3919:
2510:
3607:
3787:
3447:
3883:
2937:
3156:
1642:
2356:
3489:
2361:
3283:
3361:
2038:(single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the
2742:
129:
1964:
2834:
3682:
1816:
1054:
2588:
4277:, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called
1756:
2666:
3372:
1122:
1192:
1371:
3798:
2995:
2259:
2308:
2203:. Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.
1308:
2086:
at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the
4406:
each of which would result in the production of the same given quantity of output of some good. In this case the absolute value of the implicit derivative
1505:
is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable
2845:
4429:
4432:
between the two factors of production: how much more capital the firm must use to produce the same amount of output with one less unit of labor.
3061:
2011:
implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by
1225:
provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero
2505:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}+{\frac {dx}{dx}}+{\frac {d}{dx}}(5)&=0\,;\\{\frac {dy}{dx}}+1+0&=0\,.\end{aligned}}}
3602:{\displaystyle {\frac {dy}{dx}}=-{\frac {\,{\frac {\partial R}{\partial x}}\,}{\frac {\partial R}{\partial y}}}=-{\frac {R_{x}}{R_{y}}}\,,}
1519:
466:
530:
967:
4772:
4745:
4711:
4675:
4650:
4618:
2601:
An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function
1821:
But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as
486:
204:
471:
3212:
4833:
3294:
807:
481:
456:
138:
2681:
4838:
4823:
4212:
In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.
4828:
4798:
4556:
4356:
1869:
589:
536:
417:
4793:
243:
215:
326:
4843:
4566:
4096:
3782:{\displaystyle {\frac {\partial R}{\partial x}}{\frac {dx}{dx}}+{\frac {\partial R}{\partial y}}{\frac {dy}{dx}}=0\,,}
1230:
840:
448:
286:
258:
48:
2770:
4516:
711:
675:
452:
331:
225:
220:
210:
4561:
4463:
4183:
4000:
2091:
1222:
475:
1771:
311:
610:
170:
4251:
is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an
995:
4021:
2543:
924:
716:
605:
4441:
4221:
2951:, and implicit differentiation is the only feasible method of differentiation. An example is the equation
1402:
1269:
1226:
1139:
1131:
1061:
989:
960:
889:
850:
734:
670:
594:
3442:{\displaystyle y\neq \pm {\frac {1}{\sqrt{5}}}\quad {\text{and}}\quad y\neq \pm {\frac {i}{\sqrt{5}}}\,.}
4467:
2148:
1705:
1692:
1685:
934:
600:
491:
371:
316:
277:
183:
2618:
4489:
of various goods. When utility is being maximized, typically the resulting implicit functions are the
3878:{\displaystyle {\frac {\partial R}{\partial x}}+{\frac {\partial R}{\partial y}}{\frac {dy}{dx}}=0\,,}
1075:
4278:
3001:
1145:
939:
919:
845:
514:
433:
407:
321:
1327:
4581:
4534:
1844:
1699:. A simple example of an algebraic function is given by the left side of the unit circle equation:
1135:
914:
884:
874:
761:
615:
412:
268:
151:
146:
2957:
4789:
4455:
4322:
4274:
3631:
1696:
1496:
879:
782:
766:
706:
660:
541:
460:
366:
361:
165:
4462:
even though the objective function has not been restricted to any specific functional form. The
2222:
701:
696:
160:
2274:
4768:
4760:
4756:
4741:
4737:
4707:
4699:
4671:
4646:
4614:
1134:
that is defined by an implicit equation, that relates one of the variables, considered as the
953:
787:
565:
443:
396:
248:
4607:
4512:
4451:
4403:
4289:
The solutions of differential equations generally appear expressed by an implicit function.
3654:
2031:
1860:
1852:
1278:
1247:
797:
691:
665:
526:
438:
402:
4695:
4494:
4486:
4447:
2087:
929:
802:
756:
751:
638:
551:
496:
1451:. However, this is often not possible, or only by introducing a new notation (as in the
4529:
4470:
of the optimization define an implicit function for each element of the optimal vector
4252:
4187:
1848:
812:
620:
387:
2042:-axis and "cutting away" some unwanted function branches. Then an equation expressing
4817:
4726:
4639:
4576:
4481:. When profit is being maximized, typically the resulting implicit functions are the
3992:
792:
556:
306:
263:
1665:. This algebraic function can be written as the right side of the solution equation
4721:
4602:
4544:
4490:
4482:
546:
291:
17:
4733:
4571:
4037:
1469:
1452:
1195:
1069:
981:
909:
4549:
3918:
3650:
2764:
It is substantially easier to implicitly differentiate the original equation:
2111:
1065:
655:
579:
301:
296:
200:
2932:{\displaystyle {\frac {dy}{dx}}={\frac {-4x^{3}}{4y}}=-{\frac {x^{3}}{y}}\,.}
4539:
4501:
4303:
2747:
and then differentiate this function. This creates two derivatives: one for
584:
574:
3151:{\displaystyle 5y^{4}{\frac {dy}{dx}}-{\frac {dy}{dx}}={\frac {dx}{dx}}\,,}
27:
Mathematical relation consisting of a multi-variable function equal to zero
4363:
one must receive in order to be indifferent to a loss of one unit of
4511:— the partial derivatives of the implicit function — can be expressed as
4391:
4095:
defines an implicit function that is differentiable in some small enough
2103:
650:
392:
349:
38:
4806:
4206:
1462:
by interchanging the roles of the dependent and independent variables.
4333:
consumed of two goods, the absolute value of the implicit derivative
2034:
having a "hump" in its graph. Thus, for an implicit function to be a
1637:{\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0\,,}
3964:. (Unlike in many cases, here this function can be made explicit as
2350:
Alternatively, one can totally differentiate the original equation:
3917:
3922:
The unit circle can be defined implicitly as the set of points
2094:
provides a uniform way of handling these sorts of pathologies.
2046:
as an implicit function of the other variables can be written.
2947:
Often, it is difficult or impossible to solve explicitly for
1843:
While explicit solutions can be found for equations that are
4458:
function is to be maximized with respect to a choice vector
4115:
that is defined and differentiable in some neighbourhood of
2064:
can also have other pathologies. For example, the equation
1969:
Nevertheless, one can still refer to the implicit solution
4706:. New York: Cambridge University Press. pp. 198–211.
3278:{\displaystyle \left(5y^{4}-1\right){\frac {dy}{dx}}=1\,,}
3055:
can be obtained by differentiating the equation to obtain
2313:
where the right side is the explicit form of the function
2143:, it is not generally possible to solve it explicitly for
3356:{\displaystyle {\frac {dy}{dx}}={\frac {1}{5y^{4}-1}}\,,}
3033:
by explicit differentiation. Using the implicit method,
2737:{\displaystyle y(x)=\pm {\sqrt {\frac {8-x^{4}}{2}}}\,,}
1258:
that has a unique inverse, then the inverse function of
4442:
Mathematical economics § Mathematical optimization
1250:. Not all functions have a unique inverse function. If
4111:; in other words, there is a differentiable function
3801:
3685:
3492:
3375:
3297:
3215:
3064:
2960:
2848:
2773:
2684:
2621:
2546:
2359:
2277:
2225:
1872:
1774:
1708:
1522:
1330:
1281:
1148:
1078:
998:
51:
4515:
of the system of first-order conditions found using
4394:
showing various combinations of utilized quantities
1959:{\displaystyle y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0\,.}
3453:
General formula for derivative of implicit function
1138:of the function, with the others considered as the
4725:
4638:
4606:
3877:
3781:
3601:
3441:
3355:
3277:
3150:
2989:
2931:
2828:
2736:
2660:
2582:
2504:
2302:
2253:
1958:
1810:
1750:
1636:
1458:Intuitively, an inverse function is obtained from
1365:
1302:
1186:
1116:
1048:
123:
4799:"Implicit Differentiation, What's Going on Here?"
