2296:
1344:
12104:
9996:
11904:
1316:
11038:
25:
7787:
9112:
12325:
9796:
13024:
11812:
1081:
12099:{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})+{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
10217:
12538:
9704:
6037:
7545:
10848:
8416:
10762:
7990:
11245:
8836:
8922:
9991:{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)\times \mathbf {f} _{2}(\mathbf {v} )+\mathbf {f} _{1}(\mathbf {v} )\times \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
12188:
11682:
2259:
9309:
4319:
5853:
10080:
4795:
1311:{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=f'_{\mathbf {v} }(\mathbf {x} )=D_{\mathbf {v} }f(\mathbf {x} )=Df(\mathbf {x} )(\mathbf {v} )=\partial _{\mathbf {v} }f(\mathbf {x} )=\mathbf {v} \cdot {\nabla f(\mathbf {x} )}=\mathbf {v} \cdot {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {x} }}.}
12405:
9574:
8256:
7848:
11033:{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)~f_{2}({\boldsymbol {S}})+f_{1}({\boldsymbol {S}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
11563:
7782:{\displaystyle \mathbf {L} ={\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix}}\mathbf {i} +{\begin{pmatrix}0&0&-1\\0&0&0\\1&0&0\end{pmatrix}}\mathbf {j} +{\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}}\mathbf {k} .}
10638:
2475:
11899:
11677:
11116:
9475:
1978:
5187:
9107:{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)~f_{2}(\mathbf {v} )+f_{1}(\mathbf {v} )~\left({\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
8712:
1768:
12320:{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
3038:
11807:{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}
2580:
10525:
5040:
3423:
12183:
4683:
6789:
8615:
9190:
9791:
8073:
4104:
6517:
4606:
9569:
10212:{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {f} _{2}}}\cdot \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
7843:
8159:
5813:
4670:
11412:
12533:{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
1873:
9699:{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}+{\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }
6032:{\displaystyle ^{N}f(\mathbf {x} )=f(\mathbf {x} +N{\boldsymbol {\varepsilon }})=f(\mathbf {x} +{\boldsymbol {\lambda }})=U({\boldsymbol {\lambda }})f(\mathbf {x} )=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right)f(\mathbf {x} ),}
3639:
2896:
5275:
2723:
10075:
10843:
10633:
7532:
7271:
4429:
2345:
12400:
5544:
4099:
4898:
9324:
8411:{\displaystyle {\frac {\partial f}{\partial \mathbf {n} }}=\nabla f(\mathbf {x} )\cdot \mathbf {n} =\nabla _{\mathbf {n} }{f}(\mathbf {x} )={\frac {\partial f}{\partial \mathbf {x} }}\cdot \mathbf {n} =Df(\mathbf {x} ).}
10757:{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}
5106:
7085:
6636:
7985:{\displaystyle U(R(\delta {\boldsymbol {\theta }}))f(\mathbf {x} )=f(\mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} )=f(\mathbf {x} )-(\delta {\boldsymbol {\theta }}\times \mathbf {x} )\cdot \nabla f.}
1655:
8917:
8707:
7381:
5436:
3990:
11111:
2799:
11819:
2488:
10380:
6210:
5370:
11240:{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial f_{2}}}~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
7455:
2928:
4915:
3726:
11597:
8831:{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}+{\frac {\partial f_{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }
3324:
1550:
9185:
6890:
5215:
8251:
6641:
8470:
7995:
6329:
3192:
180:
11341:
11286:
9319:
Let f(v) be a vector valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the second order tensor defined through its dot product with any vector u being
6374:
1983:
4539:
1610:
3285:
2254:{\displaystyle {\begin{aligned}0&=\lim _{t\to 0}{\frac {f(x+tv)-f(x)-tDf(x)(v)}{t}}\\&=\lim _{t\to 0}{\frac {f(x+tv)-f(x)}{t}}-Df(x)(v)\\&=\nabla _{v}f(x)-Df(x)(v).\end{aligned}}}
8438:
3239:
7796:
8078:
12111:
5754:
4611:
9304:{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial f_{1}}{\partial f_{2}}}~{\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} }
5678:
1647:
1805:
4508:
9711:
10309:
10256:
3544:
2815:
5226:
2645:
5749:
1410:
11585:
11407:
11385:
11363:
11308:
10547:
10375:
10353:
10331:
10278:
9489:
7460:
7182:
4324:
4015:
3872:
3847:
3782:
8445:
The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.
8208:
4790:{\displaystyle U({\boldsymbol {\lambda }})f(\mathbf {x} )=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right)f(\mathbf {x} )=f(\mathbf {x} +{\boldsymbol {\lambda }}).}
1454:
1432:
3539:
3503:
6243:
5465:
1973:
7173:
6118:
3668:
4818:
4314:{\displaystyle (1+\delta '\cdot D)(1+\delta \cdot D)S^{\rho }-(1+\delta \cdot D)(1+\delta '\cdot D)S^{\rho }=\sum _{\mu ,\nu }\delta '^{\mu }\delta ^{\nu }S_{\rho }.}
3916:
3892:
3822:
3757:
1895:
6920:
1921:
8075:
Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:
10003:
4608:
By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:
6929:
6522:
4449:
3802:
5093:
4020:
7276:
5405:
10769:
10559:
3921:
13995:
12332:
11558:{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})=\left_{\alpha =0}}
2739:
6123:
5304:
13983:
9479:
for all vectors u. The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u.
3673:
1470:
6794:
2470:{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h|\mathbf {v} |}},}
8843:
8633:
14105:
13990:
12551:
11894:{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})}
6248:
11045:
276:
7175:
This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means that
1789:
1555:
14141:
13973:
13968:
11672:{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}})}
9470:{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )=\left_{\alpha =0}}
7415:
13978:
13963:
13077:
5182:{\displaystyle f(\mathbf {x} +{\boldsymbol {\varepsilon }})=\left(1+{\boldsymbol {\varepsilon }}\cdot \nabla \right)f(\mathbf {x} ).}
42:
12722:– for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector.
13265:
13958:
13028:
12615:
2728:
1763:{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}
1456:
in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.
46:
9119:
5553:
5192:
13575:
13329:
12978:
12652:
8217:
7393:
4479:
3154:
2575:{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot {\frac {\mathbf {v} }{|\mathbf {v} |}}.}
537:
517:
11313:
11258:
10520:{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})=\left_{\alpha =0}}
8619:
for all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of
5683:
3049:
3033:{\displaystyle \nabla _{\mathbf {v} }(h\circ g)(\mathbf {p} )=h'(g(\mathbf {p} ))\nabla _{\mathbf {v} }g(\mathbf {p} ).}
13127:
12864:
6791:
After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition
5035:{\displaystyle f(x+\varepsilon )=f(x)+\varepsilon \,{\frac {df}{dx}}=\left(1+\varepsilon \,{\frac {d}{dx}}\right)f(x).}
1013:
576:
3418:{\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )=\left.{\frac {d}{d\tau }}f\circ \gamma (\tau )\right|_{\tau =0}.}
3248:
99:
14073:
13932:
12955:
12914:
12889:
12839:
12814:
12785:
12752:
12581:
3201:
532:
255:
68:
12178:{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))}
13487:
13403:
8425:
Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of
522:
14146:
14068:
14000:
13625:
13480:
13448:
13207:
6784:{\displaystyle f^{a}({\bar {\xi }},\xi )=\xi ^{a}+{\bar {\xi }}^{a}+\sum _{b,c}f^{abc}{\bar {\xi }}^{b}\xi ^{c}.}
853:
527:
507:
189:
8610:{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )=\left_{\alpha =0}}
4321:
It can be argued that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
14136:
14131:
13701:
13678:
13393:
1618:
50:
9786:{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} )}
8068:{\displaystyle U(R(\delta \mathbf {\theta } ))=1-(\delta \mathbf {\theta } \times \mathbf {x} )\cdot \nabla .}
7098:
13791:
13729:
13524:
13398:
13070:
635:
582:
468:
6512:{\displaystyle U(T(\xi ))=1+i\sum _{a}\xi ^{a}t_{a}+{\frac {1}{2}}\sum _{b,c}\xi ^{b}\xi ^{c}t_{bc}+\cdots }
13277:
13255:
4601:{\displaystyle U({\boldsymbol {\lambda }})=\exp \left(-i{\boldsymbol {\lambda }}\cdot \mathbf {P} \right).}
294:
266:
14100:
12696:, the directional derivative is still defined, and a similar relation exists with the exterior derivative.
10283:
10230:
377:
14085:
13851:
13465:
13287:
2638:
2295:
886:
499:
337:
309:
9564:{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} )}
3288:
1354:
13470:
13240:
11568:
11390:
11368:
11346:
11291:
10530:
10358:
10336:
10314:
10261:
8185:
7838:{\displaystyle \mathbf {x} \rightarrow \mathbf {x} -\delta {\boldsymbol {\theta }}\times \mathbf {x} .}
2622:
757:
721:
503:
382:
271:
261:
8439:
Tensor derivative (continuum mechanics) § Derivatives with respect to vectors and second-order tensors
7789:
It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector
1975:
and using the definition of the derivative as a limit which can be calculated along this path to get:
13889:
13836:
8154:{\displaystyle U(R(\mathbf {\theta } ))=\exp(-(\mathbf {\theta } \times \mathbf {x} )\cdot \nabla ).}
13297:
5808:{\displaystyle U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).}
4665:{\displaystyle U({\boldsymbol {\lambda }})=\exp \left({\boldsymbol {\lambda }}\cdot \nabla \right).}
362:
14005:
13776:
13324:
13063:
3736:
2319:, some authors define the directional derivative to be with respect to an arbitrary nonzero vector
656:
221:
8191:
1437:
1415:
13771:
13443:
12672:
3508:
3472:
3061:
1325:
1038:
970:
762:
651:
35:
6215:
1868:{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {v} }
13899:
13781:
13602:
13356:
13334:
13202:
12732:
12606: – Time rate of change of some physical quantity of a material element in a velocity field
12587:
2307:
and the horizontal will be maximum if the cutting plane contains the direction of the gradient
1934:
1613:
1034:
that measures the rate at which a function changes in a particular direction at a given point.
1031:
1006:
935:
896:
780:
640:
6096:
3995:
3852:
3827:
3762:
3644:
14025:
13884:
13796:
13453:
13388:
13361:
13351:
13272:
13260:
13245:
13217:
4519:
3901:
3877:
3807:
3742:
3634:{\displaystyle {\mathcal {L}}_{V}W^{\mu }=(V\cdot \nabla )W^{\mu }-(W\cdot \nabla )V^{\mu }.}
2891:{\displaystyle \nabla _{\mathbf {v} }(fg)=g\nabla _{\mathbf {v} }f+f\nabla _{\mathbf {v} }g.}
1880:
980:
646:
422:
367:
328:
234:
12569:
6895:
6042:
As a technical note, this procedure is only possible because the translation group forms an
5270:{\displaystyle U({\boldsymbol {\varepsilon }})=1+{\boldsymbol {\varepsilon }}\cdot \nabla .}
2718:{\displaystyle \nabla _{\mathbf {v} }(f+g)=\nabla _{\mathbf {v} }f+\nabla _{\mathbf {v} }g.}
1906:
1331:, all other coordinates being constant. The directional derivative is a special case of the
13841:
13460:
13307:
12668:
12563:
10070:{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {f} _{2}(\mathbf {v} ))}
4533:
3895:
3195:
1042:
985:
965:
891:
560:
484:
458:
372:
13035:
8:
13861:
13786:
13673:
13630:
13381:
13366:
13197:
13185:
13172:
13132:
13112:
12736:
12603:
12592:
3148:
1650:
960:
930:
920:
807:
661:
463:
319:
202:
197:
4469:
1052:
intuitively represents the instantaneous rate of change of the function, moving through
13950:
13925:
13756:
13709:
13650:
13615:
13610:
13590:
13585:
13580:
13545:
13492:
13475:
13376:
13250:
13235:
13180:
13147:
13005:
12676:
12575:
7092:
4434:
3894:. Instead of building the directional derivative using partial derivatives, we use the
3787:
1332:
1321:
925:
828:
812:
752:
747:
742:
706:
587:
511:
417:
412:
216:
211:
10838:{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}})}
10628:{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}})}
7527:{\displaystyle U(R(\mathbf {\theta } ))=\exp(-i\mathbf {\theta } \cdot \mathbf {L} ).}
7266:{\displaystyle f_{\text{abelian}}^{a}({\bar {\xi }},\xi )=\xi ^{a}+{\bar {\xi }}^{a},}
5045:
4424:{\displaystyle S_{\rho }=\pm \sum _{\sigma }R^{\sigma }{}_{\rho \mu \nu }S_{\sigma },}
1343:
14090:
13914:
13846:
13668:
13645:
13519:
13512:
13415:
13230:
13122:
13009:
12974:
12967:
12951:
12910:
12885:
12860:
12835:
12810:
12803:
12781:
12758:
12748:
12740:
12648:
12557:
6050:
2324:
1328:
999:
833:
611:
494:
447:
304:
299:
12395:{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))}
5095:
is a translation operator. This is instantly generalized to multivariable functions
4672:
This is a translation operator in the sense that it acts on multivariable functions
3541:
is given by the difference of two directional derivatives (with vanishing torsion):
14151:
14048:
13831:
13744:
13724:
13655:
13565:
13507:
13499:
13433:
13346:
13107:
13102:
12997:
12620:
12609:
7095:. The generators for translations are partial derivative operators, which commute:
4527:
3089:
2626:
843:
737:
711:
572:
489:
453:
5539:{\displaystyle ^{N}=U(N{\boldsymbol {\varepsilon }})=U({\boldsymbol {\lambda }}).}
14110:
14095:
13879:
13734:
13714:
13683:
13660:
13640:
13534:
13190:
13137:
12719:
11250:
10222:
4452:
2590:
2316:
1465:
975:
848:
802:
797:
684:
597:
542:
6519:
is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e.,
4893:{\displaystyle {\frac {df}{dx}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}
14156:
14020:
13919:
13766:
13719:
13620:
13423:
13045:
12597:
6331:
The actual operators on the
Hilbert space are represented by unitary operators
3466:
3242:
3105:
2327:, thus being independent of its magnitude and depending only on its direction.
1773:
858:
666:
438:
13438:
14125:
13894:
13749:
13635:
13339:
13314:
12762:
8177:
6081:) should not be taken for granted. We also note that Poincaré is a connected
6043:
4530:
1772:
This definition is valid in a broad range of contexts, for example where the
838:
602:
357:
314:
13904:
13874:
13739:
13302:
8181:
7396:
also contains a directional derivative. The rotation operator for an angle
6368:
2804:
1348:
1064:
592:
342:
7080:{\displaystyle =i\sum _{a}(-f^{abc}+f^{acb})t_{a}=i\sum _{a}C^{abc}t_{a},}
6631:{\displaystyle U(T({\bar {\xi }}))U(T(\xi ))=U(T(f({\bar {\xi }},\xi ))).}
4803:
In standard single-variable calculus, the derivative of a smooth function
3735:
Directional derivatives are often used in introductory derivations of the
13152:
13094:
8462:
6923:
2598:
1924:
955:
4094:{\displaystyle 1+\sum _{\mu }\delta '^{\mu }D_{\mu }=1+\delta '\cdot D.}
13869:
13801:
13555:
13428:
13292:
13282:
13225:
13049:
13001:
12809:(Reprinted (with corr.). ed.). Cambridge : Cambridge Univ. Press.
8912:{\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} )}
8702:{\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} )}
8173:
7376:{\displaystyle U(T({\bar {\xi }}))U(T(\xi ))=U(T({\bar {\xi }}+\xi )).}
6053:) in the Poincaré algebra. In particular, the group multiplication law
5431:{\displaystyle {\boldsymbol {\lambda }}=N{\boldsymbol {\varepsilon }}.}
5223:. We have found the infinitesimal version of the translation operator:
2901:
2613:
hold for the directional derivative. These include, for any functions
2610:
2272:
701:
625:
352:
347:
251:
53: in this section. Unsourced material may be challenged and removed.
8457:(v) be a real valued function of the vector v. Then the derivative of
3985:{\displaystyle 1+\sum _{\nu }\delta ^{\nu }D_{\nu }=1+\delta \cdot D,}
14063:
13811:
13806:
13117:
13039:
6082:
3670:, the Lie derivative reduces to the standard directional derivative:
2601:. With this restriction, both the above definitions are equivalent.
630:
620:
12988:
Shapiro, A. (1990). "On concepts of directional differentiability".
12554: – Mathematical gradient operator in certain coordinate systems
11288:
be a second order tensor valued function of the second order tensor
11106:{\displaystyle f({\boldsymbol {S}})=f_{1}(f_{2}({\boldsymbol {S}}))}
9314:
8448:
8172:
is a directional derivative taken in the direction normal (that is,
2794:{\displaystyle \nabla _{\mathbf {v} }(cf)=c\nabla _{\mathbf {v} }f.}
24:
14058:
13560:
13086:
12689:
6046:
5217:
is the directional derivative along the infinitesimal displacement
1899:
696:
443:
400:
89:
8461:(v) with respect to v (or at v) is the vector defined through its
13909:
13162:
12909:(2nd ed.). New York: Kluwer Academic / Plenum. p. 318.
