:
1559:
1544:
1526:
1485:
1471:
1468:
1451:
1433:
1425:
1400:
1225:
1219:
1216:
1167:
22:Part of a series of articles about
14:
12136:
12115:Vectors (mathematics and physics)
10422:. For all piecewise smooth loops
9714:. Thus the line integrals along
11949:Introduction to Smooth Manifolds
11832:, et al.:"Bi-Bun-Seki-Bun-Gaku"
11807:Calculus – Early Transcendentals
11711:
11660:
11609:
11165:
11109:
11004:
10960:
10788:{\displaystyle \Sigma =\psi (D)}
10726:
10713:
10674:
10639:
10612:
10599:
10560:
10525:
10456:
10113:
9803:
9769:
9599:
9544:
9491:
9464:
8548:
8507:
8147:
8133:
8105:
8069:
8037:
7995:
7920:
7900:
7770:
7745:
7720:
7694:{\displaystyle \mathbb {R} ^{3}}
7502:
7479:
7437:
7192:
7175:
7000:
6968:
6953:
6845:
6800:
6655:
6640:
6629:
6612:
6590:
6546:
6480:
6399:{\displaystyle \mathbb {R} ^{3}}
6322:
6313:
6242:
6198:
6189:
6118:
6074:
6065:
5994:
5770:
5692:{\displaystyle {}^{\mathsf {T}}}
5645:
5607:
5513:
5473:
5368:
5280:
5163:
5099:
4940:
4882:
4780:
4722:
4634:
4611:
4569:
4518:
4501:
4493:
4473:
4456:
4448:
4383:
4341:
4290:
4197:
4070:
3951:
3874:
3855:
3778:
3750:
3716:
3691:
3677:
3638:
3622:
3603:
3589:
3550:
3534:
3494:
3474:
3459:
3440:
3424:
3408:
3397:
3377:
3362:
3343:
3327:
3311:
3287:
3270:
3262:
3206:Recognizing that the columns of
3102:
3083:
3067:
3000:
2980:
2963:
2955:
2868:
2853:{\displaystyle \mathbb {R} ^{3}}
2817:
2792:
2771:
2763:
2744:
2716:
2706:
2690:
2651:{\displaystyle \mathbb {R} ^{3}}
2586:
2571:
2546:
2530:
2502:{\displaystyle \mathbb {R} ^{3}}
2466:
2448:{\displaystyle \partial \Sigma }
2308:{\displaystyle \Sigma =\psi (D)}
2177:{\displaystyle \mathbb {R} ^{3}}
2084:{\displaystyle \mathbb {R} ^{2}}
1967:{\displaystyle \mathbb {R} ^{2}}
1937:and then apply the machinery of
1490:
1477:
1456:
1440:
1246:
1202:{\displaystyle \mathbb {R} ^{3}}
1180:be a smooth oriented surface in
1137:{\displaystyle \mathbb {R} ^{3}}
1066:{\displaystyle \mathbb {R} ^{3}}
1005:and the thumb is directed along
12043:
12028:
11991:
11884:Introduction to Electrodynamics
10369:Whitney's approximation theorem
9941:, with a Lamellar vector field
9038:
8951:
8870:
8789:
8164:
7878:. Then one can calculate that
6647:
5751:{\displaystyle A=(A_{ij})_{ij}}
11875:
11798:
11316:
11293:
11272:
11249:
10782:
10776:
9409:
9403:
9400:
9348:
9342:
9339:
9333:
9327:
9314:
9308:
9268:
9265:
9259:
9246:
9233:
9227:
9081:
9063:
9057:
9051:
9029:
9026:
9014:
8994:
8976:
8970:
8964:
8942:
8939:
8927:
8907:
8895:
8889:
8883:
8861:
8858:
8846:
8826:
8814:
8808:
8802:
8780:
8777:
8765:
7780:
7647:
7489:
7474:
7462:
7432:
7420:
7405:
7372:
7360:
7344:
7332:
7179:
7165:
7110:
7098:
7070:
7058:
7030:
7027:
7015:
7007:
7004:
6990:
6957:
6943:
6849:
6835:
6804:
6790:
6633:
6619:
6594:
6584:
6572:
6559:
6550:
6540:
6528:
6515:
6484:
6474:
6462:
6449:
5736:
5719:
5650:
5639:
5627:
5614:
5601:
5589:
5517:
5507:
5495:
5482:
5467:
5455:
5372:
5362:
5350:
5337:
5284:
5274:
5262:
5249:
5175:
5159:
5111:
5095:
4952:
4936:
4894:
4878:
4792:
4776:
4734:
4718:
4621:
4606:
4594:
4564:
4552:
4537:
4505:
4497:
4460:
4452:
4378:
4366:
4336:
4324:
4306:
4294:
4264:
4252:
4224:
4221:
4209:
4201:
4185:
4173:
4137:
4125:
4097:
4094:
4082:
4074:
4058:
4046:
3941:
3929:
3901:
3898:
3886:
3878:
3845:
3833:
3805:
3802:
3790:
3782:
3766:
3754:
3681:
3673:
3645:
3642:
3634:
3626:
3593:
3585:
3557:
3554:
3546:
3538:
3498:
3469:
3454:
3446:
3431:
3428:
3420:
3412:
3401:
3372:
3357:
3349:
3334:
3331:
3323:
3315:
3274:
3266:
3116:
3108:
3090:
3087:
3079:
3071:
3048:
3040:
3023:
3020:
3012:
3004:
2967:
2959:
2898:, which is stated in terms of
2711:
2686:
2480:is any smooth vector field on
2379:
2376:
2370:
2364:
2355:
2349:
2302:
2296:
2227:
2115:denote the compact part; then
2005:
2002:
1990:
1444:
1430:
1376:
1373:
1355:
1339:
1321:
1305:
1287:
1274:
1268:
1250:
1148:in which case its curl is its
107:
101:
92:
86:
70:
64:
1:
12076:American Mathematical Society
11791:
8337:with a lamellar vector field
1030:fundamental theorem for curls
407:Integral of inverse functions
10646:{\displaystyle \mathbf {B} }
10532:{\displaystyle \mathbf {E} }
9947:and a piecewise smooth loop
8638:Proof of Helmholtz's theorem
8178:) based on Stokes' theorem.
7962:{\displaystyle \mathrm {d} }
7624:Proof via differential forms
2917:
2875:{\displaystyle \mathbf {g} }
2824:{\displaystyle \mathbf {g} }
2473:{\displaystyle \mathbf {F} }
2458:With the above notation, if
1941:; for that approach see the
1929:cannot be defined for a non-
7:
12088:). See theorems 7 & 8.
12050:Edwards, Harold M. (1994).
12035:Colley, Susan Jane (2002).
12000:American Journal of Physics
11446:{\displaystyle A;n\times n}
830:Calculus on Euclidean space
248:Logarithmic differentiation
10:
12141:
11352:{\displaystyle \|\cdot \|}
10498:
10163:exists. The definition of
10159:is simply connected, such
5012:equality of mixed partials
1155:
1113:generalized Stokes theorem
1107:over the enclosed surface.
1077:, the theorem relates the
16:Theorem in vector calculus
11920:Conlon, Lawrence (2008).
11882:Griffiths, David (2013).
9970:, if there is a homotopy
8275:conservative vector field
5701:transposition of matrices
564:Summand limit (term test)
11923:Differentiable Manifolds
11781:for the specific problem
10748:
10511:Maxwell–Faraday equation
9628:is tubular(satisfying ),
8730:into four line segments
8706:{\displaystyle \ominus }
4027:are defined as follows,
2885:
1939:geometric measure theory
243:Implicit differentiation
233:Differentiation notation
160:Inverse function theorem
11963:Stewart, James (2010).
11886:. Pearson. p. 34.
11805:Stewart, James (2012).
11078:can mean the following
10923:{\displaystyle \Gamma }
10883:{\displaystyle \gamma }
10859:{\displaystyle \Gamma }
10515:Maxwell–Ampère equation
9175:By our assumption that
8682:{\displaystyle \oplus }
6688:(scalar) triple product
2425:{\displaystyle \Sigma }
2405:{\displaystyle \Gamma }
2328:{\displaystyle \Gamma }
2201:{\displaystyle \Sigma }
2148:{\displaystyle \gamma }
2091:into two components, a
2055:{\displaystyle \gamma }
1406:{\displaystyle \Sigma }
1393:in a region containing
1173:{\displaystyle \Sigma }
1144:can be considered as a
706:Helmholtz decomposition
11764:
11585:
11516:
11447:
11415:
11353:
11327:
11222:
11050:
10934:, and topologically a
10924:
10904:
10890:interacts poorly with
10884:
10860:
10833:
10813:
10789:
10737:
10647:
10623:
10533:
10486:
10406:
10202:
10165:simply connected space
10143:
9928:
9878:
9820:
9708:
9665:
9616:
9580:
9508:
9420:
9167:
9092:
8707:
8683:
8662:
8572:
8324:
8222:
8184:A smooth vector field
8156:
7963:
7933:
7853:
7695:
7666:
7609:
7379:
7298:
7140:
6910:
6678:
6492:
6400:
6340:
5931:
5752:
5693:
5665:
5564:
5002:
4643:
4400:
4276:
4149:
4021:
3994:
3965:
3729:
3133:
2876:
2854:
2825:
2803:
2652:
2619:
2618:{\displaystyle \cdot }
2597:
2503:
2474:
2449:
2426:
2406:
2386:
2329:
2309:
2274:
2246:
2202:
2178:
2149:
2129:
2109:
2085:
2056:
2024:
1968:
1913:
1501:
1407:
1383:
1232:
1203:
1174:
1138:
1067:
1010:
981:and the normal vector
840:Limit of distributions
660:Directional derivative
316:Faà di Bruno's formula
114:
11946:Lee, John M. (2002).
11872:(Written in Japanese)
11846:(Written in Japanese)
11765:
11586:
11517:
11448:
11416:
11354:
11328:
11223:
11051:
10925:
10905:
10903:{\displaystyle \psi }
10885:
10861:
10834:
10832:{\displaystyle \psi }
10814:
10790:
10738:
10648:
10624:
10534:
10487:
10407:
10203:
10144:
9929:
9879:
9821:
9709:
9666:
9617:
9560:
9509:
9421:
9168:
9093:
8708:
8684:
8645:
8608:as "homotopy between
8573:
8325:
8234:lamellar vector field
8223:
8176:lamellar vector field
8157:
7964:
7934:
7854:
7696:
7667:
7610:
7380:
7299:
7141:
6911:
6679:
6493:
6401:
6341:
5932:
5753:
5694:
5666:
5565:
5003:
4644:
4401:
4277:
4150:
4022:
4020:{\displaystyle P_{v}}
3995:
3993:{\displaystyle P_{u}}
3966:
3730:
3134:
2877:
2855:
2826:
2804:
2653:
2620:
2598:
2504:
2475:
2450:
2427:
2407:
2387:
2330:
2310:
2275:
2247:
2203:
2179:
2150:
2130:
2110:
2086:
2057:
2025:
1969:
1914:
1502:
1408:
1384:
1233:
1204:
1175:
1139:
1068:
1018:Kelvin–Stokes theorem
972:
924:Mathematical analysis
835:Generalized functions
520:arithmetico-geometric
361:Leibniz integral rule
115:
12125:Theorems in calculus
11986:mathematics course
11604:
11526:
11460:
11425:
11376:
11337:
11232:
11104:
10955:
10914:
10894:
10874:
10850:
10823:
10803:
10764:
10657:
10635:
10543:
10521:
10435:
10381:
10177:
10092:
9903:
9845:
9742:
9675:
9632:
9524:
9444:
9210:
9102:
8745:
8697:
8691:fundamental groupoid
8673:
8665:In what follows, we
8486:
8382:is piecewise smooth,
8299:
8197:
7977:
7951:
7882:
7705:
7676:
7632:
7391:
7319:
7152:
6926:
6694:
6506:
6445:
6381:
5957:
5766:
5710:
5675:
5576:
5018:
4671:
4430:
4286:
4158:
4031:
4004:
3977:
3746:
3240:
2937:
2864:
2835:
2813:
2670:
2633:
2609:
2513:
2484:
2462:
2436:
2416:
2396:
2343:
2319:
2284:
2264:
2215:
2192:
2159:
2139:
2119:
2099:
2066:
2046:
2040:Jordan curve theorem
1981:
1949:
1511:
1417:
1397:
1242:
1238:. If a vector field
1213:
1184:
1164:
1119:
1048:
1016:, also known as the
989:, and the direction
929:Nonstandard analysis
397:Lebesgue integration
267:Rules and identities
38:
10495:Maxwell's equations
9888:Conservative forces
9828:On the other hand,
8646:The definitions of
8628:typical definitions
8287:Helmholtz's theorem
8281:Helmholtz's theorem
8257:and the domain of
8170:Irrotational fields
7971:exterior derivative
5706:To be precise, let
2896:more general result
1391:partial derivatives
1150:exterior derivative
997:, are related by a
600:Cauchy condensation
402:Contour integration
128:Fundamental theorem
55:
11984:University of Bath
11760:
11751:
11700:
11649:
11581:
11512:
11443:
11411:
11349:
11323:
11218:
11046:
11037:
10993:
10920:
10900:
10880:
10856:
10829:
10809:
10785:
10733:
10643:
10619:
10529:
10505:In the physics of
10482:
10402:
10198:
10139:
9924:
9874:
9816:
9704:
9661:
9612:
9504:
9416:
9414:
9163:
9088:
9086:
8703:
8679:
8663:
8568:
8320:
8218:
8152:
7959:
7929:
7849:
7691:
7662:
7605:
7375:
7294:
7136:
7134:
6906:
6904:
6674:
6488:
6396:
6336:
6334:
6303:
6179:
6055:
5927:
5921:
5825:
5748:
5689:
5661:
5560:
5558:
4998:
4996:
4639:
4396:
4272:
4145:
4017:
3990:
3961:
3725:
3723:
3129:
2900:differential forms
2872:
2850:
2821:
2799:
2648:
2615:
2593:
2499:
2470:
2445:
2422:
2402:
2382:
2325:
2305:
2270:
2242:
2198:
2174:
2145:
2125:
2105:
2081:
2052:
2036:Jordan plane curve
2020:
1964:
1909:
1907:
1497:
1403:
1379:
1228:
1199:
1170:
1134:
1063:
1011:
772:Partial derivative
701:generalized Stokes
595:Alternating series
476:Reduction formulae
465:Heaviside's method
446:tangent half-angle
433:Cylindrical shells
356:Integral transform
351:Lists of integrals
155:Mean value theorem
110:
41:
12022:10.1119/5.0182191
11893:978-0-321-85656-2
11842:978-4-7853-1039-4
11830:Nagayoshi Iwahori
11816:978-0-538-49790-9
11600:In this article,
11578:
11556:
11546:
11533:
11509:
11491:
11480:
11467:
11393:
11383:
11314:
11270:
11213:
11193:
11157:
11137:
10951:In this article,
10812:{\displaystyle D}
10244:to a fixed point
10208:be non-empty and
7580:
7550:
7269:
7239:
7096:
7056:
6900:
6875:
6830:
6785:
6756:
6726:
6651:
5699:" represents the
5554:
5435:
5403:
5395:
5332:
5307:
5244:
5212:
5187:
5148:
5123:
5080:
5050:
4992:
4931:
4906:
4863:
4832:
4771:
4746:
4703:
4250:
4123:
3927:
3831:
3671:
3583:
2625:" represents the
2273:{\displaystyle D}
2128:{\displaystyle D}
2108:{\displaystyle D}
1772:
1742:
1684:
1654:
1596:
1566:
967:
966:
847:
846:
809:
808:
777:Multiple integral
713:
712:
617:
616:
584:Direct comparison
555:Convergence tests
493:
492:
461:Partial fractions
328:
327:
238:Second derivative
12132:
12105:Electromagnetism
12089:
12072:
12066:
12065:
12047:
12041:
12040:
12032:
12026:
12025:
12015:
11995:
11989:
11980:
11971:
11970:
11960:
11954:
11953:
11943:
11928:
11927:
11917:
11898:
11897:
11879:
11873:
11856:
11847:
11827:
11821:
11820:
11802:
11785:
11777:
11771:
11769:
11767:
11766:
11761:
11756:
11755:
11720:
11719:
11714:
11705:
11704:
11669:
11668:
11663:
11654:
11653:
11618:
11617:
11612:
11598:
11592:
11590:
11588:
11587:
11582:
11580:
11579:
11573:
11572:
11571:
11558:
11557:
11548:
11547:
11535:
11534:
11521:
11519:
11518:
11513:
11511:
11510:
11501:
11500:
11499:
11493:
11492:
11482:
11481:
11469:
11468:
11452:
11450:
11449:
11444:
11420:
11418:
11417:
11412:
11410:
11409:
11404:
11395:
11394:
11385:
11384:
11370:
11364:
11358:
11356:
11355:
11350:
11332:
11330:
11329:
11324:
11319:
11315:
11313:
11305:
11297:
11288:
11287:
11275:
11271:
11269:
11261:
11253:
11244:
11243:
11227:
11225:
11224:
11219:
11214:
11212:
11204:
11196:
11194:
11192:
11191:
11179:
11174:
11173:
11168:
11158:
11156:
11148:
11140:
11138:
11136:
11135:
11123:
11118:
11117:
11112:
11099:
11077:
11055:
11053:
11052:
11047:
11042:
11041:
11013:
11012:
11007:
10998:
10997:
10969:
10968:
10963:
10949:
10943:
10929:
10927:
10926:
10921:
10910:. Nonetheless,
10909:
10907:
10906:
10901:
10889:
10887:
10886:
10881:
10865:
10863:
10862:
10857:
10845:
10839:
10838:
10836:
10835:
10830:
10818:
10816:
10815:
10810:
10794:
10792:
10791:
10786:
10759:
10742:
10740:
10739:
10734:
10729:
10724:
10716:
10708:
10703:
10702:
10690:
10685:
10677:
10672:
10671:
10652:
10650:
10649:
10644:
10642:
10628:
10626:
10625:
10620:
10615:
10610:
10602:
10594:
10589:
10588:
10576:
10571:
10563:
10558:
10557:
10538:
10536:
10535:
10530:
10528:
10507:electromagnetism
10491:
10489:
10488:
10483:
10475:
10474:
10465:
10459:
10454:
10453:
10452:
10451:
10431:
10421:
10411:
10409:
10408:
10403:
10401:
10400:
10395:
10354:
10347:
10321:
10314:
10296:
10289:
10262:
10253:
10243:
10239:
10229:
10218:simply connected
10215:
10207:
10205:
10204:
10199:
10197:
10196:
10191:
10162:
10158:
10154:
10148:
10146:
10145:
10140:
10132:
10131:
10122:
10116:
10111:
10110:
10109:
10108:
10083:
10076:
10050:
10043:
10025:
10018:
9991:piecewise smooth
9988:
9979:
9969:
9959:
9946:
9933:
9931:
9930:
9925:
9923:
9922:
9917:
9883:
9881:
9880:
9875:
9873:
9872:
9857:
9856:
9840:
9825:
9823:
9822:
9817:
9812:
9806:
9801:
9800:
9799:
9798:
9778:
9772:
9767:
9766:
9765:
9764:
9738:cancel, leaving
9737:
9725:
9713:
9711:
9710:
9705:
9703:
9702:
9687:
9686:
9670:
9668:
9667:
9662:
9660:
9659:
9644:
9643:
9627:
9621:
9619:
9618:
9613:
9608:
9602:
9597:
9596:
9595:
9594:
9579:
9574:
9553:
9547:
9542:
9541:
9519:
9513:
9511:
9510:
9505:
9500:
9494:
9489:
9488:
9473:
9467:
9456:
9455:
9439:
9435:
9432:be the image of
9431:
9425:
9423:
9422:
9417:
9415:
9399:
9398:
9386:
9385:
9373:
9372:
9360:
9359:
9272:
9258:
9257:
9226:
9225:
9206:
9192:
9183:
9172:
9170:
9169:
9164:
9162:
9161:
9149:
9148:
9136:
9135:
9123:
9122:
9097:
9095:
9094:
9089:
9087:
9050:
9049:
9010:
9009:
8963:
8962:
8923:
8922:
8882:
8881:
8842:
8841:
8801:
8800:
8761:
8760:
8740:
8729:
8722:
8712:
8710:
8709:
8704:
8688:
8686:
8685:
8680:
8661:
8625:
8616:
8607:
8597:
8588:
8577:
8575:
8574:
8569:
8567:
8566:
8557:
8551:
8546:
8545:
8544:
8543:
8526:
8525:
8516:
8510:
8505:
8504:
8503:
8502:
8477:
8470:
8447:
8440:
8414:
8407:
8381:
8372:
8362:
8342:
8329:
8327:
8326:
8321:
8319:
8318:
8313:
8272:
8265:simply connected
8262:
8252:
8243:
8227:
8225:
8224:
8219:
8217:
8216:
8211:
8189:
8161:
8159:
8158:
8153:
8151:
8150:
8145:
8136:
8124:
8123:
8111:
8110:
8109:
8108:
8088:
8087:
8075:
8074:
8073:
8072:
8062:
8056:
8055:
8043:
8042:
8041:
8040:
8029:
8028:
8013:
8012:
8007:
7998:
7992:
7991:
7968:
7966:
7965:
7960:
7958:
7942:
7938:
7936:
7935:
7930:
7925:
7924:
7923:
7913:
7905:
7904:
7903:
7877:
7866:
7858:
7856:
7855:
7850:
7842:
7836:
7835:
7820:
7814:
7813:
7798:
7792:
7791:
7779:
7778:
7773:
7767:
7766:
7754:
7753:
7748:
7742:
7741:
7729:
7728:
7723:
7717:
7716:
7700:
7698:
7697:
7692:
7690:
7689:
7684:
7671:
7669:
7668:
7663:
7661:
7660:
7655:
7646:
7645:
7640:
7614:
7612:
7611:
7606:
7601:
7592:
7586:
7582:
7581:
7579:
7571:
7570:
7569:
7556:
7551:
7549:
7541:
7540:
7539:
7526:
7519:
7518:
7506:
7505:
7500:
7488:
7487:
7482:
7461:
7460:
7459:
7446:
7445:
7440:
7419:
7418:
7417:
7403:
7402:
7384:
7382:
7381:
7376:
7359:
7358:
7331:
7330:
7303:
7301:
7300:
7295:
7290:
7281:
7275:
7271:
7270:
7268:
7260:
7259:
7258:
7245:
7240:
7238:
7230:
7229:
7228:
7215:
7208:
7207:
7195:
7190:
7178:
7164:
7163:
7145:
7143:
7142:
7137:
7135:
7131:
7127:
7118:
7097:
7095:
7087:
7086:
7077:
7057:
7055:
7047:
7046:
7037:
7014:
7003:
6988:
6987:
6971:
6956:
6942:
6941:
6920:surface integral
6915:
6913:
6912:
6907:
6905:
6901:
6899:
6891:
6890:
6881:
6876:
6874:
6866:
6865:
6856:
6848:
6831:
6829:
6821:
6820:
6811:
6803:
6786:
6784:
6776:
6775:
6766:
6757:
6755:
6747:
6746:
6745:
6732:
6727:
6725:
6717:
6716:
6715:
6702:
6683:
6681:
6680:
6675:
6673:
6672:
6667:
6658:
6652:
6649:
6643:
6632:
6615:
6610:
6606:
6605:
6604:
6603:
6597:
6593:
6588:
6587:
6571:
6553:
6549:
6544:
6543:
6527:
6501:
6497:
6495:
6494:
6489:
6487:
6483:
6478:
6477:
6461:
6437:
6431:
6405:
6403:
6402:
6397:
6395:
6394:
6389:
6376:
6345:
6343:
6342:
6337:
6335:
6331:
6330:
6325:
6316:
6308:
6307:
6293:
6292:
6276:
6275:
6251:
6250:
6245:
6239:
6235:
6234:
6233:
6232:
6207:
6206:
6201:
6192:
6184:
6183:
6176:
6175:
6155:
6154:
6127:
6126:
6121:
6115:
6111:
6110:
6109:
6108:
6083:
6082:
6077:
6068:
6060:
6059:
6052:
6051:
6035:
6034:
6003:
6002:
5997:
5991:
5987:
5986:
5985:
5984:
5952:
5936:
5934:
5933:
5928:
5926:
5925:
5918:
5917:
5905:
5904:
5891:
5890:
5878:
5877:
5864:
5863:
5851:
5850:
5830:
5829:
5822:
5821:
5808:
5807:
5794:
5793:
5773:
5761:
5758:be an arbitrary
5757:
5755:
5754:
5749:
5747:
5746:
5734:
5733:
5698:
5696:
5695:
5690:
5688:
5687:
5686:
5680:
5670:
5668:
5667:
5662:
5660:
5659:
5658:
5648:
5643:
5642:
5626:
5610:
5605:
5604:
5588:
5569:
5567:
5566:
5561:
5559:
5555:
5553:
5545:
5544:
5535:
5533:
5529:
5528:
5527:
5526:
5520:
5516:
5511:
5510:
5494:
5476:
5471:
5470:
5454:
5436:
5434:
5426:
5425:
5416:
5408:
5404:
5401:
5398:
5396:
5394:
5386:
5385:
5376:
5371:
5366:
5365:
5349:
5333:
5331:
5323:
5322:
5313:
5308:
5306:
5298:
5297:
5288:
5283:
5278:
5277:
5261:
5245:
5243:
5235:
5234:
5225:
5217:
5213:
5211:
5203:
5202:
5193:
5188:
5186:
5178:
5174:
5166:
5154:
5149:
5147:
5139:
5138:
5129:
5124:
5122:
5114:
5110:
5102:
5090:
5081:
5079:
5071:
5070:
5069:
5056:
5051:
5049:
5041:
5040:
5039:
5026:
5007:
5005:
5004:
4999:
4997:
4993:
4991:
4976:
4975:
4970:
4969:
4959:
4951:
4943:
4932:
4930:
4922:
4921:
4912:
4907:
4905:
4897:
4893:
4885:
4873:
4864:
4862:
4854:
4853:
4852:
4839:
4833:
4831:
4816:
4815:
4810:
4809:
4799:
4791:
4783:
4772:
4770:
4762:
4761:
4752:
4747:
4745:
4737:
4733:
4725:
4713:
4704:
4702:
4694:
4693:
4692:
4679:
4648:
4646:
4645:
4640:
4638:
4637:
4632:
4620:
4619:
4614:
4593:
4592:
4591:
4578:
4577:
4572:
4551:
4550:
4549:
4535:
4534:
4522:
4521:
4516:
4504:
4496:
4490:
4489:
4477:
4476:
4471:
4459:
4451:
4445:
4444:
4425:
4417:
4405:
4403:
4402:
4397:
4392:
4391:
4386:
4365:
4364:
4363:
4350:
4349:
4344:
4323:
4322:
4321:
4293:
4281:
4279:
4278:
4273:
4271:
4267:
4251:
4249:
4241:
4240:
4231:
4208:
4200:
4172:
4171:
4170:
4154:
4152:
4151:
4146:
4144:
4140:
4124:
4122:
4114:
4113:
4104:
4081:
4073:
4045:
4044:
4043:
4026:
4024:
4023:
4018:
4016:
4015:
3999:
3997:
3996:
3991:
3989:
3988:
3970:
3968:
3967:
3962:
3960:
3959:
3954:
3948:
3944:
3928:
3926:
3918:
3917:
3908:
3885:
3877:
3864:
3863:
3858:
3852:
3848:
3832:
3830:
3822:
3821:
3812:
3789:
3781:
3753:
3734:
3732:
3731:
3726:
3724:
3720:
3719:
3714:
3705:
3701:
3700:
3699:
3694:
3688:
3684:
3680:
3672:
3670:
3662:
3661:
3652:
3641:
3633:
3625:
3612:
3611:
3606:
3600:
3596:
3592:
3584:
3582:
3574:
3573:
3564:
3553:
3545:
3537:
3521:
3520:
3505:
3501:
3497:
3492:
3483:
3482:
3477:
3468:
3467:
3462:
3453:
3445:
3444:
3443:
3427:
3419:
3411:
3400:
3395:
3386:
3385:
3380:
3371:
3370:
3365:
3356:
3348:
3347:
3346:
3330:
3322:
3314:
3308:
3307:
3291:
3290:
3285:
3273:
3265:
3259:
3258:
3235:
3229:
3221:
3202:
3196:
3171:
3156:
3148:
3138:
3136:
3135:
3130:
3128:
3124:
3115:
3107:
3106:
3105:
3086:
3078:
3070:
3064:
3063:
3051:
3047:
3039:
3034:
3019:
3011:
3003:
2997:
2996:
2984:
2983:
2978:
2966:
2958:
2952:
2951:
2932:
2928:
2906:Elementary proof
2881:
2879:
2878:
2873:
2871:
2859:
2857:
2856:
2851:
2849:
2848:
2843:
2830:
2828:
2827:
2822:
2820:
2808:
2806:
2805:
2800:
2795:
2790:
2786:
2785:
2781:
2774:
2766:
2747:
2742:
2732:
2731:
2719:
2710:
2709:
2703:
2693:
2685:
2684:
2657:
2655:
2654:
2649:
2647:
2646:
2641:
2624:
2622:
2621:
2616:
2602:
2600:
2599:
2594:
2589:
2584:
2574:
2563:
2562:
2550:
2549:
2543:
2533:
2528:
2527:
2508:
2506:
2505:
2500:
2498:
2497:
2492:
2479:
2477:
2476:
2471:
2469:
2454:
2452:
2451:
2446:
2431:
2429:
2428:
2423:
2412:the boundary of
2411:
2409:
2408:
2403:
2391:
2389:
2388:
2383:
2334:
2332:
2331:
2326:
2314:
2312:
2311:
2306:
2279:
2277:
2276:
2271:
2251:
2249:
2248:
2243:
2241:
2240:
2235:
2207:
2205:
2204:
2199:
2183:
2181:
2180:
2175:
2173:
2172:
2167:
2154:
2152:
2151:
2146:
2134:
2132:
2131:
2126:
2114:
2112:
2111:
2106:
2090:
2088:
2087:
2082:
2080:
2079:
2074:
2061:
2059:
2058:
2053:
2029:
2027:
2026:
2021:
2019:
2018:
2013:
1973:
1971:
1970:
1965:
1963:
1962:
1957:
1935:weak formulation
1918:
1916:
1915:
1910:
1908:
1901:
1900:
1891:
1885:
1884:
1869:
1863:
1862:
1847:
1841:
1840:
1831:
1830:
1824:
1823:
1805:
1801:
1797:
1793:
1784:
1778:
1774:
1773:
1771:
1763:
1762:
1761:
1748:
1743:
1741:
1733:
1732:
1731:
1718:
1705:
1696:
1690:
1686:
1685:
1683:
1675:
1674:
1673:
1660:
1655:
1653:
1645:
1644:
1643:
1630:
1617:
1608:
1602:
1598:
1597:
1595:
1587:
1586:
1585:
1572:
1567:
1565:
1557:
1556:
1555:
1542:
1530:
1529:
1517:
1506:
1504:
1503:
1498:
1493:
1488:
1480:
1475:
1474:
1459:
1454:
1443:
1429:
1428:
1412:
1410:
1409:
1404:
1388:
1386:
1385:
1380:
1354:
1353:
1320:
1319:
1286:
1285:
1249:
1237:
1235:
1234:
1229:
1208:
1206:
1205:
1200:
1198:
1197:
1192:
1179:
1177:
1176:
1171:
1143:
1141:
1140:
1135:
1133:
1132:
1127:
1099:surface integral
1072:
1070:
1069:
1064:
1062:
1061:
1056:
1008:
1004:
996:
992:
988:
984:
980:
976:
959:
952:
945:
893:
858:
824:
823:
820:
787:Surface integral
730:
729:
726:
634:
633:
630:
590:Limit comparison
510:
509:
506:
392:Riemann integral
345:
344:
341:
301:L'Hôpital's rule
258:Taylor's theorem
179:
178:
175:
119:
117:
116:
111:
63:
54:
49:
19:
18:
12140:
12139:
12135:
12134:
12133:
12131:
12130:
12129:
12120:Vector calculus
12095:
12094:
12093:
12092:
12073:
12069:
12062:
12048:
12044:
12037:Vector Calculus
12033:
12029:
11996:
11992:
11981:
11974:
11961:
11957:
11944:
11931:
11918:
11901:
11894:
11880:
11876:
11857:
11850:
11828:
11824:
11817:
11803:
11799:
11794:
11789:
11788:
11778:
11774:
11750:
11749:
11743:
11742:
11736:
11735:
11725:
11724:
11715:
11710:
11709:
11699:
11698:
11692:
11691:
11685:
11684:
11674:
11673:
11664:
11659:
11658:
11648:
11647:
11641:
11640:
11634:
11633:
11623:
11622:
11613:
11608:
11607:
11605:
11602:
11601:
11599:
11595:
11575:
11574:
11567:
11566:
11562:
11553:
11552:
11543:
11542:
11530:
11529:
11527:
11524:
11523:
11506:
11505:
11495:
11494:
11488:
11487:
11486:
11477:
11476:
11464:
11463:
11461:
11458:
11457:
11426:
11423:
11422:
11405:
11400:
11399:
11390:
11389:
11380:
11379:
11377:
11374:
11373:
11371:
11367:
11338:
11335:
11334:
11306:
11298:
11296:
11292:
11283:
11279:
11262:
11254:
11252:
11248:
11239:
11235:
11233:
11230:
11229:
11205:
11197:
11195:
11187:
11183:
11178:
11169:
11164:
11163:
11149:
11141:
11139:
11131:
11127:
11122:
11113:
11108:
11107:
11105:
11102:
11101:
11097:
11088:
11079:
11075:
11066:
11057:
11036:
11035:
11029:
11028:
11018:
11017:
11008:
11003:
11002:
10992:
10991:
10985:
10984:
10974:
10973:
10964:
10959:
10958:
10956:
10953:
10952:
10950:
10946:
10915:
10912:
10911:
10895:
10892:
10891:
10875:
10872:
10871:
10851:
10848:
10847:
10846:
10842:
10824:
10821:
10820:
10804:
10801:
10800:
10795:represents the
10765:
10762:
10761:
10760:
10756:
10751:
10745:
10725:
10720:
10712:
10704:
10698:
10694:
10686:
10681:
10673:
10664:
10660:
10658:
10655:
10654:
10638:
10636:
10633:
10632:
10611:
10606:
10598:
10590:
10584:
10580:
10572:
10567:
10559:
10550:
10546:
10544:
10541:
10540:
10524:
10522:
10519:
10518:
10503:
10497:
10470:
10466:
10461:
10455:
10447:
10443:
10442:
10438:
10436:
10433:
10432:
10423:
10417:
10396:
10391:
10390:
10382:
10379:
10378:
10349:
10327:
10316:
10302:
10291:
10272:
10260:
10245:
10241:
10231:
10221:
10213:
10192:
10187:
10186:
10178:
10175:
10174:
10160:
10156:
10152:
10127:
10123:
10118:
10112:
10104:
10100:
10099:
10095:
10093:
10090:
10089:
10078:
10056:
10045:
10031:
10020:
10012:
9998:
9986:
9971:
9961:
9954:
9948:
9942:
9918:
9913:
9912:
9904:
9901:
9900:
9890:
9868:
9864:
9852:
9848:
9846:
9843:
9842:
9839:
9835:
9829:
9808:
9802:
9794:
9790:
9789:
9785:
9774:
9768:
9760:
9756:
9755:
9751:
9743:
9740:
9739:
9731:
9727:
9719:
9715:
9698:
9694:
9682:
9678:
9676:
9673:
9672:
9655:
9651:
9639:
9635:
9633:
9630:
9629:
9625:
9604:
9598:
9590:
9586:
9585:
9581:
9575:
9564:
9549:
9543:
9537:
9533:
9525:
9522:
9521:
9515:
9496:
9490:
9484:
9480:
9469:
9463:
9451:
9447:
9445:
9442:
9441:
9437:
9433:
9429:
9413:
9412:
9394:
9390:
9381:
9377:
9368:
9364:
9355:
9351:
9317:
9302:
9301:
9271:
9253:
9249:
9236:
9221:
9217:
9213:
9211:
9208:
9207:
9194:
9191:
9185:
9182:
9176:
9157:
9153:
9144:
9140:
9131:
9127:
9118:
9114:
9103:
9100:
9099:
9085:
9084:
9045:
9041:
9039:
9005:
9001:
8998:
8997:
8958:
8954:
8952:
8918:
8914:
8911:
8910:
8877:
8873:
8871:
8837:
8833:
8830:
8829:
8796:
8792:
8790:
8756:
8752:
8748:
8746:
8743:
8742:
8739:
8731:
8724:
8717:
8698:
8695:
8694:
8674:
8671:
8670:
8660:
8653:
8647:
8640:
8624:
8618:
8615:
8609:
8599:
8596:
8590:
8587:
8581:
8562:
8558:
8553:
8547:
8539:
8535:
8534:
8530:
8521:
8517:
8512:
8506:
8498:
8494:
8493:
8489:
8487:
8484:
8483:
8472:
8453:
8442:
8434:
8420:
8409:
8401:
8387:
8379:
8364:
8357:
8350:
8344:
8338:
8314:
8309:
8308:
8300:
8297:
8296:
8283:
8268:
8258:
8248:
8237:
8212:
8207:
8206:
8198:
8195:
8194:
8185:
8172:
8167:
8146:
8141:
8132:
8125:
8119:
8115:
8104:
8097:
8093:
8089:
8083:
8079:
8068:
8067:
8063:
8058:
8057:
8051:
8047:
8036:
8035:
8031:
8030:
8021:
8017:
8008:
8003:
7994:
7993:
7984:
7980:
7978:
7975:
7974:
7954:
7952:
7949:
7948:
7940:
7919:
7918:
7914:
7909:
7899:
7892:
7888:
7883:
7880:
7879:
7876:
7868:
7862:
7838:
7831:
7827:
7816:
7809:
7805:
7794:
7787:
7783:
7774:
7769:
7768:
7762:
7758:
7749:
7744:
7743:
7737:
7733:
7724:
7719:
7718:
7712:
7708:
7706:
7703:
7702:
7685:
7680:
7679:
7677:
7674:
7673:
7656:
7651:
7650:
7641:
7636:
7635:
7633:
7630:
7629:
7626:
7597:
7588:
7572:
7565:
7561:
7557:
7555:
7542:
7535:
7531:
7527:
7525:
7524:
7520:
7514:
7510:
7501:
7496:
7483:
7478:
7477:
7455:
7451:
7450:
7441:
7436:
7435:
7413:
7409:
7408:
7404:
7398:
7394:
7392:
7389:
7388:
7354:
7350:
7326:
7322:
7320:
7317:
7316:
7313:Green's theorem
7309:
7286:
7277:
7261:
7254:
7250:
7246:
7244:
7231:
7224:
7220:
7216:
7214:
7213:
7209:
7203:
7199:
7191:
7186:
7174:
7159:
7155:
7153:
7150:
7149:
7133:
7132:
7123:
7114:
7088:
7082:
7078:
7076:
7048:
7042:
7038:
7036:
7010:
6999:
6989:
6983:
6979:
6972:
6967:
6952:
6937:
6933:
6929:
6927:
6924:
6923:
6903:
6902:
6892:
6886:
6882:
6880:
6867:
6861:
6857:
6855:
6844:
6822:
6816:
6812:
6810:
6799:
6777:
6771:
6767:
6765:
6758:
6748:
6741:
6737:
6733:
6731:
6718:
6711:
6707:
6703:
6701:
6697:
6695:
6692:
6691:
6668:
6663:
6662:
6654:
6648:
6639:
6628:
6611:
6599:
6598:
6589:
6567:
6566:
6562:
6558:
6557:
6545:
6523:
6522:
6518:
6514:
6513:
6509:
6507:
6504:
6503:
6499:
6479:
6457:
6456:
6452:
6448:
6446:
6443:
6442:
6433:
6410:
6390:
6385:
6384:
6382:
6379:
6378:
6374:
6365:
6356:
6347:
6333:
6332:
6326:
6321:
6320:
6312:
6302:
6301:
6295:
6294:
6288:
6284:
6278:
6277:
6271:
6267:
6260:
6259:
6252:
6246:
6241:
6240:
6228:
6227:
6223:
6216:
6212:
6209:
6208:
6202:
6197:
6196:
6188:
6178:
6177:
6171:
6167:
6164:
6163:
6157:
6156:
6150:
6146:
6136:
6135:
6128:
6122:
6117:
6116:
6104:
6103:
6099:
6092:
6088:
6085:
6084:
6078:
6073:
6072:
6064:
6054:
6053:
6047:
6043:
6037:
6036:
6030:
6026:
6023:
6022:
6012:
6011:
6004:
5998:
5993:
5992:
5980:
5979:
5975:
5968:
5964:
5960:
5958:
5955:
5954:
5940:
5920:
5919:
5913:
5909:
5900:
5896:
5893:
5892:
5886:
5882:
5873:
5869:
5866:
5865:
5859:
5855:
5846:
5842:
5835:
5834:
5824:
5823:
5817:
5813:
5810:
5809:
5803:
5799:
5796:
5795:
5789:
5785:
5778:
5777:
5769:
5767:
5764:
5763:
5762:matrix and let
5759:
5739:
5735:
5726:
5722:
5711:
5708:
5707:
5682:
5681:
5679:
5678:
5676:
5673:
5672:
5654:
5653:
5649:
5644:
5622:
5621:
5617:
5606:
5584:
5583:
5579:
5577:
5574:
5573:
5557:
5556:
5546:
5540:
5536:
5534:
5522:
5521:
5512:
5490:
5489:
5485:
5481:
5480:
5472:
5450:
5449:
5445:
5444:
5440:
5427:
5421:
5417:
5415:
5406:
5405:
5400:
5397:
5387:
5381:
5377:
5375:
5367:
5345:
5344:
5340:
5324:
5318:
5314:
5312:
5299:
5293:
5289:
5287:
5279:
5257:
5256:
5252:
5236:
5230:
5226:
5224:
5215:
5214:
5204:
5198:
5194:
5192:
5179:
5170:
5162:
5155:
5153:
5140:
5134:
5130:
5128:
5115:
5106:
5098:
5091:
5089:
5082:
5072:
5065:
5061:
5057:
5055:
5042:
5035:
5031:
5027:
5025:
5021:
5019:
5016:
5015:
4995:
4994:
4977:
4971:
4965:
4961:
4960:
4958:
4947:
4939:
4923:
4917:
4913:
4911:
4898:
4889:
4881:
4874:
4872:
4865:
4855:
4848:
4844:
4840:
4838:
4835:
4834:
4817:
4811:
4805:
4801:
4800:
4798:
4787:
4779:
4763:
4757:
4753:
4751:
4738:
4729:
4721:
4714:
4712:
4705:
4695:
4688:
4684:
4680:
4678:
4674:
4672:
4669:
4668:
4661:Green's theorem
4657:
4633:
4628:
4615:
4610:
4609:
4587:
4583:
4582:
4573:
4568:
4567:
4545:
4541:
4540:
4536:
4530:
4526:
4517:
4512:
4500:
4492:
4491:
4485:
4481:
4472:
4467:
4455:
4447:
4446:
4437:
4433:
4431:
4428:
4427:
4419:
4413:
4387:
4382:
4381:
4359:
4355:
4354:
4345:
4340:
4339:
4317:
4313:
4312:
4289:
4287:
4284:
4283:
4242:
4236:
4232:
4230:
4204:
4196:
4195:
4191:
4166:
4162:
4161:
4159:
4156:
4155:
4115:
4109:
4105:
4103:
4077:
4069:
4068:
4064:
4039:
4035:
4034:
4032:
4029:
4028:
4011:
4007:
4005:
4002:
4001:
3984:
3980:
3978:
3975:
3974:
3955:
3950:
3949:
3919:
3913:
3909:
3907:
3881:
3873:
3872:
3868:
3859:
3854:
3853:
3823:
3817:
3813:
3811:
3785:
3777:
3776:
3772:
3749:
3747:
3744:
3743:
3740:
3722:
3721:
3715:
3710:
3695:
3690:
3689:
3676:
3663:
3657:
3653:
3651:
3637:
3629:
3621:
3620:
3616:
3607:
3602:
3601:
3588:
3575:
3569:
3565:
3563:
3549:
3541:
3533:
3532:
3528:
3527:
3523:
3522:
3516:
3512:
3503:
3502:
3493:
3488:
3478:
3473:
3472:
3463:
3458:
3457:
3449:
3439:
3438:
3434:
3423:
3415:
3407:
3396:
3391:
3381:
3376:
3375:
3366:
3361:
3360:
3352:
3342:
3341:
3337:
3326:
3318:
3310:
3309:
3303:
3299:
3292:
3286:
3281:
3269:
3261:
3260:
3251:
3247:
3243:
3241:
3238:
3237:
3231:
3223:
3215:
3207:
3198:
3194:
3185:
3176:
3158:
3154:
3151:Jacobian matrix
3149:stands for the
3146:
3140:
3120:
3111:
3101:
3100:
3096:
3082:
3074:
3066:
3065:
3059:
3055:
3043:
3035:
3030:
3015:
3007:
2999:
2998:
2992:
2988:
2979:
2974:
2962:
2954:
2953:
2944:
2940:
2938:
2935:
2934:
2930:
2922:
2913:
2908:
2892:Green's theorem
2888:
2867:
2865:
2862:
2861:
2844:
2839:
2838:
2836:
2833:
2832:
2816:
2814:
2811:
2810:
2791:
2770:
2762:
2755:
2751:
2743:
2738:
2737:
2733:
2727:
2723:
2715:
2705:
2704:
2699:
2689:
2677:
2673:
2671:
2668:
2667:
2664:
2642:
2637:
2636:
2634:
2631:
2630:
2610:
2607:
2606:
2585:
2580:
2570:
2558:
2554:
2545:
2544:
2539:
2529:
2520:
2516:
2514:
2511:
2510:
2493:
2488:
2487:
2485:
2482:
2481:
2465:
2463:
2460:
2459:
2437:
2434:
2433:
2417:
2414:
2413:
2397:
2394:
2393:
2344:
2341:
2340:
2320:
2317:
2316:
2285:
2282:
2281:
2265:
2262:
2261:
2236:
2231:
2230:
2216:
2213:
2212:
2193:
2190:
2189:
2186:parametrization
2168:
2163:
2162:
2160:
2157:
2156:
2140:
2137:
2136:
2120:
2117:
2116:
2100:
2097:
2096:
2075:
2070:
2069:
2067:
2064:
2063:
2047:
2044:
2043:
2014:
2009:
2008:
1982:
1979:
1978:
1958:
1953:
1952:
1950:
1947:
1946:
1927:Lebesgue theory
1906:
1905:
1896:
1895:
1887:
1880:
1876:
1865:
1858:
1854:
1843:
1836:
1832:
1826:
1825:
1816:
1812:
1803:
1802:
1789:
1780:
1764:
1757:
1753:
1749:
1747:
1734:
1727:
1723:
1719:
1717:
1716:
1712:
1701:
1692:
1676:
1669:
1665:
1661:
1659:
1646:
1639:
1635:
1631:
1629:
1628:
1624:
1613:
1604:
1588:
1581:
1577:
1573:
1571:
1558:
1551:
1547:
1543:
1541:
1540:
1536:
1535:
1531:
1525:
1521:
1514:
1512:
1509:
1508:
1489:
1484:
1476:
1467:
1463:
1455:
1450:
1439:
1424:
1420:
1418:
1415:
1414:
1398:
1395:
1394:
1349:
1345:
1315:
1311:
1281:
1277:
1245:
1243:
1240:
1239:
1214:
1211:
1210:
1193:
1188:
1187:
1185:
1182:
1181:
1165:
1162:
1161:
1158:
1128:
1123:
1122:
1120:
1117:
1116:
1057:
1052:
1051:
1049:
1046:
1045:
1042:vector calculus
1014:Stokes' theorem
1006:
1002:
999:right-hand-rule
994:
990:
986:
982:
978:
977:, its boundary
974:
963:
934:
933:
919:Integration Bee
894:
891:
884:
883:
859:
856:
849:
848:
821:
818:
811:
810:
792:Volume integral
727:
722:
715:
714:
631:
626:
619:
618:
588:
507:
502:
495:
494:
486:Risch algorithm
456:Euler's formula
342:
337:
330:
329:
311:General Leibniz
194:generalizations
176:
171:
164:
150:Rolle's theorem
145:
120:
56:
50:
45:
39:
36:
35:
17:
12:
11:
5:
12138:
12128:
12127:
12122:
12117:
12112:
12107:
12091:
12090:
12067:
12060:
12054:. Birkhäuser.
12042:
12027:
12006:(5): 354–359.
11990:
11972:
11955:
11929:
11899:
11892:
11874:
11868:978-4563004415
11862:(jp)(1979/01)
11848:
11822:
11815:
11796:
11795:
11793:
11790:
11787:
11786:
11772:
11759:
11754:
11748:
11745:
11744:
11741:
11738:
11737:
11734:
11731:
11730:
11728:
11723:
11718:
11713:
11708:
11703:
11697:
11694:
11693:
11690:
11687:
11686:
11683:
11680:
11679:
11677:
11672:
11667:
11662:
11657:
11652:
11646:
11643:
11642:
11639:
11636:
11635:
11632:
11629:
11628:
11626:
11621:
11616:
11611:
11593:
11570:
11565:
11561:
11551:
11541:
11538:
11522:and therefore
11504:
11498:
11485:
11475:
11472:
11442:
11439:
11436:
11433:
11430:
11408:
11403:
11398:
11388:
11365:
11361:Euclidean norm
11348:
11345:
11342:
11322:
11318:
11312:
11309:
11304:
11301:
11295:
11291:
11286:
11282:
11278:
11274:
11268:
11265:
11260:
11257:
11251:
11247:
11242:
11238:
11217:
11211:
11208:
11203:
11200:
11190:
11186:
11182:
11177:
11172:
11167:
11162:
11155:
11152:
11147:
11144:
11134:
11130:
11126:
11121:
11116:
11111:
11093:
11084:
11071:
11062:
11045:
11040:
11034:
11031:
11030:
11027:
11024:
11023:
11021:
11016:
11011:
11006:
11001:
10996:
10990:
10987:
10986:
10983:
10980:
10979:
10977:
10972:
10967:
10962:
10944:
10940:countably many
10919:
10899:
10879:
10855:
10840:
10828:
10808:
10784:
10781:
10778:
10775:
10772:
10769:
10753:
10752:
10750:
10747:
10732:
10728:
10723:
10719:
10715:
10711:
10707:
10701:
10697:
10693:
10689:
10684:
10680:
10676:
10670:
10667:
10663:
10641:
10618:
10614:
10609:
10605:
10601:
10597:
10593:
10587:
10583:
10579:
10575:
10570:
10566:
10562:
10556:
10553:
10549:
10527:
10496:
10493:
10481:
10478:
10473:
10469:
10464:
10458:
10450:
10446:
10441:
10399:
10394:
10389:
10386:
10357:
10356:
10323:
10298:
10268:
10210:path-connected
10195:
10190:
10185:
10182:
10138:
10135:
10130:
10126:
10121:
10115:
10107:
10103:
10098:
10086:
10085:
10052:
10027:
10010:
9994:
9960:. Fix a point
9952:
9921:
9916:
9911:
9908:
9889:
9886:
9871:
9867:
9863:
9860:
9855:
9851:
9837:
9833:
9815:
9811:
9805:
9797:
9793:
9788:
9784:
9781:
9777:
9771:
9763:
9759:
9754:
9750:
9747:
9729:
9717:
9701:
9697:
9693:
9690:
9685:
9681:
9658:
9654:
9650:
9647:
9642:
9638:
9611:
9607:
9601:
9593:
9589:
9584:
9578:
9573:
9570:
9567:
9563:
9559:
9556:
9552:
9546:
9540:
9536:
9532:
9529:
9503:
9499:
9493:
9487:
9483:
9479:
9476:
9472:
9466:
9462:
9459:
9454:
9450:
9411:
9408:
9405:
9402:
9397:
9393:
9389:
9384:
9380:
9376:
9371:
9367:
9363:
9358:
9354:
9350:
9347:
9344:
9341:
9338:
9335:
9332:
9329:
9326:
9323:
9320:
9318:
9316:
9313:
9310:
9307:
9304:
9303:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9279:
9276:
9273:
9270:
9267:
9264:
9261:
9256:
9252:
9248:
9245:
9242:
9239:
9237:
9235:
9232:
9229:
9224:
9220:
9216:
9215:
9189:
9180:
9160:
9156:
9152:
9147:
9143:
9139:
9134:
9130:
9126:
9121:
9117:
9113:
9110:
9107:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9056:
9053:
9048:
9044:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9019:
9016:
9013:
9008:
9004:
9000:
8999:
8996:
8993:
8990:
8987:
8984:
8981:
8978:
8975:
8972:
8969:
8966:
8961:
8957:
8953:
8950:
8947:
8944:
8941:
8938:
8935:
8932:
8929:
8926:
8921:
8917:
8913:
8912:
8909:
8906:
8903:
8900:
8897:
8894:
8891:
8888:
8885:
8880:
8876:
8872:
8869:
8866:
8863:
8860:
8857:
8854:
8851:
8848:
8845:
8840:
8836:
8832:
8831:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8799:
8795:
8791:
8788:
8785:
8782:
8779:
8776:
8773:
8770:
8767:
8764:
8759:
8755:
8751:
8750:
8735:
8702:
8678:
8667:abuse notation
8658:
8651:
8639:
8636:
8622:
8613:
8594:
8585:
8565:
8561:
8556:
8550:
8542:
8538:
8533:
8529:
8524:
8520:
8515:
8509:
8501:
8497:
8492:
8480:
8479:
8449:
8432:
8416:
8399:
8383:
8355:
8348:
8317:
8312:
8307:
8304:
8282:
8279:
8215:
8210:
8205:
8202:
8171:
8168:
8166:
8163:
8149:
8144:
8139:
8135:
8131:
8128:
8122:
8118:
8114:
8107:
8103:
8100:
8096:
8092:
8086:
8082:
8078:
8071:
8066:
8061:
8054:
8050:
8046:
8039:
8034:
8027:
8024:
8020:
8016:
8011:
8006:
8001:
7997:
7990:
7987:
7983:
7957:
7928:
7922:
7917:
7912:
7908:
7902:
7898:
7895:
7891:
7887:
7872:
7848:
7845:
7841:
7834:
7830:
7826:
7823:
7819:
7812:
7808:
7804:
7801:
7797:
7790:
7786:
7782:
7777:
7772:
7765:
7761:
7757:
7752:
7747:
7740:
7736:
7732:
7727:
7722:
7715:
7711:
7688:
7683:
7659:
7654:
7649:
7644:
7639:
7628:The functions
7625:
7622:
7604:
7600:
7595:
7591:
7585:
7578:
7575:
7568:
7564:
7560:
7554:
7548:
7545:
7538:
7534:
7530:
7523:
7517:
7513:
7509:
7504:
7499:
7494:
7491:
7486:
7481:
7476:
7473:
7470:
7467:
7464:
7458:
7454:
7449:
7444:
7439:
7434:
7431:
7428:
7425:
7422:
7416:
7412:
7407:
7401:
7397:
7385:defined on D;
7374:
7371:
7368:
7365:
7362:
7357:
7353:
7349:
7346:
7343:
7340:
7337:
7334:
7329:
7325:
7308:
7305:
7293:
7289:
7284:
7280:
7274:
7267:
7264:
7257:
7253:
7249:
7243:
7237:
7234:
7227:
7223:
7219:
7212:
7206:
7202:
7198:
7194:
7189:
7184:
7181:
7177:
7173:
7170:
7167:
7162:
7158:
7148:So, we obtain
7130:
7126:
7121:
7117:
7112:
7109:
7106:
7103:
7100:
7094:
7091:
7085:
7081:
7075:
7072:
7069:
7066:
7063:
7060:
7054:
7051:
7045:
7041:
7035:
7032:
7029:
7026:
7023:
7020:
7017:
7013:
7009:
7006:
7002:
6998:
6995:
6992:
6986:
6982:
6978:
6975:
6973:
6970:
6966:
6962:
6959:
6955:
6951:
6948:
6945:
6940:
6936:
6932:
6931:
6898:
6895:
6889:
6885:
6879:
6873:
6870:
6864:
6860:
6854:
6851:
6847:
6843:
6840:
6837:
6834:
6828:
6825:
6819:
6815:
6809:
6806:
6802:
6798:
6795:
6792:
6789:
6783:
6780:
6774:
6770:
6764:
6761:
6759:
6754:
6751:
6744:
6740:
6736:
6730:
6724:
6721:
6714:
6710:
6706:
6700:
6699:
6671:
6666:
6661:
6657:
6646:
6642:
6638:
6635:
6631:
6627:
6624:
6621:
6618:
6614:
6609:
6602:
6596:
6592:
6586:
6583:
6580:
6577:
6574:
6570:
6565:
6561:
6556:
6552:
6548:
6542:
6539:
6536:
6533:
6530:
6526:
6521:
6517:
6512:
6486:
6482:
6476:
6473:
6470:
6467:
6464:
6460:
6455:
6451:
6393:
6388:
6370:
6361:
6352:
6329:
6324:
6319:
6315:
6311:
6306:
6300:
6297:
6296:
6291:
6287:
6283:
6280:
6279:
6274:
6270:
6266:
6265:
6263:
6258:
6255:
6253:
6249:
6244:
6238:
6231:
6226:
6222:
6219:
6215:
6211:
6210:
6205:
6200:
6195:
6191:
6187:
6182:
6174:
6170:
6166:
6165:
6162:
6159:
6158:
6153:
6149:
6145:
6142:
6141:
6139:
6134:
6131:
6129:
6125:
6120:
6114:
6107:
6102:
6098:
6095:
6091:
6087:
6086:
6081:
6076:
6071:
6067:
6063:
6058:
6050:
6046:
6042:
6039:
6038:
6033:
6029:
6025:
6024:
6021:
6018:
6017:
6015:
6010:
6007:
6005:
6001:
5996:
5990:
5983:
5978:
5974:
5971:
5967:
5963:
5962:
5924:
5916:
5912:
5908:
5903:
5899:
5895:
5894:
5889:
5885:
5881:
5876:
5872:
5868:
5867:
5862:
5858:
5854:
5849:
5845:
5841:
5840:
5838:
5833:
5828:
5820:
5816:
5812:
5811:
5806:
5802:
5798:
5797:
5792:
5788:
5784:
5783:
5781:
5776:
5772:
5745:
5742:
5738:
5732:
5729:
5725:
5721:
5718:
5715:
5685:
5657:
5652:
5647:
5641:
5638:
5635:
5632:
5629:
5625:
5620:
5616:
5613:
5609:
5603:
5600:
5597:
5594:
5591:
5587:
5582:
5552:
5549:
5543:
5539:
5532:
5525:
5519:
5515:
5509:
5506:
5503:
5500:
5497:
5493:
5488:
5484:
5479:
5475:
5469:
5466:
5463:
5460:
5457:
5453:
5448:
5443:
5439:
5433:
5430:
5424:
5420:
5414:
5411:
5409:
5407:
5399:
5393:
5390:
5384:
5380:
5374:
5370:
5364:
5361:
5358:
5355:
5352:
5348:
5343:
5339:
5336:
5330:
5327:
5321:
5317:
5311:
5305:
5302:
5296:
5292:
5286:
5282:
5276:
5273:
5270:
5267:
5264:
5260:
5255:
5251:
5248:
5242:
5239:
5233:
5229:
5223:
5220:
5218:
5216:
5210:
5207:
5201:
5197:
5191:
5185:
5182:
5177:
5173:
5169:
5165:
5161:
5158:
5152:
5146:
5143:
5137:
5133:
5127:
5121:
5118:
5113:
5109:
5105:
5101:
5097:
5094:
5088:
5085:
5083:
5078:
5075:
5068:
5064:
5060:
5054:
5048:
5045:
5038:
5034:
5030:
5024:
5023:
4990:
4987:
4983:
4980:
4974:
4968:
4964:
4957:
4954:
4950:
4946:
4942:
4938:
4935:
4929:
4926:
4920:
4916:
4910:
4904:
4901:
4896:
4892:
4888:
4884:
4880:
4877:
4871:
4868:
4866:
4861:
4858:
4851:
4847:
4843:
4837:
4836:
4830:
4827:
4823:
4820:
4814:
4808:
4804:
4797:
4794:
4790:
4786:
4782:
4778:
4775:
4769:
4766:
4760:
4756:
4750:
4744:
4741:
4736:
4732:
4728:
4724:
4720:
4717:
4711:
4708:
4706:
4701:
4698:
4691:
4687:
4683:
4677:
4676:
4656:
4653:
4636:
4631:
4626:
4623:
4618:
4613:
4608:
4605:
4602:
4599:
4596:
4590:
4586:
4581:
4576:
4571:
4566:
4563:
4560:
4557:
4554:
4548:
4544:
4539:
4533:
4529:
4525:
4520:
4515:
4510:
4507:
4503:
4499:
4495:
4488:
4484:
4480:
4475:
4470:
4465:
4462:
4458:
4454:
4450:
4443:
4440:
4436:
4395:
4390:
4385:
4380:
4377:
4374:
4371:
4368:
4362:
4358:
4353:
4348:
4343:
4338:
4335:
4332:
4329:
4326:
4320:
4316:
4311:
4308:
4305:
4302:
4299:
4296:
4292:
4270:
4266:
4263:
4260:
4257:
4254:
4248:
4245:
4239:
4235:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4207:
4203:
4199:
4194:
4190:
4187:
4184:
4181:
4178:
4175:
4169:
4165:
4143:
4139:
4136:
4133:
4130:
4127:
4121:
4118:
4112:
4108:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4080:
4076:
4072:
4067:
4063:
4060:
4057:
4054:
4051:
4048:
4042:
4038:
4014:
4010:
3987:
3983:
3958:
3953:
3947:
3943:
3940:
3937:
3934:
3931:
3925:
3922:
3916:
3912:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3884:
3880:
3876:
3871:
3867:
3862:
3857:
3851:
3847:
3844:
3841:
3838:
3835:
3829:
3826:
3820:
3816:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3788:
3784:
3780:
3775:
3771:
3768:
3765:
3762:
3759:
3756:
3752:
3739:
3736:
3718:
3713:
3708:
3704:
3698:
3693:
3687:
3683:
3679:
3675:
3669:
3666:
3660:
3656:
3650:
3647:
3644:
3640:
3636:
3632:
3628:
3624:
3619:
3615:
3610:
3605:
3599:
3595:
3591:
3587:
3581:
3578:
3572:
3568:
3562:
3559:
3556:
3552:
3548:
3544:
3540:
3536:
3531:
3526:
3519:
3515:
3511:
3508:
3506:
3504:
3500:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3452:
3448:
3442:
3437:
3433:
3430:
3426:
3422:
3418:
3414:
3410:
3406:
3403:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3355:
3351:
3345:
3340:
3336:
3333:
3329:
3325:
3321:
3317:
3313:
3306:
3302:
3298:
3295:
3293:
3289:
3284:
3279:
3276:
3272:
3268:
3264:
3257:
3254:
3250:
3246:
3245:
3211:
3190:
3181:
3142:
3127:
3123:
3118:
3114:
3110:
3104:
3099:
3095:
3092:
3089:
3085:
3081:
3077:
3073:
3069:
3062:
3058:
3054:
3050:
3046:
3042:
3038:
3033:
3028:
3025:
3022:
3018:
3014:
3010:
3006:
3002:
2995:
2991:
2987:
2982:
2977:
2972:
2969:
2965:
2961:
2957:
2950:
2947:
2943:
2918:§ Theorem
2912:
2909:
2907:
2904:
2887:
2884:
2870:
2847:
2842:
2819:
2798:
2794:
2789:
2784:
2780:
2777:
2773:
2769:
2765:
2761:
2758:
2754:
2750:
2746:
2741:
2736:
2730:
2726:
2722:
2718:
2713:
2708:
2702:
2697:
2692:
2688:
2683:
2680:
2676:
2663:
2660:
2645:
2640:
2614:
2592:
2588:
2583:
2578:
2573:
2569:
2566:
2561:
2557:
2553:
2548:
2542:
2537:
2532:
2526:
2523:
2519:
2496:
2491:
2468:
2444:
2441:
2421:
2401:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2324:
2304:
2301:
2298:
2295:
2292:
2289:
2269:
2256:smooth at the
2239:
2234:
2229:
2226:
2223:
2220:
2197:
2171:
2166:
2144:
2135:is bounded by
2124:
2104:
2078:
2073:
2051:
2017:
2012:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1961:
1956:
1943:coarea formula
1923:Koch snowflake
1904:
1899:
1894:
1890:
1883:
1879:
1875:
1872:
1868:
1861:
1857:
1853:
1850:
1846:
1839:
1835:
1829:
1822:
1819:
1815:
1811:
1808:
1806:
1804:
1800:
1796:
1792:
1787:
1783:
1777:
1770:
1767:
1760:
1756:
1752:
1746:
1740:
1737:
1730:
1726:
1722:
1715:
1711:
1708:
1704:
1699:
1695:
1689:
1682:
1679:
1672:
1668:
1664:
1658:
1652:
1649:
1642:
1638:
1634:
1627:
1623:
1620:
1616:
1611:
1607:
1601:
1594:
1591:
1584:
1580:
1576:
1570:
1564:
1561:
1554:
1550:
1546:
1539:
1534:
1528:
1524:
1520:
1518:
1516:
1496:
1492:
1487:
1483:
1479:
1473:
1470:
1466:
1462:
1458:
1453:
1449:
1446:
1442:
1438:
1435:
1432:
1427:
1423:
1402:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1352:
1348:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1318:
1314:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1284:
1280:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1248:
1227:
1224:
1221:
1218:
1209:with boundary
1196:
1191:
1169:
1157:
1154:
1131:
1126:
1109:
1108:
1060:
1055:
1032:or simply the
965:
964:
962:
961:
954:
947:
939:
936:
935:
932:
931:
926:
921:
916:
914:List of topics
911:
906:
901:
895:
890:
889:
886:
885:
882:
881:
876:
871:
866:
860:
855:
854:
851:
850:
845:
844:
843:
842:
837:
832:
822:
817:
816:
813:
812:
807:
806:
805:
804:
799:
794:
789:
784:
779:
774:
766:
765:
761:
760:
759:
758:
753:
748:
743:
735:
734:
728:
721:
720:
717:
716:
711:
710:
709:
708:
703:
698:
693:
688:
683:
675:
674:
670:
669:
668:
667:
662:
657:
652:
647:
642:
632:
625:
624:
621:
620:
615:
614:
613:
612:
607:
602:
597:
592:
586:
581:
576:
571:
566:
558:
557:
551:
550:
549:
548:
543:
538:
533:
528:
523:
508:
501:
500:
497:
496:
491:
490:
489:
488:
483:
478:
473:
471:Changing order
468:
458:
453:
435:
430:
425:
417:
416:
415:Integration by
412:
411:
410:
409:
404:
399:
394:
389:
379:
377:Antiderivative
371:
370:
366:
365:
364:
363:
358:
353:
343:
336:
335:
332:
331:
326:
325:
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
270:
269:
263:
262:
261:
260:
255:
250:
245:
240:
235:
227:
226:
222:
221:
220:
219:
218:
217:
212:
207:
197:
184:
183:
177:
170:
169:
166:
165:
163:
162:
157:
152:
146:
144:
143:
138:
132:
131:
130:
122:
121:
109:
106:
103:
100:
97:
94:
91:
88:
85:
82:
79:
76:
72:
69:
66:
62:
59:
53:
48:
44:
34:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
12137:
12126:
12123:
12121:
12118:
12116:
12113:
12111:
12108:
12106:
12103:
12102:
12100:
12087:
12084:
12081:
12077:
12071:
12063:
12061:0-8176-3707-9
12057:
12053:
12046:
12038:
12031:
12023:
12019:
12014:
12009:
12005:
12001:
11994:
11988:
11985:
11979:
11977:
11968:
11967:
11959:
11951:
11950:
11942:
11940:
11938:
11936:
11934:
11925:
11924:
11916:
11914:
11912:
11910:
11908:
11906:
11904:
11895:
11889:
11885:
11878:
11871:
11869:
11865:
11861:
11855:
11853:
11845:
11843:
11839:
11836:(jp) 1983/12
11835:
11831:
11826:
11818:
11812:
11808:
11801:
11797:
11782:
11776:
11757:
11752:
11746:
11739:
11732:
11726:
11721:
11716:
11706:
11701:
11695:
11688:
11681:
11675:
11670:
11665:
11655:
11650:
11644:
11637:
11630:
11624:
11619:
11614:
11597:
11563:
11559:
11549:
11539:
11536:
11502:
11483:
11473:
11470:
11455:
11454:square matrix
11440:
11437:
11434:
11431:
11428:
11406:
11396:
11386:
11369:
11362:
11359:" represents
11343:
11320:
11310:
11302:
11289:
11284:
11280:
11276:
11266:
11258:
11245:
11240:
11236:
11215:
11209:
11201:
11188:
11184:
11180:
11175:
11170:
11160:
11153:
11145:
11132:
11128:
11124:
11119:
11114:
11096:
11092:
11087:
11083:
11074:
11070:
11065:
11061:
11043:
11038:
11032:
11025:
11019:
11014:
11009:
10999:
10994:
10988:
10981:
10975:
10970:
10965:
10948:
10941:
10937:
10936:connected sum
10933:
10897:
10877:
10869:
10866:may not be a
10844:
10826:
10806:
10798:
10779:
10773:
10770:
10758:
10754:
10746:
10743:
10730:
10717:
10709:
10695:
10691:
10678:
10661:
10629:
10616:
10603:
10595:
10581:
10577:
10564:
10547:
10516:
10512:
10508:
10502:
10492:
10479:
10476:
10471:
10467:
10448:
10444:
10439:
10430:
10426:
10420:
10415:
10397:
10387:
10384:
10376:
10372:
10370:
10365:
10363:
10352:
10346:
10342:
10338:
10334:
10330:
10326:
10324:
10319:
10313:
10309:
10305:
10301:
10299:
10294:
10287:
10283:
10279:
10275:
10271:
10269:
10266:
10259:
10257:
10256:
10255:
10252:
10248:
10238:
10234:
10228:
10224:
10219:
10211:
10193:
10183:
10180:
10172:
10168:
10166:
10149:
10136:
10133:
10128:
10124:
10105:
10101:
10096:
10081:
10075:
10071:
10067:
10063:
10059:
10055:
10053:
10048:
10042:
10038:
10034:
10030:
10028:
10023:
10016:
10009:
10005:
10001:
9997:
9995:
9992:
9985:
9983:
9982:
9981:
9978:
9974:
9968:
9964:
9958:
9951:
9945:
9940:
9937:
9919:
9909:
9906:
9898:
9894:
9885:
9869:
9861:
9858:
9853:
9849:
9832:
9826:
9795:
9786:
9782:
9761:
9752:
9748:
9745:
9735:
9723:
9699:
9691:
9688:
9683:
9656:
9648:
9645:
9640:
9622:
9591:
9582:
9576:
9571:
9568:
9565:
9561:
9557:
9534:
9530:
9527:
9518:
9481:
9477:
9474:
9460:
9452:
9448:
9426:
9406:
9395:
9387:
9382:
9374:
9369:
9361:
9356:
9345:
9336:
9330:
9324:
9321:
9319:
9311:
9298:
9295:
9292:
9289:
9286:
9283:
9280:
9277:
9274:
9262:
9254:
9250:
9243:
9240:
9238:
9230:
9222:
9205:
9201:
9197:
9188:
9179:
9173:
9158:
9154:
9150:
9145:
9141:
9137:
9132:
9128:
9124:
9119:
9115:
9111:
9108:
9078:
9075:
9072:
9069:
9066:
9060:
9054:
9046:
9042:
9035:
9032:
9023:
9020:
9017:
9011:
9006:
9002:
8991:
8988:
8985:
8982:
8979:
8973:
8967:
8959:
8955:
8948:
8945:
8936:
8933:
8930:
8924:
8919:
8915:
8904:
8901:
8898:
8892:
8886:
8878:
8874:
8867:
8864:
8855:
8852:
8849:
8843:
8838:
8834:
8823:
8820:
8817:
8811:
8805:
8797:
8793:
8786:
8783:
8774:
8771:
8768:
8762:
8757:
8753:
8738:
8734:
8728:
8720:
8714:
8700:
8692:
8676:
8668:
8657:
8650:
8644:
8635:
8633:
8629:
8621:
8612:
8606:
8602:
8593:
8584:
8578:
8563:
8559:
8540:
8536:
8531:
8527:
8522:
8518:
8499:
8495:
8490:
8475:
8468:
8464:
8460:
8456:
8452:
8450:
8445:
8438:
8431:
8427:
8423:
8419:
8417:
8412:
8405:
8398:
8394:
8390:
8386:
8384:
8378:
8376:
8375:
8374:
8371:
8367:
8361:
8354:
8347:
8341:
8336:
8333:
8315:
8305:
8302:
8294:
8290:
8288:
8278:
8276:
8271:
8266:
8261:
8256:
8251:
8245:
8241:
8235:
8231:
8213:
8203:
8200:
8193:
8188:
8183:
8179:
8177:
8162:
8137:
8129:
8116:
8112:
8101:
8094:
8090:
8080:
8076:
8064:
8048:
8044:
8032:
8018:
8014:
8009:
7999:
7981:
7972:
7946:
7926:
7915:
7906:
7896:
7889:
7885:
7875:
7871:
7865:
7859:
7846:
7843:
7832:
7828:
7824:
7821:
7810:
7806:
7802:
7799:
7788:
7784:
7775:
7763:
7759:
7755:
7750:
7738:
7734:
7730:
7725:
7713:
7709:
7686:
7657:
7642:
7621:
7620:
7615:
7602:
7593:
7583:
7576:
7566:
7562:
7552:
7546:
7536:
7532:
7521:
7515:
7511:
7507:
7492:
7484:
7471:
7468:
7465:
7456:
7452:
7447:
7442:
7429:
7426:
7423:
7414:
7410:
7399:
7395:
7386:
7369:
7366:
7363:
7355:
7351:
7347:
7341:
7338:
7335:
7327:
7323:
7314:
7304:
7291:
7282:
7272:
7265:
7255:
7251:
7241:
7235:
7225:
7221:
7210:
7204:
7200:
7196:
7182:
7171:
7156:
7146:
7128:
7119:
7107:
7104:
7101:
7092:
7073:
7067:
7064:
7061:
7052:
7033:
7024:
7021:
7018:
6996:
6984:
6980:
6976:
6974:
6964:
6960:
6949:
6934:
6921:
6916:
6896:
6877:
6871:
6852:
6841:
6832:
6826:
6807:
6796:
6787:
6781:
6762:
6760:
6752:
6742:
6738:
6728:
6722:
6712:
6708:
6689:
6684:
6669:
6659:
6644:
6636:
6625:
6616:
6607:
6581:
6578:
6575:
6563:
6554:
6537:
6534:
6531:
6519:
6510:
6471:
6468:
6465:
6453:
6441:Substituting
6439:
6436:
6430:
6426:
6422:
6418:
6414:
6407:
6391:
6373:
6369:
6364:
6360:
6355:
6351:
6327:
6317:
6309:
6304:
6298:
6289:
6285:
6281:
6272:
6268:
6261:
6256:
6254:
6247:
6236:
6224:
6220:
6217:
6213:
6203:
6193:
6185:
6180:
6172:
6168:
6160:
6151:
6147:
6143:
6137:
6132:
6130:
6123:
6112:
6100:
6096:
6093:
6089:
6079:
6069:
6061:
6056:
6048:
6044:
6040:
6031:
6027:
6019:
6013:
6008:
6006:
5999:
5988:
5976:
5972:
5969:
5965:
5951:
5947:
5943:
5937:
5922:
5914:
5910:
5906:
5901:
5897:
5887:
5883:
5879:
5874:
5870:
5860:
5856:
5852:
5847:
5843:
5836:
5831:
5826:
5818:
5814:
5804:
5800:
5790:
5786:
5779:
5774:
5743:
5740:
5730:
5727:
5723:
5716:
5713:
5704:
5702:
5636:
5633:
5630:
5618:
5611:
5598:
5595:
5592:
5580:
5570:
5550:
5530:
5504:
5501:
5498:
5486:
5477:
5464:
5461:
5458:
5446:
5441:
5437:
5431:
5412:
5410:
5391:
5359:
5356:
5353:
5341:
5334:
5328:
5309:
5303:
5271:
5268:
5265:
5253:
5246:
5240:
5221:
5219:
5208:
5189:
5183:
5167:
5150:
5144:
5125:
5119:
5103:
5086:
5084:
5076:
5066:
5062:
5052:
5046:
5036:
5032:
5013:
5008:
4988:
4981:
4966:
4955:
4944:
4933:
4927:
4908:
4902:
4886:
4869:
4867:
4859:
4849:
4845:
4828:
4821:
4806:
4795:
4784:
4773:
4767:
4748:
4742:
4726:
4709:
4707:
4699:
4689:
4685:
4666:
4662:
4652:
4649:
4624:
4616:
4603:
4600:
4597:
4588:
4584:
4579:
4574:
4561:
4558:
4555:
4546:
4542:
4531:
4527:
4523:
4508:
4486:
4482:
4478:
4463:
4434:
4424:
4423:
4416:
4411:
4406:
4393:
4388:
4375:
4372:
4369:
4360:
4356:
4351:
4346:
4333:
4330:
4327:
4318:
4314:
4309:
4303:
4300:
4297:
4268:
4261:
4258:
4255:
4246:
4227:
4218:
4215:
4212:
4192:
4188:
4182:
4179:
4176:
4167:
4163:
4141:
4134:
4131:
4128:
4119:
4100:
4091:
4088:
4085:
4065:
4061:
4055:
4052:
4049:
4040:
4036:
4012:
4008:
3985:
3981:
3971:
3956:
3945:
3938:
3935:
3932:
3923:
3904:
3895:
3892:
3889:
3869:
3865:
3860:
3849:
3842:
3839:
3836:
3827:
3808:
3799:
3796:
3793:
3773:
3769:
3763:
3760:
3757:
3735:
3706:
3702:
3696:
3685:
3667:
3648:
3617:
3613:
3608:
3597:
3579:
3560:
3529:
3524:
3517:
3513:
3509:
3507:
3484:
3479:
3464:
3435:
3404:
3387:
3382:
3367:
3338:
3304:
3300:
3296:
3294:
3277:
3248:
3234:
3228:
3227:
3220:
3219:
3214:
3210:
3204:
3201:
3193:
3189:
3184:
3180:
3173:
3169:
3165:
3161:
3152:
3145:
3125:
3097:
3093:
3060:
3056:
3052:
3044:
3026:
3016:
2993:
2989:
2985:
2970:
2941:
2927:
2926:
2920:
2919:
2903:
2901:
2897:
2893:
2883:
2845:
2796:
2787:
2782:
2775:
2767:
2759:
2752:
2748:
2734:
2724:
2720:
2695:
2674:
2659:
2643:
2628:
2612:
2603:
2590:
2576:
2567:
2555:
2551:
2535:
2517:
2494:
2456:
2392:then we call
2373:
2367:
2361:
2358:
2352:
2338:
2299:
2293:
2290:
2267:
2259:
2255:
2237:
2224:
2221:
2218:
2209:
2187:
2169:
2142:
2122:
2102:
2094:
2076:
2049:
2042:implies that
2041:
2037:
2033:
2015:
1999:
1996:
1993:
1987:
1984:
1975:
1959:
1944:
1940:
1936:
1932:
1928:
1924:
1919:
1902:
1892:
1881:
1877:
1873:
1870:
1859:
1855:
1851:
1848:
1837:
1833:
1813:
1809:
1807:
1798:
1794:
1785:
1775:
1768:
1758:
1754:
1744:
1738:
1728:
1724:
1713:
1709:
1706:
1697:
1687:
1680:
1670:
1666:
1656:
1650:
1640:
1636:
1625:
1621:
1618:
1609:
1599:
1592:
1582:
1578:
1568:
1562:
1552:
1548:
1537:
1532:
1522:
1519:
1494:
1481:
1464:
1460:
1447:
1436:
1421:
1392:
1370:
1367:
1364:
1361:
1358:
1350:
1346:
1342:
1336:
1333:
1330:
1327:
1324:
1316:
1312:
1308:
1302:
1299:
1296:
1293:
1290:
1282:
1278:
1271:
1265:
1262:
1259:
1256:
1253:
1222:
1194:
1153:
1151:
1147:
1129:
1114:
1106:
1105:
1100:
1096:
1095:line integral
1092:
1091:
1090:
1088:
1087:line integral
1084:
1080:
1076:
1058:
1043:
1039:
1035:
1031:
1027:
1026:George Stokes
1023:
1019:
1015:
1000:
971:
960:
955:
953:
948:
946:
941:
940:
938:
937:
930:
927:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
897:
896:
888:
887:
880:
877:
875:
872:
870:
867:
865:
862:
861:
853:
852:
841:
838:
836:
833:
831:
828:
827:
826:
825:
815:
814:
803:
800:
798:
795:
793:
790:
788:
785:
783:
782:Line integral
780:
778:
775:
773:
770:
769:
768:
767:
763:
762:
757:
754:
752:
749:
747:
744:
742:
739:
738:
737:
736:
732:
731:
725:
724:Multivariable
719:
718:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
678:
677:
676:
672:
671:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
637:
636:
635:
629:
623:
622:
611:
608:
606:
603:
601:
598:
596:
593:
591:
587:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
561:
560:
559:
556:
553:
552:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
521:
517:
514:
513:
512:
511:
505:
499:
498:
487:
484:
482:
479:
477:
474:
472:
469:
466:
462:
459:
457:
454:
451:
447:
443:
442:trigonometric
439:
436:
434:
431:
429:
426:
424:
421:
420:
419:
418:
414:
413:
408:
405:
403:
400:
398:
395:
393:
390:
387:
383:
380:
378:
375:
374:
373:
372:
368:
367:
362:
359:
357:
354:
352:
349:
348:
347:
346:
340:
334:
333:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
272:
271:
268:
265:
264:
259:
256:
254:
253:Related rates
251:
249:
246:
244:
241:
239:
236:
234:
231:
230:
229:
228:
224:
223:
216:
213:
211:
210:of a function
208:
206:
205:infinitesimal
203:
202:
201:
198:
195:
191:
188:
187:
186:
185:
181:
180:
174:
168:
167:
161:
158:
156:
153:
151:
148:
147:
142:
139:
137:
134:
133:
129:
126:
125:
124:
123:
104:
98:
95:
89:
83:
80:
77:
74:
67:
60:
57:
51:
46:
42:
33:
32:
29:
26:
25:
21:
20:
12070:
12051:
12045:
12036:
12030:
12003:
11999:
11993:
11965:
11958:
11948:
11922:
11883:
11877:
11825:
11806:
11800:
11780:
11775:
11596:
11368:
11094:
11090:
11085:
11081:
11072:
11068:
11063:
11059:
10947:
10930:is always a
10870:if the loop
10868:Jordan curve
10843:
10757:
10744:
10630:
10504:
10428:
10424:
10418:
10375:Theorem 2-2.
10374:
10373:
10366:
10361:
10358:
10350:
10344:
10340:
10336:
10332:
10328:
10325:
10317:
10311:
10307:
10303:
10300:
10292:
10285:
10281:
10277:
10273:
10270:
10264:
10258:
10250:
10246:
10236:
10232:
10226:
10222:
10170:
10169:
10150:
10087:
10079:
10073:
10069:
10065:
10061:
10057:
10054:
10046:
10040:
10036:
10032:
10029:
10021:
10014:
10007:
10003:
9999:
9996:
9990:
9984:
9976:
9972:
9966:
9962:
9956:
9949:
9943:
9896:
9895:
9891:
9830:
9827:
9733:
9721:
9623:
9516:
9427:
9203:
9199:
9195:
9186:
9177:
9174:
8736:
8732:
8726:
8723:, and split
8718:
8715:
8664:
8655:
8648:
8631:
8619:
8610:
8604:
8600:
8591:
8582:
8579:
8481:
8473:
8466:
8462:
8458:
8454:
8451:
8443:
8436:
8429:
8425:
8421:
8418:
8410:
8403:
8396:
8392:
8388:
8385:
8377:
8369:
8365:
8359:
8352:
8345:
8339:
8292:
8291:
8284:
8269:
8259:
8255:irrotational
8254:
8249:
8246:
8239:
8230:irrotational
8229:
8186:
8181:
8180:
8173:
8165:Applications
7873:
7869:
7863:
7860:
7701:via the map
7627:
7616:
7387:
7310:
7147:
6917:
6685:
6502:, we obtain
6440:
6434:
6428:
6424:
6420:
6416:
6412:
6408:
6371:
6367:
6362:
6358:
6353:
6349:
5949:
5945:
5941:
5938:
5705:
5571:
5402:(chain rule)
5009:
4665:product rule
4658:
4650:
4421:
4420:
4414:
4408:This is the
4407:
3972:
3741:
3232:
3225:
3224:
3217:
3216:
3212:
3208:
3205:
3199:
3191:
3187:
3182:
3178:
3174:
3167:
3163:
3159:
3143:
2924:
2923:
2916:
2914:
2889:
2665:
2604:
2457:
2258:neighborhood
2210:
1976:
1920:
1159:
1152:, a 2-form.
1110:
1102:
1075:vector field
1034:curl theorem
1033:
1029:
1017:
1013:
1012:
690:
438:Substitution
200:Differential
173:Differential
10254:; that is,
2627:dot product
2605:Here, the "
2339:defined by
2337:space curve
1022:Lord Kelvin
899:Precalculus
892:Miscellanea
857:Specialized
764:Definitions
531:Alternating
369:Definitions
182:Definitions
12099:Categories
12085:(22 #5980
12013:2312.17268
11860:Bai-Fu-Kan
11834:Sho-Ka-Bou
11792:References
11421:, for all
10499:See also:
10362:continuous
10265:continuous
10216:is called
9980:such that
9897:Lemma 2-2.