2671:To differentiate this explicitly with respect to
2173:and then solve the resulting linear equation for
1472:is an implicit function giving the solution for
124:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
4805:. Essence of Calculus. May 3, 2017 – via
2114:to differentiate implicitly defined functions.
2829:{\displaystyle 4x^{3}+4y{\frac {dy}{dx}}=0\,,}
1691:Algebraic functions play an important role in
1386:is an implicit definition. For some functions
4632:
4630:
4609:Fundamental Methods of Mathematical Economics
2195:to explicitly get the derivative in terms of
1984:involving the multi-valued implicit function
1068:). For example, the implicit equation of the
961:
8:
4767:. New York: W. W. Norton. pp. 334–371.
4761:"Implicit Functions and Their Derivatives"
4273:. The implicit equations are the basis of
3472:, the derivative of the implicit function
1811:{\displaystyle y=\pm {\sqrt {1-x^{2}}}\,.}
968:
954:
834:
740:
644:
520:
355:
189:
29:
4613:(Third ed.). New York: McGraw-Hill.
4500:Moreover, the influence of the problem's
3953:can be expressed as an implicit function
3871:
3845:
3825:
3802:
3800:
3775:
3749:
3729:
3706:
3686:
3684:
3595:
3587:
3577:
3571:
3543:
3523:
3522:
3519:
3493:
3491:
3435:
3427:
3418:
3403:
3394:
3385:
3374:
3349:
3334:
3321:
3298:
3296:
3271:
3245:
3228:
3214:
3144:
3124:
3101:
3078:
3072:
3063:
2983:
2965:
2959:
2925:
2914:
2908:
2885:
2872:
2849:
2847:
2822:
2796:
2781:
2772:
2730:
2717:
2703:
2683:
2654:
2642:
2626:
2620:
2576:
2547:
2545:
2494:
2452:
2444:
2410:
2387:
2364:
2360:
2358:
2296:
2276:
2247:
2224:
2147:and then differentiate. Instead, one can
1952:
1925:
1909:
1893:
1877:
1871:
1804:
1796:
1784:
1773:
1744:
1726:
1713:
1707:
1630:
1609:
1584:
1559:
1546:
1527:
1521:
1359:
1341:
1329:
1280:
1246:A common type of implicit function is an
1166:
1153:
1147:
1096:
1083:
1077:
1028:
1009:
997:
84:
61:
56:
50:
3987:.) No such function exists around point
2593:the same answer as obtained previously.
1049:{\displaystyle R(x_{1},\dots ,x_{n})=0,}
4594:
4430:marginal rate of technical substitution
4371:Marginal rate of technical substitution
3649:The above formula comes from using the
1840:is the multi-valued implicit function.
1321:. This solution can then be written as
774:
743:
683:
564:
492:Differentiating under the integral sign
425:
379:
276:
235:
192:
37:
2583:{\displaystyle {\frac {dy}{dx}}=-1\,,}
2117:To differentiate an implicit function
1859:, the same is not in general true for
1863:and higher degree equations, such as
1218:is restricted to nonnegative values.
7:
4728:Principles of Mathematical Analysis
4375:Similarly, sometimes the level set
4359:of the two goods: how much more of
2264:This equation is easy to solve for
1401:can be written out explicitly as a
4645:. Brooks/Cole Publishing Company.
3836:
3828:
3813:
3805:
3740:
3732:
3697:
3689:
3555:
3547:
3534:
3526:
1751:{\displaystyle x^{2}+y^{2}-1=0\,.}
1268:, is the unique function giving a
33:Part of a series of articles about
25:
2661:{\displaystyle x^{4}+2y^{2}=8\,.}
3011:, and therefore one cannot find
1117:{\displaystyle x^{2}+y^{2}-1=0.}
3408:
3402:
1187:{\displaystyle x^{2}+y^{2}-1=0}
4641:Calculus Concepts And Contexts
3910:, gives the expression above.
2694:
2688:
2431:
2425:
1621:
1615:
1577:
1571:
1539:
1533:
1366:{\displaystyle x=g^{-1}(y)\,.}
1356:
1350:
1297:
1291:
1064:of several variables (often a
1034:
1002:
118:
112:
103:
97:
81:
75:
1:
4557:Marginal rate of substitution
4357:marginal rate of substitution
4298:Marginal rate of substitution
2324:. Differentiation then gives
418:Integral of inverse functions
3007:explicitly as a function of
2990:{\displaystyle y^{5}-y=x\,.}
1765:gives an explicit solution:
1661:are polynomial functions of
1142:. For example, the equation
4567:Logarithmic differentiation
1231:continuously differentiable
1202:as an implicit function of
841:Calculus on Euclidean space
259:Logarithmic differentiation
4860:
4765:Mathematics for Economists
4670:. Boston: Addison-Wesley.
4450:, some function such as a
4439:
3998:
2254:{\displaystyle y+x+5=0\,.}
2071:does not imply a function
1494:
4562:Implicit function theorem
4464:implicit function theorem
4293:Applications in economics
4285:In differential equations
4001:Implicit function theorem
3914:Implicit function theorem
2303:{\displaystyle y=-x-5\,,}
2128:, defined by an equation
2092:implicit function theorem
1223:implicit function theorem
575:Summand limit (term test)
4666:Kaplan, Wilfred (2003).
3288:which yields the result
2612:defined by the equation
2108:implicit differentiation
2098:Implicit differentiation
254:Implicit differentiation
244:Differentiation notation
171:Inverse function theorem
4637:Stewart, James (1998).
4142:in this neighbourhood.