8426:
6205:{\displaystyle T({\bar {\xi }})T(\xi )=T(f({\bar {\xi }},\xi )).}
5365:{\displaystyle U(\mathbf {a} )U(\mathbf {b} )=U(\mathbf {a+b} ).}
7450:{\displaystyle {\hat {\theta }}={\boldsymbol {\theta }}/\theta }
6922:
is by definition symmetric in its indices, we have the standard
14078:
13142:
13023:
12964:
12578: – Generalization of the concept of directional derivative
3050:
Tangent space § Tangent vectors as directional derivatives
1796:, then the directional derivative exists along any unit vector
11251:
Derivatives of tensor valued functions of second-order tensors
10223:
Derivatives of scalar valued functions of second-order tensors
3721:{\displaystyle {\mathcal {L}}_{V}\phi =(V\cdot \nabla )\phi .}
13157:
12731:
12667:
The applicability extends to functions over spaces without a
7539:
6093:) that are described by a continuous set of real parameters
4451:
is the
Riemann curvature tensor and the sign depends on the
1545:{\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})}
1412:, showing the gradient vector in black, and the unit vector
13055:
3739:. Consider a curved rectangle with an infinitesimal vector
3425:
This definition can be proven independent of the choice of
3359:
9180:{\displaystyle f(\mathbf {v} )=f_{1}(f_{2}(\mathbf {v} ))}
6885:{\displaystyle t_{bc}=-t_{b}t_{c}-i\sum _{a}f^{abc}t_{a}.}
12859:(Repr. ed.). Cambridge: Cambridge University Press.
12584: – Fundamental construction of differential calculus
5210:{\displaystyle {\boldsymbol {\varepsilon }}\cdot \nabla }
1434:
scaled by the directional derivative in the direction of
12566: – Differential geometry construct on fiber bundles
8433:
provides a systematic way of finding these derivatives.
8246:{\textstyle {\frac {\partial f}{\partial \mathbf {n} }}}
1324:, in which the rate of change is taken along one of the
12560: – Expression that may be integrated over a region
8420:
12780:. Princeton: Princeton University Press. p. 341.
8220:
8176:) to some surface in space, or more generally along a
7710:
7636:
7562:
6367:)). For a small neighborhood around the identity, the
4472:, we can define an infinitesimal translation operator
12408:
12335:
12191:
12114:
11907:
11822:
11685:
11600:
11571:
11415:
11393:
11371:
11349:
11316:
11294:
11261:
11119:
11048:
10851:
10772:
10641:
10562:
10533:
10383:
10361:
10339:
10317:
10286:
10264:
10258:
be a real valued function of the second order tensor
10233:
10083:
10006:
9799:
9714:
9577:
9492:
9327:
9193:
9122:
8925:
8846:
8715:
8636:
8473:
8259:
8194:
8081:
7998:
7851:
7799:
7548:
7463:
7418:
7279:
7185:
7101:
6932:
6898:
6797:
6644:
6525:
6377:
6324:{\displaystyle f^{a}(\xi ,0)=f^{a}(0,\xi )=\xi ^{a}.}
6251:
6218:
6126:
6099:
5856:
5757:
5686:
5556:
5468:
5408:
5307:
5229:
5195:
5109:
5048:
4918:
4821:
4686:
4614:
4542:
4482:
4437:
4327:
4107:
4023:
3998:
3924:
3904:
3880:
3855:
3830:
3810:
3790:
3765:
3745:
3676:
3647:
3547:
3511:
3475:
3327:
3251:
3204:
3187:{\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )}
3157:
2931:
2818:
2742:
2648:
2491:
2348:
2336:
per unit of distance moved in the direction given by
1981:
1937:
1909:
1883:
1808:
1658:
1621:
1558:
1473:
1440:
1418:
1357:
1084:
102:
12642:
11336:{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
11281:{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
1931:
is a unit vector. This follows from defining a path
1776:
of a vector (and hence a unit vector) is undefined.
12966:
12802:
12709:, Addison-Wesley Publ. Co., fifth edition, p. 593.
12705:Thomas, George B. Jr.; and Finney, Ross L. (1979)
12532:
12394:
12319:
12177:
12098:
11893:
11806:
11671:
11579:
11557:
11401:
11379:
11357:
11335:
11302:
11280:
11239:
11105:
11032:
10837:
10756:
10627:
10541:
10519:
10369:
10347:
10325:
10303:
10272:
10250:
10211:
10069:
9990:
9785:
9698:
9563:
9469:
9303:
9179:
9106:
8911:
8830:
8701:
8609:
8410:
8245:
8202:
8153:
8067:
7984:
7837:
7781:
7526:
7449:
7375:
7265:
7167:
7079:
6914:
6884:
6783:
6630:
6511:
6323:
6237:
6204:
6112:
6031:
5807:
5743:
5672:
5538:
5430:
5364:
5269:
5209:
5181:
5087:
5034:
4892:
4789:
4664:
4600:
4502:
4443:
4423:
4313:
4093:
4009:
3984:
3910:
3886:
3866:
3841:
3816:
3796:
3776:
3751:
3720:
3662:
3633:
3533:
3497:
3417:
3279:
3233:
3186:
3032:
2890:
2793:
2717:
2574:
2469:
2253:
1967:
1915:
1889:
1867:
1762:
1641:
1605:{\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}
1604:
1544:
1448:
1426:
1404:
1310:
174:
9315:Derivatives of vector valued functions of vectors
8449:Derivatives of scalar valued functions of vectors
7845:So we would expect under infinitesimal rotation:
6245:as the coordinates of the identity, we must have
3321:. Then the directional derivative is defined by
2290:
14123:
12969:Mathematical methods for physics and engineering
5372:So suppose that we take the finite displacement
5277:It is evident that the group multiplication law
3280:{\displaystyle {\mathbf {v} }_{\mathbf {p} }(f)}
2609:Many of the familiar properties of the ordinary
2381:
2098:
1997:
1691:
175:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
12990:Journal of Optimization Theory and Applications
12692:is also undefined; however, for differentiable
3289:Tangent space § Definition via derivations
3234:{\displaystyle L_{\mathbf {v} }f(\mathbf {p} )}
2584:
1779:
6120:. The group multiplication law takes the form
5546:We can now plug in our above expression for U(
2330:This definition gives the rate of increase of
1037:The directional derivative of a multivariable
13071:
12879:
12600: – A derivative in Differential Geometry
4101:The difference between the two paths is then
3437:is selected in the prescribed manner so that
1007:
12965:K.F. Riley; M.P. Hobson; S.J. Bence (2010).
12552:Del in cylindrical and spherical coordinates
6343:)). In the above notation we suppressed the
16:Instantaneous rate of change of the function
12572: – Derivative defined on normed spaces
8210:, then the normal derivative of a function
3082:is a function defined in a neighborhood of
3043:
2263:Intuitively, the directional derivative of
13078:
13064:
12945:
8429:with respect to vectors and tensors. The
1014:
1000:
14106:Regiomontanus' angle maximization problem
12929:J. E. Marsden and T. J. R. Hughes, 2000,
12923:
12834:. Princeton: Princeton University Press.
12647:(3rd ed.). Schaum's Outline Series.
12623: – Type of derivative in mathematics
5673:{\displaystyle ^{N}=\left^{N}=\left^{N}.}
5061:
4996:
4958:
3784:along the other. We translate a covector
1642:{\displaystyle \nabla _{\mathbf {v} }{f}}
1320:It therefore generalizes the notion of a
135:
69:Learn how and when to remove this message
13949:
12800:
12745:Calculus : Single and multivariable
8188:. If the normal direction is denoted by
3849:and then subtract the translation along
2294:
1342:
1078:may be denoted by any of the following:
41:Relevant discussion may be found on the
13454:Differentiating under the integral sign
12987:
12904:
12880:Larson, Ron; Edwards, Bruce H. (2010).
12616:Tensor derivative (continuum mechanics)
12521:
12510:
12494:
12465:
12435:
12424:
12382:
12368:
12343:
12308:
12297:
12281:
12252:
12235:
12220:
12209:
12199:
12165:
12151:
12136:
12124:
12116:
12087:
12076:
12060:
12037:
12023:
12011:
11997:
11983:
11972:
11956:
11936:
11925:
11915:
11884:
11870:
11858:
11844:
11832:
11824:
11800:
11784:
11768:
11750:
11734:
11714:
11703:
11693:
11662:
11648:
11636:
11622:
11610:
11602:
11573:
11531:
11517:
11509:
11474:
11463:
11455:
11444:
11433:
11423:
11409:is the fourth order tensor defined as
11395:
11373:
11351:
11326:
11318:
11296:
11271:
11263:
11228:
11217:
11146:
11135:
11093:
11056:
11021:
11010:
10973:
10949:
10923:
10912:
10878:
10867:
10828:
10804:
10780:
10750:
10734:
10702:
10668:
10657:
10618:
10594:
10570:
10535:
10493:
10479:
10438:
10427:
10410:
10399:
10377:is the second order tensor defined as
10363:
10341:
10319:
10294:
10266:
10241:
7955:
7913:
7868:
7820:
7435:
5997:
5961:
5944:
5919:
5867:
5787:
5765:
5640:
5600:
5567:
5526:
5509:
5479:
5421:
5410:
5254:
5237:
5197:
5147:
5125:
4777:
4730:
4694:
4644:
4622:
4578:
4550:
2283:with respect to time, when moving past
538:Differentiating under the integral sign
14124:
12931:Mathematical Foundations of Elasticity
12884:(9th ed.). Belmont: Brooks/Cole.
12854:
7538:is the vector operator that generates
6638:The expansion of f to second power is
4503:{\displaystyle \mathbf {P} =i\nabla .}
13330:Inverse functions and differentiation
13059:
12688:If the dot product is undefined, the
3730:
3460:
10304:{\displaystyle f({\boldsymbol {S}})}
10251:{\displaystyle f({\boldsymbol {S}})}
8421:In the continuum mechanics of solids
8163:
51:adding citations to reliable sources
18:
12829:
12775:
12612: – Tensor related to gradients
6085:. It is a group of transformations
5386:→∞ is implied everywhere), so that
3291:), can be defined as follows. Let
13:
13128:Free variables and bound variables
12948:Advanced Calculus for Applications
12506:
12489:
12460:
12445:
12420:
12412:
12293:
12276:
12247:
12230:
12205:
12195:
12072:
12055:
11968:
11951:
11921:
11911:
11780:
11763:
11746:
11729:
11699:
11689:
11429:
11419:
11213:
11198:
11171:
11156:
11131:
11123:
11006:
10991:
10908:
10893:
10863:
10855:
10730:
10715:
10698:
10683:
10653:
10645:
10395:
10387:
10185:
10168:
10139:
10122:
10097:
10087:
9964:
9947:
9860:
9843:
9813:
9803:
9672:
9655:
9638:
9621:
9591:
9581:
9341:
9331:
9282:
9267:
9245:
9230:
9205:
9197:
9080:
9065:
8982:
8967:
8937:
8929:
8804:
8789:
8772:
8757:
8727:
8719:
8485:
8477:
8355:
8347:
8314:
8285:
8271:
8263:
8232:
8224:
8142:
8059:
7973:
7138:
7134:
7113:
7109:
6004:
5794:
5744:{\displaystyle \exp(x)=\left^{N},}
5647:
5607:
5261:
5204:
5154:
4737:
4651:
4494:
4458:
3706:
3680:
3641:In particular, for a scalar field
3612:
3584:
3551:
3329:
3159:
3002:
2933:
2871:
2850:
2820:
2774:
2744:
2698:
2680:
2650:
2589:In the context of a function on a
2523:
2493:
2350:
2196:
1884:
1840:
1810:
1660:
1623:
1405:{\displaystyle f(x,y)=x^{2}+y^{2}}
1294:
1275:
1243:
1206:
1086:
84:Part of a series of articles about
14:
14168:
14142:Generalizations of the derivative
13933:The Method of Mechanical Theorems
13017:
12582:Generalizations of the derivative
11580:{\displaystyle {\boldsymbol {T}}}
11402:{\displaystyle {\boldsymbol {T}}}
11380:{\displaystyle {\boldsymbol {S}}}
11358:{\displaystyle {\boldsymbol {S}}}
11303:{\displaystyle {\boldsymbol {S}}}
10542:{\displaystyle {\boldsymbol {T}}}
10370:{\displaystyle {\boldsymbol {T}}}
10348:{\displaystyle {\boldsymbol {S}}}
10326:{\displaystyle {\boldsymbol {S}}}
10273:{\displaystyle {\boldsymbol {S}}}
2593:, some texts restrict the vector
13488:Partial fractions in integration
13404:Stochastic differential equation
13022:
10200:
10189:
10173:
10144:
10127:
10112:
10101:
10091:
10057:
10043:
10028:
10016:
10008:
9979:
9968:
9952:
9929:
9915:
9903:
9889:
9875:
9864:
9848:
9828:
9817:
9807:
9776:
9762:
9750:
9736:
9724:
9716:
9692:
9676:
9660:
9642:
9626:
9606:
9595:
9585:
9554:
9540:
9528:
9514:
9502:
9494:
9443:
9429:
9421:
9386:
9375:
9367:
9356:
9345:
9335:
9297:
9286:
9220:
9209:
9167:
9130:
9095:
9084:
9047:
9023:
8997:
8986:
8952:
8941:
8902:
8878:
8854:
8824:
8808:
8776:
8742:
8731:
8692:
8668:
8644:
8583:
8569:
8528:
8517:
8500:
8489:
8437:This section is an excerpt from
8398:
8387:
8370:
8359:
8334:
8319:
8306:
8295:
8275:
8236:
8196:
8132:
8049:
7963:
7938:
7921:
7902:
7885:
7828:
7809:
7801:
7772:
7698:
7624:
7550:
7514:
6019:
5975:
5936:
5908:
5891:
5352:
5349:
5346:
5329:
5315:
5169:
5117:
4769:
4752:
4708:
4586:
4484:
3347:
3334:
3262:
3255:
3224:
3211:
3177:
3164:
3020:
3007:
2991:
2963:
2938:
2876:
2855:
2825:
2779:
2749:
2703:
2685:
2655:
2557:
2546:
2533:
2513:
2498:
2452:
2434:
2417:
2406:
2370:
2355:
1861:
1850:
1830:
1815:
1744:
1727:
1716:
1680:
1665:
1628:
1560:
1481:
1442:
1420:
1298:
1285:
1265:
1253:
1235:
1224:
1211:
1195:
1184:
1164:
1151:
1135:
1122:
1106:
1091:
1063:The directional derivative of a
1039:differentiable (scalar) function
23:
13626:Jacobian matrix and determinant
13481:Tangent half-angle substitution
13449:Fundamental theorem of calculus
12907:Principles of quantum mechanics
12898:
12873:
12848:
12643:R. Wrede; M.R. Spiegel (2010).
4900:This can be rearranged to find
4463:
3898:. The translation operator for
3301:be a differentiable curve with
34:needs additional citations for
13702:Arithmetico-geometric sequence
13394:Ordinary differential equation
12973:. Cambridge University Press.
12832:Einstein gravity in a nutshell
12823:
12794:
12778:Einstein gravity in a nutshell
12769:
12725:
12712:
12707:Calculus and Analytic Geometry
12699:
12682:
12661:
12636:
12389:
12386:
12378:
12363:
12347:
12339:
12172:
12169:
12161:
12146:
12128:
12120:
12041:
12033:
12015:
12007:
11888:
11880:
11862:
11854:
11836:
11828:
11666:
11658:
11640:
11632:
11614:
11606:
11535:
11513:
11478:
11470:
11467:
11459:
11330:
11322:
11275:
11267:
11100:
11097:
11089:
11076:
11060:
11052:
10977:
10969:
10953:
10945:
10832:
10824:
10808:
10800:
10784:
10776:
10622:
10614:
10598:
10590:
10574:
10566:
10497:
10475:
10442:
10434:
10431:
10423:
10298:
10290:
10245:
10237:
10064:
10061:
10053:
10038:
10020:
10012:
9933:
9925:
9907:
9899:
9780:
9772:
9754:
9746:
9728:
9720:
9558:
9550:
9532:
9524:
9506:
9498:
9447:
9425:
9390:
9382:
9379:
9371:
9174:
9171:
9163:
9150:
9134:
9126:
9051:
9043:
9027:
9019:
8906:
8898:
8882:
8874:
8858:
8850:
8696:
8688:
8672:
8664:
8648:
8640:
8587:
8565:
8532:
8524:
8521:
8513:
8402:
8394:
8391:
8383:
8338:
8330:
8299:
8291:
8145:
8136:
8120:
8114:
8102:
8099:
8091:
8085:
8053:
8034:
8022:
8019:
8008:
8002:
7967:
7948:
7942:
7934:
7925:
7898:
7889:
7881:
7875:
7872:
7861:
7855:
7805:
7518:
7496:
7484:
7481:
7473:
7467:
7425:
7367:
7364:
7352:
7343:
7337:
7328:
7325:
7319:
7313:
7307:
7304:
7298:
7289:
7283:
7248:
7222:
7210:
7201:
7019:
6978:
6959:
6933:
6756:
6702:
6676:
6664:
6655:
6622:
6619:
6616:
6604:
6595:
6589:
6583:
6574:
6571:
6565:
6559:
6553:
6550:
6544:
6535:
6529:
6396:
6393:
6387:
6381:
6302:
6290:
6274:
6262:
6196:
6193:
6181:
6172:
6166:
6157:
6151:
6145:
6139:
6130:
6023:
6015:
5979:
5971:
5965:
5957:
5948:
5932:
5923:
5904:
5895:
5887:
5875:
5871:
5863:
5857:
5769:
5761:
5699:
5693:
5575:
5571:
5563:
5557:
5530:
5522:
5513:
5502:
5487:
5483:
5475:
5469:
5356:
5342:
5333:
5325:
5319:
5311:
5241:
5233:
5173:
5165:
5129:
5113:
5082:
5079:
5062:
5049:
5026:
5020:
4949:
4943:
4934:
4922:
4878:
4872:
4863:
4851:
4781:
4765:
4756:
4748:
4712:
4704:
4698:
4690:
4626:
4618:
4554:
4546:
4354:
4328:
4295:
4269:
4212:
4189:
4186:
4168:
4152:
4134:
4131:
4108:
3709:
3697:
3657:
3651:
3615:
3603:
3587:
3575:
3528:
3522:
3492:
3486:
3392:
3386:
3351:
3343:
3274:
3268:
3228:
3220:
3181:
3173:
3024:
3016:
2998:
2995:
2987:
2981:
2967:
2959:
2956:
2944:
2840:
2831:
2764:
2755:
2673:
2661:
2562:
2552:
2537:
2529:
2517:
2509:
2457:
2447:
2438:
2430:
2421:
2402:
2388:
2374:
2366:
2291:Using only direction of vector
2241:
2235:
2232:
2226:
2214:
2208:
2182:
2176:
2173:
2167:
2149:
2143:
2134:
2119:
2105:
2078:
2072:
2069:
2063:
2048:
2042:
2033:
2018:
2004:
1947:
1941:
1854:
1846:
1834:
1826:
1748:
1740:
1731:
1712:
1698:
1684:
1676:
1599:
1567:
1539:
1494:
1485:
1477:
1373:
1361:
1289:
1281:
1257:
1249:
1228:
1220:
1199:
1191:
1188:
1180:
1168:
1160:
1139:
1131:
1110:
1102:
169:
163:
154:
148:
132:
126:
1:
13525:Integro-differential equation
13399:Partial differential equation
12939:
12882:Calculus of a single variable
12855:Cahill, Kevin Cahill (2013).