8373:such that
7945:Hodge star
5939:Note that
4663:, via the
2432:, written
1073:. Given a
879:Variations
874:Stochastic
864:Fractional
733:Formalisms
696:Divergence
665:Identities
645:Divergence
190:Derivative
141:Continuity
12110:Mechanics
11560:⋅
11537:⋅
11471:⋅
11438:×
11397:∈
11347:‖
11344:⋅
11341:‖
11333:and the "
11308:∂
11303:φ
11300:∂
11264:∂
11259:φ
11256:∂
11207:∂
11202:φ
11199:∂
11151:∂
11146:φ
11143:∂
10918:Γ
10898:ψ
10878:γ
10854:Γ
10827:ψ
10797:image set
10774:ψ
10768:Σ
10718:⋅
10710:×
10706:∇
10700:Σ
10696:∬
10679:⋅
10669:Σ
10666:∂
10662:∮
10604:⋅
10596:×
10592:∇
10586:Σ
10582:∬
10565:⋅
10555:Σ
10552:∂
10548:∮
10440:∫
10388:⊆
10184:⊆
10167:follows:
10097:∫
9910:⊆
9866:Γ
9862:⊖
9814:Γ
9792:Γ
9787:∮
9780:Γ
9758:Γ
9753:∮
9696:Γ
9692:⊖
9680:Γ
9653:Γ
9649:⊖
9637:Γ
9610:Γ
9588:Γ
9583:∮
9562:∑
9555:Γ
9539:Γ
9535:∮
9502:Γ
9486:Γ
9482:∮
9461:×
9458:∇
9449:∬
9392:Γ
9388:⊕
9379:Γ
9375:⊕
9366:Γ
9362:⊕
9353:Γ
9331:γ
9306:Γ
9251:γ
9219:Γ
9155:γ
9151:⊕
9142:γ
9138:⊕
9129:γ
9125:⊕
9116:γ
9106:∂
9076:−
9043:γ
9030:→
9003:γ
8983:−
8956:γ
8943:→
8916:γ
8875:γ
8862:→
8835:γ
8794:γ
8781:→
8754:γ
8701:⊖
8677:⊕
8669:and use "
8532:∫
8491:∫
8306:⊆
8204:⊆
8148:Σ
8138:⋅
8130:×
8127:∇
8121:Σ
8117:∬
8102:×
8099:∇
8095:ω
8091:⋆
8085:Σ
8081:∫
8065:ω
8053:Σ
8049:∫
8033:ω
8026:Σ
8023:∂
8019:∮
8010:γ
8000:⋅
7989:Σ
7986:∂
7982:∮
7916:ω
7897:×
7894:∇
7890:ω
7886:⋆
7781:↦
7648:→
7574:∂
7559:∂
7553:−
7544:∂
7529:∂
7512:∬
7493:⋅
7400:γ
7396:∮
7263:∂
7248:∂
7242:−
7233:∂
7218:∂
7201:∬
7193:Σ
7183:⋅
7172:×
7169:∇
7161:Σ
7157:∬
7090:∂
7084:ψ
7080:∂
7074:×
7050:∂
7044:ψ
7040:∂
7034:⋅
7012:ψ
6997:×
6994:∇
6981:∬
6969:Σ
6961:⋅
6950:×
6947:∇
6939:Σ
6935:∬
6894:∂
6888:ψ
6884:∂
6878:×
6869:∂
6863:ψ
6859:∂
6853:⋅
6842:×
6839:∇
6824:∂
6818:ψ
6814:∂
6808:×
6797:×
6794:∇
6788:⋅
6779:∂
6773:ψ
6769:∂
6750:∂
6735:∂
6729:−
6720:∂
6705:∂
6660:∈
6637:×
6626:×
6623:∇
6569:ψ
6555:−
6525:ψ
6459:ψ
6318:×
6282:−
6221:−
6194:×
6144:−
6097:−
6070:×
6041:−
5973:−
5907:−
5880:−
5853:−
5624:ψ
5612:−
5586:ψ
5548:∂
5542:ψ
5538:∂
5492:ψ
5478:−
5452:ψ
5438:⋅
5429:∂
5423:ψ
5419:∂
5389:∂
5383:ψ
5379:∂
5347:ψ
5335:⋅
5326:∂
5320:ψ
5316:∂
5310:−
5301:∂
5295:ψ
5291:∂
5259:ψ
5247:⋅
5238:∂
5232:ψ
5228:∂
5206:∂
5200:ψ
5196:∂
5190:⋅
5181:∂
5172:ψ
5168:∘
5157:∂
5151:−
5142:∂
5136:ψ
5132:∂
5126:⋅
5117:∂
5108:ψ
5104:∘
5093:∂
5074:∂
5059:∂
5053:−
5044:∂
5029:∂
4986:∂
4979:∂
4973:ψ
4963:∂
4956:⋅
4949:ψ
4945:∘
4925:∂
4919:ψ
4915:∂
4909:⋅
4900:∂
4891:ψ
4887:∘
4876:∂
4857:∂
4842:∂
4826:∂
4819:∂
4813:ψ
4803:∂
4796:⋅
4789:ψ
4785:∘
4765:∂
4759:ψ
4755:∂
4749:⋅
4740:∂
4731:ψ
4727:∘
4716:∂
4697:∂
4682:∂
4625:⋅
4532:γ
4528:∮
4509:⋅
4487:γ
4483:∮
4464:⋅
4442:Σ
4439:∂
4435:∮
4244:∂
4238:ψ
4234:∂
4228:⋅
4206:ψ
4117:∂
4111:ψ
4107:∂
4101:⋅
4079:ψ
3921:∂
3915:ψ
3911:∂
3905:⋅
3883:ψ
3825:∂
3819:ψ
3815:∂
3809:⋅
3787:ψ
3707:⋅
3665:∂
3659:ψ
3655:∂
3649:⋅
3631:ψ
3577:∂
3571:ψ
3567:∂
3561:⋅
3543:ψ
3518:γ
3514:∮
3485:⋅
3451:ψ
3417:ψ
3388:⋅
3354:ψ
3320:ψ
3305:γ
3301:∮
3288:Γ
3278:⋅
3256:Σ
3253:∂
3249:∮
3126:γ
3113:ψ
3094:⋅
3076:ψ
3061:γ
3057:∮
3045:γ
3037:ψ
3027:⋅
3017:γ
3009:ψ
2994:γ
2990:∮
2981:Γ
2971:⋅
2949:Σ
2946:∂
2942:∮
2779:∇
2776:×
2768:−
2760:×
2757:∇
2749:⋅
2745:Σ
2729:Σ
2725:∬
2707:Γ
2696:⋅
2682:Σ
2679:∂
2675:∮
2613:⋅
2587:Σ
2577:⋅
2568:×
2565:∇
2560:Σ
2556:∬
2547:Γ
2536:⋅
2525:Σ
2522:∂
2518:∮
2443:Σ
2440:∂
2420:Σ
2400:Γ
2368:γ
2362:ψ
2347:Γ
2323:Γ
2294:ψ
2288:Σ
2254:piecewise
2228:→
2219:ψ
2196:Σ
2143:γ
2050:γ
2032:piecewise
2006:→
1985:γ
1931:Lipschitz
1821:Σ
1818:∂
1814:∮
1766:∂
1751:∂
1745:−
1736:∂
1721:∂
1678:∂
1663:∂
1657:−
1648:∂
1633:∂
1590:∂
1575:∂
1569:−
1560:∂
1545:∂
1527:Σ
1523:∬
1491:Γ
1482:⋅
1472:Σ
1469:∂
1465:∮
1457:Σ
1448:⋅
1437:×
1434:∇
1426:Σ
1422:∬
1401:Σ
1226:Γ
1223:≡
1220:Σ
1217:∂
1168:Σ
869:Malliavin
756:Geometric
655:Laplacian
605:Dirichlet
516:Geometric
96:−
43:∫
11372:For all
11317:‖
11294:‖
11273:‖
11250:‖
10513:and the
10348:for all
10315:for all
10290:for all
10235:: × →
10077:for all
10044:for all
10019:for all
9975:: × →
9440:. That
9098:so that
8603:: × →
8471:for all
8441:for all
8408:for all
8368:: × →
8343:and let
6432:for any
4410:pullback
3175:Now let
2211:Suppose
2062:divides
1079:integral
909:Glossary
819:Advanced
797:Jacobian
751:Exterior
681:Gradient
673:Theorems
640:Gradient
579:Integral
541:Binomial
526:Harmonic
386:improper
382:Integral
339:Integral
321:Reynolds
296:Quotient
225:Concepts
61:′
28:Calculus
12083:0115178
11969:. Cole.
10310:, 1) =
10280:, 0) =
10039:, 1) =
10006:, 0) =
8654:, ...,
8428:, 1) =
8395:, 0) =
8267:, then
7969:is the
7943:is the
6650:for all
5014:. So,
2860:. When
2335:is the
2280:, with
2093:compact
2034:smooth
1413:, then
1156:Theorem
1101:of its
1081:of the
1038:theorem
1036:, is a
904:History
802:Hessian
691:Stokes'
686:Green's
518: (
440: (
384: (
306:Inverse
281:Product
192: (
12058:
11890:
11866:
11840:
11813:
11228:Here,
10088:Then,
9939:subset
9934:be an
9436:under
8482:Then,
8335:subset
8330:be an
8190:on an
7939:where
7619:Q.E.D.
6346:Here,
4418:along
4282:then,
3139:where
2915:As in
2809:where
2509:, then
2038:. The
1146:1-form
1028:, the
1020:after
746:Tensor
741:Matrix
628:Vector
546:Taylor
504:Series
136:Limits
12008:arXiv
10749:Notes
10427:: →
10240:from
10225:: →
9955:: →
8721:= ×
8693:and "
8358:: →
8273:is a
8236:) if
6409:Thus
5760:3 × 3
2886:Proof
2315:. If
2030:be a
569:Ratio
536:Power
450:Euler
428:Discs
423:Parts
291:Power
286:Chain
215:total
12056:ISBN
11888:ISBN
11864:ISBN
11838:ISBN
11811:ISBN
11784:end.
10932:loop
10414:open
10377:Let
10343:) =
10339:(1,
10335:) =
10331:(0,
10173:Let
10072:) =
10068:(1,
10064:) =
10060:(0,
9936:open
9899:Let
9726:and
9671:and
9428:Let
9184:and
8716:Let
8617:and
8589:and
8465:(1,
8461:) =
8457:(0,
8332:open
8295:Let
8238:∇ ×
8192:open
7947:and
6498:for
6438:.
4000:and
2929:and
1160:Let
1104:curl
1093:The
1083:curl
1024:and
650:Curl
610:Abel
574:Root
12018:doi
10938:of
10819:by
10799:of
10412:be
10263:is
10212:.
9989:is
9836:= Γ
9624:As
8741:.
8263:is
8253:is
8242:= 0
8228:is
7867:as
4412:of
3230:at
3157:at
3153:of
2629:in
2260:of
2252:is
2188:of
1044:on
1040:in
276:Sum
12101::
12080:MR
12016:.
12004:92
12002:.
11975:^
11932:^
11902:^
11851:^
11456:,
11089:,
11067:,
10653::
10539::
10353:∈
10320:∈
10295:∈
10249:∈
10082:∈
10049:∈
10024:∈
9965:∈
9841:,
9202:→
9198::
8634:.
8476:∈
8446:∈
8413:∈
8351:,
8289:.
8277:.
8244:.
6690::
6427:×
6423:=
6415:−
6406:.
6366:,
6357:,
5948:×
5944:↦
5915:12
5902:21
5888:31
5875:13
5861:23
5848:32
5703:.
4667::
3203:.
3186:,
3172:.
3162:=
2658:.
2455:.
2208:.
1974:.
1009:).
1003:∂Σ
987:∂Σ
979:∂Σ
448:,
444:,
12064:.
12024:.
12020::
12010::
11896:.
11819:.
11758:.
11753:]
11747:1
11740:0
11733:0
11727:[
11722:=
11717:3
11712:e
11707:,
11702:]
11696:0
11689:1
11682:0
11676:[
11671:=
11666:2
11661:e
11656:,
11651:]
11645:0
11638:0
11631:1
11625:[
11620:=
11615:1
11610:e
11591:.
11577:a
11569:T
11564:A
11555:b
11550:=
11545:b
11540:A
11532:a
11508:b
11503:A
11497:T
11490:a
11484:=
11479:b
11474:A
11466:a
11441:n
11435:n
11432:;
11429:A
11407:n
11402:R
11392:b
11387:,
11382:a
11363:.
11321:,
11311:v
11290:=
11285:v
11281:h
11277:,
11267:u
11246:=
11241:u
11237:h
11216:.
11210:v
11189:v
11185:h
11181:1
11176:=
11171:v
11166:t
11161:,
11154:u
11133:u
11129:h
11125:1
11120:=
11115:u
11110:t
11098:}
11095:v
11091:t
11086:u
11082:t
11080:{
11076:}
11073:v
11069:e
11064:u
11060:e
11058:{
11044:.
11039:]
11033:1
11026:0
11020:[
11015:=
11010:v
11005:e
11000:,
10995:]
10989:0
10982:1
10976:[
10971:=
10966:u
10961:e
10807:D
10783:)
10780:D
10777:(
10771:=
10731:.
10727:S
10722:d
10714:B
10692:=
10688:l
10683:d
10675:B
10640:B
10617:.
10613:S
10608:d
10600:E
10578:=
10574:l
10569:d
10561:E
10526:E
10480:0
10477:=
10472:0
10468:c
10463:d
10457:F
10449:0
10445:c
10429:U
10425:c
10419:F
10398:3
10393:R
10385:U
10355:.
10351:s
10345:p
10341:s
10337:H
10333:s
10329:H
10322:,
10318:t
10312:p
10308:t
10306:(
10304:H
10297:,
10293:t
10288:)
10286:t
10284:(
10282:c
10278:t
10276:(
10274:H
10267:,
10261:H
10251:c
10247:p
10242:c
10237:M
10233:H
10227:M
10223:c
10214:M
10194:n
10189:R
10181:M
10161:H
10157:U
10153:H
10137:0
10134:=
10129:0
10125:c
10120:d
10114:F
10106:0
10102:c
10084:.