4022:differentiable function
3888:which, when solved for
2675:, one has first to get
1647:where the coefficients
1227:multivariable functions
717:Helmholtz decomposition
4834:Multivariable calculus
4468:first-order conditions
4428:is interpreted as the
4355:is interpreted as the
4024:of two variables, and
3996:
3879:
3783:
3651:generalized chain rule
3603:
3443:
3357:
3279:
3152:
2991:
2933:
2830:
2738:
2662:
2584:
2506:
2304:
2255:
2049:The defining equation
1960:
1812:
1752:
1680:. Written like this,
1638:
1403:closed-form expression
1367:
1304:
1303:{\displaystyle y=g(x)}
1188:
1118:
1050:
851:Limit of distributions
671:Directional derivative
327:Faà di Bruno's formula
125:
4839:Differential topology
4824:Differential calculus
4517:total differentiation
4477:of the choice vector
4306:, when the level set
4279:affine algebraic sets
4216:In algebraic geometry
4190:of implicit equation
3921:
3880:
3784:
3604:
3444:
3366:which is defined for
3358:
3280:
3153:
3002:algebraically express
2992:
2934:
2831:
2739:
2663:
2585:
2507:
2305:
2256:
2149:totally differentiate
2082:giving solutions for
1961:
1813:
1753:
1693:mathematical analysis
1639:
1509:gives a solution for
1368:
1305:
1189:
1119:
1051:
935:Mathematical analysis
846:Generalized functions
531:arithmetico-geometric
372:Leibniz integral rule
126:
4829:Theorems in analysis
4700:"Implicit Functions"
4466:guarantees that the
3799:
3683:
3490:
3373:
3295:
3213:
3062:
3000:It is impossible to
2958:
2846:
2771:
2682:
2619:
2544:
2357:
2275:
2223:
1870:
1772:
1706:
1520:
1328:
1279:
1146:
1076:
996:
940:Nonstandard analysis
408:Lebesgue integration
278:Rules and identities
49:
4582:Folium of Descartes
4535:Functional equation
4497:for various goods.
4325:for the quantities
3661:— of both sides of
3632:partial derivatives
1996:Not every equation
1688:implicit function.
1491:Algebraic functions
1405:— for instance, if
611:Cauchy condensation
413:Contour integration
139:Fundamental theorem
66:
4844:Algebraic geometry
4323:indifference curve
4275:algebraic geometry
3997:
3875:
3779:
3657:— with respect to
3599:
3439:
3353:
3275:
3148:
2987:
2929:
2826:
2734:
2658:
2580:
2502:
2500:
2300:
2251:
2106:, a method called
1956:
1808:
1748:
1697:algebraic geometry
1634:
1503:algebraic function
1497:Algebraic function
1382:as the inverse of
1363:
1300:
1184:
1114:
1046:
783:Partial derivative
712:generalized Stokes
606:Alternating series
487:Reduction formulae
476:Heaviside's method
457:tangent half-angle
444:Cylindrical shells
367:Integral transform
362:Lists of integrals
166:Mean value theorem
121:
52:
18:Implicitly defined
4668:Advanced Calculus
4513:total derivatives
4493:function and the
4485:function and the
4209:is not vertical.
3863:
3843:
3820:
3767:
3747:
3724:
3704:
3593:
3563:
3562:
3541:
3511:
3433:
3432:
3406:
3400:
3399:
3347:
3316:
3263:
3142:
3119:
3096:
2923:
2900:
2867:
2814:
2728:
2727:
2565:
2470:
2423:
2405:
2382:
2110:makes use of the
1802:
1254:is a function of
1242:Inverse functions
1128:implicit function
986:implicit equation
978:
977:
858:
857:
820:
819:
788:Multiple integral
724:
723:
628:
627:
595:Direct comparison
566:Convergence tests
504:
503:
472:Partial fractions
339:
338:
249:Second derivative
16:(Redirected from
4851:
4810:
4778:
4755:Simon, Carl P.;
4751:
4731:
4717:
4682:
4681:
4663:
4657:
4656:
4644:
4634:
4625:
4624:
4612:
4603:Chiang, Alpha C.
4599:
4510:
4495:demand functions
4487:supply functions
4480:
4476:
4461:
4452:utility function
4427:
4426:
4424:
4423:
4418:
4415:
4404:physical capital
4401:
4397:
4389:
4366:
4362:
4354:
4353:
4351:
4350:
4345:
4342:
4332:
4328:
4320:
4272:
4264:implicit surface
4261:
4250:
4246:
4204:
4181:
4169:
4167:
4165:
4164:
4158:
4155:
4141:
4137:
4118:
4114:
4110:
4094:
4079:
4077:
4075:
4074:
4068:
4065:
4054:
4035:
4019:
3990:
3986:
3985:
3984:
3963:
3952:
3948:
3944:
3933:
3909:
3908:
3906:
3905:
3900:
3897:
3884:
3882:
3881:
3876:
3864:
3862:
3854:
3846:
3844:
3842:
3834:
3826:
3821:
3819:
3811:
3803:
3788:
3786:
3785:
3780:
3768:
3766:
3758:
3750:
3748:
3746:
3738:
3730:
3725:
3723:
3715:
3707:
3705:
3703:
3695:
3687:
3675:
3660:
3655:total derivative
3645:
3641:
3638:with respect to
3637:
3629:
3620:
3608:
3606:
3605:
3600:
3594:
3592:
3591:
3582:
3581:
3572:
3564:
3561:
3553:
3545:
3544:
3542:
3540:
3532:
3524:
3520:
3512:
3510:
3502:
3494:
3482:
3471:
3448:
3446:
3445:
3440:
3434:
3431:
3423:
3419:
3407:
3404:
3401:
3398:
3390:
3386:
3362:
3360:
3359:
3354:
3348:
3346:
3339:
3338:
3322:
3317:
3315:
3307:
3299:
3284:
3282:
3281:
3276:
3264:
3262:
3254:
3246:
3244:
3240:
3233:
3232:
3205:
3204:
3202:
3201:
3196:
3193:
3184:. Factoring out
3183:
3181:
3179:
3178:
3173:
3170:
3157:
3155:
3154:
3149:
3143:
3141:
3133:
3125:
3120:
3118:
3110:
3102:
3097:
3095:
3087:
3079:
3077:
3076:
3054:
3053:
3051:
3050:
3045:
3042:
3032:
3031:
3029:
3028:
3023:
3020:
3010:
3006:
2996:
2994:
2993:
2988:
2970:
2969:
2950:
2938:
2936:
2935:
2930:
2924:
2919:
2918:
2909:
2901:
2899:
2891:
2890:
2889:
2873:
2868:
2866:
2858:
2850:
2835:
2833:
2832:
2827:
2815:
2813:
2805:
2797:
2786:
2785:
2760:
2754:and another for
2753:
2743:
2741:
2740:
2735:
2729:
2723:
2722:
2721:
2705:
2704:
2674:
2667:
2665:
2664:
2659:
2647:
2646:
2631:
2630:
2611:
2589:
2587:
2586:
2581:
2566:
2564:
2556:
2548:
2536:
2535:
2533:
2532:
2527:
2524:
2511:
2509:
2508:
2503:
2501:
2471:
2469:
2461:
2453:
2424:
2422:
2411:
2406:
2404:
2396:
2388:
2383:
2381:
2373:
2365:
2346:
2344:
2342:
2341:
2336:
2333:
2323:
2309:
2307:
2306:
2301:
2267:
2260:
2258:
2257:
2252:
2202:
2198:
2194:
2193:
2191:
2190:
2185:
2182:
2172:
2168:
2165:with respect to
2164:
2146:
2142:
2127:
2085:
2081:
2070:
2063:
2045:
2041:
2032:cubic polynomial
2029:
2025:
2010:
1987:
1983:
1965:
1963:
1962:
1957:
1930:
1929:
1914:
1913:
1898:
1897:
1882:
1881:
1858:
1839:
1835:
1817:
1815:
1814:
1809:
1803:
1801:
1800:
1785:
1764:
1757:
1755:
1754:
1749:
1731:
1730:
1718:
1717:
1683:
1679:
1664:
1660:
1643:
1641:
1640:
1635:
1614:
1613:
1595:
1594:
1570:
1569:
1551:
1550:
1532:
1531:
1512:
1508:
1486:
1476:of the equation
1475:
1461:
1455:example below).
