3841:
3292:
3836:{\displaystyle {\begin{aligned}\int _{x_{1}}^{x_{2}}\left.{\frac {dL}{d\varepsilon }}\right|_{\varepsilon =0}dx&=\int _{x_{1}}^{x_{2}}\left({\frac {\partial L}{\partial f}}\eta +{\frac {\partial L}{\partial f'}}\eta '\right)\,dx\\&=\int _{x_{1}}^{x_{2}}{\frac {\partial L}{\partial f}}\eta \,dx+\left.{\frac {\partial L}{\partial f'}}\eta \right|_{x_{1}}^{x_{2}}-\int _{x_{1}}^{x_{2}}\eta {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\,dx\\&=\int _{x_{1}}^{x_{2}}\left({\frac {\partial L}{\partial f}}\eta -\eta {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\,dx\\\end{aligned}}}
2765:
3085:
8441:
6916:
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in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The
Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions.
1630:
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the
16613:
as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding
1079:
requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing
6031:
13621:
This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in
Courant and
7547:
There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a
Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a
4185:
7062:
Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable
Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the
10432:
This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in
Courant and Hilbert (1953).
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7894:
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1603:
Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the
12923:
8273:
1120:
was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old
Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the
8278:
17475:
8827:
must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the
2957:
16974:
5009:
4277:
6729:
6514:, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.
5833:
19169:"Euler waited until Lagrange had published on the subject in 1762 ... before he committed his lecture ... to print, so as not to rob Lagrange of his glory. Indeed, it was only Lagrange's method that Euler called Calculus of Variations."
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19185:: regarding Dynamic Programming, "The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community."
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as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.
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Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.
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small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small changes in the function itself, which are called variations.
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can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes
1038:
between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as
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Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The
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2760:{\displaystyle \Phi '(0)\equiv \left.{\frac {d\Phi }{d\varepsilon }}\right|_{\varepsilon =0}=\int _{x_{1}}^{x_{2}}\left.{\frac {dL}{d\varepsilon }}\right|_{\varepsilon =0}dx=0\,.}
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at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of
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for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.
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Analogy with Fermat's principle suggests that solutions of
Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of
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8436:{\displaystyle \iint _{D}\nabla \cdot (v\nabla u)\,dx\,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u\,dx\,dy=\int _{C}v{\frac {\partial u}{\partial n}}\,ds,}
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3080:{\displaystyle {\frac {dL}{d\varepsilon }}={\frac {\partial L}{\partial y}}{\frac {dy}{d\varepsilon }}+{\frac {\partial L}{\partial y'}}{\frac {dy'}{d\varepsilon }}}
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is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable.
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Connected with the
Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.
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One may note the similarity to the sufficient condition for a minimum of a function, where the first derivative is zero and the second derivative is positive.
13696:
1645:
6911:{\displaystyle {\frac {\partial f}{\partial y}}-{\frac {d}{dx}}\left({\frac {\partial f}{\partial y'}}\right)+\dots +(-1)^{n}{\frac {d^{n}}{dx^{n}}}\left=0.}
5242:
1191:. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The
6026:{\displaystyle f(x)=mx+b\qquad {\text{with}}\ \ m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\quad {\text{and}}\quad b={\frac {x_{2}y_{1}-x_{1}y_{2}}{x_{2}-x_{1}}}}
5155:
21193:
16765:(or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral
13525:
13349:
6577:
16492:
These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification
6972:
21616:
9073:
14816:
may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form
10019:
9711:
4600:
4180:{\displaystyle \int _{x_{1}}^{x_{2}}\eta (x)\left({\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\,dx=0\,.}
15775:
13140:
9582:
17002:
16768:
11321:
15969:
In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of
1034:
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a
8645:
7167:
6301:
21186:
17228:
10731:
21151:
20506:
7904:
It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by
13982:
16249:
10362:
17894:
8839:
5353:
232:
12929:
for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.
7074:
4190:
15992:
13130:{\displaystyle Q=\iiint _{D}p(X)\nabla \varphi \cdot \nabla \varphi +q(X)\varphi ^{2}\,dx\,dy\,dz+\iint _{B}\sigma (S)\varphi ^{2}\,dS,}
6945:
requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a
10469:
6925:
The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral
21820:
20108:
17057:
9058:
Eventually it was shown that
Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for
22071:
21095:
21074:
21005:
19307:
The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–185 of
Courant & Hilbert (1953).
17575:
16332:
5464:
4862:
loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.
19038:
16390:
5611:
20931:. Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005,
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10640:
21720:
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19063:
20998:
18534:
10359:
can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless
9912:
22297:
17563:
17496:
16168:
11231:; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by
9059:
1093:
31:
13447:
11109:
11017:
2438:
21812:
21497:
21088:
21024:
21016:
20984:
20936:
20718:
Ball & Mizel (1985). "One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation".
20683:
20395:
14819:
8750:
7889:{\displaystyle \varphi _{xx}(1+\varphi _{y}^{2})+\varphi _{yy}(1+\varphi _{x}^{2})-2\varphi _{x}\varphi _{y}\varphi _{xy}=0.}
12121:{\displaystyle V_{1}={\frac {2}{R}}\left(\int _{x_{1}}^{x_{2}}\left\,dx+a_{1}u(x_{1})v(x_{1})+a_{2}u(x_{2})v(x_{2})\right),}
10888:
are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio
21254:
19828:
19285:
Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.
14615:{\displaystyle n(x,y)={\begin{cases}n_{(-)}&{\text{if}}\quad x<0,\\n_{(+)}&{\text{if}}\quad x>0,\end{cases}}}
12937:
Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain
5014:
498:
473:
17:
21287:
3280:{\displaystyle {\frac {dL}{d\varepsilon }}={\frac {\partial L}{\partial y}}\eta +{\frac {\partial L}{\partial y'}}\eta '.}
3133:
16447:
6169:
1281:, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements
21172:
21168:
18153:
15722:
13862:
6470:
21825:
20619:
20551:
20303:
19033:
6949:
of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If
5572:
3090:
974:
537:
55:
21321:
20365:
20355:
20335:
19113:
19018:
8268:{\displaystyle \left.{\frac {d}{d\varepsilon }}V\right|_{\varepsilon =0}=\iint _{D}\nabla u\cdot \nabla v\,dx\,dy=0.}
1196:
493:
211:
21052:
17572:, a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning;
17115:
22123:
20970:
20950:
17788:
17578:, a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity;
7727:
consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of
478:
11420:
This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.
22292:
22203:
22130:
21845:
21762:
21556:
21331:
19098:
9518:
will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment
1600:, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.
814:
488:
463:
145:
2199:
1909:
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4543:
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1192:
19778:
17470:{\displaystyle {\frac {\partial \psi }{\partial t}}+H\left(x,{\frac {\partial \psi }{\partial x}},t\right)=0.}
7509:
5783:
5733:
4493:
2864:
22118:
22064:
19108:
17584:
is a variational method for finding numerical solutions to boundary-value problems in differential equations;
17566:, one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states;
12918:{\displaystyle -p(x_{1})u'(x_{1})+a_{1}u(x_{1})=0,\quad {\hbox{and}}\quad p(x_{2})u'(x_{2})+a_{2}u(x_{2})=0.}
5693:
1848:
596:
543:
424:
16681:
15219:
21408:
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17383:
8528:
7285:
4319:
2829:
250:
222:
21865:
21492:
20928:
20414:
Ferguson, James (2004). "Brief Survey of the History of the Calculus of Variations and its Applications".
16969:{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial L}{\partial x}},}
1109:
333:
21400:
21120:
13638:
states that light takes a path that (locally) minimizes the optical length between its endpoints. If the
2550:
2333:
1636:
1625:
1613:
1164:
1028:
847:
455:
293:
265:
14477:
The light rays may be determined by integrating this equation. This formalism is used in the context of
7332:
21830:
21475:
21404:
21062:
20656:
18411:
18037:
16615:
16090:
Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy
11729:
10457:
10173:
2258:
718:
682:
459:
338:
227:
217:
3846:
1187:(1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of
30:"Variational method" redirects here. For the use as an approximation method in quantum mechanics, see
21870:
21860:
21546:
21459:
21388:
21038:
The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics
20789:
19073:
19013:
17587:
16338:
8940:
1609:
1592:
there. For a function space of continuous functions, extrema of corresponding functionals are called
1057:
482:
21115:
20013:
14526:
14280:
After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation
11423:
The variational problem also applies to more general boundary conditions. Instead of requiring that
8981:
7407:
3932:
2579:
1566:
1514:
318:
22057:
21850:
21835:
21677:
21480:
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21213:
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18993:
18942:
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11271:
7562:
5004:{\displaystyle {\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0}
4272:{\displaystyle {\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0}
2414:
1608:
may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a
1160:
1135:(1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.
1072:
1052:
connecting two points, which depends upon the material of the medium. One corresponding concept in
1004:
617:
177:
20514:
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18903:
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18469:
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11207:
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15510:
Note that this integral is invariant with respect to changes in the parametric representation of
6385:
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931:
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612:
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but Ball and Mizel procured the first functional that displayed Lavrentiev's Phenomenon across
6295:
does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the
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The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.
19319:
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17959:
14743:
and in fact the path is a straight line there, since the refractive index is constant. At the
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20546:. Vol. I (First English ed.). New York: Interscience Publishers, Inc. p. 169.