11565:for all second order tensors
10527:for all second order tensors
7273:and thus for abelian groups,
2604:
1338:
1056:with a velocity specified by
469:Integral of inverse functions
13085:
12805:The quantum theory of fields
8203:{\displaystyle \mathbf {n} }
7412:| about an axis parallel to
7387:
4522:) For a finite displacement
2585:Restriction to a unit vector
1780:For differentiable functions
1449:{\displaystyle \mathbf {u} }
1427:{\displaystyle \mathbf {u} }
1074:at a point (e.g., position)
7:
13679:Generalized Stokes' theorem
13466:Integration by substitution
12747:. John wiley. p. 780.
12545:
4811:) is defined by (for small
3534:{\displaystyle V^{\mu }(x)}
3498:{\displaystyle W^{\mu }(x)}
887:Calculus on Euclidean space
310:Logarithmic differentiation
10:
14173:
13208:(ε, δ)-definition of limit
12946:Hildebrand, F. B. (1976).
8623:at v, in the u direction.
8436:
8186:Neumann boundary condition
6238:{\displaystyle \xi ^{a}=0}
4800:Proof of the last equation
3047:
14101:Proof that 22/7 exceeds π
14038:
14016:
13942:
13890:Gottfried Wilhelm Leibniz
13860:
13837:e (mathematical constant)
13822:
13694:
13601:
13533:
13414:
13216:
13171:
13093:
12801:Weinberg, Steven (1999).
12718:This typically assumes a
11310:. Then the derivative of
10280:. Then the derivative of
8180:field orthogonal to some
1968:{\displaystyle h(t)=x+tv}
1897:on the right denotes the
1070:with respect to a vector
621:Summand limit (term test)
13852:Stirling's approximation
13325:Implicit differentiation
13273:Rules of differentiation
12673:differentiable manifolds
12630:
8465:with any vector u being
8214:is sometimes denoted as
7168:{\displaystyle \left=0.}
6113:{\displaystyle \xi ^{a}}
5452:times, we can construct
4010:{\displaystyle \delta '}
3867:{\displaystyle \delta '}
3842:{\displaystyle \delta '}
3777:{\displaystyle \delta '}
3737:Riemann curvature tensor
3663:{\displaystyle \phi (x)}
3044:In differential geometry
2342:. In this case, one has
305:Implicit differentiation
295:Differentiation notation
222:Inverse function theorem
14086:Euler–Maclaurin formula
13991:trigonometric functions
13444:Constant of integration
13036:Directional derivatives
12733:Hughes Hallett, Deborah
3911:{\displaystyle \delta }
3887:{\displaystyle \delta }
3817:{\displaystyle \delta }
3752:{\displaystyle \delta }
3136:, denoted variously as
3062:differentiable manifold
1890:{\displaystyle \nabla }
763:Helmholtz decomposition
14147:Multivariable calculus
14055:Differential geometry
13900:Infinitesimal calculus
13603:Multivariable calculus
13551:Directional derivative
13357:Second derivative test
13335:Logarithmic derivative
13308:General Leibniz's rule
13203:Order of approximation
13046:Directional derivative
13029:Directional derivative
12588:Semi-differentiability
12534:
12396:
12321:
12179:
12100:
11895:
11808:
11673:
11581:
11559:
11403:
11381:
11359:
11337:
11304:
11282:
11241:
11107:
11034:
10839:
10758:
10629:
10543:
10521:
10371:
10349:
10327:
10305:
10274:
10252:
10213:
10071:
9992:
9787:
9700:
9565:
9471:
9305:
9181:
9108:
8913:
8832:
8703:
8611:
8412:
8253:. In other notations,
8247:
8204:
8155:
8069:
7986:
7839:
7783:
7528:
7451:
7377:
7267:
7169:
7081:
6916:
6915:{\displaystyle t_{ab}}
6886:
6785:
6632:
6513:
6325:
6239:
6206:
6114:
6033:
5809:
5745:
5674:
5540:
5432:
5366:
5271:
5211:
5183:
5089:
5036:
4894:
4791:
4666:
4602:
4504:
4445:
4425:
4315:
4095:
4011:
3986:
3912:
3888:
3868:
3843:
3818:
3798:
3778:
3753:
3722:
3664:
3635:
3535:
3499:
3419:
3281:
3235:
3188:
3122:directional derivative
3034:
2892:
2795:
2719:
2576:
2471:
2312:
2279:, in the direction of
2255:
1969:
1917:
1916:{\displaystyle \cdot }
1891:
1869:
1764:
1643:
1606:
1546:
1462:directional derivative
1457:
1450:
1428:
1406:
1312:
1032:multivariable calculus
1028:directional derivative
897:Limit of distributions
717:Directional derivative
378:Faà di Bruno's formula
176:
14137:Differential geometry
14132:Differential calculus
13974:logarithmic functions
13969:exponential functions
13885:Generality of algebra
13763:Tests of convergence
13389:Differential equation
13373:Further applications
13362:Extreme value theorem
13352:First derivative test
13246:Differential operator
13218:Differential calculus
13031:at Wikimedia Commons
12535:
12397:
12322:
12180:
12101:
11896:
11809:
11674:
11582:
11560:
11404:
11382:
11360:
11338:
11305:
11283:
11242:
11108:
11035:
10840:
10759:
10630:
10544:
10522:
10372:
10350:
10328:
10306:
10275:
10253:
10214:
10072:
9993:
9788:
9701:
9566:
9472:
9306:
9182:
9109:
8914:
8833:
8704:
8612:
8431:directional directive
8413:
8248:
8205:
8156:
8070:
7987:
7840:
7784:
7529:
7452:
7378:
7268:
7170:
7082:
6917:
6887:
6786:
6633:
6514:
6326:
6240:
6207:
6115:
6034:
5810:
5746:
5675:
5541:
5433:
5367:
5272:
5212:
5184:
5090:
5037:
4895:
4792:
4667:
4603:
4536:for translations is
4520:self-adjoint operator
4505:
4446:
4426:
4316:
4096:
4012:
3987:
3913:
3889:
3869:
3844:
3819:
3799:
3779:
3754:
3723:
3665:
3636:
3536:
3505:along a vector field
3500:
3420:
3282:
3236:
3189:
3035:
2917:is differentiable at
2909:is differentiable at
2893:
2796:
2720:
2577:
2481:is differentiable at
2472:
2298:
2256:
1970:
1918:
1892:
1870:
1765:
1644:
1607:
1547:
1451:
1429:
1407:
1346:
1313:
981:Mathematical analysis
892:Generalized functions
577:arithmetico-geometric
423:Leibniz integral rule
177:
14039:Miscellaneous topics
13979:hyperbolic functions
13964:irrational functions
13842:Exponential function
13695:Sequences and series
13461:Integration by parts
12905:Shankar, R. (1994).
12857:Physical mathematics
12737:McCallum, William G.
12564:Ehresmann connection
12406:
12333:
12189:
12112:
11905:
11820:
11683:
11598:
11569:
11413:
11391:
11369:
11347:
11314:
11292:
11259:
11117:
11046:
10849:
10770:
10639:
10560:
10531:
10381:
10359:
10337:
10315:
10284:
10262:
10231:
10081:
10004:
9797:
9712:
9575:
9490:
9325:
9191:
9120:
8923:
8844:
8713:
8634:
8471:
8257:
8218:
8192:
8079:
7996:
7849:
7797:
7546:
7461:
7416:
7402:, i.e. by an amount
7277:
7183:
7179:is simply additive:
7099:
6930:
6896:
6795:
6642:
6523:
6375:
6249:
6216:
6124:
6097:
5854:
5755:
5684:
5554:
5466:
5406:
5305:
5227:
5193:
5107:
5046:
4916:
4819:
4684:
4612:
4540:
4480:
4435:
4325:
4105:
4021:
3996:
3922:
3902:
3896:covariant derivative
3878:
3853:
3828:
3808:
3788:
3763:
3743:
3674:
3645:
3545:
3509:
3473:
3325:
3249:
3202:
3196:Covariant derivative
3155:
2929:
2816:
2740:
2729:constant factor rule
2646:
2489:
2346:
2303:between the tangent
1979:
1935:
1907:
1881:
1806:
1656:
1619:
1556:
1471:
1438:
1416:
1355:
1082:
986:Nonstandard analysis
459:Lebesgue integration
329:Rules and identities
100:
47:improve this article
14026:List of derivatives
13862:History of calculus
13777:Cauchy condensation
13674:Exterior derivative
13631:Lagrange multiplier
13367:Maximum and minimum
13198:Limit of a sequence
13186:Limit of a function
13133:Graph of a function
13113:Continuous function
12604:Material derivative
12593:Hadamard derivative
11387:) in the direction
10355:) in the direction
7200:
5680:Using the identity
5378:and divide it into
3759:along one edge and
3149:Exterior derivative
2732:: For any constant
1130:
657:Cauchy condensation
464:Contour integration
190:Fundamental theorem
117:
13959:rational functions
13926:Method of Fluxions
13772:Alternating series
13669:Differential forms
13651:Partial derivative
13611:Divergence theorem
13493:Quadratic integral
13261:Leibniz's notation
13251:Mean value theorem
13236:Partial derivative
13181:Indeterminate form
13002:10.1007/BF00940933
12741:Gleason, Andrew M.
12677:general relativity
12576:Gateaux derivative
12570:Fréchet derivative
12530:
12392:
12317:
12175:
12096:
11891:
11804:
11669:
11577:
11555:
11399:
11377:
11355:
11333:
11300:
11278:
11237:
11103:
11030:
10835:
10754:
10625:
10539:
10517:
10367:
10345:
10323:
10301:
10270:
10248:
10209:
10067:
9988:
9783:
9696:
9561:
9467:
9301:
9177:
9104:
8909:
8828:
8699:
8607:
8408:
8243:
8200:
8184:. See for example
8151:
8065:
7982:
7835:
7779:
7765:
7691:
7617:
7524:
7447:
7373:
7263:
7186:
7165:
7093:structure constant
7077:
7047:
6977:
6912:
6882:
6852:
6781:
6732:
6628:
6509:
6469:
6420:
6321:
6235:
6202:
6110:
6029:
5805:
5741:
5670:
5536:
5428:
5402:. In other words,
5362:
5267:
5207:
5179:
5085:
5032:
4890:
4801:
4787:
4662:
4598:
4500:
4441:
4421:
4382:
4311:
4243:
4091:
4039:
4007:
3982:
3940:
3908:
3884:
3864:
3839:
3814:
3794:
3774:
3749:
3731:The Riemann tensor
3718:
3660:
3631:
3531:
3495:
3469:of a vector field
3461:The Lie derivative
3415:
3277:
3231:
3184:
3030:
2888:
2791:
2715:
2572:
2467:
2395:
2313:
2251:
2249:
2112:
2011:
1965:
1913:
1887:
1865:
1800:at x, and one has
1760:
1705:
1639:
1602:
1542:
1458:
1446:
1424:
1402:
1333:Gateaux derivative
1322:partial derivative
1308:
1116:
829:Partial derivative
758:generalized Stokes
652:Alternating series
533:Reduction formulae
508:tangent half-angle
495:Cylindrical shells
418:Integral transform
413:Lists of integrals
217:Mean value theorem
172:
103:
14119:
14118:
14045:Complex calculus
14034:
14033:
13915:Law of Continuity
13847:Natural logarithm
13832:Bernoulli numbers
13823:Special functions
13782:Direct comparison
13646:Multiple integral
13520:Integral equation
13416:Integral calculus
13347:Stationary points
13321:Other techniques
13266:Newton's notation
13231:Second derivative
13123:Finite difference
13027:Media related to
12980:978-0-521-86153-3
12950:. Prentice Hall.
12654:978-0-07-162366-7
12645:Advanced Calculus
12558:Differential form
12515:
12476:
12429:
12302:
12263:
12214:
12081:
11977:
11930:
11789:
11755:
11708:
11529:
11507:
11503:
11438:
11222:
11189:
11185:
11140:
11015:
10982:
10934:
10917:
10872:
10813:
10739:
10707:
10662:
10491:
10471:
10467:
10404:
10194:
10155:
10106:
9973:
9869:
9822:
9681:
9647:
9600:
9441:
9419:
9415:
9350:
9291:
9263:
9259:
9214:
9089:
9056:
9008:
8991:
8946:
8887:
8813:
8781:
8736:
8581:
8561:
8557:
8494:
8364:
8280:
8241:
8170:normal derivative
8164:Normal derivative
7992:It follows that
7428:
7394:rotation operator
7355:
7301:
7251:
7213:
7193:
7152:
7127:
7038:
6968:
6843:
6759:
6717:
6705:
6667:
6607:
6547:
6454:
6452:
6411:
6184:
6142:
6051:Cartan subalgebra
5725:
5654:
5438:Then by applying
5301:) takes the form
5010:
4977:
4885:
4840:
4799:
4444:{\displaystyle R}
4373:
4228:
4030:
3931:
3797:{\displaystyle S}
3375:
2567:
2462:
2380:
2156:
2097:
2085:
1996:
1755:
1690:
1329:coordinate curves
1303:
1048:at a given point
1024:
1023:
904:
903:
866:
865:
834:Multiple integral
770:
769:
674:
673:
641:Direct comparison
612:Convergence tests
550:
549:
523:Partial fractions
390:
389:
300:Second derivative
79:
78:
71:
14164:
14049:Contour integral
13947:
13946:
13797:Limit comparison
13706:Types of series
13665:Advanced topics
13656:Surface integral
13500:Trapezoidal rule
13439:Basic properties
13434:Riemann integral
13382:Taylor's theorem
13108:Concave function
13103:Binomial theorem
13080:
13073:
13066:
13057:
13056:
13026:
13013:
12984:
12972:
12961:
12934:
12927:
12921:
12920:
12902:
12896:
12895:
12877:
12871:
12870:
12852:
12846:
12845:
12830:Zee, A. (2013).
12827:
12821:
12820:
12808:
12798:
12792:
12791:
12776:Zee, A. (2013).