1450:
1444:
1442:
1441:
1438:
1435:
1419:
1400:
1389:
1385:
1381:
1372:
1370:
1369:
1364:
1349:
1348:
1320:
1316:
1309:
1307:
1306:
1301:
1272:of the equation
1267:
1261:
1257:
1253:
1248:inverse function
1217:
1213:
1205:
1201:
1193:
1191:
1190:
1185:
1171:
1170:
1158:
1157:
1123:
1121:
1120:
1115:
1101:
1100:
1088:
1087:
1059:
1055:
1053:
1052:
1047:
1033:
1032:
1014:
1013:
970:
963:
956:
904:
869:
835:
831:
798:Surface integral
741:
737:
645:
641:
601:Limit comparison
521:
517:
403:Riemann integral
356:
352:
312:L'Hôpital's rule
269:Taylor's theorem
190:
186:
130:
128:
127:
122:
74:
65:
60:
30:
21:
4859:
4858:
4854:
4853:
4852:
4850:
4849:
4848:
4814:
4813:
4797:
4794:Wayback Machine
4785:
4775:
4757:Blume, Lawrence
4754:
4748:
4720:
4714:
4694:
4691:
4689:Further reading
4686:
4685:
4678:
4665:
4664:
4660:
4653:
4636:
4635:
4628:
4621:
4601:
4600:
4596:
4591:
4586:
4525:
4505:
4478:
4471:
4459:
4448:economic theory
4444:
4438:
4419:
4416:
4411:
4410:
4408:
4407:
4399:
4395:
4376:
4373:
4364:
4360:
4346:
4343:
4338:
4337:
4335:
4334:
4330:
4326:
4307:
4300:
4295:
4287:
4267:
4256:
4248:
4244:
4235:
4225:
4218:
4191:
4171:
4159:
4156:
4150:
4149:
4147:
4146:
4139:
4120:
4116:
4112:
4100:
4081:
4069:
4066:
4060:
4059:
4057:
4056:
4041:
4025:
4006:
4003:
3988:
3979:
3977:
3971:
3965:
3954:
3950:
3946:
3945:. Around point
3935:
3923:
3916:
3901:
3898:
3893:
3892:
3890:
3889:
3855:
3847:
3835:
3827:
3812:
3804:
3797:
3796:
3759:
3751:
3739:
3731:
3716:
3708:
3696:
3688:
3681:
3680:
3662:
3658:
3643:
3639:
3635:
3627:
3622:
3618:
3613:
3583:
3573:
3554:
3546:
3533:
3525:
3521:
3503:
3495:
3488:
3487:
3473:
3458:
3455:
3371:
3370:
3330:
3326:
3308:
3300:
3293:
3292:
3255:
3247:
3224:
3220:
3216:
3211:
3210:
3197:
3194:
3189:
3188:
3186:
3185:
3174:
3171:
3166:
3165:
3163:
3162:
3134:
3126:
3111:
3103:
3088:
3080:
3068:
3060:
3059:
3046:
3043:
3038:
3037:
3035:
3034:
3024:
3021:
3016:
3015:
3013:
3012:
3008:
3004:
2961:
2956:
2955:
2948:
2945:
2910:
2892:
2881:
2874:
2859:
2851:
2844:
2843:
2806:
2798:
2777:
2769:
2768:
2755:
2748:
2713:
2706:
2680:
2679:
2672:
2638:
2622:
2617:
2616:
2602:
2599:
2557:
2549:
2542:
2541:
2528:
2525:
2520:
2519:
2517:
2516:
2499:
2498:
2484:
2462:
2454:
2449:
2448:
2434:
2415:
2397:
2389:
2374:
2366:
2355:
2354:
2337:
2334:
2329:
2328:
2326:
2325:
2314:
2273:
2272:
2265:
2221:
2220:
2214:
2209:
2200:
2196:
2186:
2183:
2178:
2177:
2175:
2174:
2170:
2166:
2151:
2144:
2129:
2118:
2100:
2083:
2072:
2065:
2050:
2043:
2039:
2027:
2012:
1997:
1994:
1985:
1970:
1921:
1905:
1889:
1873:
1868:
1867:
1856:
1837:
1822:
1792:
1770:
1769:
1762:
1722:
1709:
1704:
1703:
1681:
1666:
1662:
1653:
1648:
1605:
1580:
1555:
1542:
1523:
1518:
1517:
1513:of an equation
1510:
1506:
1499:
1493:
1477:
1473:
1459:
1439:
1436:
1433:
1432:
1430:
1421:
1406:
1391:
1387:
1383:
1377:
1337:
1326:
1325:
1318:
1314:
1277:
1276:
1263:
1259:
1255:
1251:
1244:
1239:
1215:
1207:
1203:
1199:
1162:
1149:
1144:
1143:
1092:
1079:
1074:
1073:
1057:
1024:
1005:
994:
993:
974:
945:
944:
930:Integration Bee
905:
902:
895:
894:
870:
867:
860:
859:
832:
829:
822:
821:
803:Volume integral
738:
733:
726:
725:
642:
637:
630:
629:
599:
518:
513:
506:
505:
497:Risch algorithm
467:Euler's formula
353:
348:
341:
340:
322:General Leibniz
205:generalizations
187:
182:
175:
161:Rolle's theorem
156:
131:
67:
47:
46:
28:
23:
22:
15:
12:
11:
5:
4857:
4855:
4847:
4846:
4841:
4836:
4831:
4826:
4816:
4815:
4812:
4811:
4784:
4783:External links
4781:
4780:
4779:
4773:
4752:
4746:
4718:
4712:
4696:Binmore, K. G.
