25:
1217:
585:
1713:
2161:
1027:
1949:
793:
289:
992:
1343:
1467:
837:, the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces. Two results of David Preiss (1981 and 1983) show the kind of difficulties that one can expect to encounter in this setting:
1791:
427:
1560:
2017:
1800:
649:
145:
872:
1535:
1249:
1212:{\displaystyle \lim _{r\to 0}\inf \left\{\left.{\frac {1}{\gamma {\big (}B_{s}(x){\big )}}}\int _{B_{s}(x)}f(y)\,\mathrm {d} \gamma (y)\right|x\in H,0<s<r\right\}=+\infty .}
1378:
1726:
3239:
3227:
54:
1478:
3349:
3234:
2256:
3217:
3212:
3222:
3207:
2321:
580:{\displaystyle \lim _{r\to 0}{\frac {1}{\lambda ^{n}{\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \lambda ^{n}(y)=f(x)}
2509:
3202:
1708:{\displaystyle {\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y){\xrightarrow{\gamma }}f(x),}
2819:
2573:
2237:
2156:{\displaystyle \lim _{r\to 0}{\frac {1}{\gamma {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \gamma (y)=f(x)}
2371:
1944:{\displaystyle {\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y){\xrightarrow{}}f(x),}
3317:
3176:
788:{\displaystyle \lim _{r\to 0}{\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y)=f(x)}
284:{\displaystyle \lim _{r\to 0}{\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y)=f(x)}
76:
47:
2731:
2647:
377:
3312:
3244:
2869:
2724:
2692:
2451:
987:{\displaystyle \lim _{r\to 0}{\frac {\gamma {\big (}M\cap B_{r}(x){\big )}}{\gamma {\big (}B_{r}(x){\big )}}}=1.}
3375:
2945:
2922:
2637:
815:
The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a
611:
The result for
Lebesgue measure turns out to be a special case of the following result, which is based on the
3390:
3385:
3035:
2973:
2768:
2642:
2314:
1338:{\displaystyle \langle Sx,y\rangle =\int _{H}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \gamma (z),}
2521:
2499:
612:
3344:
3329:
3095:
2709:
2531:
410:
2714:
2484:
2205:
1986:
As of 2007, it is still an open question whether there exists an infinite-dimensional
Gaussian measure
599:. It is important to note, however, that the measure zero set of "bad" points depends on the function
3133:
3080:
37:
2541:
3249:
3020:
2568:
2307:
109:
41:
33:
3380:
3015:
2687:
834:
3143:
3025:
2846:
2794:
2600:
2578:
2446:
1716:
620:
343:.) This is a natural question to ask, especially in view of the heuristic construction of the
112:
of a point approximates the value of the function at that point. More formally, given a space
105:
58:
3269:
3128:
3040:
2697:
2632:
2605:
2595:
2516:
2504:
2489:
2461:
2199:
2193:
117:
3085:
2704:
2551:
2291:
2247:
1471:
In 1981, Preiss and
Jaroslav Tišer showed that if there exists a constant 0 <
8:
3105:
3030:
2917:
2874:
2625:
2610:
2441:
2429:
2416:
2376:
2356:
2220:
Preiss, David; Tišer, Jaroslav (1982). "Differentiation of measures on
Hilbert spaces".
1462:{\displaystyle Sx=\sum _{i\in \mathbf {N} }\sigma _{i}^{2}\langle x,e_{i}\rangle e_{i}.}
3194:
3169:
3000:
2953:
2894:
2859:
2854:
2834:
2829:
2824:
2789:
2736:
2719:
2620:
2494:
2479:
2424:
2391:
2279:
3334:
3158:
3090:
2912:
2889:
2763:
2756:
2659:
2474:
2366:
2233:
1349:
3292:
3075:
2988:
2968:
2899:
2809:
2751:
2743:
2677:
2590:
2351:
2346:
2271:
2225:
2224:. Lecture Notes in Mathematics. Vol. 945. Berlin: Springer. pp. 194–207.