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17550:
17502:
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The proof for the case of one dimensional integrals may be adapted to this case to show that
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21011:
Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag,
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9272:
6411:
4039:
by definition. Also, as previously mentioned the left side of the equation is zero so that
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4452:
In order to illustrate this process, consider the problem of finding the extremal function
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9330:
9035:
8836:. However Weierstrass gave an example of a variational problem with no solution: minimize
6215:
6036:
5126:
4868:
1977:
1817:
8:
22250:
22240:
22193:
21710:
21700:
21695:
21538:
21528:
21411:
21326:
21139:
19401:
which is not valid more generally in variational calculus with non-holonomic constraints.
19028:
14493:
There is a discontinuity of the refractive index when light enters or leaves a lens. Let
12534:{\displaystyle {\frac {R}{2}}V_{1}=\int _{x_{1}}^{x_{2}}v(x)\left\,dx+v(x_{1})+v(x_{2}).}
12131:
11764:
10891:
9863:
8829:
7617:
6969:
has continuous first and second derivatives with respect to all of its arguments, and if
4401:
2524:
2006:
1254:
1176:
921:
891:
881:
768:
622:
419:
275:
158:
153:
22049:
21592:
21178:
20731:
20464:
19621:
19381:
18014:
17765:
17362:
16658:
16635:
16070:
15886:
15513:
15336:
14794:
13601:
13306:
10320:
10150:
10080:
9956:
9889:
9521:
9438:
9411:
9359:
9307:
8785:
8727:
8582:
8466:
8094:
8031:
5441:
5103:
4897:
4411:
2934:
2382:
2119:
is an arbitrary function that has at least one derivative and vanishes at the endpoints
2070:
1543:
1438:
22272:
22155:
21939:
21657:
21454:
21309:
20743:
20415:
20088:
20068:
20048:
19644:
19601:
19450:
19259:
19197:
18766:
18707:
18505:
18205:
18124:
17861:
17681:
17617:
17557:
17208:
17188:
16982:
16740:
16716:
16614:
families of solutions of the variational problem. This is the essential content of the
16596:
16370:
16050:
15974:
15909:
15702:
15287:
15187:
15134:
14774:
14482:
13962:
13942:
13641:
13505:
13329:
13237:
12960:
12940:
12714:
12694:
12564:
12544:
11426:
11301:
11277:
11167:
10997:
10968:
10939:
10919:
10711:
10620:
10463:
10342:
10202:
10103:
9999:
9979:
9544:
9501:
9481:
9461:
9015:
8810:
8707:
8625:
8605:
8562:
8508:
8488:
8446:
8117:
8074:
8054:
7991:
7981:{\displaystyle V={\frac {1}{2}}\iint _{D}\nabla \varphi \cdot \nabla \varphi \,dx\,dy.}
7730:
7597:
7067:
7037:
6952:
6928:
6705:
6525:
6450:
6391:
6367:
6278:
6094:
6065:
5222:
5083:
4290:
2495:
2362:
2285:
2037:
1485:
1461:
1418:
1412:
1341:
1312:
1284:
1269:
The calculus of variations is concerned with the maxima or minima (collectively called
1214:
886:
789:
773:
713:
708:
703:
667:
548:
467:
373:
368:
172:
167:
20483:
20448:
19548:
The first variation is also called the variation, differential, or first differential.
22245:
22208:
21757:
21084:
21020:
21012:
20980:
20966:
20932:
20679:
20615:
20547:
20488:
20434:
20391:
20361:
20331:
20299:
19180:
19118:
19083:
17509:
15970:
14478:
11665:{\displaystyle Q=\int _{x_{1}}^{x_{2}}\left\,dx+a_{1}y(x_{1})^{2}+a_{2}y(x_{2})^{2},}
7717:{\displaystyle U=\iint _{D}{\sqrt {1+\nabla \varphi \cdot \nabla \varphi }}\,dx\,dy.}
6296:
1144:
1140:
1020:
1008:
960:
794:
572:
450:
403:
260:
255:
20747:
15501:{\displaystyle A=\int _{t_{0}}^{t_{1}}n(X){\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,dt.}
14688:
are constants. Then the Euler–Lagrange equation holds as before in the region where
1249:
and F. H. Clarke developed new mathematical tools for the calculus of variations in
22198:
22178:
21910:
21855:
21746:
21741:
21597:
21444:
21434:
21336:
21297:
20914:
20770:
20735:
20478:
20468:
20351:
19043:
17591:
16758:
16627:
12684:{\displaystyle -(pu')'+qu-\lambda ru=0\quad {\hbox{for}}\quad x_{1}<x<x_{2}.}
11443:
vanish at the endpoints, we may not impose any condition at the endpoints, and set
10994:
is constant. It is shown below that the Euler–Lagrange equation for the minimizing
8275:
Provided that u has two derivatives, we may apply the divergence theorem to obtain
4847:{\displaystyle y'(x)={\frac {dy}{dx}}\,,\ \ y_{1}=f(x_{1})\,,\ \ y_{2}=f(x_{2})\,.}
2770:
1605:
1230:
1199:
1101:
1068:
804:
698:
672:
533:
445:
409:
21066:
8807:
The difficulty with this reasoning is the assumption that the minimizing function
22213:
22150:
22145:
22140:
21930:
21901:
21875:
21796:
21781:
21690:
21662:
21639:
21572:
21449:
21439:
21264:
21259:
21102:
21044:
21037:
21030:
20991:
20960:
20956:
20943:
20774:
20603:
20535:
20293:
19494:
19088:
19058:
18988:
17537:
17528:
16564:{\displaystyle {\frac {ds}{dt}}={\frac {\sqrt {{\dot {X}}\cdot {\dot {X}}}}{n}}.}
13936:
1258:
1222:
1105:
936:
809:
763:
758:
645:
558:
503:
20614:. Vol. I (First English ed.). New York: Interscience Publishers, Inc.
16889:
The Euler–Lagrange equations for this system are known as Lagrange's equations:
16156:{\displaystyle \varphi _{t}^{2}=c(X)^{2}\,\nabla \varphi \cdot \nabla \varphi .}
15605:{\displaystyle {\frac {d}{dt}}P={\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,\nabla n,}
14468:{\displaystyle -{\frac {d}{dx}}\left+n_{y}(x,f_{0}){\sqrt {1+f_{0}'(x)^{2}}}=0.}
22017:
21925:
21786:
21685:
21587:
21393:
20635:
20298:(Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 3.
20285:
16734:
15690:{\displaystyle P={\frac {n(X){\dot {X}}}{\sqrt {{\dot {X}}\cdot {\dot {X}}}}}.}
15205:
1302:
1242:
1226:
1184:
1113:
1049:
819:
627:
394:
21127:
20357:
A History of the Calculus of Variations from the 17th through the 19th Century
11184:
has two derivatives and satisfies the Euler–Lagrange equation. The associated
7054:
has two continuous derivatives, and it satisfies the Euler–Lagrange equation.
1163:(1837) have been among the contributors. An important general work is that of
22286:
22217:
22173:
21801:
21576:
21352:
21244:
21239:
20607:
20539:
20195:
if it is a bilinear functional with two argument functions that are equal. A
19157:
19053:
17515:
15986:
12711:
satisfies this condition, then the first variation will vanish for arbitrary
11298:
The next smallest eigenvalue and eigenfunction can be obtained by minimizing
6158:
In other words, the shortest distance between two points is a straight line.
5730:
is a constant and therefore that the shortest curve that connects two points
2064:
1234:
1206:
1063:
Many important problems involve functions of several variables. Solutions of
1035:
799:
563:
313:
270:
16976:
and they are equivalent to Newton's equations of motion (for such systems).
15533:
The Euler–Lagrange equations for a minimizing curve have the symmetric form
22022:
21523:
21378:
20492:
20289:
19123:
4865:
The Euler–Lagrange equation will now be used to find the extremal function
1238:
1218:
1136:
553:
298:
20473:
17484:
Further applications of the calculus of variations include the following:
22012:
22007:
21891:
21056:
20924:
20236:
Sufficient conditions for a strong minimum are given by the theorem on p.
18329:{\displaystyle \Delta J=\varphi _{1}+\varphi _{2}+\varepsilon \|h\|^{2},}
3959:
on the second term. The second term on the second line vanishes because
1210:
1188:
1027:. Functions that maximize or minimize functionals may be found using the
1016:
916:
13817:{\displaystyle A=\int _{x_{0}}^{x_{1}}n(x,f(x)){\sqrt {1+f'(x)^{2}}}dx,}
1758:{\displaystyle J=\int _{x_{1}}^{x_{2}}L\left(x,y(x),y'(x)\right)\,dx\,.}
22103:
21935:
21905:
21667:
21274:
21132:
20739:
20449:"Dynamic Programming and a new formalism in the calculus of variations"
20222:
Sufficient conditions for a weak minimum are given by the theorem on p.
20220:
5: "The Second Variation. Sufficient Conditions for a Weak Extremum" –
19527:
17607:
is defined as the linear part of the change in the functional, and the
10100:
This boundary condition is a consequence of the minimizing property of
9498:
are continuous, regularity theory implies that the minimizing function
5341:{\displaystyle {\frac {d}{dx}}\ {\frac {f'(x)}{\sqrt {1+^{2}}}}\ =0\,.}
4594:
1180:
1024:
662:
586:
308:
303:
207:
19295:
12561:
vanish at the endpoints, the first variation will vanish for all such
10936:
satisfying the endpoint conditions, which is equivalent to minimizing
7640:
plane, then its potential energy is proportional to its surface area:
4441:
1143:
also gave some early attention to the subject. To this discrimination
21144:
20420:
5212:{\displaystyle {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0\,.}
1053:
591:
581:
20915:"Weak Lower Semicontinuity of Integral Functionals and Applications"
17598:
9070:
A more general expression for the potential energy of a membrane is
22260:
22098:
22093:
21944:
21426:
21103:
Calculus of Variations with Applications to Physics and Engineering
20437:. Dynamic programming and optimal control. Athena Scientific, 2005.