12773:
12767:
12766:
12729:
12723:
12716:
12710:
12703:
12697:
12686:
12680:
12665:
12659:
12658:
12640:
12621:Total derivative
12610:Structure tensor
12539:
12537:
12536:
12531:
12529:
12525:
12524:
12516:
12514:
12513:
12504:
12503:
12502:
12497:
12487:
12477:
12475:
12474:
12473:
12468:
12458:
12457:
12456:
12443:
12438:
12430:
12428:
12427:
12418:
12410:
12401:
12399:
12398:
12393:
12385:
12377:
12376:
12371:
12362:
12361:
12346:
12326:
12324:
12323:
12318:
12316:
12312:
12311:
12303:
12301:
12300:
12291:
12290:
12289:
12284:
12274:
12264:
12262:
12261:
12260:
12255:
12245:
12244:
12243:
12238:
12228:
12223:
12215:
12213:
12212:
12203:
12202:
12193:
12184:
12182:
12181:
12176:
12168:
12160:
12159:
12154:
12145:
12144:
12139:
12127:
12119:
12105:
12103:
12102:
12097:
12095:
12091:
12090:
12082:
12080:
12079:
12070:
12069:
12068:
12063:
12053:
12040:
12032:
12031:
12026:
12014:
12006:
12005:
12000:
11991:
11987:
11986:
11978:
11976:
11975:
11966:
11965:
11964:
11959:
11949:
11939:
11931:
11929:
11928:
11919:
11918:
11909:
11900:
11898:
11897:
11892:
11887:
11879:
11878:
11873:
11861:
11853:
11852:
11847:
11835:
11827:
11813:
11811:
11810:
11805:
11803:
11795:
11791:
11790:
11788:
11787:
11778:
11777:
11776:
11771:
11761:
11756:
11754:
11753:
11744:
11743:
11742:
11737:
11727:
11717:
11709:
11707:
11706:
11697:
11696:
11687:
11678:
11676:
11675:
11670:
11665:
11657:
11656:
11651:
11639:
11631:
11630:
11625:
11613:
11605:
11586:
11584:
11583:
11578:
11576:
11564:
11562:
11561:
11556:
11554:
11553:
11542:
11538:
11534:
11527:
11520:
11512:
11505:
11504:
11502:
11491:
11477:
11466:
11458:
11447:
11439:
11437:
11436:
11427:
11426:
11417:
11408:
11406:
11405:
11400:
11398:
11386:
11384:
11383:
11378:
11376:
11364:
11362:
11361:
11356:
11354:
11343:with respect to
11342:
11340:
11339:
11334:
11329:
11321:
11309:
11307:
11306:
11301:
11299:
11287:
11285:
11284:
11279:
11274:
11266:
11246:
11244:
11243:
11238:
11236:
11232:
11231:
11223:
11221:
11220:
11211:
11210:
11209:
11196:
11187:
11186:
11184:
11183:
11182:
11169:
11168:
11167:
11154:
11149:
11141:
11139:
11138:
11129:
11121:
11112:
11110:
11109:
11104:
11096:
11088:
11087:
11075:
11074:
11059:
11039:
11037:
11036:
11031:
11029:
11025:
11024:
11016:
11014:
11013:
11004:
11003:
11002:
10989:
10980:
10976:
10968:
10967:
10952:
10944:
10943:
10932:
10931:
10927:
10926:
10918:
10916:
10915:
10906:
10905:
10904:
10891:
10881:
10873:
10871:
10870:
10861:
10853:
10844:
10842:
10841:
10836:
10831:
10823:
10822:
10811:
10807:
10799:
10798:
10783:
10763:
10761:
10760:
10755:
10753:
10745:
10741:
10740:
10738:
10737:
10728:
10727:
10726:
10713:
10708:
10706:
10705:
10696:
10695:
10694:
10681:
10671:
10663:
10661:
10660:
10651:
10643:
10634:
10632:
10631:
10626:
10621:
10613:
10612:
10597:
10589:
10588:
10573:
10548:
10546:
10545:
10540:
10538:
10526:
10524:
10523:
10518:
10516:
10515:
10504:
10500:
10496:
10489:
10482:
10469:
10468:
10466:
10455:
10441:
10430:
10413:
10405:
10403:
10402:
10393:
10385:
10376:
10374:
10373:
10368:
10366:
10354:
10352:
10351:
10346:
10344:
10332:
10330:
10329:
10324:
10322:
10311:with respect to
10310:
10308:
10307:
10302:
10297:
10279:
10277:
10276:
10271:
10269:
10257:
10255:
10254:
10249:
10244:
10218:
10216:
10215:
10210:
10208:
10204:
10203:
10195:
10193:
10192:
10183:
10182:
10181:
10176:
10166:
10156:
10154:
10153:
10152:
10147:
10137:
10136:
10135:
10130:
10120:
10115:
10107:
10105:
10104:
10095:
10094:
10085:
10076:
10074:
10073:
10068:
10060:
10052:
10051:
10046:
10037:
10036:
10031:
10019:
10011:
9997:
9995:
9994:
9989:
9987:
9983:
9982:
9974:
9972:
9971:
9962:
9961:
9960:
9955:
9945:
9932:
9924:
9923:
9918:
9906:
9898:
9897:
9892:
9883:
9879:
9878:
9870:
9868:
9867:
9858:
9857:
9856:
9851:
9841:
9831:
9823:
9821:
9820:
9811:
9810:
9801:
9792:
9790:
9789:
9784:
9779:
9771:
9770:
9765:
9753:
9745:
9744:
9739:
9727:
9719:
9705:
9703:
9702:
9697:
9695:
9687:
9683:
9682:
9680:
9679:
9670:
9669:
9668:
9663:
9653:
9648:
9646:
9645:
9636:
9635:
9634:
9629:
9619:
9609:
9601:
9599:
9598:
9589:
9588:
9579:
9570:
9568:
9567:
9562:
9557:
9549:
9548:
9543:
9531:
9523:
9522:
9517:
9505:
9497:
9476:
9474:
9473:
9468:
9466:
9465:
9454:
9450:
9446:
9439:
9432:
9424:
9417:
9416:
9414:
9403:
9389:
9378:
9370:
9359:
9351:
9349:
9348:
9339:
9338:
9329:
9310:
9308:
9307:
9302:
9300:
9292:
9290:
9289:
9280:
9279:
9278:
9265:
9261:
9260:
9258:
9257:
9256:
9243:
9242:
9241:
9228:
9223:
9215:
9213:
9212:
9203:
9195:
9186:
9184:
9183:
9178:
9170:
9162:
9161:
9149:
9148:
9133:
9113:
9111:
9110:
9105:
9103:
9099:
9098:
9090:
9088:
9087:
9078:
9077:
9076:
9063:
9054:
9050:
9042:
9041:
9026:
9018:
9017:
9006:
9005:
9001:
9000:
8992:
8990:
8989:
8980:
8979:
8978:
8965:
8955:
8947:
8945:
8944:
8935:
8927:
8918:
8916:
8915:
8910:
8905:
8897:
8896:
8885:
8881:
8873:
8872:
8857:
8837:
8835:
8834:
8829:
8827:
8819:
8815:
8814:
8812:
8811:
8802:
8801:
8800:
8787:
8782:
8780:
8779:
8770:
8769:
8768:
8755:
8745:
8737:
8735:
8734:
8725:
8717:
8708:
8706:
8705:
8700:
8695:
8687:
8686:
8671:
8663:
8662:
8647:
8616:
8614:
8613:
8608:
8606:
8605:
8594:
8590:
8586:
8579:
8572:
8559:
8558:
8556:
8545:
8531:
8520:
8503:
8495:
8493:
8492:
8483:
8475:
8417:
8415:
8414:
8409:
8401:
8390:
8373:
8365:
8363:
8362:
8353:
8345:
8337:
8329:
8324:
8323:
8322:
8309:
8298:
8281:
8279:
8278:
8269:
8261:
8252:
8250:
8249:
8244:
8242:
8240:
8239:
8230:
8222:
8209:
8207:
8206:
8201:
8199:
8160:
8158:
8157:
8152:
8135:
8127:
8098:
8074:
8072:
8071:
8066:
8052:
8044:
8018:
7991:
7989:
7988:
7983:
7966:
7958:
7941:
7924:
7916:
7905:
7888:
7871:
7844:
7842:
7841:
7836:
7831:
7823:
7812:
7804:
7788:
7786:
7785:
7780:
7775:
7770:
7769:
7701:
7696:
7695:
7627:
7622:
7621:
7553:
7533:
7531:
7530:
7525:
7517:
7509:
7480:
7456:
7454:
7453:
7448:
7443:
7438:
7430:
7429:
7421:
7382:
7380:
7379:
7374:
7357:
7356:
7348:
7303:
7302:
7294:
7272:
7270:
7269:
7264:
7259:
7258:
7253:
7252:
7244:
7237:
7236:
7215:
7214:
7206:
7199:
7194:
7191:
7174:
7172:
7171:
7166:
7158:
7154:
7153:
7151:
7150:
7149:
7133:
7128:
7126:
7125:
7124:
7108:
7086:
7084:
7083:
7078:
7073:
7072:
7063:
7062:
7046:
7031:
7030:
7018:
7017:
6999:
6998:
6976:
6958:
6957:
6945:
6944:
6921:
6919:
6918:
6913:
6911:
6910:
6891:
6889:
6888:
6883:
6878:
6877:
6868:
6867:
6851:
6836:
6835:
6826:
6825:
6810:
6809:
6790:
6788:
6787:
6782:
6777:
6776:
6767:
6766:
6761:
6760:
6752:
6748:
6747:
6731:
6713:
6712:
6707:
6706:
6698:
6691:
6690:
6669:
6668:
6660:
6654:
6653:
6637:
6635:
6634:
6629:
6609:
6608:
6600:
6549:
6548:
6540:
6518:
6516:
6515:
6510:
6502:
6501:
6489:
6488:
6479:
6478:
6468:
6453:
6445:
6440:
6439:
6430:
6429:
6419:
6371:representation
6330:
6328:
6327:
6322:
6317:
6316:
6289:
6288:
6261:
6260:
6244:
6242:
6241:
6236:
6228:
6227:
6211:
6209:
6208:
6203:
6186:
6185:
6177:
6144:
6143:
6135:
6119:
6117:
6116:
6111:
6109:
6108:
6038:
6036:
6035:
6030:
6022:
6011:
6007:
6000:
5978:
5964:
5947:
5939:
5922:
5911:
5894:
5883:
5882:
5870:
5849:
5814:
5812:
5811:
5806:
5801:
5797:
5790:
5768:
5750:
5748:
5747:
5742:
5737:
5736:
5731:
5727:
5726:
5718:
5679:
5677:
5676:
5671:
5666:
5665:
5660:
5656:
5655:
5650:
5643:
5637:
5620:
5619:
5614:
5610:
5603:
5583:
5582:
5570:
5545:
5543:
5542:
5537:
5529:
5512:
5495:
5494:
5482:
5437:
5435:
5434:
5429:
5424:
5413:
5371:
5369:
5368:
5363:
5355:
5332:
5318:
5276:
5274:
5273:
5268:
5257:
5240:
5216:
5214:
5213:
5208:
5200:
5188:
5186:
5185:
5180:
5172:
5161:
5157:
5150:
5128:
5120:
5094:
5092:
5091:
5088:{\displaystyle }
5086:
5072:
5042:It follows that
5041:
5039:
5038:
5033:
5016:
5012:
5011:
5009:
4998:
4978:
4976:
4968:
4960:
4899:
4897:
4896:
4891:
4886:
4881:
4846:
4841:
4839:
4831:
4823:
4796:
4794:
4793:
4788:
4780:
4772:
4755:
4744:
4740:
4733:
4711:
4697:
4671:
4669:
4668:
4663:
4658:
4654:
4647:
4625:
4607:
4605:
4604:
4599:
4594:
4590:
4589:
4581:
4553:
4509:
4507:
4506:
4501:
4487:
4470:Poincaré algebra
4450:
4448:
4447:
4442:
4430:
4428:
4427:
4422:
4417:
4416:
4407:
4406:
4395:
4392:
4391:
4381:
4366:
4365:
4353:
4352:
4340:
4339:
4320:
4318:
4317:
4312:
4307:
4306:
4294:
4293:
4281:
4280:
4268:
4267:
4258:
4257:
4256:
4242:
4224:
4223:
4205:
4164:
4163:
4124:
4100:
4098:
4097:
4092:
4081:
4064:
4063:
4054:
4053:
4052:
4038:
4016:
4014:
4013:
4008:
4006:
3991:
3989:
3988:
3983:
3960:
3959:
3950:
3949:
3939:
3917:
3915:
3914:
3909:
3893:
3891:
3890:
3885:
3873:
3871:
3870:
3865:
3863:
3848:
3846:
3845:
3840:
3838:
3823:
3821:
3820:
3815:
3803:
3801:
3800:
3795:
3783:
3781:
3780:
3775:
3773:
3758:
3756:
3755:
3750:
3727:
3725:
3724:
3719:
3690:
3689:
3684:
3683:
3669:
3667:
3666:
3661:
3640:
3638:
3637:
3632:
3627:
3626:
3599:
3598:
3571:
3570:
3561:
3560:
3555:
3554:
3540:
3538:
3537:
3532:
3521:
3520:
3504:
3502:
3501:
3496:
3485:
3484:
3456:
3446:
3436:
3430:
3424:
3422:
3421:
3416:
3411:
3410:
3399:
3395:
3376:
3374:
3363:
3350:
3339:
3338:
3337:
3320:
3310:
3300:
3286:
3284:
3283:
3278:
3267:
3266:
3265:
3259:
3258:
3240:
3238:
3237:
3232:
3227:
3216:
3215:
3214:
3193:
3191:
3190:
3185:
3180:
3169:
3168:
3167:
3146:
3135:
3129:
3119:
3113:
3103:
3097:
3087:
3081:
3076:. Suppose that
3075:
3069:
3059:
3039:
3037:
3036:
3031:
3023:
3012:
3011:
3010:
2994:
2980:
2966:
2943:
2942:
2941:
2897:
2895:
2894:
2889:
2881:
2880:
2879:
2860:
2859:
2858:
2830:
2829:
2828:
2800:
2798:
2797:
2792:
2784:
2783:
2782:
2754:
2753:
2752:
2724:
2722:
2721:
2716:
2708:
2707:
2706:
2690:
2689:
2688:
2660:
2659:
2658:
2581:
2579:
2578:
2573:
2568:
2566:
2565:
2560:
2555:
2549:
2544:
2536:
2516:
2508:
2503:
2502:
2501:
2476:
2474:
2473:
2468:
2463:
2461:
2460:
2455:
2450:
2441:
2437:
2420:
2409:
2397:
2394:
2373:
2365:
2360:
2359:
2358:
2341:
2335:
2260:
2258:
2257:
2252:
2250:
2204:
2203:
2188:
2157:
2152:
2114:
2111:
2090:
2086:
2081:
2013:
2010:
1974:
1972:
1971:
1966:
1922:
1920:
1919:
1914:
1896:
1894:
1893:
1888:
1874:
1872:
1871:
1866:
1864:
1853:
1833:
1825:
1820:
1819:
1818:
1784:If the function
1769:
1767:
1766:
1761:
1756:
1751:
1747:
1730:
1719:
1707:
1704:
1683:
1675:
1670:
1669:
1668:
1648:
1646:
1645:
1640:
1638:
1633:
1632:
1631:
1611:
1609:
1608:
1603:
1598:
1597:
1579:
1578:
1563:
1551:
1549:
1548:
1543:
1538:
1537:
1519:
1518:
1506:
1505:
1484:
1455:
1453:
1452:
1447:
1445:
1433:
1431:
1430:
1425:
1423:
1411:
1409:
1408:
1403:
1401:
1400:
1388:
1387:
1317:
1315:
1314:
1309:
1304:
1302:
1301:
1292:
1288:
1273:
1268:
1260:
1256:
1238:
1227:
1216:
1215:
1214:
1198:
1187:
1167:
1156:
1155:
1154:
1138:
1126:
1125:
1109:
1101:
1096:
1095:
1094:
1030:is a concept in
1016:
1009:
1002:
950:
915:
881:
880:
877:
844:Surface integral
787:
786:
783:
691:
690:
687:
647:Limit comparison
567:
566:
563:
454:Riemann integral
407:
406:
403:
363:L'Hôpital's rule
320:Taylor's theorem
241:
240:
237:
181:
179:
178:
173:
125:
116:
111:
81:
80:
74:
67:
63:
60:
54:
27:
19:
14172:
14171:
14167:
14166:
14165:
14163:
14162:
14161:
14122:
14121:
14120:
14115:
14111:Steinmetz solid
14096:Integration Bee
14030:
14012:
13938:
13880:Colin Maclaurin
13856:
13824:
13818:
13690:
13684:Tensor calculus
13661:Volume integral
13597:
13572:Basic theorems
13535:Vector calculus
13529:
13410:
13377:Newton's method
13212:
13191:One-sided limit
13167:
13148:Rolle's theorem
13138:Linear function
13089:
13084:
13020:
12981:
12958:
12942:
12937:
12928:
12924:
12917:
12903:
12899:
12892:
12878:
12874:
12867:
12853:
12849:
12842:
12828:
12824:
12817:
12799:
12795:
12788:
12774:
12770:
12755:
12730:
12726:
12720:Euclidean space
12717:
12713:
12704:
12700:
12687:
12683:
12666:
12662:
12655:
12641:
12637:
12633:
12627:
12548:
12543:
12542:
12520:
12509:
12505:
12498:
12493:
12492:
12488:
12486:
12485:
12481:
12469:
12464:
12463:
12459:
12452:
12448:
12444:
12442:
12434:
12423:
12419:
12411:
12409:
12407:
12404:
12403:
12381:
12372:
12367:
12366:
12357:
12353:
12342:
12334:
12331:
12330:
12307:
12296:
12292:
12285:
12280:
12279:
12275:
12273:
12272:
12268:
12256:
12251:
12250:
12246:
12239:
12234:
12233:
12229:
12227:
12219:
12208:
12204:
12198:
12194:
12192:
12190:
12187:
12186:
12164:
12155:
12150:
12149:
12140:
12135:
12134:
12123:
12115:
12113:
12110:
12109:
12086:
12075:
12071:
12064:
12059:
12058:
12054:
12052:
12051:
12047:
12036:
12027:
12022:
12021:
12010:
12001:
11996:
11995:
11982:
11971:
11967:
11960:
11955:
11954:
11950:
11948:
11947:
11943:
11935:
11924:
11920:
11914:
11910:
11908:
11906:
11903:
11902:
11883:
11874:
11869:
11868:
11857:
11848:
11843:
11842:
11831:
11823:
11821:
11818:
11817:
11799:
11783:
11779:
11772:
11767:
11766:
11762:
11760:
11749:
11745:
11738:
11733:
11732:
11728:
11726:
11725:
11721:
11713:
11702:
11698:
11692:
11688:
11686:
11684:
11681:
11680:
11661:
11652:
11647:
11646:
11635:
11626:
11621:
11620:
11609:
11601:
11599:
11596:
11595:
11572:
11570:
11567:
11566:
11543:
11530:
11516:
11508:
11495:
11490:
11489:
11485:
11484:
11473:
11462:
11454:
11443:
11432:
11428:
11422:
11418:
11416:
11414:
11411:
11410:
11394:
11392:
11389:
11388:
11372:
11370:
11367:
11366:
11350:
11348:
11345:
11344:
11325:
11317:
11315:
11312:
11311:
11295:
11293:
11290:
11289:
11270:
11262:
11260:
11257:
11256:
11253:
11227:
11216:
11212:
11205:
11201:
11197:
11195:
11194:
11190:
11178:
11174:
11170:
11163:
11159:
11155:
11153:
11145:
11134:
11130:
11122:
11120:
11118:
11115:
11114:
11092:
11083:
11079:
11070:
11066:
11055:
11047:
11044:
11043:
11020:
11009:
11005:
10998:
10994:
10990:
10988:
10987:
10983:
10972:
10963:
10959:
10948:
10939:
10935:
10922:
10911:
10907:
10900:
10896:
10892:
10890:
10889:
10885:
10877:
10866:
10862:
10854:
10852:
10850:
10847:
10846:
10827:
10818:
10814:
10803:
10794:
10790:
10779:
10771:
10768:
10767:
10749:
10733:
10729:
10722:
10718:
10714:
10712:
10701:
10697:
10690:
10686:
10682:
10680:
10679:
10675:
10667:
10656:
10652:
10644:
10642:
10640:
10637:
10636:
10617:
10608:
10604:
10593:
10584:
10580:
10569:
10561:
10558:
10557:
10534:
10532:
10529:
10528:
10505:
10492:
10478:
10459:
10454:
10453:
10449:
10448:
10437:
10426:
10409:
10398:
10394:
10386:
10384:
10382:
10379:
10378:
10362:
10360:
10357:
10356:
10340:
10338:
10335:
10334:
10318:
10316:
10313:
10312:
10293:
10285:
10282:
10281:
10265:
10263:
10260:
10259:
10240:
10232:
10229:
10228:
10225:
10199:
10188:
10184:
10177:
10172:
10171:
10167:
10165:
10164:
10160:
10148:
10143:
10142:
10138:
10131:
10126:
10125:
10121:
10119:
10111:
10100:
10096:
10090:
10086:
10084:
10082:
10079:
10078:
10056:
10047:
10042:
10041:
10032:
10027:
10026:
10015:
10007:
10005:
10002:
10001:
9978:
9967:
9963:
9956:
9951:
9950:
9946:
9944:
9943:
9939:
9928:
9919:
9914:
9913:
9902:
9893:
9888:
9887:
9874:
9863:
9859:
9852:
9847:
9846:
9842:
9840:
9839:
9835:
9827:
9816:
9812:
9806:
9802:
9800:
9798:
9795:
9794:
9775:
9766:
9761:
9760:
9749:
9740:
9735:
9734:
9723:
9715:
9713:
9710:
9709:
9691:
9675:
9671:
9664:
9659:
9658:
9654:
9652:
9641:
9637:
9630:
9625:
9624:
9620:
9618:
9617:
9613:
9605:
9594:
9590:
9584:
9580:
9578:
9576:
9573:
9572:
9553:
9544:
9539:
9538:
9527:
9518:
9513:
9512:
9501:
9493:
9491:
9488:
9487:
9455:
9442:
9428:
9420:
9407:
9402:
9401:
9397:
9396:
9385:
9374:
9366:
9355:
9344:
9340:
9334:
9330:
9328:
9326:
9323:
9322:
9317:
9296:
9285:
9281:
9274:
9270:
9266:
9264:
9252:
9248:
9244:
9237:
9233:
9229:
9227:
9219:
9208:
9204:
9196:
9194:
9192:
9189:
9188:
9166:
9157:
9153:
9144:
9140:
9129:
9121:
9118:
9117:
9094:
9083:
9079:
9072:
9068:
9064:
9062:
9061:
9057:
9046:
9037:
9033:
9022:
9013:
9009:
8996:
8985:
8981:
8974:
8970:
8966:
8964:
8963:
8959:
8951:
8940:
8936:
8928:
8926:
8924:
8921:
8920:
8901:
8892:
8888:
8877:
8868:
8864:
8853:
8845:
8842:
8841:
8823:
8807:
8803:
8796:
8792:
8788:
8786:
8775:
8771:
8764:
8760:
8756:
8754:
8753:
8749:
8741:
8730:
8726:
8718:
8716:
8714:
8711:
8710:
8691:
8682:
8678:
8667:
8658:
8654:
8643:
8635:
8632:
8631:
8595:
8582:
8568:
8549:
8544:
8543:
8539:
8538:
8527:
8516:
8499:
8488:
8484:
8476:
8474:
8472:
8469:
8468:
8451:
8442:
8423:
8397:
8386:
8369:
8358:
8354:
8346:
8344:
8333:
8325:
8318:
8317:
8313:
8305:
8294:
8274:
8270:
8262:
8260:
8258:
8255:
8254:
8235:
8231:
8223:
8221:
8219:
8216:
8215:
8195:
8193:
8190:
8189:
8166:
8131:
8123:
8094:
8080:
8077:
8076:
8048:
8040:
8014:
7997:
7994:
7993:
7962:
7954:
7937:
7920:
7912:
7901:
7884:
7867:
7850:
7847:
7846:
7827:
7819:
7808:
7800:
7798:
7795:
7794:
7771:
7764:
7763:
7758:
7753:
7747:
7746:
7741:
7736:
7727:
7726:
7721:
7716:
7706:
7705:
7697:
7690:
7689:
7684:
7679:
7673:
7672:
7667:
7662:
7656:
7655:
7647:
7642:
7632:
7631:
7623:
7616:
7615:
7610:
7602:
7596:
7595:
7590:
7585:
7579:
7578:
7573:
7568:
7558:
7557:
7549:
7547:
7544:
7543:
7513:
7505:
7476:
7462:
7459:
7458:
7439:
7434:
7420:
7419:
7417:
7414:
7413:
7390:
7385:
7347:
7346:
7293:
7292:
7278:
7275:
7274:
7254:
7243:
7242:
7241:
7232:
7228:
7205:
7204:
7195:
7190:
7184:
7181:
7180:
7145:
7141:
7137:
7132:
7120:
7116:
7112:
7107:
7106:
7102:
7100:
7097:
7096:
7068:
7064:
7052:
7048:
7042:
7026:
7022:
7007:
7003:
6988:
6984:
6972:
6953:
6949:
6940:
6936:
6931:
6928:
6927:
6903:
6899:
6897:
6894:
6893:
6873:
6869:
6857:
6853:
6847:
6831:
6827:
6821:
6817:
6802:
6798:
6796:
6793:
6792:
6772:
6768:
6762:
6751:
6750:
6749:
6737:
6733:
6721:
6708:
6697:
6696:
6695:
6686:
6682:
6659:
6658:
6649:
6645:
6643:
6640:
6639:
6599:
6598:
6539:
6538:
6524:
6521:
6520:
6494:
6490:
6484:
6480:
6474:
6470:
6458:
6444:
6435:
6431:
6425:
6421:
6415:
6376:
6373:
6372:
6347:; we now write
6312:
6308:
6284:
6280:
6256:
6252:
6250:
6247:
6246:
6223:
6219:
6217:
6214:
6213:
6176:
6175:
6134:
6133:
6125:
6122:
6121:
6104:
6100:
6098:
6095:
6094:
6018:
5996:
5995:
5991:
5974:
5960:
5943:
5935:
5918:
5907:
5890:
5878:
5874:
5866:
5855:
5852:
5851:
5816:
5786:
5785:
5781:
5764:
5756:
5753:
5752:
5732:
5717:
5710:
5706:
5705:
5685:
5682:
5681:
5661:
5639:
5638:
5636:
5629:
5625:
5624:
5615:
5599:
5592:
5588:
5587:
5578:
5574:
5566:
5555:
5552:
5551:
5525:
5508:
5490:
5486:
5478:
5467:
5464:
5463:
5420:
5409:
5407:
5404:
5403:
5345:
5328:
5314:
5306:
5303:
5302:
5253:
5236:
5228:
5225:
5224:
5196:
5194:
5191:
5190:
5168:
5146:
5139:
5135:
5124:
5116:
5108:
5105:
5104:
5068:
5047:
5044:
5043:
5002:
4997:
4986:
4982:
4969:
4961:
4959:
4917:
4914:
4913:
4847:
4845:
4832:
4824:
4822:
4820:
4817:
4816:
4776:
4768:
4751:
4729:
4728:
4724:
4707:
4693:
4685:
4682:
4681:
4643:
4642:
4638:
4621:
4613:
4610:
4609:
4585:
4577:
4570:
4566:
4549:
4541:
4538:
4537:
4483:
4481:
4478:
4477:
4466:
4461:
4459:In group theory
4455:of the author.
4453:sign convention
4436:
4433:
4432:
4412:
4408:
4396:
4394:
4393:
4387:
4383:
4377:
4361:
4357:
4348:
4344:
4335:
4331:
4326:
4323:
4322:
4302:
4298:
4289:
4285:
4276:
4272:
4263:
4259:
4252:
4248:
4244:
4232:
4219:
4215:
4198:
4159:
4155:
4117:
4106:
4103:
4102:
4074:
4059:
4055:
4048:
4044:
4040:
4034:
4022:
4019:
4018:
3999:
3997:
3994:
3993:
3955:
3951:
3945:
3941:
3935:
3923:
3920:
3919:
3903:
3900:
3899:
3879:
3876:
3875:
3856:
3854:
3851:
3850:
3831:
3829:
3826:
3825:
3809:
3806:
3805:
3789:
3786:
3785:
3766:
3764:
3761:
3760:
3744:
3741:
3740:
3733:
3685:
3679:
3678:
3677:
3675:
3672:
3671:
3646:
3643:
3642:
3622:
3618:
3594:
3590:
3566:
3562:
3556:
3550:
3549:
3548:
3546:
3543:
3542:
3516:
3512:
3510:
3507:
3506:
3480:
3476:
3474:
3471:
3470:
3463:
3448:
3438:
3432:
3426:
3400:
3367:
3362:
3361:
3358:
3357:
3346:
3333:
3332:
3328:
3326:
3323:
3322:
3312:
3302:
3292:
3261:
3260:
3254:
3253:
3252:
3250:
3247:
3246:
3223:
3210:
3209:
3205:
3203:
3200:
3199:
3176:
3163:
3162:
3158:
3156:
3153:
3152:
3137:
3131:
3125:
3115:
3109:
3099:
3093:
3083:
3077:
3071:
3065:
3055:
3052:
3046:
3019:
3006:
3005:
3001:
2990:
2973:
2962:
2937:
2936:
2932:
2930:
2927:
2926:
2875:
2874:
2870:
2854:
2853:
2849:
2824:
2823:
2819:
2817:
2814:
2813:
2778:
2777:
2773:
2748:
2747:
2743:
2741:
2738:
2737:
2702:
2701:
2697:
2684:
2683:
2679:
2654:
2653:
2649:
2647:
2644:
2643:
2607:
2591:Euclidean space
2587:
2561:
2556:
2551:
2550:
2545:
2543:
2532:
2512:
2504:
2497:
2496:
2492:
2490:
2487:
2486:
2456:
2451:
2446:
2442:
2433:
2416:
2405:
2398:
2396:
2384:
2369:
2361:
2354:
2353:
2349:
2347:
2344:
2343:
2337:
2331:
2317:Euclidean space
2293:
2271:represents the
2248:
2247:
2199:
2195:
2186:
2185:
2115:
2113:
2101:
2088:
2087:
2014:
2012:
2000:
1989:
1982:
1980:
1977:
1976:
1936:
1933:
1932:
1908:
1905:
1904:
1882:
1879:
1878:
1860:
1849:
1829:
1821:
1814:
1813:
1809:
1807:
1804:
1803:
1782:
1743:
1726:
1715:
1708:
1706:
1694:
1679:
1671:
1664:
1663:
1659:
1657:
1654:
1653:
1649:defined by the
1634:
1627:
1626:
1622:
1620:
1617:
1616:
1593:
1589:
1574:
1570:
1559:
1557:
1554:
1553:
1552:along a vector
1533:
1529:
1514:
1510:
1501:
1497:
1480:
1472:
1469:
1468:
1466:scalar function
1441:
1439:
1436:
1435:
1419:
1417:
1414:
1413:
1396:
1392:
1383:
1379:
1356:
1353:
1352:
1341:
1297:
1293:
1284:
1274:
1272:
1264:
1252:
1242:
1234:
1223:
1210:
1209:
1205:
1194:
1183:
1163:
1150:
1149:
1145:
1134:
1121:
1120:
1105:
1097:
1090:
1089:
1085:
1083:
1080:
1079:
1065:scalar function
1020:
991:
990:
976:Integration Bee
951:
948:
941:
940:
916:
913:
906:
905:
878:
875:
868:
867:
849:Volume integral
784:
779:
772:
771:
688:
683:
676:
675:
645:
564:
559:
552:
551:
543:Risch algorithm
518:Euler's formula
404:
399:
392:
391:
373:General Leibniz
256:generalizations
238:
233:
226:
212:Rolle's theorem
207:
182:
118:
112:
107:
101:
98:
97:
75:
64:
58:
55:
40:
28:
17:
12:
11:
5:
14170:
14160:
14159:
14154:
14149:
14144:
14139:
14134:
14117:
14116:
14114:
14113:
14108:
14103:
14098:
14093:
14091:Gabriel's horn
14088:
14083:
14082:
14081:
14076:
14071:
14066:
14061:
14053:
14052:
14051:
14042:
14040:
14036:
14035:
14032:
14031:
14029:
14028:
14023:
14021:List of limits
14017:
14014:
14013:
14011:
14010:
14009:
14008:
14003:
13998:
13988:
13987:
13986:
13976:
13971:
13966:
13961:
13955:
13953:
13944:
13940:
13939:
13937:
13936:
13929:
13922:
13920:Leonhard Euler
13917:
13912:
13907:
13902:
13897:
13892:
13887:
13882:
13877:
13872:
13866:
13864:
13858:
13857:
13855:
13854:
13849:
13844:
13839:
13834:
13828:
13826:
13820:
13819:
13817:
13816:
13815:
13814:
13809:
13804:
13799:
13794:
13789:
13784:
13779:
13774:
13769:
13761:
13760:
13759:
13754:
13753:
13752:
13747:
13737:
13732:
13727:
13722:
13717:
13712:
13704:
13698:
13696:
13692:
13691:
13689:
13688:
13687:
13686:
13681:
13676:
13671:
13663:
13658:
13653:
13648:
13643:
13638:
13633:
13628:
13623:
13621:Hessian matrix
13618:
13613:
13607:
13605:
13599:
13598:
13596:
13595:
13594:
13593:
13588:
13583:
13578:
13576:Line integrals
13570:
13569:
13568:
13563:
13558:
13553:
13548:
13539:
13537:
13531:
13530:
13528:
13527:
13522:
13517:
13516:
13515:
13510:
13502:
13497:
13496:
13495:
13485:
13484:
13483:
13478:
13473:
13463:
13458:
13457:
13456:
13446:
13441:
13436:
13431:
13426:
13424:Antiderivative
13420:
13418:
13412:
13411:
13409:
13408:
13407:
13406:
13401:
13396:
13386:
13385:
13384:
13379:
13371:
13370:
13369:
13364:
13359:
13354:
13344:
13343:
13342:
13337:
13332:
13327:
13319:
13318:
13317:
13312:
13311:
13310:
13300:
13295:
13290:
13285:
13280:
13270:
13269:
13268:
13263:
13253:
13248:
13243:
13238:
13233:
13228:
13222:
13220:
13214:
13213:
13211:
13210:
13205:
13200:
13195:
13194:
13193:
13183:
13177:
13175:
13169:
13168:
13166:
13165:
13160:
13155:
13150:
13145:
13140:
13135:
13130:
13125:
13120:
13115:
13110:
13105:
13099:
13097:
13091:
13090:
13083:
13082:
13075:
13068:
13060:
13054:
13053:
13043:
13019:
13018:External links
13016:
13015:
13014:
12996:(3): 477–487.
12985:
12979:
12962:
12956:
12941:
12938:
12936:
12935:
12922:
12915:
12897:
12890:
12872:
12866:978-1107005211
12865:
12847:
12840:
12822:
12815:
12793:
12786:
12768:
12753:
12743:(2012-01-01).