827:
392:
344:
3354:
3339:
3123:
2978:
2958:
2927:
2904:
2884:
2778:
2434:
2381:
2287:
2243:
816:
403:
3264:
3163:
3010:
2963:
2864:
2667:
381:
2682:
3369:
3138:
2993:
2879:
2583:
2558:
1346:
819:
623:
1786:{\displaystyle \sigma _{i+1}^{2}\leq {\frac {\sigma _{i}^{2}}{i^{\alpha }}}}
3148:
3118:
2983:
2546:
124:
2396:
2338:
90:
3113:
3045:
2799:
2672:
2536:
2526:
2469:
2283:
2229:
1223:
296:
1923:
3307:
3055:
3050:
2361:
2175:. However, it is conjectured that no such measure exists, since the
1910:
1670:
389:
321:
2275:
1683:
3302:
2804:
2330:
101:
3153:
2406:
98:
2257:"Differentiation theorem for Gaussian measures on Hilbert space"
3322:
2386:
366:
333:
2401:
1222:
However, there is some hope if one has good control over the
2299:
1056:
2202: – Differentiation under the integral sign formula
1530:{\displaystyle \sigma _{i+1}^{2}\leq q\sigma _{i}^{2},}
2208: – 3D generalization of the Leibniz integral rule
376:
One result on the differentiation of integrals is the
2020:
1803:
1729:
1563:
1481:
1381:
1252:
1030:
875:
652:
430:
148:
2222:
Measure theory, Oberwolfach 1981 (Oberwolfach, 1981)
2196: – Rules for computing derivatives of functions
97:
is that of determining under what circumstances the
2155:
1943:
1785:
1707:
1529:
1461:
1337:
1211:
986:
787:
579:
283:
2264:Transactions of the American Mathematical Society
3367:
2022:
1047:
1032:
877:
654:
432:
150:
46:but its sources remain unclear because it lacks
355:) is a "good representative" for the values of
2315:
2074:
2048:
1841:
1815:
1601:
1575:
1096:
1070:
970:
944:
932:
900:
706:
680:
491:
465:
202:
176:
1443:
1424:
1311:
1299:
1296:
1284:
1268:
1253:
367:Theorems on the differentiation of integrals
2219:
2322:
2308:
3350:Regiomontanus' angle maximization problem
2120:
1887:
1647:
1314:
1142:
752:
537:
307:? (Here, as in the rest of the article,
248:
77:Learn how and when to remove this message
3193:
826:, ⟨ , ⟩) equipped with a
2698:Differentiating under the integral sign
606:
3368:
1797: > 5 ⁄ 2, then
642:is locally integrable with respect to
347:, in which it is almost implicit that
2574:Inverse functions and differentiation
2303:
2254:
810:
18:
833:. As stated in the article on the
371:
13:
2372:Free variables and bound variables
2184:would have to decay very rapidly.
2122:
1889:
1649:
1316:
1203:
1144:
754:
539:
250:
14:
3402:
3177:The Method of Mechanical Theorems
1723:. In 1988, Tišer showed that if
1230:. Let the covariance operator of
2732:Partial fractions in integration
2648:Stochastic differential equation
1403:
378:Lebesgue differentiation theorem
23:
2870:Jacobian matrix and determinant
2725:Tangent half-angle substitution
2693:Fundamental theorem of calculus
2946:Arithmetico-geometric sequence
2638:Ordinary differential equation
2150:
2144:
2135:
2129:
2117:
2111:
2103:
2097:
2069:
2063:
2029:
1935:
1929:
1915:
1902:
1896:
1884:
1878:
1870:
1864:
1836:
1830:
1699:
1693:
1675:
1662:
1656:
1644:
1638:
1630:
1624:
1596:
1590:
1329:
1323:
1157:
1151:
1139:
1133:
1125:
1119:
1091:
1085:
1039:
965:
959:
927:
921:
884:
782:
776:
767:
761:
749:
743:
735:
729:
701:
695:
661:
574:
568:
559:
553:
534:
528:
520:
514:
486:
480:
439:
278:
272:
263:
257:
245:
239:
231:
225:
197:
191:
157:
130:, one asks for what functions
1:
2769:Integro-differential equation
2643:Partial differential equation
2212:
1990:on a separable Hilbert space
1000:on a separable Hilbert space
845:on a separable Hilbert space
2329:
1475: < 1 such that
996:There is a Gaussian measure
841:There is a Gaussian measure
613:Besicovitch covering theorem
95:differentiation of integrals
7:
2923:Generalized Stokes' theorem
2710:Integration by substitution
2187:
411:locally integrable function
10:
3407:
2452:(ε, δ)-definition of limit
2206:Reynolds transport theorem
3345:Proof that 22/7 exceeds π
3282:
3260:
3186:
3134:Gottfried Wilhelm Leibniz
3104:
3081:e (mathematical constant)
3066:
2938:
2845:
2777:
2658:
2460:
2415:
2337:
1715:where the convergence is
3096:Stirling's approximation
2569:Implicit differentiation
2517:Rules of differentiation
2255:Tišer, Jaroslav (1988).