19153:
17522:
17489:
15394:
be its tangent vector. The optical length of the curve is given by
13591:{\displaystyle p(S){\frac {\partial u}{\partial n}}+\sigma (S)u=0,}
13437:{\displaystyle -\nabla \cdot (p(X)\nabla u)+q(x)u-\lambda r(x)u=0,}
6695:{\displaystyle S=\int _{a}^{b}f(x,y(x),y'(x),\dots ,y^{(n)}(x))dx,}
1974:
is twice continuously differentiable with respect to its arguments
1081:
1040:
657:
399:
356:
45:
17678:
as its argument, and there is a small change in its argument from
9065:
1171:(1844). Other valuable treatises and memoirs have been written by
21725:
21161:
19114:
Measures of central tendency as solutions to variational problems
11164:
It can be shown (see Gelfand and Fomin 1963) that the minimizing
8091:
is an arbitrary smooth function that vanishes on the boundary of
1172:
20944:
Variational Methods with Applications in Science and Engineering
20176:{\displaystyle \|h\|=\displaystyle \max _{a\leq x\leq b}|h(x)|.}
10637:
is restricted to functions that satisfy the boundary conditions
7371:
Examples (in one-dimension) are traditionally manifested across
7071:
For instance the following problem, presented by Manià in 1934:
7027:{\displaystyle {\frac {\partial ^{2}L}{\partial f'^{2}}}\neq 0,}
9909:
the boundary integral vanishes, and we conclude as before that
9262:{\displaystyle V=\iint _{D}\left\,dx\,dy\,+\int _{C}\left\,ds.}
5123:
the first term in the Euler–Lagrange equation vanishes for all
1168:
19276:
a positive number that specifies the size of the neighborhood.
11270:
This variational characterization of eigenvalues leads to the
10070:{\displaystyle {\frac {\partial u}{\partial n}}+\sigma u+g=0,}
9853:{\displaystyle \iint _{D}\left\,dx\,dy+\int _{C}v\left\,ds=0.}
4702:{\displaystyle A=\int _{x_{1}}^{x_{2}}{\sqrt {1+^{2}}}\,dx\,,}
20961:
Dirichlet's principle, conformal mapping and minimal surfaces
17545:, or reformulations of Newton's laws of motion, most notably
15854:{\displaystyle A=\int _{t_{0}}^{t_{1}}P\cdot {\dot {X}}\,dt.}
13227:{\displaystyle R=\iiint _{D}r(X)\varphi (X)^{2}\,dx\,dy\,dz.}
9701:{\displaystyle \iint _{D}\left\,dx\,dy+\int _{C}\left\,ds=0.}
7751:. The Euler–Lagrange equation for this problem is nonlinear:
4440:
A sufficient condition for a minimum is given in the section
21638:
20761:
Ferriero, Alessandro (2007). "The Weak Repulsion property".
19557:
The second variation is also called the second differential.
17156:
results if the conjugate momenta are introduced in place of
17047:{\displaystyle p={\frac {\partial L}{\partial {\dot {x}}}}.}
16850:{\displaystyle S=\int _{t_{0}}^{t_{1}}L(x,{\dot {x}},t)\,dt}
12932:
11413:{\displaystyle \int _{x_{1}}^{x_{2}}r(x)u_{1}(x)y(x)\,dx=0.}
21283:
Differentiable vector–valued functions from Euclidean space
21158:
Mathematics - Calculus of Variations and Integral Equations
20234:
6: "Fields. Sufficient Conditions for a Strong Extremum" –
14608:
13658:-coordinate is chosen as the parameter along the path, and
10120:: it is not imposed beforehand. Such conditions are called
8161:
3561:
3331:
2703:
2631:
20699:
Manià, Bernard (1934). "Sopra un esempio di Lavrentieff".
8697:{\displaystyle \iint _{D}v\nabla \cdot \nabla u\,dx\,dy=0}
7273:{\displaystyle {A}=\{x\in W^{1,1}(0,1):x(0)=0,\ x(1)=1\}.}
6447:
The intuition behind this result is that, if the variable
6357:{\displaystyle L-f'{\frac {\partial L}{\partial f'}}=C\,,}
22079:
21208:
19661:
has been left out to simplify the notation. For example,
17317:{\displaystyle H(x,p,t)=p\,{\dot {x}}-L(x,{\dot {x}},t).}
15211:
4193:, the part of the integrand in parentheses is zero, i.e.
20892:
10823:{\displaystyle R=\int _{x_{1}}^{x_{2}}r(x)y(x)^{2}\,dx.}
7594:
denotes the displacement of a membrane above the domain
20641:
Lectures on the principles of demonstrative mathematics
9996:
to assume arbitrary boundary values, this implies that
1635:
is equal to zero. This leads to solving the associated
20674:
Kot, Mark (2014). "Chapter 4: Basic Generalizations".
20602:
20360:. Springer Science & Business Media. p. 110.
16618:, which applies to more general variational problems.
14271:{\displaystyle \delta A=\int _{x_{0}}^{x_{1}}\leftdx.}
12827:
12642:
20796:
20326:. In Bradley, Robert E.; Sandifer, C. Edward (eds.).
20124:
20111:
20091:
20071:
20051:
20016:
19981:
19952:
19919:
19906:{\displaystyle \varphi \left=\varphi +\varphi \left,}
19831:
19781:
19748:
19699:
19667:
19647:
19624:
19604:
19572:
19503:
19473:
19453:
19416:
19384:
19351:
19322:
19262:
19220:
19200:
18945:
18906:
18871:
18833:
18798:
18769:
18730:
18710:
18646:
18603:
18537:
18508:
18472:
18440:
18414:
18378:
18342:
18241:
18208:
18156:
18127:
18098:
18066:
18040:
18017:
17991:
17962:
17897:
17864:
17791:
17768:
17733:
17704:
17684:
17649:
17620:
17392:
17365:
17330:
17231:
17211:
17191:
17162:
17118:
17060:
17005:
16985:
16895:
16863:
16857:
is stationary with respect to variations in the path
16771:
16743:
16719:
16684:
16661:
16638:
16599:
16579:
16498:
16450:
16393:
16373:
16341:
16300:
16287:{\displaystyle \nabla \psi \cdot \nabla \psi =n^{2},}
16252:
16232:
16171:
16096:
16073:
16053:
15995:
15952:
15932:
15912:
15889:
15869:
15778:
15725:
15705:
15618:
15539:
15516:
15400:
15362:
15339:
15310:
15290:
15222:
15190:
15157:
15137:
15104:
14822:
14797:
14777:
14749:
14720:
14694:
14661:
14628:
14499:
14286:
13985:
13965:
13945:
13865:
13830:
13699:
13664:
13644:
13604:
13528:
13508:
13450:
13352:
13332:
13309:
13260:
13240:
13143:
12983:
12963:
12943:
12737:
12717:
12697:
12587:
12567:
12547:
12180:
12134:
11795:
11767:
11732:
11705:
11678:
11449:
11429:
11324:
11304:
11280:
11237:
11210:
11190:
11170:
11112:
11092:
11020:
11000:
10971:
10942:
10922:
10894:
10865:
10836:
10734:
10714:
10643:
10623:
10472:
10425:{\displaystyle \iint _{D}f\,dx\,dy+\int _{C}g\,ds=0.}
10365:
10345:
10323:
10225:
10205:
10176:
10153:
10133:
10106:
10083:
10022:
10002:
9982:
9959:
9915:
9892:
9866:
9714:
9585:
9547:
9524:
9504:
9484:
9464:
9441:
9414:
9385:
9362:
9333:
9310:
9275:
9076:
9038:
9018:
8984:
8943:
8923:
8842:
8813:
8788:
8753:
8730:
8710:
8648:
8628:
8608:
8585:
8565:
8531:
8511:
8491:
8469:
8449:
8281:
8158:
8120:
8097:
8077:
8057:
8034:
8014:
7994:
7910:
7757:
7733:
7646:
7620:
7600:
7565:
7512:
7479:
7446:
7410:
7377:
7335:
7288:
7170:
7077:
7040:
6975:
6955:
6931:
6732:
6708:
6580:
6548:
6528:
6473:
6453:
6414:
6394:
6370:
6304:
6281:
6247:
6218:
6172:
6126:
6097:
6068:
6039:
5836:
5786:
5736:
5696:
5614:
5575:
5467:
5444:
5356:
5245:
5225:
5158:
5129:
5106:
5086:
5017:
4931:
4900:
4871:
4715:
4603:
4546:
4496:
4490:
which is the shortest curve that connects two points
4458:
4414:
4374:
4322:
4293:
4283:. The left hand side of this equation is called the
4199:
4045:
4018:
3991:
3965:
3935:
3849:
3295:
3191:
3136:
3093:
2960:
2937:
2917:
2867:
2832:
2779:
2608:
2582:
2553:
2527:
2498:
2441:
2417:
2385:
2365:
2336:
2308:
2288:
2261:
2202:
2182:
2152:
2125:
2096:
2073:
2040:
2009:
1980:
1912:
1851:
1820:
1774:
1648:
1569:
1546:
1517:
1488:
1464:
1441:
1421:
1364:
1344:
1315:
1287:
1237:
applied calculus of variations in what is now called
1092:
The calculus of variations may be said to begin with
58:
20584:
20560:
17949:{\displaystyle \Delta J=\varphi +\varepsilon \|h\|,}
17594:
method for filtering high variance or noisy signals.