10080:s
10074:p
10070:s
10066:H
10062:s
10058:H
10051:,
10047:t
10041:p
10037:t
10035:(
10033:H
10026:,
10022:t
10017:)
10015:t
10013:(
10011:0
10008:c
10004:t
10002:(
10000:H
9993:,
9987:H
9977:U
9973:H
9967:U
9963:p
9957:U
9953:0
9950:c
9944:F
9920:3
9915:R
9907:U
9870:3
9859:=
9854:3
9850:c
9838:1
9834:1
9831:c
9810:d
9804:F
9796:3
9783:+
9776:d
9770:F
9762:1
9749:=
9746:0
9736:)
9734:s
9732:(
9730:4
9728:Γ
9724:)
9722:s
9720:(
9718:2
9716:Γ
9700:4
9689:=
9684:2
9657:4
9646:=
9641:2
9626:H
9606:d
9600:F
9592:i
9577:4
9572:1
9569:=
9566:i
9558:=
9551:d
9545:F
9531:=
9528:0
9517:F
9498:d
9492:F
9478:=
9475:S
9471:d
9465:F
9453:S
9438:H
9434:D
9430:S
9410:)
9407:t
9404:(
9401:)
9396:4
9383:3
9370:2
9357:1
9349:(
9346:=
9343:)
9340:)
9337:t
9334:(
9328:(
9325:H
9322:=
9315:)
9312:t
9309:(
9299:4
9296:,
9293:3
9290:,
9287:2
9284:,
9281:1
9278:=
9275:i
9269:)
9266:)
9263:t
9260:(
9255:i
9247:(
9244:H
9241:=
9234:)
9231:t
9228:(
9223:i
9204:M
9200:D
9196:H
9190:1
9187:c
9181:0
9178:c
9159:4
9146:3
9133:2
9120:1
9112:=
9109:D
9082:)
9079:s
9073:1
9070:,
9067:0
9064:(
9061:=
9058:)
9055:s
9052:(
9047:4
9036:;
9033:D
9027:]
9024:1
9021:,
9018:0
9015:[
9012::
9007:4
8995:)
8992:1
8989:,
8986:t
8980:1
8977:(
8974:=
8971:)
8968:t
8965:(
8960:3
8949:;
8946:D
8940:]
8937:1
8934:,
8931:0
8928:[
8925::
8920:3
8908:)
8905:s
8902:,
8899:1
8896:(
8893:=
8890:)
8887:s
8884:(
8879:2
8868:;
8865:D
8859:]
8856:1
8853:,
8850:0
8847:[
8844::
8839:2
8827:)
8824:0
8821:,
8818:t
8815:(
8812:=
8809:)
8806:t
8803:(
8798:1
8787:;
8784:D
8778:]
8775:1
8772:,
8769:0
8766:[
8763::
8758:1
8737:j
8733:γ
8727:D
8725:∂
8719:D
8659:4
8656:γ
8652:1
8649:γ
8623:1
8620:c
8614:0
8611:c
8605:U
8601:H
8595:1
8592:c
8586:0
8583:c
8564:1
8560:c
8555:d
8549:F
8541:1
8537:c
8528:=
8523:0
8519:c
8514:d
8508:F
8500:0
8496:c
8478:.
8474:s
8469:)
8467:s
8463:H
8459:s
8455:H
8448:,
8444:t
8439:)
8437:t
8435:(
8433:1
8430:c
8426:t
8424:(
8422:H
8415:,
8411:t
8406:)
8404:t
8402:(
8400:0
8397:c
8393:t
8391:(
8389:H
8380:H
8370:U
8366:H
8360:U
8356:1
8353:c
8349:0
8346:c
8340:F
8316:3
8311:R
8303:U
8270:F
8260:F
8250:F
8240:F
8232:(
8214:3
8209:R
8201:U
8187:F
8143:d
8134:F
8113:=
8106:F
8077:=
8070:F
8060:d
8045:=
8038:F
8015:=
8005:d
7996:F
7956:d
7941:★
7927:,
7921:F
7911:d
7907:=
7901:F
7874:F
7870:ω
7864:F
7847:.
7844:z
7840:d
7833:z
7829:F
7825:+
7822:y
7818:d
7811:y
7807:F
7803:+
7800:x
7796:d
7789:x
7785:F
7776:3
7771:e
7764:z
7760:F
7756:+
7751:2
7746:e
7739:y
7735:F
7731:+
7726:1
7721:e
7714:x
7710:F
7687:3
7682:R
7658:3
7653:R
7643:3
7638:R
7603:v
7599:d
7594:u
7590:d
7584:)
7577:v
7567:u
7563:P
7547:u
7537:v
7533:P
7522:(
7516:D
7508:=
7503:l
7498:d
7490:)
7485:v
7480:e
7475:)
7472:v
7469:,
7466:u
7463:(
7457:v
7453:P
7448:+
7443:u
7438:e
7433:)
7430:v
7427:,
7424:u
7421:(
7415:u
7411:P
7406:(
7373:)
7370:v
7367:,
7364:u
7361:(
7356:v
7352:P
7348:,
7345:)
7342:v
7339:,
7336:u
7333:(
7328:u
7324:P
7292:v
7288:d
7283:u
7279:d
7273:)
7266:v
7256:u
7252:P
7236:u
7226:v
7222:P
7211:(
7205:D
7197:=
7188:d
7180:)
7176:F
7166:(
7129:v
7125:d
7120:u
7116:d
7111:)
7108:v
7105:,
7102:u
7099:(
7093:v
7071:)
7068:v
7065:,
7062:u
7059:(
7053:u
7031:)
7028:)
7025:v
7022:,
7019:u
7016:(
7008:(
7005:)
7001:F
6991:(
6985:D
6977:=
6965:d
6958:)
6954:F
6944:(
6897:v
6872:u
6850:)
6846:F
6836:(
6833:=
6827:u
6805:)
6801:F
6791:(
6782:v
6763:=
6753:v
6743:u
6739:P
6723:u
6713:v
6709:P
6670:3
6665:R
6656:x
6645:,
6641:x
6634:)
6630:F
6620:(
6617:=
6613:x
6608:)
6601:T
6595:)
6591:F
6585:)
6582:v
6579:,
6576:u
6573:(
6564:J
6560:(
6551:)
6547:F
6541:)
6538:v
6535:,
6532:u
6529:(
6520:J
6516:(
6511:(
6500:A
6485:)
6481:F
6475:)
6472:v
6469:,
6466:u
6463:(
6454:J
6450:(
6435:x
6429:x
6425:a
6421:x
6419:)
6417:A
6413:A
6411:(
6392:3
6387:R
6375:}
6372:3
6368:e
6363:2
6359:e
6354:1
6350:e
6348:{
6328:3
6323:e
6314:a
6310:=
6305:]
6299:0
6290:1
6286:a
6273:2
6269:a
6262:[
6257:=
6248:3
6243:e
6237:)
6230:T
6225:A
6218:A
6214:(
6204:2
6199:e
6190:a
6186:=
6181:]
6173:1
6169:a
6161:0
6152:3
6148:a
6138:[
6133:=
6124:2
6119:e
6113:)
6106:T
6101:A
6094:A
6090:(
6080:1
6075:e
6066:a
6062:=
6057:]
6049:2
6045:a
6032:3
6028:a
6020:0
6014:[
6009:=
6000:1
5995:e
5989:)
5982:T
5977:A
5970:A
5966:(
5950:x
5946:a
5942:x
5923:]
5911:A
5898:A
5884:A
5871:A
5857:A
5844:A
5837:[
5832:=
5827:]
5819:3
5815:a
5805:2
5801:a
5791:1
5787:a
5780:[
5775:=
5771:a
5744:j
5741:i
5737:)
5731:j
5728:i
5724:A
5720:(
5717:=
5714:A
5684:T
5656:T
5651:)
5646:F
5640:)
5637:v
5634:,
5631:u
5628:(
5619:J
5615:(
5608:F
5602:)
5599:v
5596:,
5593:u
5590:(
5581:J
5551:u
5531:)
5524:T
5518:)
5514:F
5508:)
5505:v
5502:,
5499:u
5496:(
5487:J
5483:(
5474:F
5468:)
5465:v
5462:,
5459:u
5456:(
5447:J
5442:(
5432:v
5413:=
5392:v
5373:)
5369:F
5363:)
5360:v
5357:,
5354:u
5351:(
5342:J
5338:(
5329:u
5304:u
5285:)
5281:F
5275:)
5272:v
5269:,
5266:u
5263:(
5254:J
5250:(
5241:v
5222:=
5209:u
5184:v
5176:)
5164:F
5160:(
5145:v
5120:u
5112:)
5100:F
5096:(
5087:=
5077:v
5067:u
5063:P
5047:u
5037:v
5033:P
4989:v
4982:u
4967:2
4953:)
4941:F
4937:(
4934:+
4928:v
4903:u
4895:)
4883:F
4879:(
4870:=
4860:u
4850:v
4846:P
4829:u
4822:v
4807:2
4793:)
4781:F
4777:(
4774:+
4768:u
4743:v
4735:)
4723:F
4719:(
4710:=
4700:v
4690:u
4686:P
4635:l
4630:d
4622:)
4617:v
4612:e
4607:)
4604:v
4601:,
4598:u
4595:(
4589:v
4585:P
4580:+
4575:u
4570:e
4565:)
4562:v
4559:,
4556:u
4553:(
4547:u
4543:P
4538:(
4524:=
4519:l
4514:d
4506:)
4502:y
4498:(
4494:P
4479:=
4474:l
4469:d
4461:)
4457:x
4453:(
4449:F
4422:ψ
4415:F
4394:.
4389:v
4384:e
4379:)
4376:v
4373:,
4370:u
4367:(
4361:v
4357:P
4352:+
4347:u
4342:e
4337:)
4334:v
4331:,
4328:u
4325:(
4319:u
4315:P
4310:=
4307:)
4304:v
4301:,
4298:u
4295:(
4291:P
4269:)
4265:)
4262:v
4259:,
4256:u
4253:(
4247:v
4225:)
4222:)
4219:v
4216:,
4213:u
4210:(
4202:(
4198:F
4193:(
4189:=
4186:)
4183:v
4180:,
4177:u
4174:(
4168:v
4164:P
4142:)
4138:)
4135:v
4132:,
4129:u
4126:(
4120:u
4098:)
4095:)
4092:v
4089:,
4086:u
4083:(
4075:(
4071:F
4066:(
4062:=
4059:)
4056:v
4053:,
4050:u
4047:(
4041:u
4037:P
4013:v
4009:P
3986:u
3982:P
3957:v
3952:e
3946:)
3942:)
3939:v
3936:,
3933:u
3930:(
3924:v
3902:)
3899:)
3896:v
3893:,
3890:u
3887:(
3879:(
3875:F
3870:(
3866:+
3861:u
3856:e
3850:)
3846:)
3843:v
3840:,
3837:u
3834:(
3828:u
3806:)
3803:)
3800:v
3797:,
3794:u
3791:(
3783:(
3779:F
3774:(
3770:=
3767:)
3764:v
3761:,
3758:u
3755:(
3751:P
3717:y
3712:d
3703:)
3697:v
3692:e
3686:)
3682:)
3678:y
3674:(
3668:v
3646:)
3643:)
3639:y
3635:(
3627:(
3623:F
3618:(
3614:+
3609:u
3604:e
3598:)
3594:)
3590:y
3586:(
3580:u
3558:)
3555:)
3551:y
3547:(
3539:(
3535:F
3530:(
3525:(
3510:=
3499:)
3495:y
3490:d
3480:v
3475:e
3470:(
3465:v
3460:e
3455:)
3447:(
3441:y
3436:J
3432:)
3429:)
3425:y
3421:(
3413:(
3409:F
3405:+
3402:)
3398:y
3393:d
3383:u
3378:e
3373:(
3368:u
3363:e
3358:)
3350:(
3344:y
3339:J
3335:)
3332:)
3328:y
3324:(
3316:(
3312:F
3297:=
3283:d
3275:)
3271:x
3267:(
3263:F
3233:y
3226:ψ
3218:ψ
3213:y
3209:J
3200:R
3195:}
3192:v
3188:e
3183:u
3179:e
3177:{
3170:)
3168:t
3166:(
3164:γ
3160:y
3155:ψ
3147:ψ
3144:y
3141:J
3122:d
3117:)
3109:(
3103:y
3098:J
3091:)
3088:)
3084:y
3080:(
3072:(
3068:F
3053:=
3049:)
3041:(
3032:d
3024:)
3021:)
3013:(
3005:(
3001:F
2986:=
2976:d
2968:)
2964:x
2960:(
2956:F
2931:γ
2925:ψ
2869:g
2846:3
2841:R
2818:g
2797:,
2793:g
2788:]
2783:)
2772:F
2764:F
2753:(
2740:d
2735:[
2721:=
2717:g
2712:)
2701:d
2691:F
2687:(
2644:3
2639:R
2591:.
2582:d
2572:F
2552:=
2541:d
2531:F
2495:3
2490:R
2467:F
2380:)
2377:)
2374:t
2371:(
2365:(
2359:=
2356:)
2353:t
2350:(
2303:)
2300:D
2297:(
2291:=
2268:D
2238:3
2233:R
2225:D
2222::
2170:3
2165:R
2123:D
2103:D
2077:2
2072:R
2016:2
2011:R
2003:]
2000:b
1997:,
1994:a
1991:[
1988::
1960:2
1955:R
1903:.
1898:)
1893:z
1889:d
1882:z
1878:F
1874:+
1871:y
1867:d
1860:y
1856:F
1852:+
1849:x
1845:d
1838:x
1834:F
1828:(
1810:=
1799:)
1795:y
1791:d
1786:x
1782:d
1776:)
1769:y
1759:x
1755:F
1739:x
1729:y
1725:F
1714:(
1710:+
1707:x
1703:d
1698:z
1694:d
1688:)
1681:x
1671:z
1667:F
1651:z
1641:x
1637:F
1626:(
1622:+
1619:z
1615:d
1610:y
1606:d
1600:)
1593:z
1583:y
1579:F
1563:y
1553:z
1549:F
1538:(
1533:(
1495:.
1486:d
1478:F
1461:=
1452:d
1445:)
1441:F
1431:(
1377:)
1374:)
1371:z
1368:,
1365:y
1362:,
1359:x
1356:(
1351:z
1347:F
1343:,
1340:)
1337:z
1334:,
1331:y
1328:,
1325:x
1322:(
1317:y
1313:F
1309:,
1306:)
1303:z
1300:,
1297:y
1294:,
1291:x
1288:(
1283:x
1279:F
1275:(
1272:=
1269:)
1266:z
1263:,
1260:y
1257:,
1254:x
1251:(
1247:F
1195:3
1190:R
1130:3
1125:R
1059:3
1054:R
1007:n
995:Σ
991:n
983:n
975:Σ
958:e
951:t
944:v
522:)
467:)
463:(
452:)
388:)
196:)
108:)
105:a
102:(
99:f
93:)
90:b
87:(
84:f
81:=
78:t
75:d
71:)
68:t
65:(
58:f
52:b
47:a
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