32:This article includes a
3330:Euler–Maclaurin formula
3235:trigonometric functions
2688:Constant of integration
835:Vitali covering theorem
61:more precise citations.
3299:Differential geometry
3144:Infinitesimal calculus
2847:Multivariable calculus
2795:Directional derivative
2601:Second derivative test
2579:Logarithmic derivative
2552:General Leibniz's rule
2447:Order of approximation
2157:
1945:
1924:
1787:
1717:convergence in measure
1709:
1688:
1531:
1463:
1339:
1213:
988:
789:
581:
285:
16:Problem in mathematics
3376:Differentiation rules
3218:logarithmic functions
3213:exponential functions
3129:Generality of algebra
3007:Tests of convergence
2633:Differential equation
2617:Further applications
2606:Extreme value theorem
2596:First derivative test
2490:Differential operator
2462:Differential calculus
2200:Leibniz integral rule
2194:Differentiation rules
2158:
1946:
1906:
1788:
1710:
1666:
1532:
1464:
1340:
1214:
989:
790:
582:
291:for all (or at least
286:
3391:Theorems in calculus
3386:Theorems in analysis
3283:Miscellaneous topics
3223:hyperbolic functions
3208:irrational functions
3086:Exponential function
2939:Sequences and series
2705:Integration by parts
2018:
1801:
1727:
1561:
1479:
1379:
1250:
1028:
873:
650:
428:
146:
3270:List of derivatives
3106:History of calculus
3021:Cauchy condensation
2918:Exterior derivative
2875:Lagrange multiplier
2611:Maximum and minimum
2442:Limit of a sequence
2430:Limit of a function
2377:Graph of a function
2357:Continuous function
1983: > 1.
1922:
1770:
1750:
1687:
1682:
1523:
1502:
1423:
799:-almost all points
607:Borel measures on R
591:-almost all points
384:in 1910. Consider
3203:rational functions
3170:Method of Fluxions
3016:Alternating series
2913:Differential forms
2895:Partial derivative
2855:Divergence theorem
2737:Quadratic integral
2505:Leibniz's notation
2495:Mean value theorem
2480:Partial derivative
2425:Indeterminate form
2230:10.1007/BFb0096675
2153:
2036:
1941:
1783:
1756:
1730:
1705:
1527:
1509:
1482:
1459:
1409:
1408:
1335:
1209:
1046:
984:
891:
785:
668:
577:
446:
281:
164:
34:list of references
3363:
3362:
3289:Complex calculus
3278:
3277:
3159:Law of Continuity
3091:Natural logarithm
3076:Bernoulli numbers
3067:Special functions
3026:Direct comparison
2890:Multiple integral
2764:Integral equation
2660:Integral calculus
2591:Stationary points
2565:Other techniques
2510:Newton's notation
2475:Second derivative
2367:Finite difference
2239:978-3-540-11580-9
2080:
2021:
1994:so that, for all
1847:
1781:
1607:
1391:
1350:orthonormal basis
1102:
1031:
976:
876:
811:Gaussian measures
712:
653:
497:
431:
409:. Then, for any
208:
149:
93:, the problem of
87:
86:
79:
3398:
3293:Contour integral
3191:
3190:
3041:Limit comparison
2950:Types of series
2909:Advanced topics
2900:Surface integral
2744:Trapezoidal rule
2683:Basic properties
2678:Riemann integral
2626:Taylor's theorem
2352:Concave function
2347:Binomial theorem
2324:
2317:
2310:
2301:
2300:
2295:
2261:
2251:
2162:
2160:
2159:
2154:
2125:
2107:
2106:
2096:
2095:
2081:
2079:
2078:
2077:
2062:
2061:
2052:
2051:
2038:
2035:
1950:
1948:
1947:
1942:
1925:
1921:
1892:
1874:
1873:
1863:
1862:
1848:
1846:
1845:
1844:
1829:
1828:
1819:
1818:
1805:
1792:
1790:
1789:
1784:
1782:
1780:
1779:
1769:
1764:
1755:
1749:
1744:
1719:with respect to
1714:
1712:
1711:
1706:
1689:
1681:
1652:
1634:
1633:
1623:
1622:
1608:
1606:
1605:
1604:
1589:
1588:
1579:
1578:
1565:
1536:
1534:
1533:
1528:
1522:
1517:
1501:
1496:
1468:
1466:
1465:
1460:
1455:
1454:
1442:
1441:
1422:
1417:
1407:
1406:
1344:
1342:
1341:
1336:
1319:
1283:
1282:
1218:
1216:
1215:
1210:
1196:
1192:
1164:
1160:
1147:
1129:
1128:
1118:
1117:
1103:
1101:
1100:
1099:
1084:
1083:
1074:
1073:
1060:
1045:
993:
991:
990:
985:
977:
975:
974:
973:
958:
957:
948:
947:
937:
936:
935:
920:
919:
904:
903:
893:
890:
849:and a Borel set
828:Gaussian measure
794:
792:
791:
786:
757:
739:
738:
728:
727:
713:
711:
710:
709:
694:
693:
684:
683:
670:
667:
586:
584:
583:
578:
552:
551:
542:
524:
523:
513:
512:
498:
496:
495:
494:
479:
478:
469:
468:
462:
461:
448:
445:
393:Lebesgue measure
372:Lebesgue measure
345:Riemann integral
290:
288:
287:
282:
253:
235:
234:
224:
223:
209:
207:
206:
205:
190:
189:
180:
179:
166:
163:
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
3406:
3405:
3401:
3400:
3399:
3397:
3396:
3395:
3366:
3365:
3364:
3359:
3355:Steinmetz solid
3340:Integration Bee
3274:
3256:
3182:
3124:Colin Maclaurin
3100:
3068:
3062:
2934:
2928:Tensor calculus
2905:Volume integral
2841:
2816:Basic theorems
2779:Vector calculus
2773:
2654:
2621:Newton's method
2456:
2435:One-sided limit
2411:
2392:Rolle's theorem
2382:Linear function
2333:
2328:
2298:
2276:10.2307/2001096
2259:
2240:
2215:
2190:
2183:
2121:
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2016:
2015:
1911:
1905:
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1513:
1497:
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1480:
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1450:
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1437:
1433:
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1413:
1402:
1395:
1380:
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1376:
1370:
1360:
1315:
1278:
1274:
1251:
1248:
1247:
1143:
1113:
1109:
1108:
1104:
1095:
1094:
1079:
1075:
1069:
1068:
1064:
1059:
1058:
1055:
1054:
1050:
1035:
1029:
1026:
1025:
1004:and a function
969:
968:
953:
949:
943:
942:
938:
931:
930:
915:
911:
899:
898:
894:
892:
880:
874:
871:
870:
813:
753:
723:
719:
718:
714:
705:
704:
689:
685:
679:
678:
674:
669:
657:
651:
648:
647:
609:
547:
543:
538:
508:
504:
503:
499:
490:
489:
474:
470:
464:
463:
457:
453:
452:
447:
435:
429:
426:
425:
404:Euclidean space
380:, as proved by
374:
369:
315:
249:
219:
215:
214:
210:
201:
200:
185:
181:
175:
174:
170:
165:
153:
147:
144:
143:
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
3404:
3394:
3393:
3388:
3383:
3381:Measure theory
3378:
3361:
3360:
3358:
3357:
3352:
3347:
3342:
3337:
3335:Gabriel's horn
3332:
3327:
3326:
3325:
3320:
3315:
3310:
3305:
3297:
3296:
3295:
3286:
3284:
3280:
3279:
3276:
3275:
3273:
3272:
3267:
3265:List of limits
3261:
3258:
3257:
3255:
3254:
3253:
3252:
3247:
3242:
3232:
3231:
3230:
3220:
3215:
3210:
3205:
3199:
3197:
3188:
3184:
3183:
3181:
3180:
3173:
3166:
3164:Leonhard Euler
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3110:
3108:
3102:
3101:
3099:
3098:
3093:
3088:
3083:
3078:
3072:
3070:
3064:
3063:
3061:
3060:
3059:
3058:
3053:
3048:
3043:
3038:
3033:
3028:
3023:
3018:
3013:
3005:
3004:
3003:
2998:
2997:
2996:
2991:
2981:
2976:
2971:
2966:
2961:
2956:
2948:
2942:
2940:
2936:
2935:
2933:
2932:
2931:
2930:
2925:
2920:
2915:
2907:
2902:
2897:
2892:
2887:
2882:
2877:
2872:
2867:
2865:Hessian matrix
2862:
2857:
2851:
2849:
2843:
2842:
2840:
2839:
2838:
2837:
2832:
2827:
2822:
2820:Line integrals
2814:
2813:
2812:
2807:
2802:
2797:
2792:
2783:
2781:
2775:
2774:
2772:
2771:
2766:
2761:
2760:
2759:
2754:
2746:
2741:
2740:
2739:
2729:
2728:
2727:
2722:
2717:
2707:
2702:
2701:
2700:
2690:
2685:
2680:
2675:
2670:
2668:Antiderivative
2664:
2662:
2656:
2655:
2653:
2652:
2651:
2650:
2645:
2640:
2630:
2629:
2628:
2623:
2615:
2614:
2613:
2608:
2603:
2598:
2588:
2587:
2586:
2581:
2576:
2571:
2563:
2562:
2561:
2556:
2555:
2554:
2544:
2539:
2534:
2529:
2524:
2514:
2513:
2512:
2507:
2497:
2492:
2487:
2482:
2477:
2472:
2466:
2464:
2458:
2457:
2455:
2454:
2449:
2444:
2439:
2438:
2437:
2427:
2421:
2419:
2413:
2412:
2410:
2409:
2404:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2364:
2359:
2354:
2349:
2343:
2341:
2335:
2334:
2327:
2326:
2319:
2312:
2304:
2297:
2296:
2270:(2): 655–666.