15980:
13693:
along the path, then the optical length is given by
8910:{\displaystyle W=\int _{-1}^{1}(x\varphi ')^{2}\,dx}
6510:
implies that the Lagrangian is time-independent. By
5431:{\displaystyle {\frac {f'(x)}{\sqrt {1+^{2}}}}=c\,,}
4368:
which can be solved to obtain the extremal function
3179:{\displaystyle {\frac {dy'}{d\varepsilon }}=\eta ',}
17785:then the corresponding change in the functional is
17324:The Hamiltonian is the total energy of the system:
16655:is defined as the time integral of the Lagrangian,
15184:is the sine of angle of the refracted ray with the
7329:minimizes the functional, but we find any function
7158:{\displaystyle L=\int _{0}^{1}(x^{3}-t)^{2}x'^{6},}
6205:{\displaystyle {\frac {\partial L}{\partial x}}=0,}
1540:everywhere in an arbitrarily small neighborhood of
20979:, 3rd edition. 2014, World Scientific Publishing,
20856:
20844:
20808:
20268:
20266:
20175:
20097:
20077:
20057:
20037:
20002:
19958:
19938:
19905:
19817:
19763:
19726:
19685:
19653:
19633:
19610:
19590:
19520:
19509:
19485:
19459:
19439:
19393:
19366:
19337:
19288:
19279:
19268:
19248:
19206:
18969:
18931:
18892:
18857:
18819:
18784:
18742:
18716:
18696:
18628:
18587:
18523:
18494:
18458:
18426:
18400:
18364:
18328:
18223:
18192:
18142:
18113:
18084:
18052:
18026:
18003:
17977:
17948:
17879:
17848:
17777:
17754:
17719:
17690:
17670:
17635:
17469:
17374:
17351:
17316:
17217:
17197:
17177:
17145:
17104:
17046:
16991:
16968:
16881:
16849:
16749:
16725:
16705:
16670:
16647:
16605:
16585:
16563:
16482:
16432:
16379:
16359:
16323:
16286:
16238:
16216:
16155:
16082:
16059:
16040:{\displaystyle u_{tt}=c^{2}\nabla \cdot \nabla u,}
16039:
15961:
15938:
15918:
15898:
15875:
15863:This form suggests that if we can find a function
15853:
15762:
15711:
15689:
15604:
15525:
15500:
15386:
15348:
15325:
15296:
15276:
15196:
15176:
15143:
15131:is the sine of angle of the incident ray with the
15123:
15088:
14808:
14783:
14764:
14735:
14706:
14680:
14647:
14614:
14467:
14270:
13971:
13951:
13928:{\displaystyle f(x)=f_{0}(x)+\varepsilon f_{1}(x)}
13927:
13851:
13816:
13685:
13650:
13613:
13590:
13514:
13494:
13436:
13338:
13318:
13295:
13246:
13226:
13129:
12969:
12949:
12917:
12723:
12703:
12683:
12573:
12553:
12533:
12166:
12120:
11781:
11753:
11718:
11691:
11664:
11435:
11412:
11310:
11286:
11262:
11223:
11196:
11176:
11156:
11098:
11078:
11006:
10986:
10957:
10928:
10908:
10880:
10851:
10822:
10720:
10700:
10629:
10609:
10424:
10351:
10332:
10309:
10211:
10191:
10162:
10139:
10112:
10092:
10069:
10008:
9988:
9968:
9945:
9901:
9878:
9852:
9708:If we apply the divergence theorem, the result is
9700:
9571:
9533:
9510:
9490:
9470:
9450:
9423:
9400:
9371:
9348:
9319:
9296:
9261:
9050:
9024:
9005:
8970:
8929:
8909:
8819:
8797:
8774:
8739:
8716:
8696:
8634:
8614:
8594:
8571:
8551:
8517:
8497:
8478:
8455:
8435:
8267:
8144:
8106:
8083:
8063:
8043:
8020:
8000:
7980:
7888:
7739:
7716:
7632:
7606:
7586:
7539:
7498:
7465:
7432:
7396:
7360:
7321:
7272:
7157:
7046:
7026:
6961:
6937:
6910:
6714:
6694:
6566:
6534:
6503:{\displaystyle {\frac {\partial L}{\partial x}}=0}
6502:
6459:
6437:
6400:
6376:
6356:
6287:
6267:
6233:
6204:
6150:
6120:is a minimum. The equation for a straight line is
6112:
6083:
6054:
6025:
5822:
5772:
5722:
5682:
5600:
5561:
5453:
5430:
5340:
5231:
5211:
5144:
5115:
5092:
5070:
5003:
4918:
4886:
4846:
4701:
4585:
4532:
4482:
4432:
4392:
4354:
4308:
4271:
4179:
4031:
4004:
3977:
3947:
3921:
3835:
3279:
3178:
3122:
3079:
2946:
2923:
2903:
2853:
2818:
2759:
2594:
2568:
2539:
2513:
2484:
2426:
2403:
2371:
2351:
2320:
2294:
2270:
2245:
2188:
2168:
2138:
2111:
2082:
2055:
2023:
1995:
1966:
1897:
1835:
1800:
1757:
1584:
1555:
1532:
1503:
1470:
1450:
1427:
1403:
1350:
1330:
1293:
1202:published in 1900 encouraged further development.
130:
19296:Variations and sufficient condition for a minimum
17599:Variations and sufficient condition for a minimum
13522:must also satisfy the natural boundary condition
10610:{\displaystyle Q=\int _{x_{1}}^{x_{2}}\left\,dx,}
9431:The function that minimizes the potential energy
8028:that assume prescribed values on the boundary of
7554:
4442:Variations and sufficient condition for a minimum
3123:{\displaystyle {\frac {dy}{d\varepsilon }}=\eta }
1261:is an alternative to the calculus of variations.
1116:first elaborated the subject, beginning in 1733.
1104:(1696). It immediately occupied the attention of
999:that uses variations, which are small changes in
22284:
20880:
20717:
20528:
20126:
19560:
19447:is called the first variation of the functional
17105:{\displaystyle T={\frac {1}{2}}m{\dot {x}}^{2},}
10170:In such a case, we could allow a trial function
8642:and the first variation vanishes, the result is
131:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
20868:
20572:
20263:
17185:by a Legendre transformation of the Lagrangian
15772:Therefore, the integral may also be written as
9066:Generalization to other boundary value problems
5562:{\displaystyle {\frac {^{2}}{1+^{2}}}=c^{2}\,,}
21083:. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014,
20832:
20820:
20654:
20534:
20272:
19214:is the part of the given function space where
18372:is a linear functional (the first variation),
16678:The Lagrangian is the difference of energies,
16067:is the velocity, which generally depends upon
9269:This corresponds to an external force density
22065:
21624:
21194:
21006:An Introduction to the Calculus of Variations
20278:
19256:over the whole domain of the functions, with
16433:{\displaystyle {\frac {dP}{ds}}=n\,\nabla n,}
13303:with no condition prescribed on the boundary
8008:is to be minimized among all trial functions
7368:gives a value bounded away from the infimum.
6033:and we have thus found the extremal function
5683:{\displaystyle ^{2}={\frac {c^{2}}{1-c^{2}}}}
1612:that is used for finding weak extrema is the
1233:among others made significant contributions.
968:
20898:
20886:
20874:
20862:
20850:
20838:
20826:
20814:
20802:
20763:Journal de Mathématiques Pures et Appliquées
20676:A First Course in the Calculus of Variations
20590:
20578:
20566:
20284:
20209:
20118:
20112:
18697:{\displaystyle \delta ^{2}J\geq k\|h\|^{2},}
18682:
18675:
18447:
18441:
18314:
18307:
18073:
18067:
17998:
17992:
17940:
17934:
17762:is a function in the same function space as
17556:Geometric optics, especially Lagrangian and
15333:be the parametric representation of a curve
15216:It is expedient to use vector notation: let
13254:be the function that minimizes the quotient
10701:{\displaystyle y(x_{1})=0,\quad y(x_{2})=0.}
7264:
7179:
6920:
6166:In physics problems it may be the case that
1338:is said to have an extremum at the function
1007:, to find maxima and minima of functionals:
20720:Archive for Rational Mechanics and Analysis
20701:Bollenttino dell'Unione Matematica Italiana
20507:"Richard E. Bellman Control Heritage Award"
12174:as previously. After integration by parts,
10451:
10445:can be formulated as variational problems.
10441:Both one-dimensional and multi-dimensional
4191:fundamental lemma of calculus of variations
1167:(1842) which was condensed and improved by
22072:
22058:
21631:
21617:
21201:
21187:
21096:Introduction to the Calculus of Variations
20976:Introduction to the Calculus of Variations
20409:
20407:
20202:
19526:As a historical note, this is an axiom of
18588:{\displaystyle \delta ^{2}J=\varphi _{2}.}
16333:first-order partial differential equations
10219:is a constant. For such a trial function,
9946:{\displaystyle -\nabla \cdot \nabla u+f=0}
9433:with no restriction on its boundary values
7063:Euler–Lagrange equations in the interior.
