1600:
6483:
5079:
5067:
1773:
1566:
5869:
5252:
5240:
110:
1533:
6478:{\displaystyle {\begin{aligned}g^{(1)}(0)&=\beta _{1}f^{(1)}(1)\\g^{(2)}(0)&=\beta _{1}^{2}f^{(2)}(1)+\beta _{2}f^{(1)}(1)\\g^{(3)}(0)&=\beta _{1}^{3}f^{(3)}(1)+3\beta _{1}\beta _{2}f^{(2)}(1)+\beta _{3}f^{(1)}(1)\\g^{(4)}(0)&=\beta _{1}^{4}f^{(4)}(1)+6\beta _{1}^{2}\beta _{2}f^{(3)}(1)+(4\beta _{1}\beta _{3}+3\beta _{2}^{2})f^{(2)}(1)+\beta _{4}f^{(1)}(1)\\\end{aligned}}}
2343:
3253:
4134:
6970:
on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure
2599:
2181:
2887:
2186:
3091:
3883:
1872:
8486:
Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if the differential does not vanish on the preimage) are manifolds; this is the
5362:
5057:
continuity and its first derivative is differentiable—for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required.
2469:
6951:. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a
2059:
7294:
6974:
The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set .
3383:
2767:
8048:
7078:
3762:
4596:
3312:
2967:
1705:
7511:
4329:
4241:
8465:
717:
527:
5874:
4932:
3454:
1637:
7423:
8248:
1783:
1439:
6717:
4875:
57:
8954:
2338:{\displaystyle g'(x)={\begin{cases}-{\mathord {\cos \left({\tfrac {1}{x}}\right)}}+2x\sin \left({\tfrac {1}{x}}\right)&{\text{if }}x\neq 0,\\0&{\text{if }}x=0.\end{cases}}}
7328:
8080:
6939:(mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are
7569:
6660:
5493:
5441:
3811:
3016:
93:
8304:
7976:
5467:
5415:
4688:
1961:
8810:
3248:{\displaystyle {\frac {\partial ^{\alpha }f}{\partial x_{1}^{\alpha _{1}}\,\partial x_{2}^{\alpha _{2}}\,\cdots \,\partial x_{n}^{\alpha _{n}}}}(y_{1},y_{2},\ldots ,y_{n})}
8673:
8597:
8571:
7591:
7150:
7115:
5387:
1038:
899:
800:
4718:
1466:
1008:
973:
321:
275:
248:
6594:
6567:
6540:
6513:
2423:
8775:
7909:
7854:
5807:
2385:
8731:
8533:
8118:
7713:
7540:
4188:
1304:
7822:
7452:
2743:
1731:
1393:
1364:
650:
8699:
8647:
7378:
6743:
2710:
1766:
1163:
6908:
6875:
6848:
6812:
6773:
5861:
5834:
5702:
5675:
5644:
5611:
5582:
5532:
5201:
5172:
5143:
5114:
4482:
4403:
4376:
4161:
3838:
3699:
3652:
3501:
3043:
2669:
1493:
1331:
1248:
1221:
1190:
1133:
1098:
1071:
464:
214:
183:
8171:
7938:
5763:
5734:
4129:{\displaystyle f_{i}(x_{1},x_{2},\ldots ,x_{n})=(\pi _{i}\circ f)(x_{1},x_{2},\ldots ,x_{n})=\pi _{i}(f(x_{1},x_{2},\ldots ,x_{n})){\text{ for }}i=1,2,3,\ldots ,m}
2452:
6620:
8351:
7877:
7618:
7219:
847:
824:
740:
617:
554:
8617:
8391:
8371:
8328:
8142:
7793:
7773:
7753:
7733:
7678:
7658:
7638:
7352:
7196:
5552:
5221:
5048:
5028:
5004:
4984:
4758:
4738:
4644:
4616:
4502:
4455:
4423:
4349:
3878:
3858:
3782:
3672:
3625:
3605:
3585:
3565:
3541:
3521:
3474:
3083:
3063:
2987:
2623:
2024:
1996:
1513:
1268:
946:
926:
867:
760:
594:
574:
434:
406:
386:
362:
4964:
6940:
2680:
4763:
The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of
5257:
7022:
9140:
4514:
7227:
10029:
9220:
9069:
3317:
6787:
design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has
10024:
8938:
2594:{\displaystyle h(x)={\begin{cases}x^{4/3}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}}
9311:
7981:
7082:
Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals
2176:{\displaystyle g(x)={\begin{cases}x^{2}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}}
5704:(parametric) continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.
9335:
3711:
5836:
continuity at the point where they meet if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for
9530:
3258:
2923:
1642:
7457:
4246:
4196:
2882:{\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{1-x^{2}}}}&{\text{ if }}|x|<1,\\0&{\text{ otherwise }}\end{cases}}}
8913:
9400:
9044:
9019:
8948:
8396:
9626:
9093:
Barsky, Brian A.; DeRose, Tony D. (1989). "Geometric
Continuity of Parametric Curves: Three Equivalent Characterizations".
5053:
As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have
9679:
9207:
4884:
9136:
9963:
3388:
9173:
1908:
1606:
9728:
8307:
7383:
8176:
9711:
9320:
4824:, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve
9923:
9330:
7593:(all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any
4764:
1398:
655:
6935:
are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as
4827:
33:
9908:
9631:
9405:
9129:
8834:
6883:, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with
471:
9953:
8820:
7299:
4191:
980:
323:-function. However, it may also mean "sufficiently differentiable" for the problem under consideration.
9958:
9928:
9636:
9592:
9573:
9340:
9284:
8255:
8121:
8053:
17:
7545:
5472:
5420:
3787:
2992:
69:
10072:
9495:
9360:
8845:
8261:
5554:
being the increasing measure of smoothness. Consider the segments either side of a point on a curve:
9061:
8996:
The Beta-spline: A Local
Representation Based on Shape Parameters and Fundamental Geometric Measures
7943:
5446:
5394:
4649:
2791:
2493:
2215:
2083:
1807:
9880:
9745:
9437:
9279:
8780:
8480:
1867:{\displaystyle f(x)={\begin{cases}x&{\mbox{if }}x\geq 0,\\0&{\text{if }}x<0\end{cases}}}
338:. It is a measure of the highest order of derivative that exists and is continuous for a function.
141:
96:
8652:
8576:
8550:
7574:
7120:
7085:
6665:
5372:
1013:
874:
775:
9577:
9547:
9471:
9461:
9417:
9247:
9200:
8854: – Ratio of arc length and straight-line distance between two points on a wave-like function
8840:
7176:
7167:
theory. In contrast, sheaves of smooth functions tend not to carry much topological information.
6625:
4696:
1444:
1101:
986:
951:
299:
253:
226:
8848: – Point where a function, a curve or another mathematical object does not behave regularly
6572:
6545:
6518:
6491:
9918:
9537:
9432:
9345:
9252:
8736:
8536:
7882:
7827:
5768:
5233:
5007:
2676:
2350:
134:
8704:
8506:
8091:
7686:
7516:
4166:
1273:
9567:
9562:
7798:
7428:
7160:
6984:
6967:
2715:
1710:
1369:
1336:
622:
126:
8678:
8626:
7620:
since the smoothness requirements on the transition functions between charts ensure that if
7357:
6722:
3544:
2685:
1736:
1138:
9898:
9836:
9684:
9388:
9378:
9350:
9325:
9235:
8878:
8540:
7156:
6886:
6853:
6826:
6790:
6751:
5839:
5812:
5680:
5653:
5622:
5589:
5560:
5510:
5179:
5150:
5121:
5092:
4799:, to show that the smoothness of a curve could be measured by removing restrictions on the
4460:
4381:
4354:
4139:
3816:
3677:
3630:
3479:
3021:
2647:
2642:
2626:
2394:
1471:
1309:
1226:
1199:
1168:
1111:
1076:
1049:
442:
192:
161:
149:
8147:
7914:
5739:
5710:
2428:
8:
10036:
9718:
9596:
9581:
9510:
9269:
8934:
8829:
8544:
8468:
7164:
6599:
3086:
530:
286:
138:
10009:
8333:
7859:
7600:
7201:
7009:
can be defined globally starting from their local existence. A simple case is that of a
829:
806:
722:
599:
536:
9978:
9933:
9830:
9701:
9505:
9193:
9110:
8602:
8376:
8356:
8313:
8127:
7778:
7758:
7738:
7718:
7663:
7643:
7623:
7337:
7181:
6993:
6745:(i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).
5537:
5504:
5206:
5033:
5013:
4989:
4969:
4743:
4723:
4629:
4623:
4601:
4487:
4440:
4408:
4334:
3863:
3843:
3767:
3657:
3610:
3590:
3570:
3550:
3526:
3506:
3459:
3068:
3048:
2972:
2608:
2047:
2009:
1981:
1498:
1253:
931:
911:
852:
745:
579:
559:
419:
391:
371:
347:
9515:
4937:
9913:
9893:
9888:
9795:
9706:
9520:
9500:
9355:
9294:
9169:
9040:
9015:
8944:
8476:
7006:
6998:
6932:
6915:
6819:
904:
9114:
10051:
9845:
9800:
9723:
9694:
9552:
9485:
9480:
9475:
9465:
9257:
9240:
9102:
8905:
8488:
8479:, from one manifold to another, or down to Euclidean space where computations like
7594:
7222:
5357:{\displaystyle (1-\varepsilon ^{2})x^{2}-2px+y^{2}=0,\ p>0\ ,\varepsilon \geq 0}
4821:
1599:
4505:
9994:
9903:
9733:
9689:
9455:
8884:
7002:
6948:
4796:
2909:
118:
5071:
9860:
9785:
9755:
9653:
9646:
9586:
9557:
9427:
9422:
9383:
9034:
8251:
6944:
5078:
60:
9037:
An
Introduction to Splines for Use in Computer Graphics and Geometric Modeling
4740:
also forms a Fréchet space. One uses the same seminorms as above, except that
10066:
10046:
9870:
9865:
9850:
9840:
9790:
9767:
9641:
9601:
9542:
9490:
9289:
8866:
8085:
7011:
6936:
6911:
4768:
2762:
1768:
is differentiable. However, this function is not continuously differentiable.
1040:
976:
114:
103:
8994:
7289:{\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },}
9973:
9968:
9810:
9777:
9750:
9658:
9299:
8975:
8857:
8472:
6879:
6780:
5082:
Two Bézier curve segments attached in such a way that they are C continuous
5066:
1468:
of infinitely differentiable functions, is the intersection of the classes
979:
expansion around any point in its domain converges to the function in some
9816:
9805:
9762:
9663:
9264:
7152:
to cover the whole line, such that the sum of the functions is always 1.
4619:
3378:{\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k}
2602:
1772:
5086:
The various order of parametric continuity can be described as follows:
10041:
9999:
9825:
9738:
9185:
6952:
6784:
6776:
5646:: The curves also share a common center of curvature at the join point.
2461:
1135:
function is exactly a function whose derivative exists and is of class
983:
of the point. There exist functions that are smooth but not analytic;
335:
334:
is a classification of functions according to the properties of their
109:
9855:
9820:
9525:
9412:
9106:
8872:
8851:
5251:
1270:
to be the set of all differentiable functions whose derivative is in
1193:
365:
9039:. Morgan Kaufmann. Chapter 13. Parametric vs. Geometric Continuity.
6959:
of the real line, there exist smooth functions that are analytic on
5239:
2625:
is an example of a function that is differentiable but not locally
10019:
10014:
10004:
9395:
9216:
6955:
subset of the smooth functions. Furthermore, for every open subset
4509:
1565:
342:
7155:
From what has just been said, partitions of unity do not apply to
1523:
8043:{\displaystyle \psi \circ F\circ \phi ^{-1}:\phi (U)\to \psi (V)}
7073:{\displaystyle f(x)>0\quad {\text{ for }}\quad a<x<b.\,}
6966:
It is useful to compare the situation to that of the ubiquity of
5614:
2048:
Example: Differentiable But Not
Continuously Differentiable (not
413:
8837: – Mathematical functions which are smooth but not analytic
8494:
1395:
and there are examples to show that this containment is strict (
9611:
8869: – Integer having only small prime factors (number theory)
5677:
continuity exists if the curves can be reparameterized to have
3757:{\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}
8881: – Mathematical function defined piecewise by polynomials
2679:
are also analytic wherever they are defined, because they are
1892:
1532:
8467:
In this way smooth functions between manifolds can transport
5500:
4800:
4767:, it can sometimes be more fruitful to work instead with the
4591:{\displaystyle p_{K,m}=\sup _{x\in K}\left|f^{(m)}(x)\right|}
2748:
289:(this implies that all these derivatives are continuous).
9035:
Richard H. Bartels; John C. Beatty; Brian A. Barsky (1987).
7170:
30:"C infinity" redirects here. For the extended complex plane
9012:
Computer
Graphics and Geometric Modeling Using Beta-splines
8491:. Similarly, pushforwards along embeddings are manifolds.
6823:(with ninety degree circular arcs at the four corners) has
3307:{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}
2962:{\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} }
2875:
2587:
2331:
2169:
1860:
1700:{\displaystyle f(x)=x^{2}\sin \left({\tfrac {1}{x}}\right)}
7506:{\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} }
7019:
that takes the value 0 outside an interval and such that
6914:
are typically chosen; these curves are frequently used in
4760:
is allowed to range over all non-negative integer values.