4690:
4687:
4684:
4683:
4676:
4658:
4651:
4626:
4619:
4593:
4592:
4590:
4587:
4585:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4553:
4552:
4547:
4537:
4532:
4530:Implicit curve
4526:
4524:
4521:
4440:Main article:
4437:
4434:
4372:
4369:
4299:
4296:
4294:
4291:
4286:
4283:
4253:implicit curve
4240:
4233:
4217:
4214:
4188:implicit curve
4145:The condition
3999:Main article:
3969:
3915:
3912:
3886:
3885:
3874:
3870:
3867:
3861:
3858:
3853:
3850:
3841:
3838:
3833:
3830:
3824:
3818:
3815:
3810:
3807:
3790:
3789:
3778:
3774:
3771:
3765:
3762:
3757:
3754:
3745:
3742:
3737:
3734:
3728:
3722:
3719:
3714:
3711:
3702:
3699:
3694:
3691:
3653:to obtain the
3625:
3616:
3610:
3609:
3598:
3590:
3586:
3580:
3576:
3570:
3567:
3560:
3557:
3552:
3549:
3539:
3536:
3531:
3528:
3518:
3515:
3509:
3506:
3501:
3498:
3454:
3451:
3450:
3449:
3438:
3430:
3426:
3422:
3417:
3414:
3411:
3397:
3393:
3389:
3384:
3381:
3378:
3364:
3363:
3352:
3345:
3342:
3337:
3333:
3329:
3325:
3320:
3314:
3311:
3306:
3303:
3286:
3285:
3274:
3270:
3267:
3261:
3258:
3253:
3250:
3243:
3239:
3236:
3231:
3227:
3223:
3219:
3159:
3158:
3147:
3140:
3137:
3132:
3129:
3123:
3117:
3114:
3109:
3106:
3100:
3094:
3091:
3086:
3083:
3075:
3071:
3067:
2998:
2997:
2986:
2982:
2979:
2976:
2973:
2968:
2964:
2944:
2941:
2940:
2939:
2928:
2922:
2917:
2913:
2907:
2904:
2898:
2895:
2888:
2884:
2880:
2877:
2871:
2865:
2862:
2857:
2854:
2837:
2836:
2825:
2821:
2818:
2812:
2809:
2804:
2801:
2795:
2792:
2789:
2784:
2780:
2776:
2745:
2744:
2733:
2726:
2720:
2716:
2712:
2709:
2702:
2699:
2696:
2693:
2690:
2687:
2669:
2668:
2657:
2653:
2650:
2645:
2641:
2637:
2634:
2629:
2625:
2598:
2595:
2591:
2590:
2579:
2575:
2572:
2569:
2563:
2560:
2555:
2552:
2513:
2512:
2497:
2493:
2490:
2487:
2485:
2483:
2480:
2477:
2474:
2468:
2465:
2460:
2457:
2451:
2450:
2447:
2443:
2440:
2437:
2435:
2433:
2430:
2427:
2421:
2418:
2414:
2409:
2403:
2400:
2395:
2392:
2386:
2380:
2377:
2372:
2369:
2363:
2362:
2311:
2310:
2299:
2295:
2292:
2289:
2286:
2283:
2280:
2262:
2261:
2250:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2213:
2210:
2208:
2205:
2099:
2096:
1993:
1990:
1967:
1966:
1955:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1928:
1924:
1920:
1917:
1912:
1908:
1904:
1901:
1896:
1892:
1888:
1885:
1880:
1876:
1819:
1818:
1807:
1799:
1795:
1791:
1788:
1783:
1780:
1777:
1759:
1758:
1747:
1743:
1740:
1737:
1734:
1729:
1725:
1721:
1716:
1712:
1651:
1645:
1644:
1633:
1629:
1626:
1623:
1620:
1617:
1612:
1608:
1604:
1601:
1598:
1593:
1590:
1587:
1583:
1579:
1576:
1573:
1568:
1565:
1562:
1558:
1554:
1549:
1545:
1541:
1538:
1535:
1530:
1526:
1495:Main article:
1492:
1489:
1374:
1373:
1362:
1358:
1355:
1352:
1347:
1344:
1340:
1336:
1333:
1311:
1310:
1299:
1296:
1293:
1290:
1287:
1284:
1243:
1240:
1238:
1235:
1183:
1180:
1177:
1174:
1169:
1165:
1161:
1156:
1152:
1113:
1110:
1107:
1104:
1099:
1095:
1091:
1086:
1082:
1045:
1042:
1039:
1036:
1031:
1027:
1023:
1020:
1017:
1012:
1008:
1004:
1001:
976:
975:
973:
972:
965:
958:
950:
947:
946:
943:
942:
937:
932:
927:
925:List of topics
922:
917:
912:
906:
901:
900:
897:
896:
893:
892:
887:
882:
877:
871:
866:
865:
862:
861:
856:
855:
854:
853:
848:
843:
833:
828:
827:
824:
823:
818:
817:
816:
815:
810:
805:
800:
795:
790:
785:
777:
776:
772:
771:
770:
769:
764:
759:
754:
746:
745:
739:
732:
731:
728:
727:
722:
721:
720:
719:
714:
709:
704:
699:
694:
686:
685:
681:
680:
679:
678:
673:
668:
663:
658:
653:
643:
636:
635:
632:
631:
626:
625:
624:
623:
618:
613:
608:
603:
597:
592:
587:
582:
577:
569:
568:
562:
561:
560:
559:
554:
549:
544:
539:
534:
519:
512:
511:
508:
507:
502:
501:
500:
499:
494:
489:
484:
482:Changing order
479:
469:
464:
446:
441:
436:
428:
427:
426:Integration by
423:
422:
421:
420:
415:
410:
405:
400:
390:
388:Antiderivative
382:
381:
377:
376:
375:
374:
369:
364:
354:
347:
346:
343:
342:
337:
336:
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
281:
280:
274:
273:
272:
271:
266:
261:
256:
251:
246:
238:
237:
233:
232:
231:
230:
229:
228:
223:
218:
208:
195:
194:
188:
181:
180:
177:
176:
174:
173:
168:
163:
157:
155:
154:
149:
143:
142:
141:
133:
132:
120:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
83:
80:
77:
73:
70:
64:
59:
55:
45:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4856:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4821:
4819:
4808:
4804:
4800:
4795:
4791:
4787:
4786:
4782:
4776:
4774:0-393-95733-0
4770:
4766:
4762:
4758:
4753:
4749:
4747:0-07-054235-X
4743:
4739:
4735:
4730:
4729:
4723:
4722:Rudin, Walter
4719:
4715:
4713:0-521-28952-1
4709:
4705:
4701:
4697:
4693:
4692:
4688:
4679:
4677:0-201-79937-5
4673:
4669:
4662:
4659:
4654:
4652:0-534-34330-9
4648:
4643:
4642:
4633:
4631:
4627:
4622:
4620:0-07-010813-7
4616:
4611:
4610:
4604:
4598:
4595:
4588:
4583:
4580:
4578:
4577:Related rates
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4551:
4548:
4546:
4543:
4542:
4541:
4538:
4536:
4533:
4531:
4528:
4527:
4522:
4520:
4518:
4514:
4508:
4503:
4498:
4496:
4492:
4488:
4484:
4474:
4469:
4465:
4457:
4453:
4449:
4443:
4435:
4433:
4431:
4422:
4414:
4405:
4398:of labor and
4393:
4387:
4383:
4379:
4370:
4368:
4358:
4349:
4341:
4324:
4318:
4314:
4310:
4305:
4297:
4292:
4290:
4284:
4282:
4280:
4276:
4270:
4265:
4259:
4254:
4243:
4239:
4232:
4228:
4223:
4215:
4213:
4210:
4208:
4202:
4198:
4194:
4189:
4185:
4184:regular point
4179:
4175:
4163:
4154:
4143:
4135:
4131:
4127:
4123:
4108:
4104:
4098:
4097:neighbourhood
4092:
4088:
4084:
4073:
4064:
4052:
4048:
4044:
4039:
4036:be a pair of
4033:
4029:
4023:
4017:
4013:
4009:
4002:
3994:
3993:tangent space
3983:
3975:
3968:
3961:
3957:
3942:
3938:
3931:
3927:
3920:
3913:
3911:
3904:
3896:
3872:
3868:
3865:
3859:
3856:
3851:
3848:
3839:
3831:
3822:
3816:
3808:
3795:
3794:
3793:
3776:
3772:
3769:
3763:
3760:
3755:
3752:
3743:
3735:
3726:
3720:
3717:
3712:
3709:
3700:
3692:
3679:
3678:
3677:
3673:
3669:
3665:
3656:
3652:
3647:
3633:
3630:indicate the
3628:
3619:
3596:
3588:
3584:
3578:
3574:
3568:
3565:
3558:
3550:
3537:
3529:
3516:
3513:
3507:
3504:
3499:
3496:
3486:
3485:
3484:
3480:
3476:
3469:
3465:
3461:
3452:
3436:
3428:
3424:
3420:
3415:
3412:
3409:
3395:
3391:
3387:
3382:
3379:
3376:
3369:
3368:
3367:
3350:
3343:
3340:
3335:
3331:
3327:
3323:
3318:
3312:
3309:
3304:
3301:
3291:
3290:
3289:
3272:
3268:
3265:
3259:
3256:
3251:
3248:
3241:
3237:
3234:
3229:
3225:
3221:
3217:
3209:
3208:
3207:
3200:
3192:
3177:
3169:
3145:
3138:
3135:
3130:
3127:
3121:
3115:
3112:
3107:
3104:
3098:
3092:
3089:
3084:
3081:
3073:
3069:
3065:
3058:
3057:
3056:
3049:
3041:
3027:
3019:
3003:
2984:
2980:
2977:
2974:
2971:
2966:
2962:
2954:
2953:
2952:
2942:
2926:
2920:
2915:
2911:
2905:
2902:
2896:
2893:
2886:
2882:
2878:
2875:
2869:
2863:
2860:
2855:
2852:
2842:
2841:
2840:
2823:
2819:
2816:
2810:
2807:
2802:
2799:
2793:
2790:
2787:
2782:
2778:
2774:
2767:
2766:
2765:
2762:
2758:
2751:
2731:
2724:
2718:
2714:
2710:
2707:
2700:
2697:
2691:
2685:
2678:
2677:
2676:
2655:
2651:
2648:
2643:
2639:
2635:
2632:
2627:
2623:
2615:
2614:
2613:
2609:
2605:
2596:
2594:
2577:
2573:
2570:
2567:
2561:
2558:
2553:
2550:
2540:
2539:
2538:
2531:
2523:
2495:
2491:
2488:
2486:
2481:
2478:
2475:
2472:
2466:
2463:
2458:
2455:
2445:
2441:
2438:
2436:
2428:
2419:
2416:
2412:
2407:
2401:
2398:
2393:
2390:
2384:
2378:
2375:
2370:
2367:
2353:
2352:
2351:
2348:
2340:
2332:
2321:
2317:
2297:
2293:
2290:
2287:
2284:
2281:
2278:
2271:
2270:
2269:
2248:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2219:
2218:
2217:
2211:
2206:
2204:
2189:
2181:
2162:
2158:
2154:
2150:
2140:
2136:
2132:
2125:
2121:
2115:
2113:
2109:
2105:
2097:
2095:
2093:
2089:
2079:
2075:
2068:
2061:
2057:
2053:
2047:
2037:
2033:
2023:
2019:
2015:
2008:
2004:
2000:
1991:
1989:
1981:
1977:
1973:
1953:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1926:
1922:
1918:
1915:
1910:
1906:
1902:
1899:
1894:
1890:
1886:
1883:
1878:
1874:
1866:
1865:
1864:
1862:
1854:
1850:
1846:
1841:
1833:
1829:
1825:
1805:
1797:
1793:
1789:
1786:
1781:
1778:
1775:
1768:
1767:
1766:
1745:
1741:
1738:
1735:
1732:
1727:
1723:
1719:
1714:
1710:
1702:
1701:
1700:
1698:
1694:
1689:
1687:
1677:
1673:
1669:
1658:
1654:
1631:
1627:
1624:
1618:
1610:
1606:
1602:
1599:
1596:
1591:
1588:
1585:
1581:
1574:
1566:
1563:
1560:
1556:
1552:
1547:
1543:
1536:
1528:
1524:
1516:
1515:
1514:
1504:
1498:
1490:
1488:
1484:
1480:
1471:
1467:
1463:
1456:
1454:
1448:
1428:
1424:
1417:
1413:
1409:
1404:
1398:
1394:
1380:
1360:
1353:
1345:
1342:
1338:
1334:
1331:
1324:
1323:
1322:
1294:
1288:
1285:
1282:
1275:
1274:
1273:
1271:
1266:
1249:
1241:
1236:
1234:
1232:
1228:
1224:
1219:
1211:
1197:
1181:
1178:
1175:
1172:
1167:
1163:
1159:
1154:
1150:
1141:
1137:
1133:
1129:
1124:
1111:
1108:
1105:
1102:
1097:
1093:
1089:
1084:
1080:
1071:
1067:
1063:
1043:
1040:
1037:
1029:
1025:
1021:
1018:
1015:
1010:
1006:
999:
991:
987:
983:
971:
966:
964:
959:
957:
952:
951:
949:
948:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
907:
899:
898:
891:
888:
886:
883:
881:
878:
876:
873:
872:
864:
863:
852:
849:
847:
844:
842:
839:
838:
837:
836:
826:
825:
814:
811:
809:
806:
804:
801:
799:
796:
794:
793:Line integral
791:
789:
786:
784:
781:
780:
779:
778:
773:
768:
765:
763:
760:
758:
755:
753:
750:
749:
748:
747:
742:
736:
735:Multivariable
730:
729:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
689:
688:
687:
682:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
648:
647:
646:
640:
634:
633:
622:
619:
617:
614:
612:
609:
607:
604:
602:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
572:
571:
570:
567:
563:
558:
555:
553:
550:
548:
545:
543:
540:
538:
535:
532:
528:
525:
524:
523:
522:
516:
510:
509:
498:
495:
493:
490:
488:
485:
483:
480:
477:
473:
470:
468:
465:
462:
458:
454:
453:trigonometric
450:
447:
445:
442:
440:
437:
435:
432:
431:
430:
429:
424:
419:
416:
414:
411:
409:
406:
404:
401:
398:
394:
391:
389:
386:
385:
384:
383:
378:
373:
370:
368:
365:
363:
360:
359:
358:
357:
351:
345:
344:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
283:
282:
279:
275:
270:
267:
265:
264:Related rates
262:
260:
257:
255:
252:
250:
247:
245:
242:
241:
240:
239:
234:
227:
224:
222:
221:of a function
219:
217:
216:infinitesimal
214:
213:
212:
209:
206:
202:
199:
198:
197:
196:
191:
185:
179:
178:
172:
169:
167:
164:
162:
159:
158:
153:
150:
148:
145:
144:
140:
137:
136:
135:
134:
115:
109:
106:
100:
94:
91:
88:
85:
78:
71:
68:
62:
57:
53:
44:
43:
40:
36:
32:
31:
19:
4802:
4790:Ghostarchive
4788:Archived at
4764:
4727:
4703:
4667:
4661:
4640:
4608:
4597:
4545:Contour line
4506:
4499:
4491:labor supply
4483:labor demand
4472:
4445:
4436:Optimization
4420:
4412:
4385:
4381:
4377:
4374:
4347:
4339:
4316:
4312:
4308:
4301:
4288:
4268:
4263:
4257:
4241:
4237:
4230:
4226:
4224:of the form
4219:
4211:
4200:
4196:
4192:
4177:
4173:
4161:
4152:
4144:
4133:
4129:
4125:
4121:
4119:, such that
4106:
4102:
4090:
4086:
4082:
4071:
4062:
4050:
4046:
4042:
4038:real numbers
4031:
4027:
4015:
4011:
4007:
4004:
3995:is vertical.