1967:{\displaystyle L\left(x,y(x),y'(x)\right)}
1619:
975:
961:
20482:
20472:
20419:
20385:
20350:
20185:
19598:and the variations below, depend on both
19551:
17576:Variational methods in general relativity
17262:
16840:
16420:
16217:{\displaystyle \varphi (t,X)=t-\psi (X).}
16134:
15841:
15592:
15488:
13326:The Euler–Lagrange equation satisfied by
13214:
13207:
13200:
13117:
13075:
13068:
13061:
12933:Eigenvalue problems in several dimensions
12312:
12001:
11574:
11397:
10810:
10597:
10409:
10386:
10379:
10292:
10269:
10262:
9837:
9769:
9762:
9685:
9637:
9630:
9249:
9175:
9168:
9161:
8900:
8681:
8674:
8423:
8380:
8373:
8320:
8313:
8252:
8245:
7968:
7961:
7704:
7697:
6517:
6384:is a constant. The left hand side is the
6350:
5555:
5424:
5334:
5205:
5064:
4840:
4798:
4756:
4695:
4688:
4586:{\displaystyle \left(x_{2},y_{2}\right).}
4173:
4160:
3822:
3697:
3549:
3480:
2753:
2478:
2239:
1751:
1744:
91:
21498:No infinite-dimensional Lebesgue measure
20760:
20413:
20344:
20317:
20315:
20208:For other sufficient conditions, see in
19818:{\displaystyle \varphi =\alpha \varphi }
19542:
19294:For a sufficient condition, see section
19039:Inverse problem for Lagrangian mechanics
16593:is the value of the minimizing integral
13495:{\displaystyle \lambda ={\frac {Q}{R}}.}
11157:{\displaystyle \lambda ={\frac {Q}{R}}.}
11079:{\displaystyle -(pu')'+qu-\lambda ru=0,}
10127:The preceding reasoning is not valid if
7899:
7540:{\displaystyle 1\leq p<q<\infty .}
7057:
5823:{\displaystyle \left(x_{2},y_{2}\right)}
5773:{\displaystyle \left(x_{1},y_{1}\right)}
4533:{\displaystyle \left(x_{1},y_{1}\right)}
2904:{\displaystyle y'=f'+\varepsilon \eta '}
2485:{\displaystyle \Phi (\varepsilon )=J\,.}
1435:in an arbitrarily small neighborhood of
27:Differential calculus on function spaces
21721:Locally convex topological vector space
21508:Structure theorem for Gaussian measures
21047:, 3rd edition. Wiley-Interscience, 2006
20634:
20446:
20440:
20404:
20381:
20379:
20377:
20245:
19064:Direct method in calculus of variations
17479:
15089:{\displaystyle \delta A=f_{1}(0)\left.}
9060:elliptic partial differential equations
8775:{\displaystyle \nabla \cdot \nabla u=0}
6212:meaning the integrand is a function of
1898:{\displaystyle y'(x)={\frac {dy}{dx}},}
1080:surface, and they may have non-trivial
499:Differentiating under the integral sign
14:
22285:
20929:Lectures on the Calculus of Variations
20657:"Euler–Lagrange Differential Equation"
20644:. p. 58 – via Google Books.
20324:"Euler and the Calculus of Variations"
20321:
19736:
17564:Variational method (quantum mechanics)
16165:We may look for solutions in the form
15277:{\displaystyle X=(x_{1},x_{2},x_{3}),}
15212:Fermat's principle in three dimensions
10436:
6161:
32:Variational method (quantum mechanics)
22053:
21612:
21384:infinite-dimensional Gaussian measure
21182:
21040:, World Scientific, 2003, pages 1–98.
20953:, Interscience Publishers Inc., 1968.
20698:
20328:Leonhard Euler: Life, Work and Legacy
20312:
20292:(2000). Silverman, Richard A. (ed.).
19440:{\displaystyle \varepsilon \Phi '(0)}
13859:depends upon the material. If we try
8552:{\displaystyle \partial u/\partial n}
7322:{\displaystyle x(t)=t^{\frac {1}{3}}}
6467:is actually time, then the statement
5071:{\displaystyle L={\sqrt {1+^{2}}}\,.}
4364:In general this gives a second-order
4355:{\displaystyle \delta J/\delta f(x).}
2854:{\displaystyle y=f+\varepsilon \eta }
1843:is twice continuously differentiable,
1048:: light follows the path of shortest
1019:. Functionals are often expressed as
21255:Infinite-dimensional vector function
21167:Selected papers on Geodesic Fields.
21105:, Dover, 1974 (reprint of 1952 ed.).
21058:Optimization for Engineering Systems
20787:
20374:
20105:is its maximum absolute value, i.e.
19969:
16632:In classical mechanics, the action,
15699:It follows from the definition that
11761:, the first variation for the ratio
10016:must satisfy the boundary condition
1273:) of functionals. A functional maps
1058:principle of least/stationary action
21053:"Chapter 8: Calculus of Variations"
21033:. Cambridge University Press, 1998.
20946:, Cambridge University Press, 2013.
20673:
18759:Sufficient condition for a minimum:
17497:Newton's minimal resistance problem
17382:This function is a solution of the
16483:{\displaystyle {\frac {dX}{ds}}=P.}
2569:{\displaystyle \Phi (\varepsilon )}
2352:{\displaystyle f+\varepsilon \eta }
1482:function or extremal. The extremum
1094:Newton's minimal resistance problem
24:
20907:
20511:American Automatic Control Council
19700:
19668:
19573:
19421:
18242:
18193:{\displaystyle \delta J=\varphi .}
17898:
17792:
17643:is a functional with the function
17611:is defined as the quadratic part.
17441:
17433:
17404:
17396:
17023:
17015:
16954:
16946:
16922:
16914:
16421:
16348:
16262:
16253:
16144:
16135:
16028:
16022:
15763:{\displaystyle P\cdot P=n(X)^{2}.}
15593:
13552:
13544:
13377:
13356:
13030:
13021:
12977:in three dimensions we may define
10466:involves a general quadratic form
10034:
10026:
9925:
9919:
9808:
9800:
9742:
9736:
9610:
9601:
9126:
9117:
8760:
8754:
8668:
8662:
8543:
8532:
8414:
8406:
8367:
8361:
8349:
8340:
8304:
8292:
8239:
8230:
7955:
7946:
7689:
7680:
7531:
7422:
7361:{\displaystyle x\in W^{1,\infty }}
7353:
6994:
6980:
6876:
6868:
6786:
6778:
6744:
6736:
6485:
6477:
6330:
6322:
6184:
6176:
5185:
5177:
4981:
4973:
4943:
4935:
4249:
4241:
4211:
4203:
4141:
4133:
4103:
4095:
3803:
3795:
3759:
3751:
3683:
3675:
3575:
3567:
3537:
3529:
3453:
3445:
3427:
3419:
3252:
3244:
3226:
3218:
3038:
3030:
2995:
2987:
2639:
2610:
2554:
2442:
1570:
1518:
1365:
40:Part of a series of articles about
25:
22309:
21322:Generalizations of the derivative
21288:Differentiation in Fréchet spaces
21109:
21080:Mathematical Tools for Physicists
20678:. American Mathematical Society.
19019:Infinite-dimensional optimization
18427:{\displaystyle \varepsilon \to 0}
18053:{\displaystyle \varepsilon \to 0}
15981:Connection with the wave equation
15151:axis, and the factor multiplying
11754:{\displaystyle y=u+\varepsilon v}
10192:{\displaystyle \varphi \equiv c,}
6542:depends on higher-derivatives of
5601:{\displaystyle 0\leq c^{2}<1.}
4400:The Euler–Lagrange equation is a
2271:{\displaystyle \varepsilon \eta }
21050:
12925:These latter conditions are the
11318:under the additional constraint
9379:and elastic forces with modulus
7896:See Courant (1950) for details.
3922:{\displaystyle L\left\to L\left}
1044:. A related problem is posed by
22204:Least-squares spectral analysis
22131:Fundamental theorem of calculus
21826:Ekeland's variational principle
21557:Holomorphic functional calculus
21036:Lebedev, L.P. and Cloud, M.J.:
20781:
20754:
20711:
20692:
20667:
20648:
20628:
20612:Methods of Mathematical Physics
20596:
20544:Methods of Mathematical Physics
20499:
19533:
19497:differently by leaving out the
19404:
19310:
19301:
19188:
19172:
19034:Ekeland's variational principle
18408:is a quadratic functional, and
16360:{\displaystyle P=\nabla \psi ,}
15989:for an inhomogeneous medium is
14592:
14552:
13625:
12833:
12825:
12648:
12640:
10672:
9062:; see Jost and Li–Jost (1998).
8971:{\displaystyle \varphi (-1)=-1}
8724:that vanish on the boundary of
8071:is the minimizing function and
5942:
5936:
5864:
2911:are considered as functions of
1031:of the calculus of variations.