1104:
whose derivative is continuous; such functions are called
8084:
Smooth maps between manifolds induce linear maps between
5116:: zeroth derivative is continuous (curves are continuous)
4324:{\displaystyle \pi _{i}(x_{1},x_{2},\ldots ,x_{m})=x_{i}}
4236:{\displaystyle \pi _{i}:\mathbb {R} ^{m}\to \mathbb {R} }
2915:
1223:
to be the set of all continuous functions, and declaring
4378:
if it is continuous, or equivalently, if all components
826:(So all these derivatives are continuous functions over
8825:
Pages displaying short descriptions of redirect targets
8460:{\displaystyle F^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M).}
2601:
is differentiable but its derivative is unbounded on a
9028:
8940:
4888:
2526:
2271:
2234:
2108:
1816:
1682:
8823: – Mathematical analysis of discontinuous points
8783:
8739:
8707:
8681:
8655:
8629:
8605:
8579:
8553:
8509:
8399:
8379:
8359:
8336:
8316:
8264:
8179:
8150:
8130:
8094:
8056:
7984:
7946:
7917:
7885:
7862:
7830:
7801:
7781:
7761:
7741:
7721:
7689:
7666:
7646:
7626:
7603:
7577:
7548:
7519:
7460:
7431:
7386:
7360:
7340:
7302:
7230:
7204:
7184:
7159:; their different behavior relative to existence and
7123:
7088:
7025:
6889:
6856:
6829:
6793:
6754:
6725:
6668:
6628:
6602:
6575:
6548:
6521:
6494:
5872:
5842:
5815:
5771:
5742:
5713:
5683:
5656:
5625:
5592:
5563:
5540:
5513:
5475:
5449:
5423:
5397:
5375:
5260:
5209:
5182:
5174:: zeroth, first and second derivatives are continuous
5153:
5124:
5095:
5036:
5016:
4992:
4972:
4940:
4887:
4830:
4746:
4726:
4699:
4652:
4632:
4604:
4517:
4490:
4463:
4443:
4411:
4384:
4357:
4337:
4249:
4199:
4169:
4142:
3886:
3866:
3846:
3819:
3790:
3770:
3714:
3680:
3660:
3633:
3613:
3593:
3573:
3553:
3529:
3509:
3482:
3462:
3391:
3320:
3261:
3094:
3071:
3051:
3024:
2995:
2975:
2926:
2770:
2718:
2688:
2650:
2611:
2472:
2431:
2397:
2353:
2189:
2062:
2012:
1984:
1911:
1786:
1739:
1713:
1645:
1609:
1501:
1474:
1447:
1401:
1372:
1339:
1312:
1276:
1256:
1229:
1202:
1171:
1141:
1114:
1079:
1052:
1016:
989:
954:
934:
914:
877:
855:
832:
809:
778:
748:
725:
658:
625:
602:
582:
562:
539:
474:
445:
422:
394:
374:
350:
302:
256:
229:
195:
164:
72:
36:
8862:
Pages displaying wikidata descriptions as a fallback
2681:
linear combinations of complex exponential functions
2462:
Example: Differentiable But Not
Lipschitz Continuous
9130:"Geometry and Algorithms for Computer Aided Design"
6748:While it may be obvious that a curve would require
4927:{\displaystyle \textstyle {\frac {d^{k}s}{dt^{k}}}}
4457:be an open subset of the real line. The set of all
8999:(Ph.D.). University of Utah, Salt Lake City, Utah.
8804:
8769:
8725:
8693:
8667:
8641:
8611:
8591:
8565:
8527:
8459:
8385:
8365:
8345:
8322:
8298:
8242:
8165:
8136:
8112:
8074:
8042:
7970:
7932:
7903:
7871:
7848:
7816:
7787:
7767:
7747:
7727:
7707:
7672:
7652:
7632:
7612:
7585:
7563:
7534:
7505:
7446:
7417:
7372:
7346:
7322:
7288:
7213:
7190:
7144:
7109:
7072:
6902:
6869:
6842:
6806:
6767:
6737:
6711:
6654:
6614:
6588:
6561:
6534:
6507:
6477:
5855:
5828:
5801:
5757:
5728:
5696:
5669:
5638:
5605:
5576:
5546:
5526:
5487:
5461:
5435:
5409:
5381:
5356:
5215:
5195:
5166:
5137:
5108:
5042:
5022:
4998:
4978:
4958:
4926:
4869:
4752:
4732:
4712:
4682:
4638:
4610:
4590:
4496:
4476:
4449:
4417:
4397:
4370:
4343:
4323:
4235:
4182:
4155:
4128:
3872:
3852:
3832:
3805:
3776:
3756:
3693:
3666:
3646:
3619:
3599:
3579:
3559:
3535:
3515:
3495:
3468:
3448:
3377:
3306:
3247:
3077:
3057:
3037:
3010:
2981:
2961:
2881:
2737:
2704:
2663:
2617:
2593:
2446:
2417:
2379:
2337:
2175:
2018:
1990:
1955:
1866:
1760:
1725:
1699:
1631:
1507:
1487:
1460:
1433:
1387:
1358:
1325:
1298:
1262:
1242:
1215:
1184:
1157:
1127:
1092:
1065:
1032:
1002:
967:
940:
920:
893:
861:
841:
818:
794:
754:
734:
711:
644:
611:
588:
568:
548:
521:
458:
428:
400:
380:
356:
315:
285:, that is, a function that has derivatives of all
269:
242:
208:
177:
87:
51:
8875: – Fitting an approximating function to data
5061:
4803:, with which the parameter traces out the curve.
220:th derivative that is continuous in its domain.
27:Number of derivatives of a function (mathematics)
10064:
9163:
4538:
3449:{\displaystyle (y_{1},y_{2},\ldots ,y_{n})\in U}
1073:consists of all continuous functions. The class
9088:
9086:
9009:
1632:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1043:are examples of functions with this property.
9201:
8495:Smooth functions between subsets of manifolds
7418:{\displaystyle (U,\phi )\in {\mathfrak {U}},}
7015:on the real line, that is, a smooth function
6978:
5145:: zeroth and first derivatives are continuous
9092:
9083:
9059:
8974:
8243:{\displaystyle F_{*,p}:T_{p}M\to T_{F(p)}N,}
7513:is a smooth function from a neighborhood of
7274:
7241:
4428:
2671:is analytic, and hence falls into the class
326:
6596:is constrained to be positive. In the case
5074:segments attached that is only C continuous
3567:exists and is continuous at every point of
9208:
9194:
8124:(or differential) maps tangent vectors at
6926:
9164:Guillemin, Victor; Pollack, Alan (1974).
9003:
8986:
8059:
7579:
7551:
7499:
7316:
7171:Smooth functions on and between manifolds
7069:
4857:
4229:
4215:
3793:
3744:
3729:
3168:
3164:
3138:
2998:
2955:
2941:
2632:
1874:is continuous, but not differentiable at
1625:
1617:
75:
39:
9215:
9127:
8503:for arbitrary subsets of manifolds. If
5250:
5238:
5077:
5065:
4806:
2908:is an example of a smooth function with
1893:Example: Finitely-times Differentiable (
1771:
1598:
1564:
1531:
137:is a property measured by the number of
108:
9095:IEEE Computer Graphics and Applications
8927:
7001:); these are essential in the study of
5227:
1776:A smooth function that is not analytic.
1515:varies over the non-negative integers.
1434:{\displaystyle C^{k}\subsetneq C^{k-1}}
803:if it has derivatives of all orders on
712:{\displaystyle f',f'',\dots ,f^{(k-1)}}
14:
10065:
8992:
8933:
6775:continuity to appear smooth, for good
4966:, where derivatives at the end-points
4870:{\displaystyle s:\to \mathbb {R} ^{n}}
4618:varies over an increasing sequence of
2916:Multivariate differentiability classes
2425:is not continuous at zero. Therefore,
52:{\displaystyle \mathbb {C} _{\infty }}
9189:
8903:
5584:: The curves touch at the join point.
522:{\displaystyle f',f'',\dots ,f^{(k)}}
185:is a function of smoothness at least
102:For smoothness in number theory, see
7660:in one chart it will be smooth near
6941:smooth but not analytic at any point
3255:exist and are continuous, for every
283:infinitely differentiable function
9168:. Englewood Cliffs: Prentice-Hall.
8499:There is a corresponding notion of
8306:The dual to the pushforward is the
7407:
7323:{\displaystyle f:M\to \mathbb {R} }
7233:
6983:Smooth functions with given closed
6877:continuity. The same is true for a
5707:Equivalently, two vector functions
2454:is differentiable but not of class
2183:is differentiable, with derivative
436:is said to be of differentiability
24:
8436:
8414:
7136:
7095:
4705:
3169:
3139:
3113:
3099:
1453:
1022:
960:
784:
308:
262:
235:
44:
25:
10084:
9068:. University of Toronto, Canada.
8075:{\displaystyle \mathbb {R} ^{n}.}
6921:
5613:: The curves also share a common
4484:real-valued functions defined on
3314:non-negative integers, such that
1046:To put it differently, the class
619:then it is at least in the class
9137:Technische Universität Darmstadt
7564:{\displaystyle \mathbb {R} ^{m}}
6987:are used in the construction of
6971:(their complements are meagre).
5488:{\displaystyle \varepsilon =1.2}
5436:{\displaystyle \varepsilon =0.8}
3806:{\displaystyle \mathbb {R} ^{n}}
3011:{\displaystyle \mathbb {R} ^{n}}
189:; that is, a function of class
88:{\displaystyle \mathbb {C} ^{n}}
9157:
9146:from the original on 2020-10-23
9072:from the original on 2020-11-26
9014:. Springer-Verlag, Heidelberg.
8957:from the original on 2015-10-01
8916:from the original on 2019-12-16
8299:{\displaystyle F_{*}:TM\to TN.}
7050:
7044:
4508:, with the countable family of
9248:Differentiable/Smooth manifold
9121:
9060:van de Panne, Michiel (1996).
9053:
8968:
8897:
8764:
8758:
8749:
8743:
8717:
8519:
8451:
8445:
8432:
8429:
8423:
8284:
8229:
8223:
8212:
8160:
8154:
8104:
8037:
8031:
8025:
8022:
8016:
7971:{\displaystyle F(U)\subset V,}
7956:
7950:
7927:
7921:
7898:
7886:
7843:
7831:
7699:
7529:
7523:
7495:
7492:
7486:
7399:
7387:
7312:
7270:
7244:
7139:
7124:
7104:
7089:
7035:
7029:
6850:continuity, but does not have
6779:, such as those aspired to in
6706:
6700:
6683:
6677:
6643:
6637:
6468:
6462:
6457:
6451:
6430:
6424:
6419:
6413:
6405:
6358:
6352:
6346:
6341:
6335:
6296:
6290:
6285:
6279:
6249:
6243:
6238:
6232:
6220:
6214:
6209:
6203:
6182:
6176:
6171:
6165:
6131:
6125:
6120:
6114:
6084:
6078:
6073:
6067:
6055:
6049:
6044:
6038:
6017:
6011:
6006:
6000:
5970:
5964:
5959:
5953:
5941:
5935:
5930:
5924:
5899:
5893:
5888:
5882:
5796:
5790:
5781:
5775:
5752:
5746:
5723:
5717:
5462:{\displaystyle \varepsilon =1}
5410:{\displaystyle \varepsilon =0}
5365:pencil of conic sections with
5280:
5261:
5223:-th derivatives are continuous
5062:Order of parametric continuity
4953:
4941:
4852:
4849:
4837:
4765:partial differential equations
4683:{\displaystyle m=0,1,\dots ,k}
4580:
4574:
4569:
4563:
4305:
4260:
4225:
4085:
4082:
4037:
4031:
4015:
3970:
3967:
3948:
3942:
3897:
3739:
3437:
3392:
3242:
3197:
2951:
2845:
2837:
2780:
2774:
2482:
2476:
2441:
2435:
2412:
2406:
2374:
2360:
2204:
2198:
2072:
2066:
1956:{\displaystyle f(x)=|x|^{k+1}}
1937:
1928:
1921:
1915:
1796:
1790:
1749:
1743:
1655:
1649:
1621:
1010:is thus strictly contained in
704:
692:
514:
508:
13:
1:
8890:
8805:{\displaystyle p\in U\cap X.}
8310:, which "pulls" covectors on
6910:continuity is required, then
4774:
8980:Cours de calcul différentiel
8835:Non-analytic smooth function
8668:{\displaystyle U\subseteq M}
8592:{\displaystyle Y\subseteq N}
8566:{\displaystyle X\subseteq M}
7586:{\displaystyle \mathbb {R} }
7145:{\displaystyle [d,+\infty )}
7110:{\displaystyle (-\infty ,c]}
6712:{\displaystyle f'(1)=kg'(0)}
5617:direction at the join point.
5382:{\displaystyle \varepsilon }
4934:exists and is continuous on
4331:. It is said to be of class
2900:, and hence is not of class
2893:, but it is not analytic at
1967:times differentiable at all
1033:{\displaystyle C^{\infty }.}
894:{\displaystyle C^{\omega },}
795:{\displaystyle C^{\infty },}
7:
9954:Classification of manifolds
8814:
7597:of the atlas that contains
7005:, for example to show that
6655:{\displaystyle f'(1)\neq 0}
5507:can be described as having
4713:{\displaystyle C^{\infty }}
3880:, if all of its components
3703:continuously differentiable
1518:
1461:{\displaystyle C^{\infty }}
1106:continuously differentiable
1003:{\displaystyle C^{\omega }}
968:{\displaystyle C^{\infty }}
316:{\displaystyle C^{\infty }}
270:{\displaystyle C^{\infty }}
243:{\displaystyle C^{\infty }}
10:
10089:
8256:vector bundle homomorphism
8050:is a smooth function from
6989:smooth partitions of unity
6979:Smooth partitions of unity
6589:{\displaystyle \beta _{1}}
6562:{\displaystyle \beta _{4}}
6535:{\displaystyle \beta _{3}}
6508:{\displaystyle \beta _{2}}
5231:
4820:) is a concept applied to
117:is a smooth function with
101:
66:"C^n" redirects here. For
65:
29:
10030:over commutative algebras
9987:
9946:
9879:
9776:
9672:
9619:
9610:
9446:
9369:
9308:
9228:
8993:Barsky, Brian A. (1981).
8846:Singularity (mathematics)
8770:{\displaystyle F(p)=f(p)}
8547:are subsets of manifolds
7904:{\displaystyle (V,\psi )}
7849:{\displaystyle (U,\phi )}
6947:; another example is the
5802:{\displaystyle f(1)=g(0)}
3860:, for a positive integer
3813:, is said to be of class
3764:, defined on an open set
3065:, for a positive integer
2380:{\displaystyle \cos(1/x)}
2006:times differentiable, so
1250:for any positive integer
764:infinitely differentiable
327:Differentiability classes
216:is a function that has a
9746:Riemann curvature tensor
9128:Hartmann, Erich (2003).
9010:Brian A. Barsky (1988).
8943:. Springer. p. 5 .