3991:, where the
3981:
3973:
3966:
3959:
3955:
3940:
3936:
3929:
3925:
3902:
3894:
3887:
3791:
3671:
3667:
3663:
3648:
3623:
3614:
3611:
3483:is given by
3478:
3474:
3467:
3463:
3459:
3456:
3365:
3287:
3198:
3190:
3175:
3167:
3160:
3047:
3039:
3025:
3017:
2999:
2946:
2838:
2763:
2756:
2749:
2746:
2670:
2607:
2603:
2600:
2592:
2529:
2521:
2515:Solving for
2514:
2349:
2338:
2330:
2319:
2315:
2312:
2263:
2215:
2187:
2179:
2160:
2156:
2152:
2138:
2134:
2130:
2123:
2119:
2116:
2107:
2101:
2077:
2073:
2066:
2059:
2055:
2051:
2048:
2035:
2021:
2017:
2013:
2006:
2002:
1998:
1995:
1979:
1975:
1971:
1968:
1842:
1831:
1827:
1823:
1820:
1761:Solving for
1760:
1690:
1686:multi-valued
1675:
1671:
1667:
1656:
1649:
1646:
1502:
1500:
1482:
1478:
1465:
1464:
1457:
1446:
1426:
1422:
1415:
1411:
1407:
1396:
1392:
1378:
1375:
1317:in terms of
1312:
1264:
1245:
1220:
1209:
1127:
1125:
992:of the form
985:
979:
449:Substitution
253:
211:Differential
184:Differential
4803:3Blue1Brown
4736:. pp.
4734:McGraw-Hill
4572:Polygonizer
4220:Consider a
4170:means that
3934:satisfying
3206:shows that
1470:product log
1453:product log
1196:unit circle
1070:unit circle
982:mathematics
910:Precalculus
903:Miscellanea
868:Specialized
775:Definitions
542:Alternating
380:Definitions
193:Definitions
4818:Categories
4732:. Boston:
4589:References
4550:Isosurface
4502:parameters
4205:where the
4040:such that
2112:chain rule
1066:polynomial
890:Variations
885:Stochastic
875:Fractional
744:Formalisms
707:Divergence
676:Identities
656:Divergence
201:Derivative
152:Continuity
4540:Level set
4446:Often in
4304:economics
3837:∂
3829:∂
3814:∂
3806:∂
3741:∂
3733:∂
3698:∂
3690:∂
3569:−
3556:∂
3548:∂
3535:∂
3527:∂
3517:−
3416:±
3413:≠
3383:±
3380:≠
3341:−
3235:−
3099:−
2972:−
2943:Example 3
2906:−
2876:−
2711:−
2701:±
2597:Example 2
2571:−
2291:−
2285:−
2268:, giving
2216:Consider
2212:Example 1
1941:−
1932:−
1900:−
1845:quadratic
1790:−
1782:±
1733:−
1600:⋯
1589:−
1564:−
1376:Defining
1343:−
1262:, called
1229:that are
1173:−
1140:arguments
1103:−
1019:…
880:Malliavin
767:Geometric
666:Laplacian
616:Dirichlet
527:Geometric
107:−
54:∫
4792:and the
4759:(1994).
4724:(1976).
4704:Calculus
4698:(1983).
4605:(1984).
4523:See also
4392:isoquant
4247:, where
4222:relation
2207:Examples
2104:calculus
1836:, where
1466:Example:
1270:solution
1237:Examples
1198:defines
1132:function
1062:function
990:relation
920:Glossary
830:Advanced
808:Jacobian
762:Exterior
692:Gradient
684:Theorems
651:Gradient
590:Integral
552:Binomial
537:Harmonic
397:improper
393:Integral
350:Integral
332:Reynolds
307:Quotient
236:Concepts
72:′
39:Calculus
4807:YouTube
4738:223–228
4425:
4409:
4352:
4336:
4262:and an
4207:tangent
4186:of the
4166:
4148:
4080:, then
4076:
4058:
3978:√
3907:
3891:
3203:
3187:
3180:
3164:
3052:
3036:
3030:
3014:
2839:giving
2534:
2518:
2343:
2327:
2192:
2176:
1992:Caveats
1861:quintic
1853:quartic
1443:
1431:
1420:, then
1194:of the
915:History
813:Hessian
702:Stokes'
697:Green's
529: (
451: (
395: (
317:Inverse
292:Product
203: (
4771:
4744:
4710:
4674:
4649:
4617:
4456:profit
4390:is an
4321:is an
4136:)) = 0
3792:hence
3612:where
3161:where
2759:< 0
2537:gives
2090:. The
2088:domain
2026:where
1851:, and
1214:, and
1056:where
757:Tensor
752:Matrix
639:Vector
557:Taylor
515:Series
147:Limits
4454:or a
4319:) = 0
4245:) = 0
4236:, …,
4203:) = 0
4182:is a
4093:) = 0
4055:. If
4053:) = 0
4020:be a
3674:) = 0
3470:) = 0
2163:) = 0
2141:) = 0
2062:) = 0
2030:is a
2024:) = 0
2009:) = 0
1849:cubic
1684:is a
1414:) = 2
1208:−1 ≤
1136:value
1130:is a
1060:is a
988:is a
984:, an
580:Ratio
547:Power
461:Euler
439:Discs
434:Parts
302:Power
297:Chain
226:total
4769:ISBN
4742:ISBN
4708:ISBN
4672:ISBN
4647:ISBN
4615:ISBN
4329:and
4138:for
4005:Let
3980:1 −
3976:) =
3642:and
3621:and
2345:= −1
2199:and
2169:and
2036:true
1695:and
1468:The
1449:+ 1)
1429:) =
1313:for
1221:The
661:Curl
621:Abel
585:Root
4504:on
4402:of
4302:In
4271:= 3
4266:if
4260:= 2
4255:if
4168:≠ 0
4099:of
4078:≠ 0
3943:= 1
3634:of
3457:If
3405:and
3182:= 1
2752:≥ 0
2102:In
2069:= 0
1855:in
1501:An
1485:= 0
1418:− 1
1212:≤ 1
1206:if
1126:An
1072:is
980:In
287:Sum
4820::
4801:.
4796::
4763:.
4740:.
4702:.
4629:^
4519:.
4421:dL
4413:dK
4384:,
4367:.
4348:dx
4340:dy
4315:,
4281:.
4199:,
4176:,
4128:,
4105:,
4089:,
4049:,
4030:,
4014:,
3949:,
3939:+
3928:,
3903:dx
3895:dy
3676::
3670:,
3646:.
3466:,
3199:dx
3191:dy
3176:dx
3168:dx
3048:dx
3040:dy
3026:dx
3018:dy
2761:.
2530:dx
2522:dy
2347:.
2339:dx
2331:dy
2188:dx
2180:dy
2159:,
2137:,
2058:,
2016:−
2005:,
1988:.