21552:Continuous functional calculus
20428:
20165:
20161:
20155:
20148:
20085:are real numbers, the norm of
20038:{\displaystyle a\leq x\leq b,}
19997:
19991:
19873:
19867:
19812:
19806:
19794:
19785:
19758:
19752:
19718:
19706:
19680:
19674:
19585:
19579:
19434:
19428:
19361:
19355:
19332:
19326:
19236:
19222:
19163:
19146:
18958:
18926:
18920:
18884:
18846:
18840:
18811:
18779:
18773:
18666:
18660:
18623:
18617:
18579:
18573:
18557:
18551:
18518:
18512:
18489:
18483:
18450:
18418:
18395:
18389:
18359:
18353:
18298:
18292:
18276:
18270:
18254:
18248:
18218:
18212:
18184:
18178:
18169:
18163:
18137:
18131:
18108:
18102:
18076:
18044:
17972:
17966:
17925:
17919:
17910:
17904:
17874:
17868:
17840:
17834:
17825:
17813:
17804:
17798:
17749:
17743:
17665:
17659:
17630:
17624:
17308:
17281:
17253:
17235:
17146:{\displaystyle p=m{\dot {x}}.}
16873:
16867:
16837:
16810:
16573:We conclude that the function
16208:
16202:
16187:
16175:
16125:
16118:
15926:is given by the difference of
15788:
15782:
15748:
15741:
15637:
15631:
15454:
15448:
15410:
15404:
15381:
15375:
15320:
15314:
15268:
15229:
15169:
15163:
15116:
15110:
15066:
15052:
15028:
15015:
14994:
14988:
14968:
14954:
14930:
14917:
14896:
14890:
14877:
14871:
14855:
14829:
14673:
14667:
14640:
14634:
14580:
14574:
14540:
14534:
14515:
14503:
14488:
14448:
14441:
14417:
14398:
14334:
14315:
14243:
14236:
14202:
14183:
14158:
14151:
14127:
14121:
14105:
14099:
14083:
14064:
14018:
13992:
13922:
13916:
13897:
13891:
13875:
13869:
13846:
13834:
13794:
13787:
13768:
13765:
13759:
13747:
13709:
13703:
13680:
13674:
13573:
13567:
13538:
13532:
13483:
13477:
13469:
13463:
13419:
13413:
13398:
13392:
13383:
13374:
13368:
13362:
13287:
13281:
13270:
13264:
13191:
13184:
13178:
13172:
13153:
13147:
13104:
13098:
13048:
13042:
13018:
13012:
12993:
12987:
12906:
12893:
12874:
12861:
12850:
12837:
12813:
12800:
12781:
12768:
12757:
12744:
12606:
12591:
12525:
12522:
12509:
12490:
12477:
12466:
12453:
12447:
12444:
12431:
12422:
12419:
12406:
12387:
12374:
12363:
12350:
12341:
12338:
12325:
12279:
12264:
12253:
12247:
12193:
12187:
12161:
12155:
12144:
12138:
12128:where λ is given by the ratio
12107:
12094:
12088:
12075:
12056:
12043:
12037:
12024:
11993:
11987:
11981:
11975:
11969:
11963:
11951:
11945:
11939:
11933:
11927:
11921:
11912:
11906:
11895:
11889:
11878:
11872:
11824:
11818:
11650:
11636:
11611:
11597:
11560:
11553:
11547:
11541:
11526:
11519:
11508:
11502:
11459:
11453:
11394:
11388:
11382:
11376:
11363:
11357:
11254:
11248:
11145:
11139:
11131:
11125:
11039:
11024:
10981:
10975:
10952:
10946:
10875:
10869:
10846:
10840:
10801:
10794:
10788:
10782:
10744:
10738:
10689:
10676:
10660:
10647:
10583:
10576:
10570:
10564:
10549:
10542:
10531:
10525:
10482:
10476:
10235:
10229:
9566:
9551:
9395:
9389:
9343:
9337:
9291:
9279:
9238:
9232:
9213:
9207:
9150:
9138:
9086:
9080:
9006:{\displaystyle \varphi (1)=1.}
8994:
8988:
8956:
8947:
8891:
8876:
8852:
8846:
8834:Peter Gustav Lejeune Dirichlet
8310:
8298:
8197:
8182:
8139:
8124:
7920:
7914:
7838:
7814:
7795:
7771:
7656:
7650:
7581:
7569:
7555:Functions of several variables
7433:{\displaystyle W^{1,\infty },}
7298:
7292:
7255:
7249:
7231:
7225:
7216:
7204:
7128:
7108:
7087:
7081:
6890:
6884:
6823:
6813:
6680:
6677:
6671:
6666:
6660:
6643:
6637:
6623:
6617:
6605:
6558:
6552:
6429:
6423:
6262:
6256:
6228:
6222:
6142:
6136:
6107:
6101:
6078:
6072:
6062:that minimizes the functional
6049:
6043:
5846:
5840:
5711:
5705:
5636:
5632:
5626:
5615:
5530:
5526:
5520:
5509:
5492:
5488:
5482:
5471:
5406:
5402:
5396:
5385:
5374:
5368:
5313:
5309:
5303:
5292:
5281:
5275:
5139:
5133:
5100:does not appear explicitly in
5053:
5049:
5043:
5032:
4910:
4904:
4894:that minimizes the functional
4881:
4875:
4837:
4824:
4795:
4782:
4730:
4724:
4677:
4673:
4667:
4656:
4613:
4607:
4474:
4468:
4424:
4418:
4384:
4378:
4366:ordinary differential equation
4346:
4340:
4303:
4297:
4084:
4078:
3948:{\displaystyle \varepsilon =0}
3883:
2623:
2617:
2595:{\displaystyle \varepsilon =0}
2563:
2557:
2508:
2502:
2475:
2460:
2451:
2445:
2395:
2389:
2236:
2221:
2212:
2206:
2106:
2100:
2050:
2044:
1956:
1950:
1936:
1930:
1866:
1860:
1830:
1824:
1736:
1730:
1716:
1710:
1658:
1652:
1585:{\displaystyle \Delta J\geq 0}
1533:{\displaystyle \Delta J\leq 0}
1498:
1492:
1398:
1392:
1383:
1377:
1325:
1319:
1023:involving functions and their
125:
119:
110:
104:
88:
82:
13:
1:
22298:Optimization in vector spaces
20913:Benesova, B. and Kruzik, M.:
20257:
19109:Category:Variational analysts
18970:{\displaystyle y={\hat {y}}.}
17849:{\displaystyle \Delta J=J-J.}
15387:{\displaystyle {\dot {X}}(t)}
7587:{\displaystyle \varphi (x,y)}
7548:small class of functionals.
2427:{\displaystyle \varepsilon ,}
1511:is called a local maximum if
425:Integral of inverse functions
20994:, Pergamon Press Ltd., 1962.
20775:10.1016/j.matpur.2007.06.002
20447:Bellman, Richard E. (1954).
19510:{\displaystyle \varepsilon }
19099:Luke's variational principle
19079:Variational Bayesian methods
18932:{\displaystyle \delta ^{2}J}
18893:{\displaystyle y={\hat {y}}}
18820:{\displaystyle y={\hat {y}}}
18629:{\displaystyle \delta ^{2}J}
18495:{\displaystyle \varphi _{2}}
18401:{\displaystyle \varphi _{2}}
18365:{\displaystyle \varphi _{1}}
17570:Variational Bayesian methods
16621:
11224:{\displaystyle \lambda _{1}}
10728:be a normalization integral
8559:is the normal derivative of
8114:then the first variation of
4408:, condition for an extremum
2924:{\displaystyle \varepsilon }
2411:the result is a function of
2189:{\displaystyle \varepsilon }
1404:{\displaystyle \Delta J=J-J}
1127:Elementa Calculi Variationum
7:
21846:Hermite–Hadamard inequality
21121:Encyclopedia of Mathematics
20191:A functional is said to be
19493:Some references define the
18981:
18502:is the second variation of
18459:{\displaystyle \|h\|\to 0.}
18085:{\displaystyle \|h\|\to 0.}
16737:of a mechanical system and
16331:According to the theory of
15883:whose gradient is given by
13824:where the refractive index
12927:natural boundary conditions
10122:natural boundary conditions
7747:; the solutions are called
5239:and taking the derivative,
1801:{\displaystyle x_{1},x_{2}}
848:Calculus on Euclidean space
266:Logarithmic differentiation
10:
22314:
21063:Louisiana State University
20388:The Calculus of Variations
20273:Courant & Hilbert 1953
19530:. See e.g. Kelland (1843).
19249:{\displaystyle |y-f|<h}
18858:{\displaystyle \delta J=0}
18121:is the first variation of
17178:{\displaystyle {\dot {x}}}
16625:
16440:along a system of curves (
11274:: choose an approximating
10965:under the constraint that
10455:
9401:{\displaystyle \sigma (s)}
4447:
2246:{\displaystyle J\leq J\,.}
1623:
1264:
1087:
29:
22269:
22169:
22088:
22031:
21998:
21953:
21884:
21810:
21734:
21676:
21650:
21565:
21547:Borel functional calculus
21537:
21516:
21468:
21425:
21345:
21273:
21220:
21214:topological vector spaces
21029:Jost, J. and X. Li-Jost:
20386:van Brunt, Bruce (2004).
20330:. Elsevier. p. 249.