8726:{\displaystyle F:U\to N}
8528:{\displaystyle f:X\to Y}
8250:and on the level of the
8113:{\displaystyle F:M\to N}
7795:is smooth if, for every
7708:{\displaystyle F:M\to N}
7535:{\displaystyle \phi (p)}
6943:can be made by means of
5232:Not to be confused with
4183:{\displaystyle \pi _{i}}
1528:) But Not Differentiable
1299:{\displaystyle C^{k-1}.}
1165:In general, the classes
1102:differentiable functions
146:differentiability class)
97:Complex coordinate space
8841:Quasi-analytic function
8254:, the pushforward is a
7817:{\displaystyle p\in M,}
7447:{\displaystyle p\in U,}
7163:is one of the roots of
6927:Relation to analyticity
5247:-contact (circles,line)
4877:is said to be of class
3654:if it is continuous on
3607:is said to be of class
3018:is said to be of class
2969:defined on an open set
2889:is smooth, so of class
2738:{\displaystyle e^{-ix}}
2677:trigonometric functions
1726:{\displaystyle x\neq 0}
1388:{\displaystyle k>0,}
1359:{\displaystyle C^{k-1}}
645:{\displaystyle C^{k-1}}
332:Differentiability class
9538:Manifold with boundary
9253:Differential structure
9066:Fall 1996 Online Notes
8860: – type of scheme
8806:
8771:
8727:
8701:and a smooth function
8695:
8694:{\displaystyle x\in U}
8669:
8643:
8642:{\displaystyle x\in X}
8613:
8593:
8567:
8529:
8461:
8387:
8367:
8347:
8324:
8300:
8244:
8167:
8144:to tangent vectors at
8138:
8114:
8076:
8044:
7972:
7934:
7905:
7873:
7850:
7818:
7789:
7769:
7755:-dimensional manifold
7749:
7729:
7709:
7674:
7654:
7634:
7614:
7587:
7565:
7536:
7507:
7448:
7419:
7374:
7373:{\displaystyle p\in M}
7348:
7324:
7290:
7215:
7192:
7146:
7111:
7074:
6968:transcendental numbers
6904:
6871:
6844:
6808:
6769:
6739:
6738:{\displaystyle k>0}
6713:
6656:
6616:
6590:
6563:
6536:
6509:
6479:
5857:
5830:
5803:
5759:
5730:
5698:
5671:
5640:
5607:
5578:
5548:
5528:
5496:
5489:
5463:
5437:
5411:
5383:
5358:
5248:
5234:Geometrical continuity
5217:
5197:
5168:
5139:
5110:
5083:
5075:
5044:
5024:
5000:
4980:
4960:
4928:
4871:
4754:
4734:
4714:
4684:
4640:
4612:
4592:
4498:
4478:
4451:
4419:
4399:
4372:
4345:
4325:
4237:
4184:
4157:
4130:
3874:
3854:
3834:
3807:
3778:
3758:
3695:
3668:
3648:
3621:
3601:
3581:
3561:
3537:
3517:
3497:
3470:
3450:
3379:
3308:
3249:
3079:
3059:
3039:
3012:
2983:
2963:
2883:
2739:
2706:
2705:{\displaystyle e^{ix}}
2665:
2619:
2595:
2448:
2419:
2381:
2339:
2177:
2020:
1992:
1957:
1901:For each even integer
1868:
1777:
1769:
1762:
1761:{\displaystyle f(0)=0}
1727:
1701:
1633:
1596:
1562:
1509:
1489:
1462:
1435:
1389:
1360:
1327:
1300:
1264:
1244:
1217:
1186:
1159:
1158:{\displaystyle C^{0}.}
1129:
1094:
1067:
1034:
1004:
969:
942:
922:
895:
863:
843:
820:
796:
756:
736:
713:
646:
613:
590:
570:
550:
523:
460:
430:
408:with real values. Let
402:
382:
358:
317:
277:-function (pronounced
271:
244:
210:
179:
122:
89:
53:
9166:Differential Topology
8910:mathworld.wolfram.com
8807:
8772:
8728:
8696:
8670:
8649:there is an open set
8644:
8614:
8594:
8568:
8530:
8483:are well understood.
8462:
8388:
8368:
8348:
8330:back to covectors on
8325:
8301:
8245:
8168:
8139:
8115:
8077:
8045:
7973:
7935:
7906:
7874:
7851:
7819:
7790:
7770:
7750:
7730:
7710:
7675:
7655:
7635:
7615:
7588:
7566:
7537:
7508:
7449:
7420:
7380:there exists a chart
7375:
7349:
7325:
7291:
7216:
7193:
7161:analytic continuation
7157:holomorphic functions
7147:
7112:
7075:
6905:
6903:{\displaystyle G^{2}}
6872:
6870:{\displaystyle G^{2}}
6845:
6843:{\displaystyle G^{1}}
6809:
6807:{\displaystyle G^{2}}
6770:
6768:{\displaystyle G^{1}}
6740:
6714:
6657:
6617:
6591:
6564:
6537:
6510:
6480:
5858:
5856:{\displaystyle G^{4}}
5831:
5829:{\displaystyle G^{n}}
5804:
5760:
5731:
5699:
5697:{\displaystyle C^{n}}
5672:
5670:{\displaystyle G^{n}}
5641:
5639:{\displaystyle G^{2}}
5608:
5606:{\displaystyle G^{1}}
5579:
5577:{\displaystyle G^{0}}
5549:
5529:
5527:{\displaystyle G^{n}}
5490:
5464:
5438:
5412:
5384:
5359:
5254:
5242:
5218:
5198:
5196:{\displaystyle C^{n}}
5169:
5167:{\displaystyle C^{2}}
5140:
5138:{\displaystyle C^{1}}
5111:
5109:{\displaystyle C^{0}}
5081:
5069:
5045:
5030:and from the left at
5025:
5008:one sided derivatives
5001:
4981:
4961:
4929:
4872:
4812:Parametric continuity
4807:Parametric continuity
4795:) were introduced by
4781:parametric continuity
4755:
4735:
4715:
4685:
4641:
4613:
4593:
4499:
4479:
4477:{\displaystyle C^{k}}
4452:
4420:
4400:
4398:{\displaystyle f_{i}}
4373:
4371:{\displaystyle C^{0}}
4346:
4326:
4238:
4185:
4158:
4156:{\displaystyle C^{k}}
4131:
3875:
3855:
3835:
3833:{\displaystyle C^{k}}
3808:
3779:
3759:
3696:
3694:{\displaystyle C^{1}}
3674:. Functions of class
3669:
3649:
3647:{\displaystyle C^{0}}
3622:
3602:
3582:
3562:
3538:
3518:
3498:
3496:{\displaystyle C^{k}}
3471:
3451:
3380:
3309:
3250:
3080:
3060:
3040:
3038:{\displaystyle C^{k}}
3013:
2984:
2964:
2884:
2869: otherwise
2740:
2707:
2666:
2664:{\displaystyle e^{x}}
2620:
2596:
2449:
2420:
2418:{\displaystyle g'(x)}
2382:
2340:
2178:
2021:
1993:
1958:
1869:
1775:
1763:
1728:
1702:
1634:
1602:
1568:
1535:
1524:Example: Continuous (
1510:
1490:
1488:{\displaystyle C^{k}}
1463:
1436:
1390:
1361:
1328:
1326:{\displaystyle C^{k}}
1301:
1265:
1245:
1243:{\displaystyle C^{k}}
1218:
1216:{\displaystyle C^{0}}
1187:
1185:{\displaystyle C^{k}}
1160:
1130:
1128:{\displaystyle C^{1}}
1095:
1093:{\displaystyle C^{1}}
1068:
1066:{\displaystyle C^{0}}
1035:
1005:
970:
943:
923:
896:
864:
844:
821:
797:
757:
737:
714:
647:
614:
591:
571:
551:
524:
461:
459:{\displaystyle C^{k}}
431:
403:
383:
359:
318:
272:
245:
211:
209:{\displaystyle C^{k}}
180:
178:{\displaystyle C^{k}}
127:mathematical analysis
112:
90:
54:
9685:Covariant derivative
9236:Topological manifold
8781:
8737:
8705:
8679:
8653:
8627:
8603:
8577:
8551:
8507:
8397:
8377:
8357:
8334:
8314:
8262:
8177:
8166:{\displaystyle F(p)}
8148:
8128:
8120:, at each point the
8092:
8054:
7982:
7944:
7933:{\displaystyle F(p)}
7915:
7883:
7860:
7828:
7799:
7779:
7759:
7739:
7719:
7687:
7680:in any other chart.
7664:
7644:
7624:
7601:
7575:
7546:
7517:
7458:
7429:
7384:
7358:
7338:
7300:
7228:
7202:
7182:
7121:
7086:
7023:
6887:
6854:
6827:
6791:
6752:
6723:
6666:
6626:
6600:
6573:
6546:
6519:
6492:
5870:
5840:
5813:
5769:
5758:{\displaystyle g(t)}
5740:
5729:{\displaystyle f(t)}
5711:
5681:
5654:
5623:
5590:
5561:
5538:
5511:
5473:
5447:
5421:
5395:
5373:
5258:
5228:Geometric continuity
5207:
5180:
5151:
5122:
5093:
5034:
5014:
4990:
4970:
4938:
4885:
4828:
4789:geometric continuity
4744:
4724:
4697:
4650:
4630:
4602:
4515:
4506:Fréchet vector space
4488:
4461:
4441:
4409:
4382:
4355:
4335:
4247:
4197:
4167:
4140:
3884:
3864:
3844:
3817:
3788:
3768:
3712:
3701:are also said to be
3678:
3658:
3631:
3611:
3591:
3571:
3551:
3527:
3507:
3480:
3460:
3389:
3318:
3259:
3092:
3069:
3049:
3022:
2993:
2973:
2924:
2768:
2753:) but not Analytic (
2716:
2686:
2648:
2643:exponential function
2627:Lipschitz continuous
2609:
2470:
2447:{\displaystyle g(x)}
2429:
2395:
2351:
2187:
2060:
2010:
1982:
1909:
1881:, so it is of class
1784:
1737:
1711:
1643:
1607:
1499:
1472:
1445:
1399:
1370:
1337:
1310:
1274:
1254:
1227:
1200:
1169:
1139:
1112:
1077:
1050:
1014:
987:
952:
932:
912:
875:
853:
830:
807:
776:
746:
723:
656:
623:
600:
580:
560:
537:
472:
443:
420:
392:
372:
348:
300:
292:Generally, the term
254:
227:
223:A function of class
193:
162:
70:
34:
9719:Exterior derivative
9321:Atiyah–Singer index
9270:Riemannian manifold
9062:"Parametric Curves"
8904:Weisstein, Eric W.
6963:and nowhere else .
6622:, this reduces to
6615:{\displaystyle n=1}
6569:are arbitrary, but
6404:
6319:
6273:
6108:
5994:
5010:(from the right at
4405:are continuous, on
3193:
3163:
3137:
3087:partial derivatives
2633:Example: Analytic (
2030:, but not of class
1885:, but not of class
596:-differentiable on
468:if the derivatives
279:C-infinity function
10025:Secondary calculus
9979:Singularity theory
9934:Parallel transport
9702:De Rham cohomology
9341:Generalized Stokes
8802:
8767:
8723:
8691:
8665:
8639:
8609:
8589:
8563:
8525:
8477:differential forms
8457:
8383:
8363:
8346:{\displaystyle M,}
8343:
8320:
8296:
8240:
8163:
8134:
8110:
8072:
8040:
7968:
7930:
7901:
7872:{\displaystyle p,}
7869:
7846:
7814:
7785:
7765:
7745:
7725:
7705:
7670:
7650:
7630:
7613:{\displaystyle p,}
7610:
7583:
7561:
7532:
7503:
7444:
7415:
7370:
7344:
7320:
7286:
7214:{\displaystyle m,}
7211:
7188:
7142:
7107:
7070:
7007:Riemannian metrics
6994:partition of unity
6933:analytic functions
6900:
6867:
6840:
6804:
6765:
6735:
6709:
6652:
6612:
6586:
6559:
6532:
6505:
6475:
6473:
6390:
6305:
6259:
6094:
5980:
5853:
5826:
5799:
5755:
5726:
5694:
5667:
5636:
5603:
5574:
5544:
5524:
5497:
5485:
5459:
5433:
5407:
5379:
5354:
5249:
5213:
5193:
5164:
5135:
5106:
5084:
5076:
5040:
5020:
4996:
4976:
4956:
4924:
4923:
4867:
4750:
4730:
4710:
4680:
4636:
4608:
4588:
4552:
4494:
4474:
4447:
4415:
4395:
4368:
4341:
4321:
4233:
4180:
4153:
4126:
3870:
3850:
3830:
3803:
3774:
3754:
3691:
3664:
3644:
3617:
3597:
3577:
3557:
3545:Fréchet derivative
3533:
3513:
3493:
3466:
3446:
3375:
3304:
3245:
3172:
3142:
3116:
3075:
3055:
3035:
3008:
2979:
2959:
2879:
2874:
2735:
2702:
2661:
2615:
2591:
2586:
2535:
2444:
2415:
2377:
2335:
2330:
2280:
2243:
2173:
2168:
2117:
2016:
1988:
1963:is continuous and
1953:
1864:
1859:
1820:
1778:
1770:
1758:
1723:
1697:
1691:
1629:
1597:
1563:
1505:
1485:
1458:
1431:
1385:
1356:
1323:
1296:
1260:
1240:
1213:
1182:
1155:
1125:
1090:
1063:
1030:
1000:
965:
938:
918:
891:
859:
842:{\displaystyle U.}
839:
819:{\displaystyle U.}
816:
792:
752:
735:{\displaystyle U.}
732:
719:are continuous on
709:
642:
612:{\displaystyle U,}
609:
586:
566:
549:{\displaystyle U.}
546:
519:
456:
426:
412:be a non-negative
398:
378:
354:
313:
267:
240:
206:
175:
123:
85:
49:
10060:
10059:
9942:
9941:
9707:Differential form
9361:Whitney embedding
9295:Differential form
9046:978-1-55860-400-1
9021:978-3-642-72294-3
8982:. Paris: Hermann.
8950:978-0-387-90894-6
8906:"Smooth Function"
8612:{\displaystyle f}
8386:{\displaystyle k}
8366:{\displaystyle k}
8323:{\displaystyle N}
8137:{\displaystyle p}
7824:there is a chart
7788:{\displaystyle F}
7768:{\displaystyle N}
7748:{\displaystyle n}
7728:{\displaystyle M}
7673:{\displaystyle p}
7653:{\displaystyle p}
7633:{\displaystyle f}
7347:{\displaystyle M}
7191:{\displaystyle M}
7048:
6999:topology glossary
6916:industrial design
6820:rounded rectangle
5547:{\displaystyle n}
5534:continuity, with
5369:-contact: p fix,
5341:
5329:
5216:{\displaystyle n}
5043:{\displaystyle 1}
5023:{\displaystyle 0}
4999:{\displaystyle 1}
4979:{\displaystyle 0}
4921:
4822:parametric curves
4753:{\displaystyle m}
4733:{\displaystyle D}
4639:{\displaystyle D}
4611:{\displaystyle K}
4537:
4497:{\displaystyle D}
4450:{\displaystyle D}
4418:{\displaystyle U}
4344:{\displaystyle C}
4091:
3873:{\displaystyle k}
3853:{\displaystyle U}
3777:{\displaystyle U}
3667:{\displaystyle U}
3620:{\displaystyle C}
3600:{\displaystyle f}
3580:{\displaystyle U}
3560:{\displaystyle f}
3536:{\displaystyle k}
3516:{\displaystyle U}
3469:{\displaystyle f}
3195:
3078:{\displaystyle k}
3058:{\displaystyle U}
2982:{\displaystyle U}
2870:
2834:
2825:
2749:Example: Smooth (
2618:{\displaystyle h}
2573:
2547:
2534:
2317:
2291:
2279:
2242:
2155:
2129:
2116:
2019:{\displaystyle f}
1991:{\displaystyle f}
1846:
1819:
1690:
1508:{\displaystyle k}
1263:{\displaystyle k}
941:{\displaystyle f}
928:is smooth (i.e.,
921:{\displaystyle f}
869:is said to be of
862:{\displaystyle f}
755:{\displaystyle f}
589:{\displaystyle k}
569:{\displaystyle f}
429:{\displaystyle f}
401:{\displaystyle U}
381:{\displaystyle f}
357:{\displaystyle U}
16:(Redirected from
10080:
10073:Smooth functions
10052:Stratified space
10010:Fréchet manifold
9724:Interior product
9617:
9616:
9314:
9210:
9203:
9196:
9187:
9186:
9180:
9179:
9161:
9155:
9154:
9152:
9151:
9145:
9134:
9125:
9119:
9118:
9107:10.1109/38.41470
9090:
9081:
9080:
9078:
9077:
9057:
9051:
9050:
9032:
9026:
9025:
9007:
9001:
9000:
8990:
8984:
8983:
8972:
8966:
8965:
8963:
8962:
8935:Warner, Frank W.