12724:
12711:
12698:
12681:
12660:
12653:
12634:
12632:
12629:
12625:
12624:
12618:
12613:
12607:
12601:
12598:Lie derivative
12595:
12590:
12585:
12579:
12573:
12567:
12561:
12555:
12547:
12544:
12541:
12540:
12528:
12523:
12519:
12512:
12508:
12501:
12496:
12491:
12484:
12480:
12472:
12467:
12462:
12455:
12451:
12447:
12441:
12437:
12433:
12426:
12422:
12417:
12414:
12391:
12388:
12384:
12380:
12375:
12370:
12365:
12360:
12356:
12352:
12349:
12345:
12341:
12338:
12327:
12315:
12310:
12306:
12299:
12295:
12288:
12283:
12278:
12271:
12267:
12259:
12254:
12249:
12242:
12237:
12232:
12226:
12222:
12218:
12211:
12207:
12201:
12197:
12174:
12171:
12167:
12163:
12158:
12153:
12148:
12143:
12138:
12133:
12130:
12126:
12122:
12118:
12106:
12094:
12089:
12085:
12078:
12074:
12067:
12062:
12057:
12050:
12046:
12043:
12039:
12035:
12030:
12025:
12020:
12017:
12013:
12009:
12004:
11999:
11994:
11990:
11985:
11981:
11974:
11970:
11963:
11958:
11953:
11946:
11942:
11938:
11934:
11927:
11923:
11917:
11913:
11890:
11886:
11882:
11877:
11872:
11867:
11864:
11860:
11856:
11851:
11846:
11841:
11838:
11834:
11830:
11826:
11814:
11802:
11798:
11794:
11786:
11782:
11775:
11770:
11765:
11759:
11752:
11748:
11741:
11736:
11731:
11724:
11720:
11716:
11712:
11705:
11701:
11695:
11691:
11668:
11664:
11660:
11655:
11650:
11645:
11642:
11638:
11634:
11629:
11624:
11619:
11616:
11612:
11608:
11604:
11575:
11552:
11549:
11546:
11541:
11537:
11533:
11526:
11523:
11519:
11515:
11511:
11501:
11498:
11494:
11488:
11483:
11480:
11476:
11472:
11469:
11465:
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11457:
11453:
11450:
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11442:
11435:
11431:
11425:
11421:
11397:
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11353:
11332:
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11320:
11298:
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11137:
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11099:
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10986:
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10947:
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10704:
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10693:
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10678:
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10666:
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10650:
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10587:
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10576:
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10568:
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10537:
10514:
10511:
10508:
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10499:
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10412:
10408:
10401:
10397:
10392:
10389:
10365:
10343:
10321:
10300:
10296:
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10289:
10268:
10247:
10243:
10239:
10236:
10224:
10221:
10220:
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10198:
10191:
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10175:
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9917:
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9909:
9905:
9901:
9896:
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9877:
9873:
9866:
9862:
9855:
9850:
9845:
9838:
9834:
9830:
9826:
9819:
9815:
9809:
9805:
9782:
9778:
9774:
9769:
9764:
9759:
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9748:
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9738:
9733:
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9726:
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9706:
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9667:
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9640:
9633:
9628:
9623:
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9608:
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9597:
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9464:
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9406:
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9337:
9333:
9316:
9313:
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9311:
9299:
9295:
9288:
9284:
9277:
9273:
9269:
9255:
9251:
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9240:
9236:
9232:
9226:
9222:
9218:
9211:
9207:
9202:
9199:
9176:
9173:
9169:
9165:
9160:
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9147:
9143:
9139:
9136:
9132:
9128:
9125:
9114:
9102:
9097:
9093:
9086:
9082:
9075:
9071:
9067:
9060:
9053:
9049:
9045:
9040:
9036:
9032:
9029:
9025:
9021:
9016:
9012:
9004:
8999:
8995:
8988:
8984:
8977:
8973:
8969:
8962:
8958:
8954:
8950:
8943:
8939:
8934:
8931:
8908:
8904:
8900:
8895:
8891:
8884:
8880:
8876:
8871:
8867:
8863:
8860:
8856:
8852:
8849:
8838:
8826:
8822:
8818:
8810:
8806:
8799:
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8791:
8785:
8778:
8774:
8767:
8763:
8759:
8752:
8748:
8744:
8740:
8733:
8729:
8724:
8721:
8698:
8694:
8690:
8685:
8681:
8677:
8674:
8670:
8666:
8661:
8657:
8653:
8650:
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8639:
8604:
8601:
8598:
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8575:
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8567:
8564:
8555:
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8537:
8534:
8530:
8526:
8523:
8519:
8515:
8512:
8509:
8506:
8502:
8498:
8491:
8487:
8482:
8479:
8450:
8447:
8443:
8435:
8422:
8419:
8407:
8404:
8400:
8396:
8393:
8389:
8385:
8382:
8379:
8376:
8372:
8368:
8361:
8357:
8352:
8349:
8343:
8340:
8336:
8332:
8328:
8321:
8316:
8312:
8308:
8304:
8301:
8297:
8293:
8290:
8287:
8284:
8277:
8273:
8268:
8265:
8238:
8234:
8229:
8226:
8198:
8165:
8162:
8150:
8147:
8144:
8141:
8138:
8134:
8130:
8126:
8122:
8119:
8116:
8113:
8110:
8107:
8104:
8101:
8097:
8093:
8090:
8087:
8084:
8064:
8061:
8058:
8055:
8051:
8047:
8043:
8039:
8036:
8033:
8030:
8027:
8024:
8021:
8017:
8013:
8010:
8007:
8004:
8001:
7981:
7978:
7975:
7972:
7969:
7965:
7961:
7957:
7953:
7950:
7947:
7944:
7940:
7936:
7933:
7930:
7927:
7923:
7919:
7915:
7911:
7908:
7904:
7900:
7897:
7894:
7891:
7887:
7883:
7880:
7877:
7874:
7870:
7866:
7863:
7860:
7857:
7854:
7834:
7830:
7826:
7822:
7818:
7815:
7811:
7807:
7803:
7778:
7774:
7768:
7762:
7759:
7757:
7754:
7752:
7749:
7748:
7745:
7742:
7740:
7737:
7735:
7732:
7729:
7728:
7725:
7722:
7720:
7717:
7715:
7712:
7711:
7709:
7704:
7700:
7694:
7688:
7685:
7683:
7680:
7678:
7675:
7674:
7671:
7668:
7666:
7663:
7661:
7658:
7657:
7654:
7651:
7648:
7646:
7643:
7641:
7638:
7637:
7635:
7630:
7626:
7620:
7614:
7611:
7609:
7606:
7603:
7601:
7598:
7597:
7594:
7591:
7589:
7586:
7584:
7581:
7580:
7577:
7574:
7572:
7569:
7567:
7564:
7563:
7561:
7556:
7552:
7523:
7520:
7516:
7512:
7508:
7504:
7501:
7498:
7495:
7492:
7489:
7486:
7483:
7479:
7475:
7472:
7469:
7466:
7446:
7442:
7437:
7433:
7427:
7424:
7389:
7386:
7372:
7369:
7366:
7363:
7360:
7354:
7351:
7345:
7342:
7339:
7336:
7333:
7330:
7327:
7324:
7321:
7318:
7315:
7312:
7309:
7306:
7300:
7297:
7291:
7288:
7285:
7282:
7262:
7257:
7250:
7247:
7240:
7235:
7231:
7227:
7224:
7221:
7218:
7212:
7209:
7203:
7198:
7189:
7164:
7161:
7157:
7148:
7144:
7140:
7136:
7131:
7123:
7119:
7115:
7111:
7105:
7076:
7071:
7067:
7061:
7058:
7055:
7051:
7045:
7041:
7037:
7034:
7029:
7025:
7021:
7016:
7013:
7010:
7006:
7002:
6997:
6994:
6991:
6987:
6983:
6980:
6975:
6971:
6967:
6964:
6961:
6956:
6952:
6948:
6943:
6939:
6935:
6909:
6906:
6902:
6881:
6876:
6872:
6866:
6863:
6860:
6856:
6850:
6846:
6842:
6839:
6834:
6830:
6824:
6820:
6816:
6813:
6808:
6805:
6801:
6780:
6775:
6771:
6765:
6758:
6755:
6746:
6743:
6740:
6736:
6730:
6727:
6724:
6720:
6716:
6711:
6704:
6701:
6694:
6689:
6685:
6681:
6678:
6675:
6672:
6666:
6663:
6657:
6652:
6648:
6627:
6624:
6621:
6618:
6615:
6612:
6606:
6603:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6555:
6552:
6546:
6543:
6537:
6534:
6531:
6528:
6508:
6505:
6500:
6497:
6493:
6487:
6483:
6477:
6473:
6467:
6464:
6461:
6457:
6451:
6448:
6443:
6438:
6434:
6428:
6424:
6418:
6414:
6410:
6407:
6404:
6401:
6398:
6395:
6392:
6389:
6386:
6383:
6380:
6320:
6315:
6311:
6307:
6304:
6301:
6298:
6295:
6292:
6287:
6283:
6279:
6276:
6273:
6270:
6267:
6264:
6259:
6255:
6234:
6231:
6226:
6222:
6201:
6198:
6195:
6192:
6189:
6183:
6180:
6174:
6171:
6168:
6165:
6162:
6159:
6156:
6153:
6150:
6147:
6141:
6138:
6132:
6129:
6107:
6103:
6028:
6025:
6021:
6017:
6014:
6010:
6006:
6003:
5999:
5994:
5990:
5987:
5984:
5981:
5977:
5973:
5970:
5967:
5963:
5959:
5956:
5953:
5950:
5946:
5942:
5938:
5934:
5931:
5928:
5925:
5921:
5917:
5914:
5910:
5906:
5903:
5900:
5897:
5893:
5889:
5886:
5881:
5877:
5873:
5869:
5865:
5862:
5859:
5804:
5800:
5796:
5793:
5789:
5784:
5780:
5777:
5774:
5771:
5767:
5763:
5760:
5740:
5735:
5730:
5724:
5721:
5716:
5713:
5709:
5704:
5701:
5698:
5695:
5692:
5689:
5669:
5664:
5659:
5653:
5649:
5646:
5642:
5635:
5632:
5628:
5623:
5618:
5613:
5609:
5606:
5602:
5598:
5595:
5591:
5586:
5581:
5577:
5573:
5569:
5565:
5562:
5559:
5535:
5532:
5528:
5524:
5521:
5518:
5515:
5511:
5507:
5504:
5501:
5498:
5493:
5489:
5485:
5481:
5477:
5474:
5471:
5427:
5423:
5419:
5416:
5412:
5361:
5358:
5354:
5351:
5348:
5344:
5341:
5338:
5335:
5331:
5327:
5324:
5321:
5317:
5313:
5310:
5266:
5263:
5260:
5256:
5252:
5249:
5246:
5243:
5239:
5235:
5232:
5206:
5203:
5199:
5178:
5175:
5171:
5167:
5164:
5160:
5156:
5153:
5149:
5145:
5142:
5138:
5134:
5131:
5127:
5123:
5119:
5115:
5112:
5084:
5081:
5078:
5075:
5071:
5067:
5064:
5060:
5057:
5054:
5051:
5031:
5028:
5025:
5022:
5019:
5015:
5008:
5005:
5001:
4995:
4992:
4989:
4985:
4981:
4975:
4972:
4967:
4964:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4889:
4884:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4844:
4838:
4835:
4830:
4827:
4798:
4786:
4783:
4779:
4775:
4771:
4767:
4764:
4761:
4758:
4754:
4750:
4747:
4743:
4739:
4736:
4732:
4727:
4723:
4720:
4717:
4714:
4710:
4706:
4703:
4700:
4696:
4692:
4689:
4661:
4657:
4653:
4650:
4646:
4641:
4637:
4634:
4631:
4628:
4624:
4620:
4617:
4597:
4593:
4588:
4584:
4580:
4576:
4573:
4569:
4565:
4562:
4559:
4556:
4552:
4548:
4545:
4534:representation
4499:
4496:
4493:
4490:
4486:
4465:
4462:
4460:
4457:
4440:
4420:
4415:
4411:
4405:
4402:
4399:
4390:
4386:
4380:
4376:
4372:
4369:
4364:
4360:
4356:
4351:
4347:
4343:
4338:
4334:
4330:
4310:
4305:
4301:
4297:
4292:
4288:
4284:
4279:
4275:
4271:
4266:
4262:
4255:
4251:
4247:
4241:
4238:
4235:
4231:
4227:
4222:
4218:
4214:
4211:
4208:
4204:
4201:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4162:
4158:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4123:
4120:
4116:
4113:
4110:
4090:
4087:
4084:
4080:
4077:
4073:
4070:
4067:
4062:
4058:
4051:
4047:
4043:
4037:
4033:
4029:
4026:
4005:
4002:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3958:
3954:
3948:
3944:
3938:
3934:
3930:
3927:
3907:
3883:
3862:
3859:
3837:
3834:
3813:
3793:
3772:
3769:
3748:
3732:
3729:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3688:
3682:
3659:
3656:
3653:
3650:
3630:
3625:
3621:
3617:
3614:
3611:
3608:
3605:
3602:
3597:
3593:
3589:
3586:
3583:
3580:
3577:
3574:
3569:
3565:
3559:
3553:
3530:
3527:
3524:
3519:
3515:
3494:
3491:
3488:
3483:
3479:
3467:Lie derivative
3462:
3459:
3414:
3409:
3406:
3403:
3398:
3394:
3391:
3388:
3385:
3382:
3379:
3373:
3370:
3366:
3360:
3356:
3353:
3349:
3345:
3342:
3336:
3331:
3276:
3273:
3270:
3264:
3257:
3243:Lie derivative
3230:
3226:
3222:
3219:
3213:
3208:
3183:
3179:
3175:
3172:
3166:
3161:
3106:tangent vector
3090:differentiable
3045:
3042:
3041:
3040:
3029:
3026:
3022:
3018:
3015:
3009:
3004:
3000:
2997:
2993:
2989:
2986:
2983:
2979:
2976:
2972:
2969:
2965:
2961:
2958:
2955:
2952:
2949:
2946:
2940:
2935:
2898:
2887:
2884:
2878:
2873:
2869:
2866:
2863:
2857:
2852:
2848:
2845:
2842:
2839:
2836:
2833:
2827:
2822:
2810:Leibniz's rule
2801:
2790:
2787:
2781:
2776:
2772:
2769:
2766:
2763:
2760:
2757:
2751:
2746:
2725:
2714:
2711:
2705:
2700:
2696:
2693:
2687:
2682:
2678:
2675:
2672:
2669:
2666:
2663:
2657:
2652:
2627:differentiable
2606:
2603:
2586:
2583:
2571:
2564:
2559:
2554:
2548:
2542:
2539:
2535:
2531:
2528:
2525:
2522:
2519:
2515:
2511:
2507:
2500:
2495:
2466:
2459:
2454:
2449:
2445:
2440:
2436:
2432:
2429:
2426:
2423:
2419:
2415:
2412:
2408:
2404:
2401:
2393:
2390:
2387:
2383:
2379:
2376:
2372:
2368:
2364:
2357:
2352:
2292:
2289:
2273:rate of change
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2202:
2198:
2194:
2191:
2189:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2110:
2107:
2104:
2100:
2096:
2093:
2091:
2089:
2084:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2009:
2006:
2003:
1999:
1995:
1992:
1990:
1988:
1985:
1984:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1912:
1886:
1863:
1859:
1856:
1852:
1848:
1845:
1842:
1839:
1836:
1832:
1828:
1824:
1817:
1812:
1790:differentiable
1781:
1778:
1759:
1754:
1750:
1746:
1742:
1739:
1736:
1733:
1729:
1725:
1722:
1718:
1714:
1711:
1703:
1700:
1697:
1693:
1689:
1686:
1682:
1678:
1674:
1667:
1662:
1637:
1630:
1625:
1601:
1596:
1592:
1588:
1585:
1582:
1577:
1573:
1569:
1566:
1562:
1541:
1536:
1532:
1528:
1525:
1522:
1517:
1513:
1509:
1504:
1500:
1496:
1493:
1490:
1487:
1483:
1479:
1476:
1444:
1422:
1399:
1395:
1391:
1386:
1382:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1340:
1337:
1307:
1300:
1296:
1291:
1287:
1283:
1280:
1277:
1271:
1267:
1263:
1259:
1255:
1251:
1248:
1245:
1241:
1237:
1233:
1230:
1226:
1222:
1219:
1213:
1208:
1204:
1201:
1197:
1193:
1190:
1186:
1182:
1179:
1176:
1173:
1170:
1166:
1162:
1159:
1153:
1148:
1144:
1141:
1137:
1133:
1129:
1124:
1119:
1115:
1112:
1108:
1104:
1100:
1093:
1088:
1041:along a given
1022:
1021:
1019:
1018:
1011:
1004:
996:
993:
992:
989:
988:
983:
978:
973:
971:List of topics
968:
963:
958:
952:
947:
946:
943:
942:
939:
938:
933:
928:
923:
917:
912:
911:
908:
907:
902:
901:
900:
899:
894:
889:
879:
874:
873:
870:
869:
864:
863:
862:
861:
856:
851:
846:
841:
836:
831:
823:
822:
818:
817:
816:
815:
810:
805:
800:
792:
791:
785:
778:
777:
774:
773:
768:
767:
766:
765:
760:
755:
750:
745:
740:
732:
731:
727:
726:
725:
724:
719:
714:
709:
704:
699:
689:
682:
681:
678:
677:
672:
671:
670:
669:
664:
659:
654:
649:
643:
638:
633:
628:
623:
615:
614:
608:
607:
606:
605:
600:
595:
590:
585:
580:
565:
558:
557:
554:
553:
548:
547:
546:
545:
540:
535:
530:
528:Changing order
525:
520:
515:
497:
492:
487:
479:
478:
477:Integration by
474:
473:
472:
471:
466:
461:
456:
451:
441:
439:Antiderivative
433:
432:
428:
427:
426:
425:
420:
415:
405:
398:
397:
394:
393:
388:
387:
386:
385:
380:
375:
370:
365:
360:
355:
350:
345:
340:
332:
331:
325:
324:
323:
322:
317:
312:
307:
302:
297:
289:
288:
284:
283:
282:
281:
280:
279:
274:
269:
259:
246:
245:
239:
232:
231:
228:
227:
225:
224:
219:
214:
208:
206:
205:
200:
194:
193:
192:
184:
183:
171:
168:
165:
162:
159:
156:
153:
150:
147:
144:
141:
138:
134:
131:
128:
124:
121:
115:
110:
106:
96:
93:
92:
86:
85:
77:
76:
45:. Please help
31:
29:
22:
15:
9:
6:
4:
3:
2:
14169:
14158:
14155:
14153:
14150:
14148:
14145:
14143:
14140:
14138:
14135:
14133:
14130:
14129:
14127:
14112:
14109:
14107:
14104:
14102:
14099:
14097:
14094:
14092:
14089:
14087:
14084:
14080:
14077:
14075:
14072:
14070:
14067:
14065:
14062:
14060:
14057:
14056:
14054:
14050:
14047:
14046:
14044:
14043:
14041:
14037:
14027:
14024:
14022:
14019:
14018:
14015:
14007:
14004:
14002:
13999:
13997:
13994:
13993:
13992:
13989:
13985:
13982:
13981:
13980:
13977:
13975:
13972:
13970:
13967:
13965:
13962:
13960:
13957:
13956:
13954:
13952:
13948:
13945:
13941:
13935:
13934:
13930:
13928:
13927:
13923:
13921:
13918:
13916:
13913:
13911:
13908:
13906:
13903:
13901:
13898:
13896:
13895:Infinitesimal
13893:
13891:
13888:
13886:
13883:
13881:
13878:
13876:
13873:
13871:
13868:
13867:
13865:
13863:
13859:
13853:
13850:
13848:
13845:
13843:
13840:
13838:
13835:
13833:
13830:
13829:
13827:
13821:
13813:
13810:
13808:
13805:
13803:
13800:
13798:
13795:
13793:
13790:
13788:
13785:
13783:
13780:
13778:
13775:
13773:
13770:
13768:
13765:
13764:
13762:
13758:
13755:
13751:
13748:
13746:
13743:
13742:
13741:
13738:
13736:
13733:
13731:
13728:
13726:
13723:
13721:
13718:
13716:
13713:
13711:
13708:
13707:
13705:
13703:
13700:
13699:
13697:
13693:
13685:
13682:
13680:
13677:
13675:
13672:
13670:
13667:
13666:
13664:
13662:
13659:
13657:
13654:
13652:
13649:
13647:
13644:
13642:
13639:
13637:
13636:Line integral
13634:
13632:
13629:
13627:
13624:
13622:
13619:
13617:
13614:
13612:
13609:
13608:
13606:
13604:
13600:
13592:
13589:
13587:
13584:
13582:
13579:
13577:
13574:
13573:
13571:
13567:
13564:
13562:
13559:
13557:
13554:
13552:
13549:
13547:
13544:
13543:
13541:
13540:
13538:
13536:
13532:
13526:
13523:
13521:
13518:
13514:
13511:
13509:
13508:Washer method
13506:
13505:
13503:
13501:
13498:
13494:
13491:
13490:
13489:
13486:
13482:
13479:
13477:
13474:
13472:
13471:trigonometric
13469:
13468:
13467:
13464:
13462:
13459:
13455:
13452:
13451:
13450:
13447:
13445:
13442:
13440:
13437:
13435:
13432:
13430:
13427:
13425:
13422:
13421:
13419:
13417:
13413:
13405:
13402:
13400:
13397:
13395:
13392:
13391:
13390:
13387:
13383:
13380:
13378:
13375:
13374:
13372:
13368:
13365:
13363:
13360:
13358:
13355:
13353:
13350:
13349:
13348:
13345:
13341:
13340:Related rates
13338:
13336:
13333:
13331:
13328:
13326:
13323:
13322:
13320:
13316:
13313:
13309:
13306:
13305:
13304:
13301:
13299:
13296:
13294:
13291:
13289:
13286:
13284:
13281:
13279:
13276:
13275:
13274:
13271:
13267:
13264:
13262:
13259:
13258:
13257:
13254:
13252:
13249:
13247:
13244:
13242:
13239:
13237:
13234:
13232:
13229:
13227:
13224:
13223:
13221:
13219:
13215:
13209:
13206:
13204:
13201:
13199:
13196:
13192:
13189:
13188:
13187:
13184:
13182:
13179:
13178:
13176:
13174:
13170:
13164:
13161:
13159:
13156:
13154:
13151:
13149:
13146:
13144:
13141:
13139:
13136:
13134:
13131:
13129:
13126:
13124:
13121:
13119:
13116:
13114:
13111:
13109:
13106:
13104:
13101:
13100:
13098:
13096:
13092:
13088:
13081:
13076:
13074:
13069:
13067:
13062:
13061:
13058:
13051:
13047:
13044:
13041:
13037:
13034:
13033:
13032:
13030:
13025:
13011:
13007:
13003:
12999:
12995:
12991:
12986:
12982:
12976:
12971:
12970:
12963:
12959:
12957:0-13-011189-9
12953:
12949:
12944:
12943:
12932:
12926:
12918:
12916:9780306447907
12912:
12908:
12901:
12893:
12891:9780547209982
12887:
12883:
12876:
12868:
12862:
12858:
12851:
12843:
12841:9780691145587
12837:
12833:
12826:
12818:
12816:9780521550017
12812:
12807:
12806:
12797:
12789:
12787:9780691145587
12783:
12779:
12772:
12764:
12760:
12756:
12754:9780470888612
12750:
12746:
12742:
12738:
12734:
12728:
12721:
12715:
12708:
12702:
12695:
12691:
12685:
12678:
12675:, such as in
12674:
12670:
12664:
12656:
12650:
12646:
12639:
12635:
12628:
12622:
12619:
12617:
12614:
12611:
12608:
12605:
12602:
12599:
12596:
12594:
12591:
12589:
12586:
12583:
12580:
12577:
12574:
12571:
12568:
12565:
12562:
12559:
12556:
12553:
12550:
12549:
12526:
12517:
12499:
12482:
12478:
12470:
12453:
12449:
12439:
12431:
12415:
12373:
12358:
12354:
12350:
12336:
12328:
12313:
12304:
12286:
12269:
12265:
12257:
12240:
12224:
12216:
12156:
12141:
12131:
12107:
12092:
12083:
12065:
12048:
12044:
12028:
12018:
12002:
11992:
11988:
11979:
11961:
11944:
11940:
11932:
11875:
11865:
11849:
11839:
11815:
11796:
11792:
11773:
11757:
11739:
11722:
11718:
11710:
11653:
11643:
11627:
11617:
11593:
11592:
11591:
11588:
11550:
11547:
11544:
11539:
11524:
11521:
11499:
11496:
11492:
11486:
11481:
11451:
11448:
11440:
11233:
11224:
11206:
11202:
11191:
11179:
11175:
11164:
11160:
11150:
11142:
11126:
11084:
11080:
11071:
11067:
11063:
11049:
11041:
11026:
11017:
10999:
10995:
10984:
10964:
10960:
10956:
10940:
10936:
10928:
10919:
10901:
10897:
10886:
10882:
10874:
10858:
10819:
10815:
10795:
10791:
10787:
10773:
10765:
10746:
10742:
10723:
10719:
10709:
10691:
10687:
10676:
10672:
10664:
10648:
10609:
10605:
10601:
10585:
10581:
10577:
10563:
10555:
10554:
10553:
10550:
10512:
10509:
10506:
10501:
10486:
10483:
10472:
10463:
10460:
10456:
10450:
10445:
10420:
10417:
10414:
10406:
10390:
10287:
10234:
10205:
10196:
10178:
10161:
10157:
10149:
10132:
10116:
10108:
10048:
10033:
10023:
9999:
9984:
9975:
9957:
9940:
9936:
9920:
9910:
9894:
9884:
9880:
9871:
9853:
9836:
9832:
9824:
9767:
9757:
9741:
9731:
9707:
9688:
9684:
9665:
9649:
9631:
9614:
9610:
9602:
9545:
9535:
9519:
9509:
9485:
9484:
9483:
9480:
9477:
9462:
9459:
9456:
9451:
9436:
9433:
9411:
9408:
9404:
9398:
9393:
9363:
9360:
9352:
9320:
9293:
9275:
9271:
9253:
9249:
9238:
9234:
9224:
9216:
9200:
9158:
9154:
9145:
9141:
9137:
9123:
9115:
9100:
9091:
9073:
9069:
9058:
9038:
9034:
9030:
9014:
9010:
9002:
8993:
8975:
8971:
8960:
8956:
8948:
8932:
8893:
8889:
8869:
8865:
8861:
8847:
8839:
8820:
8816:
8797:
8793:
8783:
8765:
8761:
8750:
8746:
8738:
8722:
8683:
8679:
8675:
8659:
8655:
8651:
8637:
8629:
8628:
8627:
8624:
8622:
8617:
8602:
8599:
8596:
8591:
8576:
8573:
8562:
8553:
8550:
8546:
8540:
8535:
8510:
8507:
8504:
8496:
8480:
8466:
8464:
8460:
8456:
8446:
8440:
8434:
8432:
8428:
8418:
8405:
8380:
8377:
8374:
8366:
8350:
8341:
8326:
8310:
8302:
8288:
8282:
8266:
8227:
8213:
8187:
8183:
8179:
8178:normal vector
8175:
8171:
8161:
8148:
8139:
8128:
8124:
8117:
8111:
8108:
8105:
8095:
8088:
8082:
8062:
8056:
8045:
8041:
8037:
8031:
8028:
8025:
8015:
8011:
8005:
7999:
7979:
7976:
7970:
7959:
7951:
7945:
7931:
7928:
7917:
7909:
7906:
7895:
7892:
7878:
7864:
7858:
7852:
7832:
7824:
7816:
7813:
7792:
7776:
7766:
7760:
7755:
7750:
7743:
7738:
7733:
7730:
7723:
7718:
7713:
7707:
7702:
7692:
7686:
7681:
7676:
7669:
7664:
7659:
7652:
7649:
7644:
7639:
7633:
7628:
7618:
7612:
7607:
7604:
7599:
7592:
7587:
7582:
7575:
7570:
7565:
7559:
7554:
7541:
7537:
7521:
7510:
7506:
7502:
7499:
7493:
7490:
7487:
7477:
7470:
7464:
7444:
7440:
7431:
7422:
7411:
7410:
7405:
7401:
7400:
7395:
7384:
7370:
7361:
7358:
7349:
7340:
7334:
7331:
7322:
7316:
7310:
7295:
7286:
7280:
7260:
7255:
7245:
7238:
7233:
7229:
7225:
7219:
7216:
7207:
7196:
7187:
7178:
7162:
7159:
7155:
7146:
7142:
7129:
7121:
7117:
7103:
7094:
7090:
7074:
7069:
7065:
7059:
7056:
7053:
7049:
7043:
7039:
7035:
7032:
7027:
7023:
7014:
7011:
7008:
7004:
7000:
6995:
6992:
6989:
6985:
6981:
6973:
6969:
6965:
6962:
6954:
6950:
6946:
6941:
6937:
6925:
6907:
6904:
6900:
6879:
6874:
6870:
6864:
6861:
6858:
6854:
6848:
6844:
6840:
6837:
6832:
6828:
6822:
6818:
6814:
6811:
6806:
6803:
6799:
6778:
6773:
6769:
6763:
6753:
6744:
6741:
6738:
6734:
6728:
6725:
6722:
6718:
6714:
6709:
6699:
6692:
6687:
6683:
6679:
6673:
6670:
6661:
6650:
6646:
6625:
6613:
6610:
6601:
6592:
6586:
6580:
6577:
6568:
6562:
6556:
6541:
6532:
6526:
6506:
6503:
6498:
6495:
6491:
6485:
6481:
6475:
6471:
6465:
6462:
6459:
6455:
6449:
6446:
6441:
6436:
6432:
6426:
6422:
6416:
6412:
6408:
6405:
6402:
6399:
6390:
6384:
6378:
6370:
6366:
6362:
6358:
6354:
6350:
6346:
6342:
6338:
6334:
6318:
6313:
6309:
6305:
6299:
6296:
6293:
6285:
6281:
6277:
6271:
6268:
6265:
6257:
6253:
6232:
6229:
6224:
6220:
6199:
6190:
6187:
6178:
6169:
6163:
6160:
6154:
6148:
6136:
6127:
6105:
6101:
6092:
6088:
6084:
6080:
6076:
6072:
6068:
6064:
6060:
6056:
6052:
6048:
6045:
6040:
6026:
6012:
6008:
6001:
5992:
5988:
5985:
5982:
5968:
5954:
5951:
5940:
5929:
5926:
5915:
5912:
5901:
5898:
5884:
5879:
5860:
5847:
5846:
5841:
5837:
5833:
5829:
5825:
5824:
5819:
5802:
5798:
5791:
5782:
5778:
5775:
5772:
5758:
5738:
5733:
5728:
5722:
5719:
5714:
5711:
5707:
5702:
5696:
5690:
5687:
5667:
5662:
5657:
5651:
5644:
5633:
5630:
5626:
5621:
5616:
5611:
5604:
5596:
5593:
5589:
5584:
5579:
5560:
5549:
5533:
5519:
5516:
5505:
5499:
5496:
5491:
5472:
5461:
5460:
5455:
5451:
5447:
5446:
5441:
5425:
5417:
5414:
5401:
5400:
5395:
5391:
5390:
5385:
5381:
5377:
5376:
5359:
5339:
5336:
5322:
5308:
5300:
5296:
5292:
5288:
5284:
5280:
5264:
5258:
5250:
5247:
5244:
5230:
5222:
5221:
5201:
5176:
5162:
5158:
5151:
5143:
5140:
5136:
5132:
5121:
5110:
5102:
5098:
5076:
5073:
5069:
5065:
5058:
5055:
5052:
5029:
5023:
5017:
5013:
5006:
5003:
4999:
4993:
4990:
4987:
4983:
4979:
4973:
4970:
4965:
4962:
4955:
4952:
4946:
4940:
4937:
4931:
4928:
4925:
4919:
4911:
4907:
4903:
4887:
4882:
4875:
4869:
4866:
4860:
4857:
4854:
4848:
4842:
4836:
4833:
4828:
4825:
4814:
4810:
4806:
4797:
4784:
4773:
4762:
4759:
4745:
4741:
4734:
4725:
4721:
4718:
4715:
4701:
4687:
4679:
4675:
4659:
4655:
4648:
4639:
4635:
4632:
4629:
4615:
4595:
4591:
4582:
4574:
4571:
4567:
4563:
4560:
4557:
4543:
4535:
4532:
4531:Hilbert space
4529:
4525:
4521:
4517:
4514:ensures that
4513:
4497:
4491:
4488:
4475:
4471:
4456:
4454:
4438:
4418:
4413:
4409:
4403:
4400:
4397:
4388:
4384:
4378:
4374:
4370:
4367:
4362:
4358:
4349:
4345:
4341:
4336:
4332:
4308:
4303:
4299:
4290:
4286:
4282:
4277:
4273:
4264:
4260:
4253:
4249:
4245:
4239:
4236:
4233:
4229:
4225:
4220:
4216:
4209:
4206:
4202:
4199:
4195:
4192:
4183:
4180:
4177:
4174:
4171:
4165:
4160:
4156:
4149:
4146:
4143:
4140:
4137:
4128:
4125:
4121:
4118:
4114:
4111:
4088:
4085:
4082:
4078:
4075:
4071:
4068:
4065:
4060:
4056:
4049:
4045:
4041:
4035:
4031:
4027:
4024:
4003:
4000:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3956:
3952:
3946:
3942:
3936:
3932:
3928:
3925:
3905:
3897:
3881:
3860:
3857:
3835:
3832:
3811:
3791:
3770:
3767:
3746:
3738:
3728:
3715:
3712:
3703:
3700:
3694:
3691:
3686:
3654:
3648:
3628:
3623:
3619:
3609:
3606:
3600:
3595:
3591:
3581:
3578:
3572:
3567:
3563:
3557:
3525:
3517:
3513:
3489:
3481:
3477:
3468:
3458:
3455:
3451:
3445:
3441:
3435:
3429:
3412:
3407:
3404:
3401:
3396:
3389:
3383:
3380:
3377:
3371:
3368:
3364:
3354:
3340:
3319:
3315:
3309:
3305:
3299:
3295:
3290:
3271:
3244:
3217:
3206:
3197:
3170:
3150:
3144:
3140:
3134:
3128:
3123:
3118:
3112:
3107:
3102:
3096:
3091:
3086:
3080:
3074:
3068:
3063:
3058:
3051:
3027:
3013:
2984:
2977:
2974:
2970:
2953:
2950:
2947:
2924:
2920:
2916:
2912:
2908:
2904:
2903:
2899:
2885:
2882:
2867:
2864:
2861:
2846:
2843:
2837:
2834:
2811:
2807:
2806:
2802:
2788:
2785:
2770:
2767:
2761:
2758:
2735:
2731:
2730:
2726:
2712:
2709:
2694:
2691:
2676:
2670:
2667:
2664:
2641:
2640:
2636:
2635:
2634:
2632:
2628:
2624:
2621:defined in a
2620:
2616:
2612:
2602:
2600:
2596:
2592:
2582:
2569:
2540:
2526:
2520:
2505:
2484:
2480:
2464:
2443:
2427:
2424:
2413:
2410:
2399:
2391:
2385:
2377:
2362:
2340:
2334:
2328:
2326:
2325:normalization
2322:
2318:
2310:
2306:
2302:
2297:
2288:
2286:
2282:
2278:
2274:
2270:
2266:
2261:
2244:
2238:
2229:
2223:
2220:
2217:
2211:
2205:
2200:
2192:
2190:
2179:
2170:
2164:
2161:
2158:
2153:
2146:
2140:
2137:
2131:
2128:
2125:
2122:
2116:
2108:
2102:
2094:
2092:
2082:
2075:
2066:
2060:
2057:
2054:
2051:
2045:
2039:
2036:
2030:
2027:
2024:
2021:
2015:
2007:
2001:
1993:
1991:
1986:
1962:
1959:
1956:
1953:
1950:
1944:
1938:
1930:
1926:
1910:
1902:
1901:
1875:
1857:
1843:
1837:
1822:
1801:
1799:
1795:
1791:
1787:
1777:
1775:
1770:
1757:
1752:
1737:
1734:
1723:
1720:
1709:
1701:
1695:
1687:
1672:
1652:
1635:
1615:
1594:
1590:
1586:
1583:
1580:
1575:
1571:
1564:
1534:
1530:
1526:
1523:
1520:
1515:
1511:
1507:
1502:
1498:
1491:
1488:
1474:
1467:
1463:
1397:
1393:
1389:
1384:
1380:
1376:
1370:
1367:
1364:
1358:
1350:
1345:
1336:
1334:
1330:
1327:
1323:
1318:
1305:
1278:
1269:
1261:
1246:
1239:
1231:
1217:
1202:
1177:
1174:
1171:
1157:
1146:
1142:
1127:
1117:
1113:
1098:
1077:
1073:
1069:
1066:
1061:
1059:
1055:
1051:
1047:
1044:
1040:
1035:
1033:
1029:
1017:
1012:
1010:
1005:
1003:
998:
997:
995:
994:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
953:
945:
944:
937:
934:
932:
929:
927:
924:
922:
919:
918:
910:
909:
898:
895:
893:
890:
888:
885:
884:
883:
882:
872:
871:
860:
857:
855:
852:
850:
847:
845:
842:
840:
839:Line integral
837:
835:
832:
830:
827:
826:
825:
824:
820:
819:
814:
811:
809:
806:
804:
801:
799:
796:
795:
794:
793:
789:
788:
782:
781:Multivariable
776:
775:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
735:
734:
733:
729:
728:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
694:
693:
692:
686:
680:
679:
668:
665:
663:
660:
658:
655:
653:
650:
648:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
618:
617:
616:
613:
610:
609:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
578:
574:
571:
570:
569:
568:
562:
556:
555:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
519:
516:
513:
509:
505:
504:trigonometric
501:
498:
496:
493:
491:
488:
486:
483:
482:
481:
480:
476:
475:
470:
467:
465:
462:
460:
457:
455:
452:
449:
445:
442:
440:
437:
436:
435:
434:
430:
429:
424:
421:
419:
416:
414:
411:
410:
409:
408:
402:
396:
395:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
335:
334:
333:
330:
327:
326:
321:
318:
316:
315:Related rates
313:
311:
308:
306:
303:
301:
298:
296:
293:
292:
291:
290:
286:
285:
278:
275:
273:
272:of a function
270:
268:
267:infinitesimal
265:
264:
263:
260:
257:
253:
250:
249:
248:
247:
243:
242:
236:
230:
229:
223:
220:
218:
215:
213:
210:
209:
204:
201:
199:
196:
195:
191:
188:
187:
186:
185:
166:
160:
157:
151:
145:
142:
139:
136:
129:
122:
119:
113:
108:
104:
95:
94:
91:
88:
87:
83:
82:
73:
70:
62:
52:
48:
44:
38:
37:
32:This section
30:
26:
21:
20:
14006:Secant cubed
13931:
13924:
13905:Isaac Newton
13875:Brook Taylor
13550:
13542:Derivatives
13513:Shell method
13241:Differential
13021:
12993:
12989:
12968:
12947:
12930:
12925:
12906:
12900:
12881:
12875:
12856:
12850:
12831:
12825:
12804:
12796:
12777:
12771:
12744:
12727:
12714:
12706:
12701:
12693:
12684:
12663:
12644:
12638:
12626:
11590:Properties:
11589:
11254:
10552:Properties:
10551:
10226:
9482:Properties:
9481:
9478:
9321:
9318:
8626:Properties:
8625:
8620:
8618:
8467:
8458:
8454:
8452:
8444:
8430:
8424:
8211:
8182:hypersurface
8169:
8167:
7790:
7535:
7408:
7407:
7403:
7398:
7397:
7391:
7176:
7088:
6926:commutator:
6369:power series
6364:
6360:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6090:
6086:
6078:
6074:
6070:
6066:
6062:
6058:
6054:
6041:
5844:
5843:
5839:
5835:
5831:
5827:
5822:
5821:
5817:
5547:
5458:
5457:
5453:
5449:
5444:
5443:
5439:
5398:
5397:
5393:
5388:
5387:
5383:
5379:
5374:
5373:
5298:
5294:
5290:
5286:
5282:
5278:
5219:
5218:
5100:
5096:
4909:
4905:
4901:
4812:
4808:
4804:
4802:
4677:
4673:
4523:
4515:
4511:
4473:
4467:
4464:Translations
3734:
3464:
3453:
3449:
3443:
3439:
3433:
3427:
3317:
3313:
3307:
3303:
3297:
3293:
3142:
3138:
3132:
3126:
3121:
3116:
3110:
3100:
3094:
3084:
3078:
3072:
3066:
3056:
3053:
2922:
2918:
2914:
2910:
2906:
2900:
2809:
2805:product rule
2803:
2733:
2727:
2637:
2630:
2623:neighborhood
2618:
2614:
2608:
2594:
2588:
2482:
2478:
2338:
2332:
2329:
2320:
2314:
2308:
2304:
2300:
2284:
2280:
2276:
2268:
2264:
2262:
1928:
1898:
1876:
1802:
1797:
1793:
1785:
1783:
1771:
1461:
1459:
1349:contour plot
1319:
1075:
1071:
1067:
1062:
1057:
1053:
1049:
1045:
1036:
1027:
1025:
716:
500:Substitution
262:Differential
235:Differential
65:
59:October 2012
56:
36:verification
33:
14074:of surfaces
13825:and numbers
13787:Dirichlet's
13757:Telescoping
13710:Alternating
13298:L'Hôpital's
13095:Precalculus
8463:dot product
6924:Lie algebra
3431:, provided
3296: : →
3120:, then the
3070:a point of
2599:unit vector
2597:to being a
2477:or in case
2267:at a point
1925:dot product
1326:curvilinear
956:Precalculus
949:Miscellanea
914:Specialized
821:Definitions
588:Alternating
431:Definitions
244:Definitions
14126:Categories
13870:Adequality
13556:Divergence
13429:Arc length
13226:Derivative
13050:PlanetMath
12940:References
8174:orthogonal
5815:And since
3048:See also:
2902:chain rule
2611:derivative
2605:Properties
2299:The angle
1877:where the
1339:Definition
936:Variations
931:Stochastic
921:Fractional
790:Formalisms
753:Divergence
722:Identities
702:Divergence
252:Derivative
203:Continuity
14069:of curves
14064:Curvature
13951:Integrals
13745:Maclaurin
13725:Geometric
13616:Geometric
13566:Laplacian
13278:linearity
13118:Factorial
13040:MathWorld
13010:120253580
12763:828768012
12507:∂
12490:∂
12461:∂
12446:∂
12421:∂
12413:∂
12294:∂
12277:∂
12248:∂
12231:∂
12206:∂
12196:∂
12073:∂
12056:∂
12045:⋅
11993:⋅
11969:∂
11952:∂
11922:∂
11912:∂
11866:⋅
11781:∂
11764:∂
11747:∂
11730:∂
11700:∂
11690:∂
11545:α
11525:α
11500:α
11430:∂
11420:∂
11214:∂
11199:∂
11172:∂
11157:∂
11132:∂
11124:∂
11007:∂
10992:∂
10909:∂
10894:∂
10864:∂
10856:∂
10731:∂
10716:∂
10699:∂
10684:∂
10654:∂
10646:∂
10507:α
10487:α
10464:α
10396:∂
10388:∂
10197:⋅
10186:∂
10169:∂
10158:⋅
10140:∂
10123:∂
10109:⋅
10098:∂
10088:∂
9976:⋅
9965:∂
9948:∂
9937:×
9885:×
9872:⋅
9861:∂
9844:∂
9825:⋅
9814:∂
9804:∂
9758:×
9689:⋅
9673:∂
9656:∂
9639:∂
9622:∂
9603:⋅
9592:∂
9582:∂
9457:α
9437:α
9412:α
9353:⋅
9342:∂
9332:∂
9294:⋅
9283:∂
9268:∂
9246:∂
9231:∂
9217:⋅
9206:∂
9198:∂
9092:⋅
9081:∂
9066:∂
8994:⋅
8983:∂
8968:∂
8949:⋅
8938:∂
8930:∂
8821:⋅
8805:∂
8790:∂
8773:∂
8758:∂
8739:⋅
8728:∂
8720:∂
8597:α
8577:α
8554:α
8497:⋅
8486:∂
8478:∂
8367:⋅
8356:∂
8348:∂
8315:∇
8303:⋅
8286:∇
8272:∂
8264:∂
8233:∂
8225:∂
8143:∇
8140:⋅
8129:×
8125:θ
8118:−
8112:
8096:θ
8060:∇
8057:⋅
8046:×
8042:θ
8038:δ
8032:−
8016:θ
8012:δ
7974:∇
7971:⋅
7960:×
7956:θ
7952:δ
7946:−
7918:×
7914:θ
7910:δ
7907:−
7869:θ
7865:δ
7825:×
7821:θ
7817:δ
7814:−
7806:→
7731:−
7650:−
7605:−
7511:⋅
7507:θ
7500:−
7494:
7478:θ
7445:θ
7436:θ
7426:^
7423:θ
7388:Rotations
7362:ξ
7353:¯
7350:ξ
7323:ξ
7299:¯
7296:ξ
7249:¯
7246:ξ
7230:ξ
7220:ξ
7211:¯
7208:ξ
7139:∂
7135:∂
7114:∂
7110:∂
7040:∑
6982:−
6970:∑
6845:∑
6838:−
6815:−
6770:ξ
6757:¯
6754:ξ
6719:∑
6703:¯
6700:ξ
6684:ξ
6674:ξ
6665:¯
6662:ξ
6614:ξ
6605:¯
6602:ξ
6569:ξ
6545:¯
6542:ξ
6507:⋯
6482:ξ
6472:ξ
6456:∑
6423:ξ
6413:∑
6391:ξ
6310:ξ
6300:ξ
6266:ξ
6221:ξ
6191:ξ
6182:¯
6179:ξ
6155:ξ
6140:¯
6137:ξ
6102:ξ
6083:Lie group
6005:∇
6002:⋅
5998:λ
5989:
5962:λ
5945:λ
5920:ε
5868:ε
5795:∇
5792:⋅
5788:λ
5779:
5766:λ
5691:
5648:∇
5645:⋅
5641:λ
5608:∇
5605:⋅
5601:ε
5568:ε
5527:λ
5510:ε
5480:ε
5422:ε
5411:λ
5262:∇
5259:⋅
5255:ε
5238:ε
5205:∇
5202:⋅
5198:ε
5155:∇
5152:⋅
5148:ε
5126:ε
5059:ε
4994:ε
4956:ε
4932:ε
4883:ε
4867:−
4861:ε
4778:λ
4738:∇
4735:⋅
4731:λ
4722:
4695:λ
4652:∇
4649:⋅
4645:λ
4636:
4623:λ
4583:⋅
4579:λ
4572:−
4564:
4551:λ
4495:∇
4414:σ
4404:ν
4401:μ
4398:ρ
4389:σ
4379:σ
4375:∑
4371:±
4363:ρ
4350:ν
4337:μ
4304:ρ
4291:ν
4278:μ
4265:ν
4261:δ
4254:μ
4246:δ
4240:ν
4234:μ
4230:∑
4221:ρ
4207:⋅
4200:δ
4181:⋅
4178:δ
4166:−
4161:ρ
4147:⋅
4144:δ
4126:⋅
4119:δ
4083:⋅
4076:δ
4061:μ
4050:μ
4042:δ
4036:μ
4032:∑
4001:δ
3974:⋅
3971:δ
3957:ν
3947:ν
3943:δ
3937:ν
3933:∑
3906:δ
3882:δ
3874:and then
3858:δ
3833:δ
3812:δ
3768:δ
3747:δ
3713:ϕ
3707:∇
3704:⋅
3692:ϕ
3649:ϕ
3624:μ
3613:∇
3610:⋅
3601:−
3596:μ
3585:∇
3582:⋅
3568:μ
3518:μ
3482:μ
3402:τ
3390:τ
3384:γ
3381:∘
3372:τ
3330:∇
3160:∇
3003:∇
2951:∘
2934:∇
2872:∇
2851:∇
2821:∇
2775:∇
2745:∇
2699:∇
2681:∇
2651:∇
2541:⋅
2524:∇
2494:∇
2425:−
2389:→
2351:∇
2218:−
2197:∇
2159:−
2138:−
2106:→
2052:−
2037:−
2005:→
1911:⋅
1885:∇
1858:⋅
1841:∇
1811:∇
1735:−
1699:→
1661:∇
1624:∇
1584:…
1524:…
1295:∂
1276:∂
1270:⋅
1244:∇
1240:⋅
1207:∂
1087:∇
926:Malliavin
813:Geometric
712:Laplacian
662:Dirichlet
573:Geometric
158:−
105:∫
43:talk page
14059:Manifold
13792:Integral
13735:Infinite
13730:Harmonic
13715:Binomial
13561:Gradient
13504:Volumes
13315:Quotient
13256:Notation
13087:Calculus
12933:, Dover.
12690:gradient
12546:See also
6047:subgroup
5850:we have
5751:we have
4250:′
4203:′
4122:′
4079:′
4046:′
4004:′
3992:and for
3918:is thus
3861:′
3836:′
3771:′
2978:′
2925:), then
2639:sum rule
2625:of, and
1900:gradient
1614:function
1128:′
966:Glossary
876:Advanced
854:Jacobian
808:Exterior
738:Gradient
730:Theorems
697:Gradient
636:Integral
598:Binomial
583:Harmonic
448:improper
444:Integral
401:Integral
383:Reynolds
358:Quotient
287:Concepts
123:′
90:Calculus
14152:Scalars
13996:inverse
13984:inverse
13910:Fluxion
13720:Fourier
13586:Stokes'
13581:Green's
13303:Product
13163:Tangent
12671:and to
11365:(or at
10333:(or at
8427:tensors
7383:Q.E.D.
7192:abelian
6212:Taking
6044:Abelian
6039:Q.E.D.
5382:parts (
4528:unitary
4468:In the
3452:′(0) =
3316:′(0) =
1923:is the
1612:is the
961:History
859:Hessian
748:Stokes'
743:Green's
575: (
502: (
446: (
368:Inverse
343:Product
254: (
14079:Tensor
14001:Secant
13767:Abel's
13750:Taylor
13641:Matrix
13591:Gauss'
13173:Limits
13153:Secant
13143:Radian
13008:
12977:
12954:
12913:
12888:
12863:
12838:
12813:
12784:
12761:
12751:
12669:metric
12651:
11528:
11506:
11188:
10981:
10933:
10812:
10490:
10470:
9440:
9418:
9262:
9055:
9007:
8886:
8580:
8560:
6892:Since
4526:, the
4431:where
3804:along
3442:(0) =
3306:(0) =
3245:), or
3130:along
3098:. If
3088:, and
2323:after
1043:vector
803:Tensor
798:Matrix
685:Vector
603:Taylor
561:Series
198:Limits
14157:Rates
13943:Lists
13802:Ratio
13740:Power
13476:Euler
13293:Chain
13283:Power
13158:Slope
13006:S2CID
12631:Notes
12402:then
12185:then
11901:then
11679:then
11113:then
10845:then
10635:then
10077:then
9793:then
9571:then
9187:then
8919:then
8709:then
7540:SO(3)
7534:Here
7087:with
6355:) as
5189:Here
4680:) as
4518:is a
4510:(the
3824:then
3287:(see
3241:(see
3194:(see
3147:(see
3104:is a
3060:be a
2905:: If
2315:In a
1651:limit
1464:of a
626:Ratio
593:Power
512:Euler
490:Discs
485:Parts
353:Power
348:Chain
277:total
13812:Term
13807:Root
13546:Curl
12975:ISBN
12952:ISBN
12911:ISBN
12886:ISBN
12861:ISBN
12836:ISBN
12811:ISBN
12782:ISBN
12759:OCLC
12749:ISBN
12649:ISBN
11255:Let
10227:Let
8453:Let
7392:The
7091:the
6069:) =
5834:) =
3465:The
3447:and
3311:and
3064:and
3054:Let
2913:and
2808:(or
2629:at,
2617:and
1927:and
1774:norm
1460:The
1060:.
707:Curl
667:Abel
631:Root
13288:Sum
13048:at
13038:at
12998:doi
12329:If
12108:If
11816:If
11594:If
11042:If
10766:If
10556:If
10000:If
9708:If
9486:If
9116:If
8840:If
8630:If
8109:exp
7793:by
7491:exp
7457:is
7406:= |
5986:exp
5776:exp
5688:exp
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4912:):
4719:exp
4633:exp
4561:exp
4476:as
3198:),
3151:),
3124:of
3114:at
3108:to
3092:at
2812:):
2633::
2382:lim
2275:of
2099:lim
1998:lim
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1692:lim
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338:Sum
49:by
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11475:T
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11468:)
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11460:(
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11449:=
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11441::
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11094:S
11090:(
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11057:S
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11027:)
11022:T
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11000:2
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10978:)
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10929:)
10924:T
10920::
10913:S
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10879:T
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10868:S
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10829:S
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10747::
10743:)
10735:S
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10669:T
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10610:2
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10571:S
10567:(
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10510:=
10502:]
10498:)
10494:T
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10480:S
10476:(
10473:f
10461:d
10457:d
10451:[
10446:=
10443:]
10439:T
10435:[
10432:)
10428:S
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10421:f
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10415:=
10411:T
10407::
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10364:T
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10190:v
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10117:=
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10021:)
10017:v
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9980:u
9969:v
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9941:(
9934:)
9930:v
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9747:(
9742:1
9737:f
9732:=
9729:)
9725:v
9721:(
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9460:=
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9399:[
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9380:)
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9225:=
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8600:=
8592:]
8588:)
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8574:+
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8547:d
8541:[
8536:=
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8399:n
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7217:,
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7143:x
7130:,
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3493:)
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3413:.
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2838:g
2835:f
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2789:.
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2245:.
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2236:(
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2230:x
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2221:D
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2212:x
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2171:x
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2147:x
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2067:x
2064:(
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2058:D
2055:t
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2046:x
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2031:v
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2019:(
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2002:t
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1954:x
1951:=
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1945:t
1942:(
1939:h
1929:v
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1851:x
1847:(
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1732:)
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1636:f
1629:v
1600:)
1595:n
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1581:,
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1521:,
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1512:x
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1503:1
1499:x
1495:(
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1482:x
1478:(
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1443:u
1421:u
1398:2
1394:y
1390:+
1385:2
1381:x
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1371:y
1368:,
1365:x
1362:(
1359:f
1306:.
1299:x
1290:)
1286:x
1282:(
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1225:x
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1172:=
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1161:(
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1143:=
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1136:x
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1114:=
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1107:x
1103:(
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1076:x
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1001:v
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514:)
450:)
258:)
170:)
167:a
164:(
161:f
155:)
152:b
149:(
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143:=
140:t
137:d
133:)
130:t
127:(
120:f
114:b
109:a
72:)
66:(
61:)
57:(
39:.
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