19727:{\displaystyle \Delta J.}
19486:{\displaystyle \delta J.}
19014:Principle of least action
18900:and its second variation
18466:The quadratic functional
17588:Total variation denoising
13630:
12541:If we first require that
11726:are arbitrary. If we set
11263:{\displaystyle u_{1}(x).}
10317:By appropriate choice of
10310:{\displaystyle V=c\left.}
8704:for all smooth functions
6921:Du Bois-Reymond's theorem
4597:of the curve is given by
2321:{\displaystyle \delta f.}
1096:in 1687, followed by the
582:Summand limit (term test)
22032:Applications and related
21836:Fenchel-Young inequality
21481:Inverse function theorem
21368:Classical Wiener measure
20901:, p. 100, Theorem 2
20899:Gelfand & Fomin 2000
20887:Gelfand & Fomin 2000
20875:Gelfand & Fomin 2000
20863:Gelfand & Fomin 2000
20851:Gelfand & Fomin 2000
20839:Gelfand & Fomin 2000
20827:Gelfand & Fomin 2000
20817:, p. 12, footnote 6
20815:Gelfand & Fomin 2000
20803:Gelfand & Fomin 2000
20663:. Wolfram. Eq. (5).
20591:Gelfand & Fomin 2000
20579:Gelfand & Fomin 2000
20567:Gelfand & Fomin 2000
20322:Thiele, Rüdiger (2007).
20210:Gelfand & Fomin 2000
19764:{\displaystyle \varphi }
19693:could have been written
19686:{\displaystyle \Delta J}
19591:{\displaystyle \Delta J}
19338:{\displaystyle \eta (x)}
19140:
19134:Variational vector field
18994:Isoperimetric inequality
18939:is strongly positive at
18114:{\displaystyle \varphi }
17985:is a linear functional,
17978:{\displaystyle \varphi }
17384:Hamilton–Jacobi equation
14791:must be continuous, but
11197:{\displaystyle \lambda }
11099:{\displaystyle \lambda }
10452:Sturm–Liouville problems
10147:vanishes identically on
8930:{\displaystyle \varphi }
8832:in honor of his teacher
8021:{\displaystyle \varphi }
2112:{\displaystyle \eta (x)}
1642:Consider the functional
261:Implicit differentiation
251:Differentiation notation
178:Inverse function theorem
21792:Legendre transformation
21716:Legendre transformation
21583:Convenient vector space
19959:{\displaystyle \alpha }
19939:{\displaystyle h,h_{2}}
19129:Convenient vector space
18827:if its first variation
15177:{\displaystyle n_{(+)}}
15124:{\displaystyle n_{(-)}}
15098:The factor multiplying
14736:{\displaystyle x>0,}
14681:{\displaystyle n_{(+)}}
14648:{\displaystyle n_{(-)}}
10140:{\displaystyle \sigma }
9541:The first variation of
7499:{\displaystyle W^{1,q}}
7466:{\displaystyle W^{1,p}}
7397:{\displaystyle W^{1,1}}
6722:must satisfy the Euler–
6386:Legendre transformation
6151:{\displaystyle y=f(x).}
5723:{\displaystyle f'(x)=m}
4483:{\displaystyle y=f(x),}
4281:Euler–Lagrange equation
3978:{\displaystyle \eta =0}
2819:{\displaystyle L\left,}
1637:Euler–Lagrange equation
1626:Euler–Lagrange equation
1620:Euler–Lagrange equation
1614:Euler–Lagrange equation
1563:and a local minimum if
1065:boundary value problems
1029:Euler–Lagrange equation
724:Helmholtz decomposition
22293:Calculus of variations
22136:Calculus of variations
22109:Differential equations
22039:Convexity in economics
21973:(lower) ideally convex
21831:Fenchel–Moreau theorem
21821:Carathéodory's theorem
21476:Cameron–Martin theorem
21233:Classical Wiener space
21152:Calculus of variations
21140:Calculus of Variations
21128:calculus of variations
21075:Calculus of variations
21031:Calculus of Variations
20999:Calculus of Variations
20992:Calculus of Variations
20951:Calculus of Variations
20921:59(4) (2017), 703–766.
20513:. 2004. Archived from
20295:Calculus of variations
20177:
20099:
20079:
20059:
20039:
20004:
20003:{\displaystyle h=h(x)}
19960:
19940:
19907:
19819:
19765:
19728:
19687:
19655:
19635:
19612:
19592:
19511:
19487:
19461:
19441:
19395:
19368:
19339:
19270:
19250:
19208:
18978:
18971:
18933:
18894:
18859:
18821:
18786:
18744:
18743:{\displaystyle k>0}
18724:and for some constant
18718:
18698:
18630:
18589:
18525:
18496:
18460:
18428:
18402:
18366:
18330:
18225:
18194:
18144:
18115:
18092:The linear functional
18086:
18054:
18028:
18005:
17979:
17950:
17881:
17850:
17779:
17756:
17755:{\displaystyle h=h(x)}
17721:
17692:
17672:
17671:{\displaystyle y=y(x)}
17637:
17488:The derivation of the
17471:
17376:
17353:
17352:{\displaystyle H=T+U.}
17318:
17219:
17199:
17179:
17147:
17106:
17048:
16993:
16979:The conjugate momenta
16970:
16883:
16851:
16751:
16727:
16707:
16706:{\displaystyle L=T-U,}
16672:
16649:
16616:Hamilton–Jacobi theory
16607:
16587:
16565:
16484:
16434:
16381:
16361:
16325:
16324:{\displaystyle n=1/c.}
16288:
16240:
16218:
16157:
16084:
16061:
16041:
15963:
15962:{\displaystyle \psi .}
15940:
15920:
15900:
15877:
15855:
15764:
15713:
15691:
15606:
15527:
15502:
15388:
15350:
15327:
15298:
15278:
15198:
15178:
15145:
15125:
15090:
14810:
14785:
14766:
14737:
14708:
14707:{\displaystyle x<0}
14682:
14649:
14616:
14469:
14272:
13979:with respect to ε) is
13973:
13953:
13929:
13853:
13852:{\displaystyle n(x,y)}
13818:
13687:
13686:{\displaystyle y=f(x)}
13652:
13615:
13592:
13516:
13496:
13438:
13340:
13320:
13297:
13248:
13228:
13131:
12971:
12951:
12919:
12725:
12705:
12685:
12575:
12555:
12535:
12168:
12122:
11783:
11755:
11720:
11693:
11666:
11437:
11414:
11312:
11288:
11264:
11225:
11198:
11178:
11158:
11100:
11080:
11008:
10988:
10959:
10930:
10910:
10882:
10853:
10824:
10722:
10702:
10631:
10611:
10458:Sturm–Liouville theory
10426:
10353:
10334:
10311:
10213:
10193:
10164:
10141:
10114:
10094:
10071:
10010:
9990:
9970:
9947:
9903:
9880:
9854:
9702:
9573:
9535:
9512:
9492:
9472:
9452:
9425:
9402:
9373:
9350:
9321:
9298:
9297:{\displaystyle f(x,y)}
9263:
9052:
9026:
9007:
8972:
8931:
8911:
8821:
8799:
8776:
8741:
8718:
8698:
8636:
8616:
8596:
8573:
8553:
8519:
8499:
8480:
8457:
8437:
8269:
8146:
8108:
8085:
8065:
8045:
8022:
8002:
7982:
7890:
7741:
7718:
7634:
7608:
7588:
7541:
7500:
7467:
7434:
7398:
7362:
7323:
7274:
7159:
7048:
7028:
6963:
6939:
6912:
6716:
6696:
6568:
6536:
6518:Euler–Poisson equation
6504:
6461:
6439:
6438:{\displaystyle f'(x).}
6402:
6378:
6358:
6289:
6269:
6235:
6206:
6152:
6114:
6085:
6056:
6027:
5824:
5774:
5724:
5684:
5602:
5563:
5455:
5432:
5342:
5233:
5213:
5146:
5117:
5094:
5072:
5005:
4920:
4888:
4848:
4703:
4587:
4534:
4484:
4434:
4394:
4356:
4310:
4273:
4181:
4033:
4006:
3979:
3949:
3923:
3837:
3281:
3180:
3124:
3081:
2948:
2925:
2905:
2855:
2820:
2761:
2596:
2570:
2541:
2515:
2486:
2428:
2405:
2373:
2353:
2322:
2296:
2272:
2247:
2190:
2170:
2169:{\displaystyle x_{2},}
2140:
2113:
2084:
2057:
2025:
1997:
1968:
1899:
1837:
1802:
1759:
1586:
1557:
1534:
1505:
1472:
1452:
1429:
1405:
1352:
1332:
1295:
1251:optimal control theory
1123:calculus of variations
989:calculus of variations
858:Limit of distributions
678:Directional derivative
334:Faà di Bruno's formula
132:
22229:Representation theory
22188:quaternionic analysis
22184:Hypercomplex analysis
22082:mathematical analysis
21961:Convex series related
21861:Shapley–Folkman lemma
21493:Feldman–Hájek theorem
21305:Functional derivative
21228:Abstract Wiener space
20963:. Interscience, 1950.