8931:
8925:
8924:
8922:
8921:
8901:
8863:
8830:Hadamard's lemma
8826:
8811:
8809:
8808:
8803:
8776:
8774:
8773:
8768:
8732:
8730:
8729:
8724:
8700:
8698:
8697:
8692:
8674:
8672:
8671:
8666:
8648:
8646:
8645:
8640:
8618:
8616:
8615:
8610:
8598:
8596:
8595:
8590:
8572:
8570:
8569:
8564:
8534:
8532:
8531:
8526:
8489:preimage theorem
8466:
8464:
8463:
8458:
8444:
8443:
8422:
8421:
8409:
8408:
8392:
8390:
8389:
8384:
8372:
8370:
8369:
8364:
8352:
8350:
8349:
8344:
8329:
8327:
8326:
8321:
8305:
8303:
8302:
8297:
8274:
8273:
8249:
8247:
8246:
8241:
8233:
8232:
8208:
8207:
8195:
8194:
8172:
8170:
8169:
8164:
8143:
8141:
8140:
8135:
8119:
8117:
8116:
8111:
8081:
8079:
8078:
8073:
8068:
8067:
8062:
8049:
8047:
8046:
8041:
8009:
8008:
7977:
7975:
7974:
7969:
7939:
7937:
7936:
7931:
7910:
7908:
7907:
7902:
7878:
7876:
7875:
7870:
7855:
7853:
7852:
7847:
7823:
7821:
7820:
7815:
7794:
7792:
7791:
7786:
7774:
7772:
7771:
7766:
7754:
7752:
7751:
7746:
7734:
7732:
7731:
7726:
7714:
7712:
7711:
7706:
7679:
7677:
7676:
7671:
7659:
7657:
7656:
7651:
7639:
7637:
7636:
7631:
7619:
7617:
7616:
7611:
7592:
7590:
7589:
7584:
7582:
7570:
7568:
7567:
7562:
7560:
7559:
7554:
7541:
7539:
7538:
7533:
7512:
7510:
7509:
7504:
7502:
7479:
7478:
7453:
7451:
7450:
7445:
7424:
7422:
7421:
7416:
7411:
7410:
7379:
7377:
7376:
7371:
7353:
7351:
7350:
7345:
7329:
7327:
7326:
7321:
7319:
7295:
7293:
7292:
7287:
7282:
7281:
7269:
7268:
7256:
7255:
7237:
7236:
7220:
7218:
7217:
7212:
7197:
7195:
7194:
7189:
7151:
7149:
7148:
7143:
7116:
7114:
7113:
7108:
7079:
7077:
7076:
7071:
7049:
7046:
7003:smooth manifolds
6909:
6907:
6906:
6901:
6899:
6898:
6876:
6874:
6873:
6868:
6866:
6865:
6849:
6847:
6846:
6841:
6839:
6838:
6813:
6811:
6810:
6805:
6803:
6802:
6774:
6772:
6771:
6766:
6764:
6763:
6744:
6742:
6741:
6736:
6718:
6716:
6715:
6710:
6699:
6676:
6661:
6659:
6658:
6653:
6636:
6621:
6619:
6618:
6613:
6595:
6593:
6592:
6587:
6585:
6584:
6568:
6566:
6565:
6560:
6558:
6557:
6541:
6539:
6538:
6533:
6531:
6530:
6514:
6512:
6511:
6506:
6504:
6503:
6484:
6482:
6481:
6476:
6474:
6461:
6460:
6445:
6444:
6423:
6422:
6403:
6398:
6383:
6382:
6373:
6372:
6345:
6344:
6329:
6328:
6318:
6313:
6289:
6288:
6272:
6267:
6242:
6241:
6213:
6212:
6197:
6196:
6175:
6174:
6159:
6158:
6149:
6148:
6124:
6123:
6107:
6102:
6077:
6076:
6048:
6047:
6032:
6031:
6010:
6009:
5993:
5988:
5963:
5962:
5934:
5933:
5918:
5917:
5892:
5891:
5863:continuity are:
5862:
5860:
5859:
5854:
5852:
5851:
5835:
5833:
5832:
5827:
5825:
5824:
5808:
5806:
5805:
5800:
5764:
5762:
5761:
5756:
5735:
5733:
5732:
5727:
5703:
5701:
5700:
5695:
5693:
5692:
5676:
5674:
5673:
5668:
5666:
5665:
5645:
5643:
5642:
5637:
5635:
5634:
5612:
5610:
5609:
5604:
5602:
5601:
5583:
5581:
5580:
5575:
5573:
5572:
5553:
5551:
5550:
5545:
5533:
5531:
5530:
5525:
5523:
5522:
5494:
5492:
5491:
5486:
5468:
5466:
5465:
5460:
5442:
5440:
5439:
5434:
5416:
5414:
5413:
5408:
5388:
5386:
5385:
5380:
5363:
5361:
5360:
5355:
5339:
5327:
5317:
5316:
5292:
5291:
5279:
5278:
5222:
5220:
5219:
5214:
5202:
5200:
5199:
5194:
5192:
5191:
5173:
5171:
5170:
5165:
5163:
5162:
5144:
5142:
5141:
5136:
5134:
5133:
5115:
5113:
5112:
5107:
5105:
5104:
5049:
5047:
5046:
5041:
5029:
5027:
5026:
5021:
5006:are taken to be
5005:
5003:
5002:
4997:
4985:
4983:
4982:
4977:
4965:
4963:
4962:
4959:{\displaystyle }
4957:
4933:
4931:
4930:
4925:
4922:
4920:
4919:
4918:
4905:
4901:
4900:
4890:
4876:
4874:
4873:
4868:
4866:
4865:
4860:
4759:
4757:
4756:
4751:
4739:
4737:
4736:
4731:
4719:
4717:
4716:
4711:
4709:
4708:
4689:
4687:
4686:
4681:
4645:
4643:
4642:
4637:
4617:
4615:
4614:
4609:
4597:
4595:
4594:
4589:
4587:
4583:
4573:
4572:
4551:
4533:
4532:
4503:
4501:
4500:
4495:
4483:
4481:
4480:
4475:
4473:
4472:
4456:
4454:
4453:
4448:
4424:
4422:
4421:
4416:
4404:
4402:
4401:
4396:
4394:
4393:
4377:
4375:
4374:
4369:
4367:
4366:
4350:
4348:
4347:
4342:
4330:
4328:
4327:
4322:
4320:
4319:
4304:
4303:
4285:
4284:
4272:
4271:
4259:
4258:
4242:
4240:
4239:
4234:
4232:
4224:
4223:
4218:
4209:
4208:
4190:are the natural
4189:
4187:
4186:
4181:
4179:
4178:
4162:
4160:
4159:
4154:
4152:
4151:
4135:
4133:
4132:
4127:
4092:
4089:
4081:
4080:
4062:
4061:
4049:
4048:
4030:
4029:
4014:
4013:
3995:
3994:
3982:
3981:
3960:
3959:
3941:
3940:
3922:
3921:
3909:
3908:
3896:
3895:
3879:
3877:
3876:
3871:
3859:
3857:
3856:
3851:
3839:
3837:
3836:
3831:
3829:
3828:
3812:
3810:
3809:
3804:
3802:
3801:
3796:
3783:
3781:
3780:
3775:
3763:
3761:
3760:
3755:
3753:
3752:
3747:
3738:
3737:
3732:
3700:
3698:
3697:
3692:
3690:
3689:
3673:
3671:
3670:
3665:
3653:
3651:
3650:
3645:
3643:
3642:
3626:
3624:
3623:
3618:
3606:
3604:
3603:
3598:
3586:
3584:
3583:
3578:
3566:
3564:
3563:
3558:
3542:
3540:
3539:
3534:
3522:
3520:
3519:
3514:
3502:
3500:
3499:
3494:
3492:
3491:
3475:
3473:
3472:
3467:
3456:. Equivalently,
3455:
3453:
3452:
3447:
3436:
3435:
3417:
3416:
3404:
3403:
3384:
3382:
3381:
3376:
3368:
3367:
3349:
3348:
3336:
3335:
3313:
3311:
3310:
3305:
3303:
3302:
3284:
3283:
3271:
3270:
3254:
3252:
3251:
3246:
3241:
3240:
3222:
3221:
3209:
3208:
3196:
3194:
3192:
3191:
3190:
3180:
3162:
3161:
3160:
3150:
3136:
3135:
3134:
3124:
3111:
3107:
3106:
3096:
3084:
3082:
3081:
3076:
3064:
3062:
3061:
3056:
3044:
3042:
3041:
3036:
3034:
3033:
3017:
3015:
3014:
3009:
3007:
3006:
3001:
2988:
2986:
2985:
2980:
2968:
2966:
2965:
2960:
2958:
2950:
2949:
2944:
2907:
2899:
2897:
2888:
2886:
2885:
2880:
2878:
2877:
2871:
2868:
2848:
2840:
2835:
2832:
2828:
2827:
2826:
2824:
2823:
2822:
2803:
2744:
2742:
2741:
2736:
2734:
2733:
2711:
2709:
2708:
2703:
2701:
2700:
2670:
2668:
2667:
2662:
2660:
2659:
2624:
2622:
2621:
2616:
2600:
2598:
2597:
2592:
2590:
2589:
2574:
2571:
2548:
2545:
2541:
2540:
2536:
2527:
2513:
2512:
2508:
2453:
2451:
2450:
2445:
2424:
2422:
2421:
2416:
2405:
2390:
2386:
2384:
2383:
2378:
2370:
2344:
2342:
2341:
2336:
2334:
2333:
2318:
2315:
2292:
2289:
2285:
2281:
2272:
2250:
2249:
2248:
2244:
2235:
2197:
2182:
2180:
2179:
2174:
2172:
2171:
2156:
2153:
2130:
2127:
2123:
2122:
2118:
2109:
2095:
2094:
2043:
2042:
2038:
2025:
2023:
2022:
2017:
2005:
2003:
1997:
1995:
1994:
1989:
1977:
1975:
1970:
1966:
1962:
1960:
1959:
1954:
1952:
1951:
1940:
1931:
1904:
1880:
1878:
1873:
1871:
1870:
1865:
1863:
1862:
1847:
1844:
1821:
1817:
1767:
1765:
1764:
1759:
1732:
1730:
1729:
1724:
1706:
1704:
1703:
1698:
1696:
1692:
1683:
1670:
1669:
1638:
1636:
1635:
1630:
1628:
1620:
1594:
1592:
1587:
1585:
1581:
1577:
1573:
1561:and 0 otherwise.