2252:
2238:
2216:
2214:
2211:
2210:
2209:
2203:
2197:
2189:
2186:
2179:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2124:
2119:
2116:
2113:
2110:
2105:
2102:
2099:
2094:
2090:
2085:
2076:
2071:
2068:
2065:
2060:
2056:
2050:
2045:
2041:
2034:
2031:
2028:
2024:
1940:
1937:
1934:
1931:
1928:
1920:
1917:
1914:
1909:
1904:
1901:
1898:
1895:
1891:
1886:
1883:
1880:
1877:
1872:
1869:
1866:
1861:
1857:
1852:
1843:
1838:
1835:
1832:
1827:
1823:
1817:
1812:
1808:
1778:
1774:
1768:
1763:
1759:
1753:
1748:
1743:
1740:
1737:
1733:
1704:
1701:
1698:
1695:
1692:
1686:
1680:
1677:
1674:
1669:
1664:
1661:
1658:
1655:
1651:
1646:
1643:
1640:
1637:
1632:
1629:
1626:
1621:
1617:
1612:
1603:
1598:
1595:
1592:
1587:
1583:
1577:
1572:
1568:
1537:then, for all
1526:
1521:
1516:
1512:
1508:
1505:
1500:
1495:
1492:
1489:
1485:
1458:
1453:
1449:
1445:
1440:
1436:
1432:
1429:
1426:
1421:
1416:
1412:
1405:
1401:
1398:
1394:
1390:
1387:
1384:
1362:
1356:
1334:
1331:
1328:
1325:
1322:
1318:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1281:
1277:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1220:
1219:
1208:
1205:
1202:
1199:
1195:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1163:
1159:
1156:
1153:
1150:
1146:
1141:
1138:
1135:
1132:
1127:
1124:
1121:
1116:
1112:
1107:
1098:
1093:
1090:
1087:
1082:
1078:
1072:
1067:
1063:
1057:
1053:
1049:
1044:
1041:
1038:
1034:
994:
983:
980:
972:
967:
964:
961:
956:
952:
946:
941:
934:
929:
926:
923:
918:
914:
910:
907:
902:
897:
889:
886:
883:
879:
812:
809:
784:
781:
778:
775:
772:
769:
766:
763:
760:
756:
751:
748:
745:
742:
737:
734:
731:
726:
722:
717:
708:
703:
700:
697:
692:
688:
682:
677:
673:
666:
663:
660:
656:
621:locally finite
608:
605:
576:
573:
570:
567:
564:
561:
558:
555:
550:
546:
541:
536:
533:
530:
527:
522:
519:
516:
511:
507:
502:
493:
488:
485:
482:
477:
473:
467:
460:
456:
451:
444:
441:
438:
434:
382:Henri Lebesgue
373:
370:
368:
365:
320:) denotes the
311:
280:
277:
274:
271:
268:
265:
262:
259:
256:
252:
247:
244:
241:
238:
233:
230:
227:
222:
218:
213:
204:
199:
196:
193:
188:
184:
178:
173:
169:
162:
159:
156:
152:
104:of a suitable
85:
84:
42:external links
31:
29:
22:
15:
9:
6:
4:
3:
2:
3403:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3373:
3371:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3300:
3298:
3294:
3291:
3290:
3288:
3287:
3285:
3281:
3271:
3268:
3266:
3263:
3262:
3259:
3251:
3248:
3246:
3243:
3241:
3238:
3237:
3236:
3233:
3229:
3226:
3225:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3200:
3198:
3196:
3192:
3189:
3185:
3179:
3178:
3174:
3172:
3171:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3139:Infinitesimal
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3111:
3109:
3107:
3103:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3073:
3071:
3065:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3008:
3006:
3002:
2999:
2995:
2992:
2990:
2987:
2986:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2951:
2949:
2947:
2944:
2943:
2941:
2937:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2910:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2880:Line integral
2878:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2852:
2850:
2848:
2844:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2817:
2815:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2787:
2785:
2784:
2782:
2780:
2776:
2770:
2767:
2765:
2762:
2758:
2755:
2753:
2752:Washer method
2750:
2749:
2747:
2745:
2742:
2738:
2735:
2734:
2733:
2730:
2726:
2723:
2721:
2718:
2716:
2715:trigonometric
2713:
2712:
2711:
2708:
2706:
2703:
2699:
2696:
2695:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2665:
2663:
2661:
2657:
2649:
2646:
2644:
2641:
2639:
2636:
2635:
2634:
2631:
2627:
2624:
2622:
2619:
2618:
2616:
2612:
2609:
2607:
2604:
2602:
2599:
2597:
2594:
2593:
2592:
2589:
2585:
2584:Related rates
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2566:
2564:
2560:
2557:
2553:
2550:
2549:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2519:
2518:
2515:
2511:
2508:
2506:
2503:
2502:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2467:
2465:
2463:
2459:
2453:
2450:
2448:
2445:
2443:
2440:
2436:
2433:
2432:
2431:
2428:
2426:
2423:
2422:
2420:
2418:
2414:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2344:
2342:
2340:
2336:
2332:
2325:
2320:
2318:
2313:
2311:
2306:
2305:
2302:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2265:
2258:
2253:
2249:
2245:
2241:
2235:
2231:
2227:
2223:
2218:
2217:
2207:
2204:
2201:
2198:
2195:
2192:
2191:
2185:
2182:
2178:
2174:
2171: ∈
2170:
2166:
2147:
2141:
2138:
2132:
2126:
2114:
2108:
2100:
2092:
2088:
2083:
2066:
2058:
2054:
2043:
2039:
2032:
2026:
2013:
2009:
2005:
2001:
1998: ∈
1997:
1993:
1989:
1984:
1982:
1978:
1974:
1970:
1966:
1963: ∈
1962:
1958:
1954:
1938:
1932:
1926:
1918:
1912:
1907:
1899:
1893:
1881:
1875:
1867:
1859:
1855:
1850:
1833:
1825:
1821:
1810:
1806:
1796:
1776:
1772:
1766:
1761:
1757:
1751:
1746:
1741:
1738:
1735:
1731:
1722:
1718:
1702:
1696:
1690:
1684:
1678:
1672:
1667:
1659:
1653:
1641:
1635:
1627:
1619:
1615:
1610:
1593:
1585:
1581:
1570:
1566:
1556:
1552:
1548:
1544:
1541: ∈
1540:
1524:
1519:
1514:
1510:
1506:
1503:
1498:
1493:
1490:
1487:
1483:
1474:
1469:
1456:
1451:
1447:
1438:
1434:
1430:
1427:
1419:
1414:
1410:
1399:
1396:
1392:
1388:
1385:
1382:
1374:
1369:
1365:
1359:
1355:
1351:
1348:
1345:or, for some
1332:
1326:
1320:
1308:
1305:
1302:
1293:
1290:
1287:
1279:
1275:
1271:
1265:
1262:
1259:
1256:
1245:
1242: →
1241:
1238: :
1237:
1233:
1229:
1225:
1206:
1200:
1197:
1193:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1161:
1154:
1148:
1136:
1130:
1122:
1114:
1110:
1105:
1088:
1080:
1076:
1065:
1061:
1051:
1042:
1036:
1023:
1019:
1015:
1011:
1008: ∈
1007:
1003:
999:
995:
981:
978:
962:
954:
950:
939:
924:
916:
912:
908:
905:
895:
887:
881:
868:
865: ∈
864:
860:
857:so that, for
856:
853: ⊆
852:
848:
844:
840:
839:
838:
836:
832:
829:
825:
821:
820:Hilbert space
818:
808:
806:
803: ∈
802:
798:
779:
773:
770:
764:
758:
746:
740:
732:
724:
720:
715:
698:
690:
686:
675:
671:
664:
658:
645:
641:
638: →
637:
634: :
633:
629:
625:
624:Borel measure
622:
618:
614:
604:
602:
598:
595: ∈
594:
590:
571:
565:
562:
556:
548:
544:
531:
525:
517:
509:
505:
500:
483:
475:
471:
458:
454:
449:
442:
436:
423:
420: →
419:
416: :
415:
412:
408:
405:
402:-dimensional
401:
397:
394:
391:
387:
383:
379:
364:
362:
358:
354:
350:
346:
342:
338:
335:
331:
327:
323:
319:
314:
310:
306:
303: ∈
302:
298:
294:
275:
269:
266:
260:
254:
242:
236:
228:
220:
216:
211:
194:
186:
182:
171:
167:
160:
154:
141:
138: →
137:
134: :
133:
129:
126:
122:
119:
115:
111:
110:neighbourhood
107:
103:
100:
96:
92:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
3250:Secant cubed
3175:
3168:
3149:Isaac Newton
3119:Brook Taylor
2786:Derivatives
2757:Shell method
2485:Differential
2267:
2263:
2221:
2180:
2176:
2172:
2168:
2167:-almost all
2164:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1985:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1955:-almost all
1952:
1794:
1720:
1554:
1550:
1546:
1542:
1538:
1472:
1470:
1372:
1367:
1363:
1357:
1353:
1243:
1239:
1235:
1231:
1227:
1221:
1024:) such that
1021:
1017:
1013:
1009:
1005:
1001:
997:
866:
862:
861:-almost all
858:
854:
850:
846:
842:
830:
823:
814:
804:
800:
796:
643:
639:
635:
631:
627:
616:
610:
600:
596:
592:
588:
421:
417:
413:
406:
399:
395:
385:
375:
360:
356:
352:
348:
340:
336:
329:
325:
317:
312:
308:
304:
300:
292:
139:
135:
131:
127:
120:
113:
94:
88:
73:
64:
53:Please help
45:
3318:of surfaces
3069:and numbers
3031:Dirichlet's
3001:Telescoping
2954:Alternating
2542:L'Hôpital's
2339:Precalculus
390:dimensional
339:and centre
108:on a small
91:mathematics
59:introducing
3370:Categories
3114:Adequality
2800:Divergence
2673:Arc length
2470:Derivative
2213:References
1224:covariance
424:, one has
297:almost all
99:mean value
3313:of curves
3308:Curvature
3195:Integrals
2989:Maclaurin
2969:Geometric
2860:Geometric
2810:Laplacian
2522:linearity
2362:Factorial
2127:γ
2084:∫
2044:γ
2030:→
1916:→
1894:μ
1851:∫
1811:μ
1793:for some
1777:α
1758:σ
1752:≤
1732:σ
1685:γ
1676:→
1654:μ
1611:∫
1571:μ
1511:σ
1504:≤
1484:σ
1444:⟩
1425:⟨
1411:σ
1400:∈
1393:∑
1347:countable
1321:γ
1312:⟩
1300:⟨
1297:⟩
1285:⟨
1276:∫
1269:⟩
1254:⟨
1246:given by
1204:∞
1169:∈
1149:γ
1106:∫
1066:γ
1040:→
940:γ
909:∩
896:γ
885:→
817:separable
759:μ
716:∫
676:μ
662:→
545:λ
501:∫
455:λ
440:→
322:open ball
255:μ
212:∫
172:μ
158:→
67:June 2020
3303:Manifold
3036:Integral
2979:Infinite
2974:Harmonic
2959:Binomial
2805:Gradient
2748:Volumes
2559:Quotient
2500:Notation
2331:Calculus
2188:See also
1959:and all
1908:→
1668:→
106:function
102:integral
3240:inverse
3228:inverse
3154:Fluxion
2964:Fourier
2830:Stokes'
2825:Green's
2547:Product
2407:Tangent
2292:0951621
2284:2001096
2248:0675283
2010:;
2006:,
1975:;
1971:,
1553:;
1549:,
1020:;
1016:,
646:, then
619:is any
118:measure
116:with a
55:improve
3323:Tensor
3245:Secant
3011:Abel's
2994:Taylor
2885:Matrix
2835:Gauss'
2417:Limits
2397:Secant
2387:Radian
2290:
2282:
2246:
2236:
334:radius
125:metric
123:and a
3187:Lists
3046:Ratio
2984:Power
2720:Euler
2537:Chain
2527:Power
2402:Slope
2280:JSTOR
2260:(PDF)
615:: if
359:near
328:with
142:does
40:, or
3056:Term
3051:Root
2790:Curl
2234:ISBN
2163:for
1951:for
1187:<
1181:<
795:for
630:and
587:for
2532:Sum
2272:doi
2268:308
2226:doi
2023:lim
2014:),
1979:),
1557:),
1371:of
1234:be
1226:of
1048:inf
1033:lim
878:lim
655:lim
626:on
433:lim
398:on
324:in
151:lim
89:In
3372::
2288:MR
2286:.
2278:.
2266:.
2262:.
2244:MR
2242:.
2232:.
1375:,
982:1.