1974:=
1847:,
1826:=
1670:=
1487:.
1483:xe
1481:−
1390:,
1233:.
1112:0.
459:,
455:,
4809:.
4777:.
4750:.
4716:.
4680:.
4655:.
4623:.
4509:*
4507:x
4479:x
4475:*
4473:x
4460:x
4417:/
4400:K
4396:L
4388:)
4386:K
4382:L
4380:(
4378:R
4365:x
4361:y
4344:/
4331:y
4327:x
4317:y
4313:x
4311:(
4309:R
4269:n
4258:n
4249:R
4242:n
4238:x
4234:1
4231:x
4229:(
4227:R
4201:y
4197:x
4195:(
4193:R
4180:)
4178:b
4174:a
4172:(
4162:y
4160:∂
4157:/
4153:R
4151:∂
4140:x
4134:x
4132:(
4130:f
4126:x
4124:(
4122:R
4117:a
4113:f
4109:)
4107:b
4103:a
4101:(
4091:y
4087:x
4085:(
4083:R
4072:y
4070:∂
4067:/
4063:R
4061:∂
4051:b
4047:a
4045:(
4043:R
4034:)
4032:b
4028:a
4026:(
4018:)
4016:y
4012:x
4010:(
4008:R
3989:B
3982:x
3974:x
3972:(
3970:1
3967:g
3962:)
3960:x
3958:(
3956:y
3951:y
3947:A
3941:y
3937:x
3932:)
3930:y
3926:x
3924:(
3899:/
3873:,
3869:0
3866:=
3860:x
3857:d
3852:y
3849:d
3840:y
3832:R
3823:+
3817:x
3809:R
3777:,
3773:0
3770:=
3764:x
3761:d
3756:y
3753:d
3744:y
3736:R
3727:+
3721:x
3718:d
3713:x
3710:d
3701:x
3693:R
3672:y
3668:x
3666:(
3664:R
3659:x
3644:y
3640:x
3636:R
3626:y
3624:R
3617:x
3615:R
3597:,
3589:y
3585:R
3579:x
3575:R
3566:=
3559:y
3551:R
3538:x
3530:R
3514:=
3508:x
3505:d
3500:y
3497:d
3481:)
3479:x
3477:(
3475:y
3468:y
3464:x
3462:(
3460:R
3437:.
3429:4
3425:5
3421:i
3410:y
3396:4
3392:5
3388:1
3377:y
3351:,
3344:1
3336:4
3332:y
3328:5
3324:1
3319:=
3313:x
3310:d
3305:y
3302:d
3273:,
3269:1
3266:=
3260:x
3257:d
3252:y
3249:d
3242:)
3238:1
3230:4
3226:y
3222:5
3218:(
3195:/
3172:/
3146:,
3139:x
3136:d
3131:x
3128:d
3122:=
3116:x
3113:d
3108:y
3105:d
3093:x
3090:d
3085:y
3082:d
3074:4
3070:y
3066:5
3044:/
3022:/
3009:x
3005:y
2985:.
2981:x
2978:=
2975:y
2967:5
2963:y
2949:y
2927:.
2921:y
2916:3
2912:x
2903:=
2897:y
2894:4
2887:3
2883:x
2879:4
2870:=
2864:x
2861:d
2856:y
2853:d
2824:,
2820:0
2817:=
2811:x
2808:d
2803:y
2800:d
2794:y
2791:4
2788:+
2783:3
2779:x
2775:4
2757:y
2750:y
2732:,
2725:2
2719:4
2715:x
2708:8
2698:=
2695:)
2692:x
2689:(
2686:y
2673:x
2656:.
2652:8
2649:=
2644:2
2640:y
2636:2
2633:+
2628:4
2624:x
2610:)
2608:x
2606:(
2604:y
2578:,
2574:1
2568:=
2562:x
2559:d
2554:y
2551:d
2526:/
2496:.
2492:0
2489:=
2482:0
2479:+
2476:1
2473:+
2467:x
2464:d
2459:y
2456:d
2446:;
2442:0
2439:=
2432:)
2429:5
2426:(
2420:x
2417:d
2413:d
2408:+
2402:x
2399:d
2394:x
2391:d
2385:+
2379:x
2376:d
2371:y
2368:d
2335:/
2322:)
2320:x
2318:(
2316:y
2298:,
2294:5
2288:x
2282:=
2279:y
2266:y
2249:.
2245:0
2242:=
2239:5
2236:+
2233:x
2230:+
2227:y
2201:y
2197:x
2184:/
2171:y
2167:x
2161:y
2157:x
2155:(
2153:R
2145:y
2139:y
2135:x
2133:(
2131:R
2126:)
2124:x
2122:(
2120:y
2084:y
2080:)
2078:x
2076:(
2074:f
2067:x
2060:y
2056:x
2054:(
2052:R
2044:y
2040:x
2028:C
2022:y
2020:(
2018:C
2014:x
2007:y
2003:x
2001:(
1999:R
1986:f
1982:)
1980:x
1978:(
1976:f
1972:y
1954:.
1950:0
1947:=
1944:x
1938:y
1935:6
1927:2
1923:y
1919:3
1916:+
1911:3
1907:y
1903:7
1895:4
1891:y
1887:2
1884:+
1879:5
1875:y
1857:y
1838:f
1834:)
1832:x
1830:(
1828:f
1824:y
1806:.
1798:2
1794:x
1787:1
1779:=
1776:y
1763:y
1746:.
1742:0
1739:=
1736:1
1728:2
1724:y
1720:+
1715:2
1711:x
1682:f
1678:)
1676:x
1674:(
1672:f
1668:y
1663:x
1659:)
1657:x
1655:(
1652:i
1650:a
1632:,
1628:0
1625:=
1622:)
1619:x
1616:(
1611:0
1607:a
1603:+
1597:+
1592:1
1586:n
1582:y
1578:)
1575:x
1572:(
1567:1
1561:n
1557:a
1553:+
1548:n
1544:y
1540:)
1537:x
1534:(
1529:n
1525:a
1511:y
1507:x
1479:y
1474:x
1460:g
1447:y
1445:(
1440:2
1437:/
1434:1
1427:y
1425:(
1423:g
1416:x
1412:x
1410:(
1408:g
1399:)
1397:y
1395:(
1393:g
1388:g
1384:g
1379:g
1361:.
1357:)
1354:y
1351:(
1346:1
1339:g
1335:=
1332:x
1319:y
1315:x
1298:)
1295:x
1292:(
1289:g
1286:=
1283:y
1265:g
1260:g
1256:x
1252:g
1216:y
1210:x
1204:x
1200:y
1182:0
1179:=
1176:1
1168:2
1164:y
1160:+
1155:2
1151:x
1109:=
1106:1
1098:2
1094:y
1090:+
1085:2
1081:x
1058:R
1044:,
1041:0
1038:=
1035:)
1030:n
1026:x
1022:,
1016:,
1011:1
1007:x
1003:(
1000:R
969:e
962:t
955:v
533:)
478:)
474:(
463:)
399:)
207:)
119:)
116:a
113:(
110:f
104:)
101:b
98:(
95:f
92:=
89:t
86:d
82:)
79:t
76:(
69:f
63:b
58:a
20:)
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