1560:
1558:
1553:
1552:
1548:
1544:
1514:
1512:
1511:
1506:
1494:
1492:
1491:
1486:
1484:
1483:
1467:
1465:
1464:
1459:
1457:
1456:
1440:
1438:
1437:
1432:
1430:
1429:
1411:
1410:
1394:
1392:
1391:
1386:
1365:
1363:
1362:
1357:
1355:
1354:
1333:is contained in
1332:
1330:
1329:
1324:
1322:
1321:
1305:
1303:
1302:
1297:
1292:
1291:
1269:
1267:
1266:
1261:
1249:
1247:
1246:
1241:
1239:
1238:
1222:
1220:
1219:
1214:
1212:
1211:
1191:
1189:
1188:
1183:
1181:
1180:
1164:
1162:
1161:
1156:
1151:
1150:
1134:
1132:
1131:
1126:
1124:
1123:
1100:consists of all
1099:
1097:
1096:
1091:
1089:
1088:
1072:
1070:
1069:
1064:
1062:
1061:
1039:
1037:
1036:
1031:
1026:
1025:
1009:
1007:
1006:
1001:
999:
998:
974:
972:
971:
966:
964:
963:
948:is in the class
947:
945:
944:
939:
927:
925:
924:
919:
900:
898:
897:
892:
887:
886:
868:
866:
865:
860:
848:
846:
845:
840:
825:
823:
822:
817:
801:
799:
798:
793:
788:
787:
761:
759:
758:
753:
741:
739:
738:
733:
718:
716:
715:
710:
708:
707:
677:
666:
651:
649:
648:
643:
641:
640:
618:
616:
615:
610:
595:
593:
592:
587:
575:
573:
572:
567:
555:
553:
552:
547:
528:
526:
525:
520:
518:
517:
493:
482:
465:
463:
462:
457:
455:
454:
435:
433:
432:
427:
407:
405:
404:
399:
387:
385:
384:
379:
363:
361:
360:
355:
322:
320:
319:
314:
312:
311:
276:
274:
273:
268:
266:
265:
249:
247:
246:
241:
239:
238:
219:
215:
213:
212:
207:
205:
204:
188:
184:
182:
181:
176:
174:
173:
148:it has over its
94:
92:
91:
86:
84:
83:
78:
58:
56:
55:
50:
48:
47:
42:
21:
10088:
10087:
10083:
10082:
10081:
10079:
10078:
10077:
10063:
10062:
10061:
10056:
9995:Banach manifold
9988:Generalizations
9983:
9938:
9875:
9772:
9734:Ricci curvature
9690:Cotangent space
9668:
9606:
9448:
9442:
9401:Exponential map
9365:
9310:
9304:
9224:
9214:
9184:
9183:
9176:
9162:
9158:
9149:
9147:
9143:
9132:
9126:
9122:
9091:
9084:
9075:
9073:
9058:
9054:
9047:
9033:
9029:
9022:
9008:
9004:
8991:
8987:
8973:
8969:
8960:
8958:
8951:
8932:
8928:
8919:
8917:
8902:
8898:
8893:
8885:Sobolev mapping
8861:
8824:
8817:
8782:
8779:
8778:
8738:
8735:
8734:
8706:
8703:
8702:
8680:
8677:
8676:
8654:
8651:
8650:
8628:
8625:
8624:
8604:
8601:
8600:
8578:
8575:
8574:
8552:
8549:
8548:
8508:
8505:
8504:
8497:
8439:
8435:
8417:
8413:
8404:
8400:
8398:
8395:
8394:
8378:
8375:
8374:
8358:
8355:
8354:
8335:
8332:
8331:
8315:
8312:
8311:
8269:
8265:
8263:
8260:
8259:
8219:
8215:
8203:
8199:
8184:
8180:
8178:
8175:
8174:
8149:
8146:
8145:
8129:
8126:
8125:
8093:
8090:
8089:
8063:
8058:
8057:
8055:
8052:
8051:
8001:
7997:
7983:
7980:
7979:
7945:
7942:
7941:
7916:
7913:
7912:
7884:
7881:
7880:
7861:
7858:
7857:
7829:
7826:
7825:
7800:
7797:
7796:
7780:
7777:
7776:
7760:
7757:
7756:
7740:
7737:
7736:
7720:
7717:
7716:
7688:
7685:
7684:
7665:
7662:
7661:
7645:
7642:
7641:
7640:is smooth near
7625:
7622:
7621:
7602:
7599:
7598:
7578:
7576:
7573:
7572:
7555:
7550:
7549:
7547:
7544:
7543:
7518:
7515:
7514:
7498:
7471:
7467:
7459:
7456:
7455:
7430:
7427:
7426:
7406:
7405:
7385:
7382:
7381:
7359:
7356:
7355:
7339:
7336:
7335:
7315:
7301:
7298:
7297:
7277:
7273:
7264:
7260:
7251:
7247:
7232:
7231:
7229:
7226:
7225:
7203:
7200:
7199:
7198:, of dimension
7183:
7180:
7179:
7177:smooth manifold
7173:
7122:
7119:
7118:
7087:
7084:
7083:
7047: for
7045:
7024:
7021:
7020:
6981:
6949:Fabius function
6929:
6924:
6894:
6890:
6888:
6885:
6884:
6861:
6857:
6855:
6852:
6851:
6834:
6830:
6828:
6825:
6824:
6798:
6794:
6792:
6789:
6788:
6759:
6755:
6753:
6750:
6749:
6724:
6721:
6720:
6719:, for a scalar
6692:
6669:
6667:
6664:
6663:
6629:
6627:
6624:
6623:
6601:
6598:
6597:
6580:
6576:
6574:
6571:
6570:
6553:
6549:
6547:
6544:
6543:
6526:
6522:
6520:
6517:
6516:
6499:
6495:
6493:
6490:
6489:
6472:
6471:
6450:
6446:
6440:
6436:
6412:
6408:
6399:
6394:
6378:
6374:
6368:
6364:
6334:
6330:
6324:
6320:
6314:
6309:
6278:
6274:
6268:
6263:
6252:
6231:
6227:
6224:
6223:
6202:
6198:
6192:
6188:
6164:
6160:
6154:
6150:
6144:
6140:
6113:
6109:
6103:
6098:
6087:
6066:
6062:
6059:
6058:
6037:
6033:
6027:
6023:
5999:
5995:
5989:
5984:
5973:
5952:
5948:
5945:
5944:
5923:
5919:
5913:
5909:
5902:
5881:
5877:
5873:
5871:
5868:
5867:
5847:
5843:
5841:
5838:
5837:
5820:
5816:
5814:
5811:
5810:
5770:
5767:
5766:
5741:
5738:
5737:
5712:
5709:
5708:
5688:
5684:
5682:
5679:
5678:
5661:
5657:
5655:
5652:
5651:
5630:
5626:
5624:
5621:
5620:
5597:
5593:
5591:
5588:
5587:
5568:
5564:
5562:
5559:
5558:
5539:
5536:
5535:
5518:
5514:
5512:
5509:
5508:
5474:
5471:
5470:
5448:
5445:
5444:
5422:
5419:
5418:
5396:
5393:
5392:
5390:
5374:
5371:
5370:
5364:
5312:
5308:
5287:
5283:
5274:
5270:
5259:
5256:
5255:
5237:
5230:
5208:
5205:
5204:
5203:: 0-th through
5187:
5183:
5181:
5178:
5177:
5158:
5154:
5152:
5149:
5148:
5129:
5125:
5123:
5120:
5119:
5100:
5096:
5094:
5091:
5090:
5064:
5035:
5032:
5031:
5015:
5012:
5011:
4991:
4988:
4987:
4971:
4968:
4967:
4939:
4936:
4935:
4914:
4910:
4906:
4896:
4892:
4891:
4889:
4886:
4883:
4882:
4861:
4856:
4855:
4829:
4826:
4825:
4809:
4777:
4745:
4742:
4741:
4725:
4722:
4721:
4720:functions over
4704:
4700:
4698:
4695:
4694:
4651:
4648:
4647:
4631:
4628:
4627:
4603:
4600:
4599:
4562:
4558:
4557:
4553:
4541:
4522:
4518:
4516:
4513:
4512:
4489:
4486:
4485:
4468:
4464:
4462:
4459:
4458:
4442:
4439:
4438:
4435:
4410:
4407:
4406:
4389:
4385:
4383:
4380:
4379:
4362:
4358:
4356:
4353:
4352:
4336:
4333:
4332:
4315:
4311:
4299:
4295:
4280:
4276:
4267:
4263:
4254:
4250:
4248:
4245:
4244:
4228:
4219:
4214:
4213:
4204:
4200:
4198:
4195:
4194:
4174:
4170:
4168:
4165:
4164:
4147:
4143:
4141:
4138:
4137:
4090: for
4088:
4076:
4072:
4057:
4053:
4044:
4040:
4025:
4021:
4009:
4005:
3990:
3986:
3977:
3973:
3955:
3951:
3936:
3932:
3917:
3913:
3904:
3900:
3891:
3887:
3885:
3882:
3881:
3865:
3862:
3861:
3845:
3842:
3841:
3824:
3820:
3818:
3815:
3814:
3797:
3792:
3791:
3789:
3786:
3785:
3769:
3766:
3765:
3748:
3743:
3742:
3733:
3728:
3727:
3713:
3710:
3709:
3685:
3681:
3679:
3676:
3675:
3659:
3656:
3655:
3638:
3634:
3632:
3629:
3628:
3612:
3609:
3608:
3592:
3589:
3588:
3587:. The function
3572:
3569:
3568:
3552:
3549:
3548:
3528:
3525:
3524:
3508:
3505:
3504:
3487:
3483:
3481:
3478:
3477:
3461:
3458:
3457:
3431:
3427:
3412:
3408:
3399:
3395:
3390:
3387:
3386:
3363:
3359:
3344:
3340:
3331:
3327:
3319:
3316:
3315:
3298:
3294:
3279:
3275:
3266:
3262:
3260:
3257:
3256:
3236:
3232:
3217:
3213:
3204:
3200:
3186:
3182:
3181:
3176:
3156:
3152:
3151:
3146:
3130:
3126:
3125:
3120:
3112:
3102:
3098:
3097:
3095:
3093:
3090:
3089:
3070:
3067:
3066:
3050:
3047:
3046:
3029:
3025:
3023:
3020:
3019:
3002:
2997:
2996:
2994:
2991:
2990:
2974:
2971:
2970:
2954:
2945:
2940:
2939:
2925:
2922:
2921:
2918:
2910:compact support
2905:
2904:. The function
2895:
2894:
2873:
2872:
2867:
2865:
2859:
2858:
2844:
2836:
2831:
2829:
2818:
2814:
2807:
2802:
2798:
2794:
2787:
2786:
2769:
2766:
2765:
2759:
2723:
2719:
2717:
2714:
2713:
2693:
2689:
2687:
2684:
2683:
2655:
2651:
2649:
2646:
2645:
2639:
2610:
2607:
2606:
2585:
2584:
2570:
2568:
2562:
2561:
2544:
2542:
2525:
2521:
2520:
2504:
2500:
2496:
2489:
2488:
2471:
2468:
2467:
2464:
2430:
2427:
2426:
2398:
2396:
2393:
2392:
2388:
2366:
2352:
2349:
2348:
2329:
2328:
2314:
2312:
2306:
2305:
2288:
2286:
2270:
2266:
2233:
2229:
2222:
2221:
2211:
2210:
2190:
2188:
2185:
2184:
2167:
2166:
2152:
2150:
2144:
2143:
2126:
2124:
2107:
2103:
2102:
2090:
2086:
2079:
2078:
2061:
2058:
2057:
2054:
2040:
2036:
2035:
2011:
2008:
2007:
2001:
1999:
1983:
1980:
1979:
1973:
1972:
1968:
1964:
1941:
1936:
1935:
1927:
1910:
1907:
1906:
1905:, the function
1902:
1899:
1876:
1875:
1858:
1857:
1843:
1841:
1835:
1834:
1815:
1813:
1803:
1802:
1785:
1782:
1781:
1738:
1735:
1734:
1712:
1709:
1708:
1681:
1677:
1665:
1661:
1644:
1641:
1640:
1624:
1616:
1608:
1605:
1604:
1590:
1589:
1583:
1579:
1575:
1571:
1570:
1556:
1555:
1550:
1546:
1542:
1541:
1530:
1521:
1500:
1497:
1496:
1479:
1475:
1473:
1470:
1469:
1452:
1448:
1446:
1443:
1442:
1419:
1415:
1406:
1402:
1400:
1397:
1396:
1371:
1368:
1367:
1344:
1340:
1338:
1335:
1334:
1317:
1313:
1311:
1308:
1307:
1306:In particular,
1281:
1277:
1275:
1272:
1271:
1255:
1252:
1251:
1234:
1230:
1228:
1225:
1224:
1207:
1203:
1201:
1198:
1197:
1192:can be defined
1176:
1172:
1170:
1167:
1166:
1146:
1142:
1140:
1137:
1136:
1119:
1115:
1113:
1110:
1109:
1084:
1080:
1078:
1075:
1074:
1057:
1053:
1051:
1048:
1047:
1021:
1017:
1015:
1012:
1011:
994:
990:
988:
985:
984:
959:
955:
953:
950:
949:
933:
930:
929:
913:
910:
909:
882:
878:
876:
873:
872:
854:
851:
850:
849:) The function
831:
828:
827:
808:
805:
804:
783:
779:
777:
774:
773:
747:
744:
743:
724:
721:
720:
691:
687:
670:
659:
657:
654:
653:
630:
626:
624:
621:
620:
601:
598:
597:
581:
578:
577:
561:
558:
557:
538:
535:
534:
507:
503:
486:
475:
473:
470:
469:
450:
446:
444:
441:
440:
421:
418:
417:
416:. The function
393:
390:
389:
373:
370:
369:
368:and a function
349:
346:
345:
329:
307:
303:
301:
298:
297:
294:smooth function
261:
257:
255:
252:
251:
234:
230:
228:
225:
224:
217:
200:
196:
194:
191:
190:
186:
169:
165:
163:
160:
159:
119:compact support
107:
100:
79:
74:
73:
71:
68:
67:
64:
43:
38:
37:
35:
32:
31:
28:
23:
22:
15:
12:
11:
5:
10086:
10076:
10075:
10058:
10057:
10055:
10054:
10049:
10044:
10039:
10034:
10033:
10032:
10022:
10017:
10012:
10007:
10002:
9997:
9991:
9989:
9985:
9984:
9982:
9981:
9976:
9971:
9966:
9961:
9956:
9950:
9948:
9944:
9943:
9940:
9939:
9937:
9936:
9931:
9926:
9921:
9916:
9911:
9906:
9901:
9896:
9891:
9885:
9883:
9877:
9876:
9874:
9873:
9868:
9863:
9858:
9853:
9848:
9843:
9833:
9828:
9823:
9813:
9808:
9803:
9798:
9793:
9788:
9782:
9780:
9774:
9773:
9771:
9770:
9765:
9760:
9759:
9758:
9748:
9743:
9742:
9741:
9731:
9726:
9721:
9716:
9715:
9714:
9704:
9699:
9698:
9697:
9687:
9682:
9676:
9674:
9670:
9669:
9667:
9666:
9661:
9656:
9651:
9650:
9649:
9639:
9634:
9629:
9623:
9621:
9614:
9608:
9607:
9605:
9604:
9599:
9589:
9584:
9570:
9565:
9560:
9555:
9550:
9548:Parallelizable
9545:
9540:
9535:
9534:
9533:
9523:
9518:
9513:
9508:
9503:
9498:
9493:
9488:
9483:
9478:
9468:
9458:
9452:
9450:
9444:
9443:
9441:
9440:
9435:
9430:
9428:Lie derivative
9425:
9423:Integral curve
9420:
9415:
9410:
9409:
9408:
9398:
9393:
9392:
9391:
9384:Diffeomorphism
9381:
9375:
9373:
9367:
9366:
9364:
9363:
9358:
9353:
9348:
9343:
9338:
9333:
9328:
9323:
9317:
9315:
9306:
9305:
9303:
9302:
9297:
9292:
9287:
9282:
9277:
9272:
9267:
9262:
9261:
9260:
9255:
9245:
9244:
9243:
9232:
9230:
9229:Basic concepts
9226:
9225:
9213:
9212:
9205:
9198:
9190:
9182:
9181:
9174:
9156:
9139:. p. 55.
9120:
9082:
9052:
9045:
9027:
9020:
9002:
8985:
8967:
8949:
8926:
8895:
8894:
8892:
8889:
8888:
8887:
8882:
8876:
8870:
8864:
8855:
8849:
8843:
8838:
8832:
8827:
8816:
8813:
8801:
8798:
8795:
8792:
8789:
8786:
8766:
8763:
8760:
8757:
8754:
8751:
8748:
8745:
8742:
8722:
8719:
8716:
8713:
8710:
8690:
8687:
8684:
8664:
8661:
8658:
8638:
8635:
8632:
8619:is said to be
8608:
8599:respectively.