869:,
807:.
603:.
363:.
299:)
44:,
36:,
2323:e
2316:t
2309:v
2294:.
2274::
2250:.
2228::
2181:i
2177:σ
2173:H
2169:x
2165:γ
2151:)
2148:x
2145:(
2142:f
2139:=
2136:)
2133:y
2130:(
2123:d
2118:)
2115:y
2112:(
2109:f
2104:)
2101:x
2098:(
2093:r
2089:B
2075:)
2070:)
2067:x
2064:(
2059:r
2055:B
2049:(
2040:1
2033:0
2027:r
2012:R
2008:γ
2004:H
2002:(
2000:L
1996:f
1992:H
1988:γ
1981:p
1977:R
1973:γ
1969:H
1967:(
1965:L
1961:f
1957:x
1953:γ
1939:,
1936:)
1933:x
1930:(
1927:f
1919:0
1913:r
1903:)
1900:y
1897:(
1890:d
1885:)
1882:y
1879:(
1876:f
1871:)
1868:x
1865:(
1860:r
1856:B
1842:)
1837:)
1834:x
1831:(
1826:r
1822:B
1816:(
1807:1
1795:α
1773:i
1767:2
1762:i
1747:2
1742:1
1739:+
1736:i
1721:γ
1703:,
1700:)
1697:x
1694:(
1691:f
1679:0
1673:r
1663:)
1660:y
1657:(
1650:d
1645:)
1642:y
1639:(
1636:f
1631:)
1628:x
1625:(
1620:r
1616:B
1602:)
1597:)
1594:x
1591:(
1586:r
1582:B
1576:(
1567:1
1555:R
1551:γ
1547:H
1545:(
1543:L
1539:f
1525:,
1520:2
1515:i
1507:q
1499:2
1494:1
1491:+
1488:i
1473:q
1457:.
1452:i
1448:e
1439:i
1435:e
1431:,
1428:x
1420:2
1415:i
1404:N
1397:i
1389:=
1386:x
1383:S
1373:H
1368:N
1366:∈
1364:i
1361:)
1358:i
1354:e
1352:(
1333:,
1330:)
1327:z
1324:(
1317:d
1309:z
1306:,
1303:y
1294:z
1291:,
1288:x
1280:H
1272:=
1266:y
1263:,
1260:x
1257:S
1244:H
1240:H
1236:S
1232:γ
1228:γ
1207:.
1201:+
1198:=
1194:}
1190:r
1184:s
1178:0
1175:,
1172:H
1166:x
1162:|
1158:)
1155:y
1152:(
1145:d
1140:)
1137:y
1134:(
1131:f
1126:)
1123:x
1120:(
1115:s
1111:B
1097:)
1092:)
1089:x
1086:(
1081:s
1077:B
1071:(
1062:1
1052:{
1043:0
1037:r
1022:R
1018:γ
1014:H
1012:(
1010:L
1006:f
1002:H
998:γ
979:=
971:)
966:)
963:x
960:(
955:r
951:B
945:(
933:)
928:)
925:x
922:(
917:r
913:B
906:M
901:(
888:0
882:r
867:H
863:x
859:γ
855:H
851:M
847:H
843:γ
831:γ
824:H
822:(
805:R
801:x
797:μ
783:)
780:x
777:(
774:f
771:=
768:)
765:y
762:(
755:d
750:)
747:y
744:(
741:f
736:)
733:x
730:(
725:r
721:B
707:)
702:)
699:x
696:(
691:r
687:B
681:(
672:1
665:0
659:r
644:μ
640:R
636:R
632:f
628:R
617:μ
601:f
597:R
593:x
589:λ
575:)
572:x
569:(
566:f
563:=
560:)
557:y
554:(
549:n
540:d
535:)
532:y
529:(
526:f
521:)
518:x
515:(
510:r
506:B
492:)
487:)
484:x
481:(
476:r
472:B
466:(
459:n
450:1
443:0
437:r
422:R
418:R
414:f
407:R
400:n
396:λ
388:-
386:n
361:x
357:f
353:x
351:(
349:f
341:x
337:r
332:-
330:d
326:X
318:x
316:(
313:r
309:B
305:X
301:x
295:-
293:μ
279:)
276:x
273:(
270:f
267:=
264:)
261:y
258:(
251:d
246:)
243:y
240:(
237:f
232:)
229:x
226:(
221:r
217:B
203:)
198:)
195:x
192:(
187:r
183:B
177:(
168:1
161:0
155:r
140:R
136:X
132:f
128:d
121:μ
114:X
80:)
74:(
69:)
65:(
51:.
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