8588:
8585:
8582:
8562:
8559:
8556:
8524:
8521:
8518:
8515:
8512:
8496:
8493:
8456:
8453:
8450:
8447:
8442:
8438:
8434:
8431:
8428:
8425:
8420:
8416:
8412:
8407:
8403:
8382:
8362:
8342:
8339:
8319:
8295:
8292:
8289:
8286:
8283:
8280:
8277:
8272:
8268:
8252:tangent bundle
8239:
8236:
8231:
8228:
8225:
8222:
8218:
8214:
8211:
8206:
8202:
8198:
8193:
8190:
8187:
8183:
8162:
8159:
8156:
8153:
8133:
8109:
8106:
8103:
8100:
8097:
8086:tangent spaces
8071:
8066:
8061:
8039:
8036:
8033:
8030:
8027:
8024:
8021:
8018:
8015:
8012:
8007:
8004:
8000:
7996:
7993:
7990:
7987:
7967:
7964:
7961:
7958:
7955:
7952:
7949:
7929:
7926:
7923:
7920:
7900:
7897:
7894:
7891:
7888:
7868:
7865:
7845:
7842:
7839:
7836:
7833:
7813:
7810:
7807:
7804:
7784:
7764:
7744:
7724:
7715:is a map from
7704:
7701:
7698:
7695:
7692:
7669:
7649:
7629:
7609:
7606:
7581:
7558:
7553:
7531:
7528:
7525:
7522:
7501:
7497:
7494:
7491:
7488:
7485:
7482:
7477:
7474:
7470:
7466:
7463:
7443:
7440:
7437:
7434:
7414:
7409:
7404:
7401:
7398:
7395:
7392:
7389:
7369:
7366:
7363:
7343:
7318:
7314:
7311:
7308:
7305:
7285:
7280:
7276:
7272:
7267:
7263:
7259:
7254:
7250:
7246:
7243:
7240:
7235:
7210:
7207:
7187:
7172:
7169:
7141:
7138:
7135:
7132:
7129:
7126:
7106:
7103:
7100:
7097:
7094:
7091:
7068:
7065:
7062:
7059:
7056:
7053:
7043:
7040:
7037:
7034:
7031:
7028:
6980:
6977:
6945:Fourier series
6937:bump functions
6928:
6925:
6923:
6922:Other concepts
6920:
6897:
6893:
6882:
6864:
6860:
6837:
6833:
6822:
6801:
6797:
6762:
6758:
6734:
6731:
6728:
6708:
6705:
6702:
6698:
6695:
6691:
6688:
6685:
6682:
6679:
6675:
6672:
6651:
6648:
6645:
6642:
6639:
6635:
6632:
6611:
6608:
6605:
6583:
6579:
6556:
6552:
6529:
6525:
6502:
6498:
6486:
6485:
6470:
6467:
6464:
6459:
6456:
6453:
6449:
6443:
6439:
6435:
6432:
6429:
6426:
6421:
6418:
6415:
6411:
6407:
6402:
6397:
6393:
6389:
6386:
6381:
6377:
6371:
6367:
6363:
6360:
6357:
6354:
6351:
6348:
6343:
6340:
6337:
6333:
6327:
6323:
6317:
6312:
6308:
6304:
6301:
6298:
6295:
6292:
6287:
6284:
6281:
6277:
6271:
6266:
6262:
6258:
6255:
6253:
6251:
6248:
6245:
6240:
6237:
6234:
6230:
6226:
6225:
6222:
6219:
6216:
6211:
6208:
6205:
6201:
6195:
6191:
6187:
6184:
6181:
6178:
6173:
6170:
6167:
6163:
6157:
6153:
6147:
6143:
6139:
6136:
6133:
6130:
6127:
6122:
6119:
6116:
6112:
6106:
6101:
6097:
6093:
6090:
6088:
6086:
6083:
6080:
6075:
6072:
6069:
6065:
6061:
6060:
6057:
6054:
6051:
6046:
6043:
6040:
6036:
6030:
6026:
6022:
6019:
6016:
6013:
6008:
6005:
6002:
5998:
5992:
5987:
5983:
5979:
5976:
5974:
5972:
5969:
5966:
5961:
5958:
5955:
5951:
5947:
5946:
5943:
5940:
5937:
5932:
5929:
5926:
5922:
5916:
5912:
5908:
5905:
5903:
5901:
5898:
5895:
5890:
5887:
5884:
5880:
5876:
5875:
5850:
5846:
5823:
5819:
5798:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5754:
5751:
5748:
5745:
5725:
5722:
5719:
5716:
5691:
5687:
5664:
5660:
5648:
5647:
5633:
5629:
5618:
5600:
5596:
5585:
5571:
5567:
5543:
5521:
5517:
5484:
5481:
5478:
5458:
5455:
5452:
5432:
5429:
5426:
5406:
5403:
5400:
5378:
5353:
5350:
5347:
5344:
5338:
5335:
5332:
5326:
5323:
5320:
5315:
5311:
5307:
5304:
5301:
5298:
5295:
5290:
5286:
5282:
5277:
5273:
5269:
5266:
5263:
5229:
5226:
5225:
5224:
5212:
5190:
5186:
5175:
5161:
5157:
5146:
5132:
5128:
5117:
5103:
5099:
5063:
5060:
5039:
5019:
4995:
4975:
4955:
4952:
4949:
4946:
4943:
4917:
4913:
4909:
4904:
4899:
4895:
4864:
4859:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4808:
4805:
4776:
4773:
4769:Sobolev spaces
4749:
4729:
4707:
4703:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4635:
4607:
4586:
4582:
4579:
4576:
4571:
4568:
4565:
4561:
4556:
4550:
4547:
4544:
4540:
4536:
4531:
4528:
4525:
4521:
4493:
4471:
4467:
4446:
4434:
4427:
4414:
4392:
4388:
4365:
4361:
4340:
4318:
4314:
4310:
4307:
4302:
4298:
4294:
4291:
4288:
4283:
4279:
4275:
4270:
4266:
4262:
4257:
4253:
4231:
4227:
4222:
4217:
4212:
4207:
4203:
4177:
4173:
4150:
4146:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4087:
4084:
4079:
4075:
4071:
4068:
4065:
4060:
4056:
4052:
4047:
4043:
4039:
4036:
4033:
4028:
4024:
4020:
4017:
4012:
4008:
4004:
4001:
3998:
3993:
3989:
3985:
3980:
3976:
3972:
3969:
3966:
3963:
3958:
3954:
3950:
3947:
3944:
3939:
3935:
3931:
3928:
3925:
3920:
3916:
3912:
3907:
3903:
3899:
3894:
3890:
3869:
3849:
3827:
3823:
3800:
3795:
3773:
3751:
3746:
3741:
3736:
3731:
3726:
3723:
3720:
3717:
3688:
3684:
3663:
3641:
3637:
3616:
3596:
3576:
3556:
3532:
3512:
3490:
3486:
3465:
3445:
3442:
3439:
3434:
3430:
3426:
3423:
3420:
3415:
3411:
3407:
3402:
3398:
3394:
3374:
3371:
3366:
3362:
3358:
3355:
3352:
3347:
3343:
3339:
3334:
3330:
3326:
3323:
3301:
3297:
3293:
3290:
3287:
3282:
3278:
3274:
3269:
3265:
3244:
3239:
3235:
3231:
3228:
3225:
3220:
3216:
3212:
3207:
3203:
3199:
3189:
3185:
3179:
3175:
3171:
3167:
3159:
3155:
3149:
3145:
3141:
3133:
3129:
3123:
3119:
3115:
3110:
3105:
3101:
3074:
3054:
3032:
3028:
3005:
3000:
2978:
2957:
2953:
2948:
2943:
2938:
2935:
2932:
2929:
2917:
2914:
2876:
2866:
2864:
2861:
2860:
2857:
2854:
2851:
2847:
2843:
2839:
2833: if
2830:
2821:
2817:
2813:
2810:
2806:
2801:
2797:
2793:
2792:
2790:
2785:
2782:
2779:
2776:
2773:
2758:
2747:
2732:
2729:
2726:
2722:
2699:
2696:
2692:
2658:
2654:
2638:
2631:
2614:
2588:
2583:
2580:
2577:
2569:
2567:
2564:
2563:
2560:
2557:
2554:
2551:
2543:
2539:
2533:
2530:
2524:
2519:
2516:
2511:
2507:
2503:
2499:
2495:
2494:
2492:
2487:
2484:
2481:
2478:
2475:
2463:
2460:
2443:
2440:
2437:
2434:
2414:
2411:
2408:
2404:
2401:
2387:oscillates as
2376:
2373:
2369:
2365:
2362:
2359:
2356:
2332:
2327:
2324:
2321:
2313:
2311:
2308:
2307:
2304:
2301:
2298:
2295:
2287:
2284:
2278:
2275:
2269:
2265:
2262:
2259:
2256:
2253:
2247:
2241:
2238:
2232:
2228:
2225:
2220:
2217:
2216:
2214:
2209:
2206:
2203:
2200:
2196:
2193:
2170:
2165:
2162:
2159:
2151:
2149:
2146:
2145:
2142:
2139:
2136:
2133:
2125:
2121:
2115:
2112:
2106:
2101:
2098:
2093:
2089:
2085:
2084:
2082:
2077:
2074:
2071:
2068:
2065:
2053:
2046:
2015:
1987:
1950:
1947:
1944:
1939:
1934:
1930:
1926:
1923:
1920:
1917:
1914:
1898:
1891:
1861:
1856:
1853:
1850:
1842:
1840:
1837:
1836:
1833:
1830:
1827:
1824:
1814:
1812:
1809:
1808:
1806:
1801:
1798:
1795:
1792:
1789:
1757:
1754:
1751:
1748:
1745:
1742:
1722:
1719:
1716:
1695:
1689:
1686:
1680:
1676:
1673:
1668:
1664:
1660:
1657:
1654:
1651:
1648:
1627:
1623:
1619:
1615:
1612:
1529:
1522:
1520:
1517:
1504:
1482:
1478:
1455:
1451:
1428:
1425:
1422:
1418:
1414:
1409:
1405:
1384:
1381:
1378:
1375:
1353:
1350:
1347:
1343:
1320:
1316:
1295:
1290:
1287:
1284:
1280:
1259:
1237:
1233:
1210:
1206:
1179:
1175:
1154:
1149:
1145:
1122:
1118:
1087:
1083:
1060:
1056:
1041:Bump functions
1029:
1024:
1020:
997:
993:
962:
958:
937:
917:
890:
885:
881:
858:
838:
835:
815:
812:
791:
786:
782:
762:is said to be
751:
731:
728:
706:
703:
700:
697:
694:
690:
686:
683:
680:
676:
673:
669:
665:
662:
639:
636:
633:
629:
608:
605:
585:
565:
545:
542:
529:exist and are
516:
513:
510:
506:
502:
499:
496:
492:
489:
485:
481:
478:
453:
449:
425:
397:
377:
353:
328:
325:
310:
306:
264:
260:
237:
233:
203:
199:
172:
168:
155:A function of
82:
77:
61:Riemann sphere
46:
41:
26:
9:
6:
4:
3:
2:
10085:
10074:
10071:
10070:
10068:
10053:
10050:
10048:
10047:Supermanifold
10045:
10043:
10040:
10038:
10035:
10031:
10028:
10027:
10026:
10023:
10021:
10018:
10016:
10013:
10011:
10008:
10006:
10003:
10001:
9998:
9996:
9993:
9992:
9990:
9986:
9980:
9977:
9975:
9972:
9970:
9967:
9965:
9962:
9960:
9957:
9955:
9952:
9951:
9949:
9945:
9935:
9932:
9930:
9927:
9925:
9922:
9920:
9917:
9915:
9912:
9910:
9907:
9905:
9902:
9900:
9897:
9895:
9892:
9890:
9887:
9886:
9884:
9882:
9878:
9872:
9869:
9867:
9864:
9862:
9859:
9857:
9854:
9852:
9849:
9847:
9844:
9842:
9838:
9834:
9832:
9829:
9827:
9824:
9822:
9818:
9814:
9812:
9809:
9807:
9804:
9802:
9799:
9797:
9794:
9792:
9789:
9787:
9784:
9783:
9781:
9779:
9775:
9769:
9768:Wedge product
9766:
9764:
9761:
9757:
9754:
9753:
9752:
9749:
9747:
9744:
9740:
9737:
9736:
9735:
9732:
9730:
9727:
9725:
9722:
9720:
9717:
9713:
9712:Vector-valued
9710:
9709:
9708:
9705:
9703:
9700:
9696:
9693:
9692:
9691:
9688:
9686:
9683:
9681:
9678:
9677:
9675:
9671:
9665:
9662:
9660:
9657:
9655:
9652:
9648:
9645:
9644:
9643:
9642:Tangent space
9640:
9638:
9635:
9633:
9630:
9628:
9625:
9624:
9622:
9618:
9615:
9613:
9609:
9603:
9600:
9598:
9594:
9590:
9588:
9585:
9583:
9579:
9575:
9571:
9569:
9566:
9564:
9561:
9559:
9556:
9554:
9551:
9549:
9546:
9544:
9541:
9539:
9536:
9532:
9529:
9528:
9527:
9524:
9522:
9519:
9517:
9514:
9512:
9509:
9507:
9504:
9502:
9499:
9497:
9494:
9492:
9489:
9487:
9484:
9482:
9479:
9477:
9473:
9469:
9467:
9463:
9459:
9457:
9454:
9453:
9451:
9445:
9439:
9436:
9434:
9431:
9429:
9426:
9424:
9421:
9419:
9416:
9414:
9411:
9407:
9406:in Lie theory
9404:
9403:
9402:
9399:
9397:
9394:
9390:
9387:
9386:
9385:
9382:
9380:
9377:
9376:
9374:
9372:
9368:
9362:
9359:
9357:
9354:
9352:
9349:
9347:
9344:
9342:
9339:
9337:
9334:
9332:
9329:
9327:
9324:
9322:
9319:
9318:
9316:
9313:
9309:Main results
9307:
9301:
9298:
9296:
9293:
9291:
9290:Tangent space
9288:
9286:
9283:
9281:
9278:
9276:
9273:
9271:
9268:
9266:
9263:
9259:
9256:
9254:
9251:
9250:
9249:
9246:
9242:
9239:
9238:
9237:
9234:
9233:
9231:
9227:
9222:
9218:
9211:
9206:
9204:
9199:
9197:
9192:
9191:
9188:
9177:
9175:0-13-212605-2
9171:
9167:
9160:
9142:
9138:
9131:
9124:
9116:
9112:
9108:
9104:
9100:
9096:
9089:
9087:
9071:
9067:
9063:
9056:
9048:
9042:
9038:
9031:
9023:
9017:
9013:
9006:
8998:
8997:
8989:
8981:
8977:
8971:
8956:
8952:
8946:
8942:
8941:
8936:
8930:
8915:
8911:
8907:
8900:
8896:
8886:
8883:
8880:
8877:
8874:
8871:
8868:
8867:Smooth number
8865:
8859:
8858:Smooth scheme
8856:
8853:
8850:
8847:
8844:
8842:
8839:
8836:
8833:
8831:
8828:
8822:
8821:Discontinuity
8819:
8818:
8812:
8799:
8796:
8793:
8790:
8787:
8784:
8761:
8755:
8752:
8746:
8740:
8720:
8714:
8711:
8708:
8688:
8685:
8682:
8662:
8659:
8656:
8636:
8633:
8630:
8622:
8606:
8586:
8583:
8580:
8560:
8557:
8554:
8546:
8542:
8538:
8522:
8516:
8513:
8510:
8502:
8492:
8490:
8484:
8482:
8478:
8474:
8473:vector fields
8470:
8454:
8448:
8440:
8426:
8418:
8410:
8405:
8401:
8380:
8360:
8340:
8337:
8317:
8309:
8293:
8290:
8287:
8281:
8278:
8275:
8270:
8266:
8257:
8253:
8237:
8234:
8226:
8220:
8216:
8209:
8204:
8200:
8196:
8191:
8188:
8185:
8181:
8157:
8151:
8131:
8123:
8107:
8101:
8098:
8095:
8087:
8082:
8069:
8064:
8034:
8028:
8019:
8013:
8010:
8005:
8002:
7998:
7994:
7991:
7988:
7985:
7965:
7962:
7959:
7953:
7947:
7924:
7918:
7895:
7892:
7889:
7866:
7863:
7840:
7837:
7834:
7811:
7808:
7805:
7802:
7782:
7762:
7742:
7722:
7702:
7696:
7693:
7690:
7681:
7667:
7647:
7627:
7607:
7604:
7596:
7556:
7526:
7520:
7489:
7483:
7480:
7475:
7472:
7468:
7464:
7461:
7441:
7438:
7435:
7432:
7412:
7402:
7396:
7393:
7390:
7367:
7364:
7361:
7341:
7333:
7309:
7306:
7303:
7283:
7278:
7265:
7261:
7257:
7252:
7248:
7238:
7224:
7208:
7205:
7185:
7178:
7168:
7166:
7162:
7158:
7153:
7133:
7130:
7127:
7101:
7098:
7092:
7080:
7066:
7063:
7060:
7057:
7054:
7051:
7041:
7038:
7032:
7026:
7018:
7014:
7013:
7012:bump function
7008:
7004:
7000:
6996:
6995:
6990:
6986:
6976:
6972:
6969:
6964:
6962:
6958:
6954:
6950:
6946:
6942:
6938:
6934:
6919:
6917:
6913:
6912:cubic splines
6895:
6891:
6881:
6878:
6862:
6858:
6835:
6831:
6821:
6818:
6815:
6799:
6795:
6786:
6782:
6778:
6760:
6756:
6746:
6732:
6729:
6726:
6703:
6696:
6693:
6689:
6686:
6680:
6673:
6670:
6649:
6646:
6640:
6633:
6630:
6609:
6606:
6603:
6581:
6577:
6554:
6550:
6527:
6523:
6500:
6496:
6465:
6454:
6447:
6441:
6437:
6433:
6427:
6416:
6409:
6400:
6395:
6391:
6387:
6384:
6379:
6375:
6369:
6365:
6361:
6355:
6349:
6338:
6331:
6325:
6321:
6315:
6310:
6306:
6302:
6299:
6293:
6282:
6275:
6269:
6264:
6260:
6256:
6254:
6246:
6235:
6228:
6217:
6206:
6199:
6193:
6189:
6185:
6179:
6168:
6161:
6155:
6151:
6145:
6141:
6137:
6134:
6128:
6117:
6110:
6104:
6099:
6095:
6091:
6089:
6081:
6070:
6063:
6052:
6041:
6034:
6028:
6024:
6020:
6014:
6003:
5996:
5990:
5985:
5981:
5977:
5975:
5967:
5956:
5949:
5938:
5927:
5920:
5914:
5910:
5906:
5904:
5896:
5885:
5878:
5866:
5865:
5864:
5848:
5844:
5821:
5817:
5793:
5787:
5784:
5778:
5772:
5749:
5743:
5720:
5714:
5705:
5689:
5685:
5662:
5658:
5631:
5627:
5619:
5616:
5598:
5594:
5586:
5569:
5565:
5557:
5556:
5555:
5541:
5519:
5515:
5506:
5502:
5482:
5479:
5476:
5456:
5453:
5450:
5430:
5427:
5424:
5404:
5401:
5398:
5376:
5368:
5351:
5348:
5345:
5342:
5336:
5333:
5330:
5324:
5321:
5318:
5313:
5309:
5305:
5302:
5299:
5296:
5293:
5288:
5284:
5275:
5271:
5267:
5264:
5253:
5246:
5241:
5235:
5210:
5188:
5184:
5176:
5159:
5155:
5147:
5130:
5126:
5118:
5101:
5097:
5089:
5088:
5087:
5080:
5073:
5068:
5059:
5056:
5051:
5037:
5017:
5009:
4993:
4973:
4950:
4947:
4944:
4915:
4911:
4907:
4902:
4897:
4893:
4880:
4862:
4846:
4843:
4840:
4834:
4831:
4823:
4819:
4818:
4813:
4804:
4802:
4798:
4794:
4790:
4786:
4782:
4772:
4770:
4766:
4761:
4747:
4727:
4701:
4691:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4633:
4625:
4621:
4605:
4584:
4577:
4566:
4559:
4554:
4548:
4545:
4542:
4534:
4529:
4526:
4523:
4519:
4511:
4507:
4491:
4469:
4465:
4444:
4432:
4429:The space of
4426:
4412:
4390:
4386:
4363:
4359:
4338:
4316:
4312:
4308:
4300:
4296:
4292:
4289:
4286:
4281:
4277:
4273:
4268:
4264:
4255:
4251:
4220:
4210:
4205:
4201:
4193:
4175:
4171:
4148:
4144:
4136:are of class
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4077:
4073:
4069:
4066:
4063:
4058:
4054:
4050:
4045:
4041:
4034:
4026:
4022:
4018:
4010:
4006:
4002:
3999:
3996:
3991:
3987:
3983:
3978:
3974:
3964:
3961:
3956:
3952:
3945:
3937:
3933:
3929:
3926:
3923:
3918:
3914:
3910:
3905:
3901:
3892:
3888:
3867:
3847:
3825:
3821:
3798:
3771:
3749:
3734:
3724:
3721:
3718:
3715:
3706:
3704:
3686:
3682:
3661:
3639:
3635:
3614:
3594:
3574:
3554:
3546:
3530:
3510:
3488:
3484:
3463:
3443:
3440:
3432:
3428:
3424:
3421:
3418:
3413:
3409:
3405:
3400:
3396:
3372:
3369:
3364:
3360:
3356:
3353:
3350:
3345:
3341:
3337:
3332:
3328:
3324:
3321:
3299:
3295:
3291:
3288:
3285:
3280:
3276:
3272:
3267:
3263:
3237:
3233:
3229:
3226:
3223:
3218:
3214:
3210:
3205:
3201:
3187:
3183:
3177:
3173:
3165:
3157:
3153:
3147:
3143:
3131:
3127:
3121:
3117:
3108:
3103:
3088:
3072:
3052:
3030:
3026:
3003:
2976:
2946:
2936:
2933:
2930:
2927:
2913:
2911:
2903:
2892:
2862:
2855:
2852:
2849:
2841:
2819:
2815:
2811:
2808:
2804:
2799:
2795:
2788:
2783:
2777:
2771:
2764:
2763:bump function
2756:
2752:
2746:
2730:
2727:
2724:
2720:
2697:
2694:
2690:
2682:
2678:
2674:
2656:
2652:
2644:
2636:
2630:
2628:
2612:
2605:. Therefore,
2604:
2581:
2578:
2575:
2565:
2558:
2555:
2552:
2549:
2537:
2531:
2528:
2522:
2517:
2514:
2509:
2505:
2501:
2497:
2490:
2485:
2479:
2473:
2466:The function
2459:
2457:
2438:
2432:
2409:
2402:
2399:
2371:
2367:
2363:
2357:
2354:
2345:
2325:
2322:
2319:
2309:
2302:
2299:
2296:
2293:
2282:
2276:
2273:
2267:
2263:
2260:
2257:
2254:
2251:
2245:
2239:
2236:
2230:
2226:
2223:
2218:
2212:
2207:
2201:
2194:
2191:
2163:
2160:
2157:
2147:
2140:
2137:
2134:
2131:
2119:
2113:
2110:
2104:
2099:
2096:
2091:
2087:
2080:
2075:
2069:
2063:
2056:The function
2051:
2045:
2033:
2029:
2013:
1985:
1948:
1945:
1942:
1932:
1924:
1918:
1912:
1896:
1890:
1888:
1884:
1854:
1851:
1848:
1838:
1831:
1828:
1825:
1822:
1810:
1804:
1799:
1793:
1787:
1780:The function
1774:
1755:
1752:
1746:
1740:
1720:
1717:
1714:
1693:
1687:
1684:
1678:
1674:
1671:
1666:
1662:
1658:
1652:
1646:
1613:
1610:
1603:The function
1601:
1569:The function
1567:
1539:
1534:
1527:
1516:
1502:
1480:
1476:
1449:
1441:). The class
1426:
1423:
1420:
1416:
1412:
1407:
1403:
1382:
1379:
1376:
1373:
1351:
1348:
1345:
1341:
1318:
1314:
1293:
1288:
1285:
1282:
1278:
1257:
1235:
1231:
1208:
1204:
1196:by declaring
1195:
1177:
1173:
1152:
1147:
1143:
1120:
1116:
1107:
1103:
1085:
1081:
1058:
1054:
1044:
1042:
1027:
1018:
995:
991:
982:
978:
977:Taylor series
956:
935:
915:
907:
906:
901:
888:
883:
879:
856:
836:
833:
813:
810:
802:
789:
780:
769:
765:
749:
742:The function
729:
726:
701:
698:
695:
688:
684:
681:
678:
674:
671:
667:
663:
660:
637:
634:
631:
627:
606:
603:
583:
563:
543:
540:
532:
511:
504:
500:
497:
494:
490:
487:
483:
479:
476:
467:
466:
451:
447:
423:
415:
411:
395:
375:
367:
351:
344:
339:
337:
333:
324:
304:
295:
290:
288:
284:
280:
258:
231:
221:
201:
197:
170:
166:
158:
153:
151:
147:
143:
140:
136:
132:
128:
120:
116:
115:bump function
111:
105:
104:smooth number
98:
80:
62:
19:
9974:Moving frame
9969:Morse theory
9959:Gauge theory
9751:Tensor field
9680:Closed/Exact
9659:Vector field
9627:Distribution
9568:Hypercomplex
9563:Quaternionic
9370:
9300:Vector field
9274:
9258:Smooth atlas
9165:
9159:
9148:. Retrieved
9123:
9101:(6): 60–68.
9098:
9094:
9074:. Retrieved
9065:
9055:
9036:
9030:
9011:
9005:
8995:
8988:
8979:
8976:Henri Cartan
8970:
8959:. Retrieved
8939:
8929:
8918:. Retrieved
8909:
8899:
8620:
8500:
8498:
8485:
8083:
7879:and a chart
7682:
7331:
7174:
7154:
7081:
7016:
7010:
6992:
6988:
6982:
6973:
6965:
6960:
6956:
6930:
6880:rounded cube
6816:
6814:continuity.
6781:architecture
6747:
6487:
5706:
5650:In general,
5649:
5498:
5495:: hyperbola)
5469:: parabola,
5366:
5244:
5243:Curves with
5085:
5072:Bézier curve
5054:
5052:
4878:
4816:
4815:
4811:
4810:
4797:Brian Barsky
4792:
4788:
4784:
4780:
4778:
4762:
4692:
4620:compact sets
4436:
4430:
3707:
3702:
3476:is of class
3385:, and every
2919:
2901:
2890:
2760:
2754:
2750:
2672:
2640:
2634:
2465:
2455:
2346:
2055:
2049:
2031:
2027:
2026:is of class
1900:
1894:
1886:
1882:
1779:
1537:
1525:
1105:
1045:
981:neighborhood
903:
870:
771:
767:
763:
439:
437:
409:
341:Consider an
340:
331:
330:
296:refers to a
293:
291:
282:
278:
222:
156:
154:
145:
130:
124:
9919:Levi-Civita
9909:Generalized
9881:Connections
9831:Lie algebra
9763:Volume form
9664:Vector flow
9637:Pushforward
9632:Lie bracket
9531:Lie algebra
9496:G-structure
9285:Pushforward
9265:Submanifold
8623:if for all
8481:integration
8122:pushforward
7911:containing
7856:containing
7354:if for all
7296:then a map
5443:: ellipse,
4693:The set of
4243:defined by
4192:projections
3708:A function
2920:A function
2603:compact set
1978:, however,
1194:recursively
388:defined on
336:derivatives
142:derivatives
10042:Stratifold
10000:Diffeology
9796:Associated
9597:Symplectic
9582:Riemannian
9511:Hyperbolic
9438:Submersion
9346:Hopf–Rinow
9280:Submersion
9275:Smooth map
9150:2019-08-31
9076:2019-09-01
8961:2014-11-28
8920:2019-12-13
8891:References
8733:such that
8501:smooth map
8469:local data
8373:-forms to
7940:such that
7425:such that
6931:While all
6785:sports car
6777:aesthetics
5765:such that
4779:The terms
4775:Continuity
3543:-th order
1366:for every
1108:. Thus, a
975:) and its
531:continuous
139:continuous
131:smoothness
18:Smooth map
9924:Principal
9899:Ehresmann
9856:Subbundle
9846:Principal
9821:Fibration
9801:Cotangent
9673:Covectors
9526:Lie group
9506:Hermitian
9449:manifolds
9418:Immersion
9413:Foliation
9351:Noether's
9336:Frobenius
9331:De Rham's
9326:Darboux's
9217:Manifolds
8873:Smoothing
8852:Sinuosity
8794:∩
8788:∈
8718:→
8686:∈
8660:⊆
8634:∈
8584:⊆
8558:⊆
8520:→
8437:Ω
8433:→
8415:Ω
8406:∗
8285:→
8271:∗
8213:→
8186:∗
8105:→
8029:ψ
8026:→
8014:ϕ
8003:−
7999:ϕ
7995:∘
7989:∘
7986:ψ
7960:⊂
7896:ψ
7841:ϕ
7806:∈
7700:→
7521:ϕ
7496:→
7484:ϕ
7473:−
7469:ϕ
7465:∘
7436:∈
7403:∈
7397:ϕ
7365:∈
7313:→
7279:α
7266:α
7262:ϕ
7253:α
7137:∞
7096:∞
7093:−
6647:≠
6578:β
6551:β
6524:β
6497:β
6438:β
6392:β
6376:β
6366:β
6322:β
6307:β
6261:β
6190:β
6152:β
6142:β
6096:β
6025:β
5982:β
5911:β
5477:ε
5451:ε
5425:ε
5417:: circle,
5399:ε
5389:variable
5377:ε
5349:≥
5346:ε
5294:−
5272:ε
5268:−
4853:→
4706:∞
4672:…
4546:∈
4510:seminorms
4433:functions
4290:…
4252:π
4226:→
4202:π
4172:π
4118:…
4067:…
4023:π
4000:…
3962:∘
3953:π
3927:…
3740:→
3725:⊂
3441:∈
3422:…
3370:≤
3361:α
3354:⋯
3342:α
3329:α
3322:α
3296:α
3289:…
3277:α
3264:α
3227:…
3184:α
3170:∂
3166:⋯
3154:α
3140:∂
3128:α
3114:∂
3104:α
3100:∂
3085:, if all
2952:→
2937:⊂
2812:−
2800:−
2725:−
2553:≠
2518:
2358:
2297:≠
2264:
2227:
2219:−
2135:≠
2100:
1826:≥
1718:≠
1675:
1622:→
1540:function
1454:∞
1424:−
1413:⊊
1349:−
1286:−
1023:∞
996:ω
961:∞
884:ω
785:∞
699:−
682:…
635:−
498:…
366:real line
309:∞
263:∞
236:∞
45:∞
10067:Category
10020:Orbifold
10015:K-theory
10005:Diffiety
9729:Pullback
9543:Oriented
9521:Kenmotsu
9501:Hadamard
9447:Types of
9396:Geodesic
9221:Glossary
9141:Archived
9115:17893586
9070:Archived
8978:(1977).
8955:Archived
8937:(1983).
8914:Archived
8815:See also
8777:for all
8537:function
8393:-forms:
8308:pullback
7175:Given a
6697:′
6674:′
6634:′
4163:, where
2572:if
2546:if
2403:′
2347:Because
2316:if
2290:if
2195:′
2154:if
2128:if
1845:if
1818:if
1519:Examples
905:analytic
770:, or of
675:″
664:′
491:″
480:′
343:open set
281:) is an
135:function
9964:History
9947:Related
9861:Tangent
9839:)
9819:)
9786:Adjoint
9778:Bundles
9756:density
9654:Torsion
9620:Vectors
9612:Tensors
9595:)
9580:)
9576:,
9574:Pseudo−
9553:Poisson
9486:Finsler
9481:Fibered
9476:Contact
9474:)
9466:Complex
9464:)
9433:Section
8471:, like
7775:, then
7221:and an
6985:support
5615:tangent
5505:surface
3523:if the
1998:is not
414:integer
364:on the
9929:Vector
9914:Koszul
9894:Cartan
9889:Affine
9871:Vector
9866:Tensor
9851:Spinor
9841:Normal
9837:Stable
9791:Affine
9695:bundle
9647:bundle
9593:Almost
9516:Kähler
9472:Almost
9462:Almost
9456:Closed
9356:Sard's
9312:(list)
9172:
9113:
9043:
9018:
8947:
8879:Spline
8621:smooth
8541:domain
8539:whose
8088:: for
7735:to an
7332:smooth
6953:meagre
6542:, and
6488:where
5340:
5328:
4787:) and
4646:, and
4622:whose
4598:where
2675:. The
2034:where
1593:> 0
1582:sin(1/
871:class
772:class
768:smooth
652:since
438:class
287:orders
150:domain
129:, the
95:, see
59:, see
10037:Sheaf
9811:Fiber
9587:Rizza
9558:Prime
9389:Local
9379:Curve
9241:Atlas
9144:(PDF)
9133:(PDF)
9111:S2CID
8675:with
8545:range
8535:is a
7595:chart
7223:atlas
7165:sheaf
6991:(see
5809:have
5501:curve
4881:, if
4801:speed
4624:union
4504:is a
2391:→ 0,
2039:>
1971:. At
1639:with
908:, if
157:class
133:of a
9904:Form
9806:Dual
9739:flow
9602:Tame
9578:Sub−
9491:Flat
9371:Maps
9170:ISBN
9041:ISBN
9016:ISBN
8945:ISBN
8573:and
8543:and
8475:and
8353:and
7978:and
7454:and
7117:and
7061:<
7055:<
7039:>
6997:and
6783:and
6730:>
6662:and
5736:and
5334:>
5070:Two
4986:and
4437:Let
2898:= ±1
2850:<
2761:The
2712:and
2641:The
2004:+ 1)
1852:<
1733:and
1707:for
1588:for
1578:) =
1554:for
1549:) =
1536:The
1377:>
9826:Jet
9103:doi
7683:If
7571:to
7542:in
7334:on
7330:is
5503:or
5483:1.2
5431:0.8
5050:).
4626:is
4539:sup
4351:or
3840:on
3784:of
3627:or
3547:of
3503:on
3045:on
2989:of
2515:sin
2355:cos
2261:sin
2224:cos
2097:sin
1976:= 0
1879:= 0
1672:sin
1559:≥ 0
1495:as
902:or
576:is
556:If
533:on
250:or
152:.
125:In
10069::
9817:Co
9135:.
9109:.
9097:.
9085:^
9064:.
8953:.
8912:.
8908:.
8258::
8173::
6918:.
6817:A
6515:,
5499:A
4771:.
4690:.
4425:.
3705:.
2912:.
2745:.
2629:.
2458:.
2326:0.
2044:.
1889:.
766:,
113:A
9835:(
9815:(
9591:(
9572:(
9470:(
9460:(
9223:)
9219:(
9209:e
9202:t
9195:v
9178:.
9153:.
9117:.
9105::
9099:9
9079:.
9049:.
9024:.
8964:.
8923:.
8800:.
8797:X
8791:U
8785:p
8765:)
8762:p
8759:(
8756:f
8753:=
8750:)
8747:p
8744:(
8741:F
8721:N
8715:U
8712::
8709:F
8689:U
8683:x
8663:M
8657:U
8637:X
8631:x
8607:f
8587:N
8581:Y
8561:M
8555:X
8523:Y
8517:X
8514::
8511:f
8455:.
8452:)
8449:M
8446:(
8441:k
8430:)
8427:N
8424:(
8419:k
8411::
8402:F
8381:k
8361:k
8341:,
8338:M
8318:N
8294:.
8291:N
8288:T
8282:M
8279:T
8276::
8267:F
8238:,
8235:N
8230:)
8227:p
8224:(
8221:F
8217:T
8210:M
8205:p
8201:T
8197::
8192:p
8189:,
8182:F
8161:)
8158:p
8155:(
8152:F
8132:p
8108:N
8102:M
8099::
8096:F
8070:.
8065:n
8060:R
8038:)
8035:V
8032:(
8023:)
8020:U
8017:(
8011::
8006:1
7992:F
7966:,
7963:V
7957:)
7954:U
7951:(
7948:F
7928:)
7925:p
7922:(
7919:F
7899:)
7893:,
7890:V
7887:(
7867:,
7864:p
7844:)
7838:,
7835:U
7832:(
7812:,
7809:M
7803:p
7783:F
7763:N
7743:n
7723:M
7703:N
7697:M
7694::
7691:F
7668:p
7648:p
7628:f
7608:,
7605:p
7580:R
7557:m
7552:R
7530:)
7527:p
7524:(
7500:R
7493:)
7490:U
7487:(
7481::
7476:1
7462:f
7442:,
7439:U
7433:p
7413:,
7408:U
7400:)
7394:,
7391:U
7388:(
7368:M
7362:p
7342:M
7317:R
7310:M
7307::
7304:f
7284:,
7275:}
7271:)
7258:,
7249:U
7245:(
7242:{
7239:=
7234:U
7209:,
7206:m
7186:M
7140:)
7134:+
7131:,
7128:d
7125:[
7105:]
7102:c
7099:,
7090:(
7067:.
7064:b
7058:x
7052:a
7042:0
7036:)
7033:x
7030:(
7027:f
7017:f
6961:A
6957:A
6896:2
6892:G
6863:2
6859:G
6836:1
6832:G
6800:2
6796:G
6761:1
6757:G
6733:0
6727:k
6707:)
6704:0
6701:(
6694:g
6690:k
6687:=
6684:)
6681:1
6678:(
6671:f
6650:0
6644:)
6641:1
6638:(
6631:f
6610:1
6607:=
6604:n
6582:1
6555:4
6528:3
6501:2
6469:)
6466:1
6463:(
6458:)
6455:1
6452:(
6448:f
6442:4
6434:+
6431:)
6428:1
6425:(
6420:)
6417:2
6414:(
6410:f
6406:)
6401:2
6396:2
6388:3
6385:+
6380:3
6370:1
6362:4
6359:(
6356:+
6353:)
6350:1
6347:(
6342:)
6339:3
6336:(
6332:f
6326:2
6316:2
6311:1
6303:6
6300:+
6297:)
6294:1
6291:(
6286:)
6283:4
6280:(
6276:f
6270:4
6265:1
6257:=
6250:)
6247:0
6244:(
6239:)
6236:4
6233:(
6229:g
6221:)
6218:1
6215:(
6210:)
6207:1
6204:(
6200:f
6194:3
6186:+
6183:)
6180:1
6177:(
6172:)
6169:2
6166:(
6162:f
6156:2
6146:1
6138:3
6135:+
6132:)
6129:1
6126:(
6121:)
6118:3
6115:(
6111:f
6105:3
6100:1
6092:=
6085:)
6082:0
6079:(
6074:)
6071:3
6068:(
6064:g
6056:)
6053:1
6050:(
6045:)
6042:1
6039:(
6035:f
6029:2
6021:+
6018:)
6015:1
6012:(
6007:)
6004:2
6001:(
5997:f
5991:2
5986:1
5978:=
5971:)
5968:0
5965:(
5960:)
5957:2
5954:(
5950:g
5942:)
5939:1
5936:(
5931:)
5928:1
5925:(
5921:f
5915:1
5907:=
5900:)
5897:0
5894:(
5889:)
5886:1
5883:(
5879:g
5849:4
5845:G
5822:n
5818:G
5797:)
5794:0
5791:(
5788:g
5785:=
5782:)
5779:1
5776:(
5773:f
5753:)
5750:t
5747:(
5744:g
5724:)
5721:t
5718:(
5715:f
5690:n
5686:C
5663:n
5659:G
5632:2
5628:G
5599:1
5595:G
5570:0
5566:G
5542:n
5520:n
5516:G
5480:=
5457:1
5454:=
5428:=
5405:0
5402:=
5391:(
5367:G
5352:0
5343:,
5337:0
5331:p
5325:,
5322:0
5319:=
5314:2
5310:y
5306:+
5303:x
5300:p
5297:2
5289:2
5285:x
5281:)
5276:2
5265:1
5262:(
5245:G
5236:.
5211:n
5189:n
5185:C
5160:2
5156:C
5131:1
5127:C
5102:0
5098:C
5055:C
5038:1
5018:0
4994:1
4974:0
4954:]
4951:1
4948:,
4945:0
4942:[
4916:k
4912:t
4908:d
4903:s
4898:k
4894:d
4879:C
4863:n
4858:R
4850:]
4847:1
4844:,
4841:0
4838:[
4835::
4832:s
4817:C
4814:(
4793:G
4791:(
4785:C
4783:(
4748:m
4728:D
4702:C
4678:k
4675:,
4669:,
4666:1
4663:,
4660:0
4657:=
4654:m
4634:D
4606:K
4585:|
4581:)
4578:x
4575:(
4570:)
4567:m
4564:(
4560:f
4555:|
4549:K
4543:x
4535:=
4530:m
4527:,
4524:K
4520:p
4492:D
4470:k
4466:C
4445:D
4431:C
4413:U
4391:i
4387:f
4364:0
4360:C
4339:C
4317:i
4313:x
4309:=
4306:)
4301:m
4297:x
4293:,
4287:,
4282:2
4278:x
4274:,
4269:1
4265:x
4261:(
4256:i
4230:R
4221:m
4216:R
4211::
4206:i
4176:i
4149:k
4145:C
4124:m
4121:,
4115:,
4112:3
4109:,
4106:2
4103:,
4100:1
4097:=
4094:i
4086:)
4083:)
4078:n
4074:x
4070:,
4064:,
4059:2
4055:x
4051:,
4046:1
4042:x
4038:(
4035:f
4032:(
4027:i
4019:=
4016:)
4011:n
4007:x
4003:,
3997:,
3992:2
3988:x
3984:,
3979:1
3975:x
3971:(
3968:)
3965:f
3957:i
3949:(
3946:=
3943:)
3938:n
3934:x
3930:,
3924:,
3919:2
3915:x
3911:,
3906:1
3902:x
3898:(
3893:i
3889:f
3868:k
3848:U
3826:k
3822:C
3799:n
3794:R
3772:U
3750:m
3745:R
3735:n
3730:R
3722:U
3719::
3716:f
3687:1
3683:C
3662:U
3640:0
3636:C
3615:C
3595:f
3575:U
3555:f
3531:k
3511:U
3489:k
3485:C
3464:f
3444:U
3438:)
3433:n
3429:y
3425:,
3419:,
3414:2
3410:y
3406:,
3401:1
3397:y
3393:(
3373:k
3365:n
3357:+
3351:+
3346:2
3338:+
3333:1
3325:=
3300:n
3292:,
3286:,
3281:2
3273:,
3268:1
3243:)
3238:n
3234:y
3230:,
3224:,
3219:2
3215:y
3211:,
3206:1
3202:y
3198:(
3188:n
3178:n
3174:x
3158:2
3148:2
3144:x
3132:1
3122:1
3118:x
3109:f
3073:k
3053:U
3031:k
3027:C
3004:n
2999:R
2977:U
2956:R
2947:n
2942:R
2934:U
2931::
2928:f
2906:f
2902:C
2896:x
2891:C
2863:0
2856:,
2853:1
2846:|
2842:x
2838:|
2820:2
2816:x
2809:1
2805:1
2796:e
2789:{
2784:=
2781:)
2778:x
2775:(
2772:f
2757:)
2755:C
2751:C
2731:x
2728:i
2721:e
2698:x
2695:i
2691:e
2673:C
2657:x
2653:e
2637:)
2635:C
2613:h
2582:0
2579:=
2576:x
2566:0
2559:,
2556:0
2550:x
2538:)
2532:x
2529:1
2523:(
2510:3
2506:/
2502:4
2498:x
2491:{
2486:=
2483:)
2480:x
2477:(
2474:h
2456:C
2442:)
2439:x
2436:(
2433:g
2413:)
2410:x
2407:(
2400:g
2389:x
2375:)
2372:x
2368:/
2364:1
2361:(
2323:=
2320:x
2310:0
2303:,
2300:0
2294:x
2283:)
2277:x
2274:1
2268:(
2258:x
2255:2
2252:+
2246:)
2240:x
2237:1
2231:(
2213:{
2208:=
2205:)
2202:x
2199:(
2192:g
2164:0
2161:=
2158:x
2148:0
2141:,
2138:0
2132:x
2120:)
2114:x
2111:1
2105:(
2092:2
2088:x
2081:{
2076:=
2073:)
2070:x
2067:(
2064:g
2052:)
2050:C
2041:k
2037:j
2032:C
2028:C
2014:f
2002:k
2000:(
1986:f
1974:x
1969:x
1965:k
1949:1
1946:+
1943:k
1938:|
1933:x
1929:|
1925:=
1922:)
1919:x
1916:(
1913:f
1903:k
1897:)
1895:C
1887:C
1883:C
1877:x
1855:0
1849:x
1839:0
1832:,
1829:0
1823:x
1811:x
1805:{
1800:=
1797:)
1794:x
1791:(
1788:f
1756:0
1753:=
1750:)
1747:0
1744:(
1741:f
1721:0
1715:x
1694:)
1688:x
1685:1
1679:(
1667:2
1663:x
1659:=
1656:)
1653:x
1650:(
1647:f
1626:R
1618:R
1614::
1611:f
1595:.
1591:x
1586:)
1584:x
1580:x
1576:x
1574:(
1572:g
1557:x
1551:x
1547:x
1545:(
1543:f
1538:C
1526:C
1503:k
1481:k
1477:C
1450:C
1427:1
1421:k
1417:C
1408:k
1404:C
1383:,
1380:0
1374:k
1352:1
1346:k
1342:C
1319:k
1315:C
1294:.
1289:1
1283:k
1279:C
1258:k
1236:k
1232:C
1209:0
1205:C
1178:k
1174:C
1153:.
1148:0
1144:C
1121:1
1117:C
1086:1
1082:C
1059:0
1055:C
1028:.
1019:C
992:C
957:C
936:f
916:f
889:,
880:C
857:f
837:.
834:U
814:.
811:U
790:,
781:C
750:f
730:.
727:U
705:)
702:1
696:k
693:(
689:f
685:,
679:,
672:f
668:,
661:f
638:1
632:k
628:C
607:,
604:U
584:k
564:f
544:.
541:U
515:)
512:k
509:(
505:f
501:,
495:,
488:f
484:,
477:f
452:k
448:C
424:f
410:k
396:U
376:f
352:U
305:C
259:C
232:C
218:k
202:k
198:C
187:k
171:k
167:C
144:(
121:.
106:.
99:.
81:n
76:C
63:.
40:C
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
.svg)
.svg)