20661:mathworld.wolfram.com
20474:10.1073/pnas.40.4.231
20453:Proc. Natl. Acad. Sci
20178:
20100:
20080:
20060:
20040:
20005:
19961:
19941:
19908:
19820:
19766:
19729:
19688:
19656:
19636:
19613:
19593:
19512:
19488:
19462:
19442:
19396:
19374:are evaluated at the
19369:
19340:
19271:
19251:
19209:
19104:Mountain pass theorem
19074:De Donder–Weyl theory
19024:Finite element method
19004:Variational bicomplex
18999:Variational principle
18972:
18934:
18895:
18860:
18822:
18787:
18756:
18745:
18719:
18699:
18631:
18597:The second variation
18590:
18526:
18497:
18461:
18429:
18403:
18367:
18331:
18226:
18195:
18145:
18116:
18087:
18055:
18029:
18006:
18004:{\displaystyle \|h\|}
17980:
17951:
17882:
17851:
17780:
17757:
17722:
17693:
17673:
17638:
17582:Finite element method
17551:Hamiltonian mechanics
17472:
17377:
17354:
17319:
17220:
17205:into the Hamiltonian
17200:
17180:
17154:Hamiltonian mechanics
17148:
17107:
17049:
16994:
16971:
16884:
16882:{\displaystyle x(t).}
16852:
16752:
16728:
16708:
16673:
16650:
16608:
16588:
16586:{\displaystyle \psi }
16566:
16485:
16435:
16382:
16362:
16326:
16289:
16241:
16239:{\displaystyle \psi }
16219:
16158:
16085:
16062:
16042:
15964:
15941:
15939:{\displaystyle \psi }
15921:
15901:
15878:
15876:{\displaystyle \psi }
15856:
15765:
15714:
15692:
15607:
15528:
15503:
15389:
15351:
15328:
15304:be a parameter, let
15299:
15279:
15199:
15179:
15146:
15126:
15091:
14811:
14786:
14767:
14738:
14709:
14683:
14650:
14617:
14470:
14273:
13974:
13954:
13930:
13854:
13819:
13688:
13653:
13616:
13593:
13517:
13497:
13439:
13341:
13321:
13298:
13249:
13229:
13132:
12972:
12952:
12920:
12726:
12706:
12686:
12576:
12556:
12536:
12169:
12123:
11784:
11756:
11721:
11719:{\displaystyle a_{2}}
11694:
11692:{\displaystyle a_{1}}
11667:
11438:
11415:
11313:
11289:
11265:
11226:
11199:
11179:
11159:
11101:
11081:
11009:
10989:
10960:
10931:
10911:
10883:
10854:
10825:
10723:
10703:
10632:
10612:
10427:
10354:
10335:
10312:
10214:
10194:
10165:
10142:
10115:
10095:
10072:
10011:
9991:
9971:
9948:
9904:
9881:
9855:
9703:
9574:
9536:
9513:
9493:
9473:
9453:
9426:
9403:
9374:
9351:
9322:
9299:
9264:
9053:
9027:
9008:
8973:
8932:
8912:
8822:
8800:
8777:
8742:
8719:
8699:
8637:
8617:
8597:
8574:
8554:
8520:
8500:
8481:
8458:
8438:
8270:
8147:
8109:
8086:
8066:
8046:
8023:
8003:
7983:
7900:Dirichlet's principle
7891:
7742:
7719:
7635:
7609:
7589:
7542:
7501:
7468:
7435:
7399:
7363:
7324:
7275:
7160:
7058:Lavrentiev phenomenon
7049:
7029:
6964:
6940:
6913:
6717:
6697:
6569:
6567:{\displaystyle y(x),}
6537:
6505:
6462:
6440:
6403:
6379:
6359:
6290:
6270:
6268:{\displaystyle f'(x)}
6236:
6207:
6153:
6115:
6086:
6057:
6028:
5825:
5775:
5725:
5685:
5603:
5564:
5456:
5433:
5343:
5234:
5214:
5147:
5118:
5095:
5073:
5006:
4921:
4889:
4849:
4704:
4588:
4535:
4485:
4435:
4395:
4393:{\displaystyle f(x).}
4357:
4311:
4285:functional derivative
4274:
4182:
4034:
4032:{\displaystyle x_{2}}
4007:
4005:{\displaystyle x_{1}}
3980:
3950:
3924:
3838:
3282:
3181:
3125:
3082:
2949:
2926:
2906:
2856:
2821:
2762:
2597:
2571:
2542:
2516:
2492:Since the functional
2487:
2429:
2406:
2374:
2354:
2323:
2297:
2273:
2248:
2191:
2171:
2141:
2139:{\displaystyle x_{1}}
2114:
2085:
2058:
2026:
1998:
1969:
1900:
1838:
1803:
1760:
1633:functional derivative
1587:
1558:
1535:
1506:
1473:
1453:
1430:
1406:
1353:
1333:
1305:defined over a given
1296:
1098:brachistochrone curve
1073:Dirichlet's principle
997:mathematical analysis
942:Mathematical analysis
853:Generalized functions
538:arithmetico-geometric
379:Leibniz integral rule
133:
22161:Table of derivatives
21851:Krein–Milman theorem
21644:variational analysis
21417:Radonifying function
21358:Cylinder set measure
21250:Cylinder set measure
21116:Variational calculus
21008:, Dover Publ., 1987.
20805:, pp. 11–12, 99
20792:. UK: U. St. Andrew.
20352:Goldstine, Herman H.
20109:
20089:
20069:
20049:
20014:
20010:that is defined for
19979:
19950:
19917:
19829:
19779:
19746:
19697:
19665:
19645:
19622:
19602:
19570:
19501:
19471:
19451:
19414:
19382:
19367:{\displaystyle f(x)}
19349:
19320:
19260:
19218:
19198:
19194:The neighborhood of
19094:Hu–Washizu principle
19049:Perturbation methods
18943:
18904:
18869:
18831:
18796:
18767:
18728:
18708:
18644:
18601:
18535:
18506:
18470:
18438:
18412:
18376:
18340:
18239:
18233:twice differentiable
18206:
18154:
18125:
18096:
18064:
18038:
18015:
17989:
17960:
17895:
17862:
17789:
17766:
17731:
17720:{\displaystyle y+h,}
17702:
17682:
17647:
17618:
17543:Analytical mechanics
17480:Further applications
17390:
17363:
17328:
17229:
17209:
17189:
17160:
17116:
17058:
17003:
16983:
16893:
16861:
16769:
16763:Hamilton's principle
16741:
16717:
16682:
16659:
16636:
16597:
16577:
16496:
16448:
16444:) that are given by
16391:
16371:
16339:
16298:
16250:
16230:
16169:
16094:
16071:
16051:
15993:
15950:
15930:
15910:
15887:
15867:
15776:
15723:
15703:
15616:
15537:
15514:
15398:
15360:
15337:
15326:{\displaystyle X(t)}
15308:
15288:
15220:
15188:
15155:
15135:
15102:
14820:
14795:
14775:
14765:{\displaystyle x=0,}
14747:
14718:
14692:
14659:
14626:
14497:
14284:
13983:
13963:
13943:
13863:
13828:
13697:
13662:
13642:
13602:
13526:
13506:
13448:
13350:
13330:
13307:
13296:{\displaystyle Q/R,}
13258:
13238:
13141:
12981:
12961:
12941:
12735:
12715:
12695:
12585:
12565:
12545:
12178:
12132:
11793:
11765:
11730:
11703:
11676:
11447:
11427:
11322:
11302:
11278:
11272:Rayleigh–Ritz method
11235:
11208:
11188:
11168:
11110:
11090:
11018:
10998:
10969:
10940:
10920:
10892:
10881:{\displaystyle r(x)}
10863:
10852:{\displaystyle p(x)}
10834:
10732:
10712:
10641:
10621:
10470:
10462:The Sturm–Liouville
10363:
10343:
10321:
10223:
10203:
10174:
10151:
10131:
10104:
10081:
10020:
10000:
9980:
9957:
9913:
9890:
9864:
9712:
9583:
9545:
9522:
9502:
9482:
9462:
9439:
9412:
9383:
9360:
9349:{\displaystyle g(s)}
9331:
9308:
9273:
9074:
9051:{\displaystyle W=0.}
9036:
9016:
8982:
8941:
8921:
8917:among all functions
8840:
8811:
8786:
8751:
8728:
8708:
8646:
8626:
8606:
8583:
8563:
8529:
8509:
8489:
8467:
8447:
8279:
8156:
8118:
8095:
8075:
8055:
8032:
8012:
7992:
7908:
7755:
7731:
7644:
7618:
7598:
7563:
7510:
7477:
7444:
7408:
7375:
7333:
7286:
7168:
7075:
7038:
6973:
6953:
6929:
6730:
6706:
6578:
6546:
6526:
6471:
6451:
6412:
6392:
6368:
6302:
6279:
6245:
6234:{\displaystyle f(x)}
6216:
6170:
6124:
6095:
6066:
6055:{\displaystyle f(x)}
6037:
5834:
5784:
5734:
5694:
5612:
5573:
5465:
5442:
5354:
5243:
5223:
5156:
5145:{\displaystyle f(x)}
5127:
5104:
5084:
5015:
4929:
4898:
4887:{\displaystyle f(x)}
4869:
4713:
4601:
4544:
4494:
4456:
4412:
4372:
4320:
4291:
4279:which is called the
4197:
4043:
4016:
3989:
3963:
3957:integration by parts
3933:
3847:
3293:
3189:
3134:
3091:
2958:
2935:
2915:
2865:
2830:
2777:
2606:
2580:
2551:
2525:
2496:
2439:
2415:
2383:
2363:
2334:
2306:
2286:
2259:
2200:
2180:
2176:then for any number
2150:
2123:
2094:
2071:
2038:
2007:
1996:{\displaystyle x,y,}
1978:
1910:
1849:
1836:{\displaystyle y(x)}
1818:
1772:
1646:
1567:
1544:
1515:
1486:
1462:
1439:
1419:
1362:
1342:
1313:
1285:
1205:In the 20th century
1157:Mikhail Ostrogradsky
1149:Carl Friedrich Gauss
1125:in his 1756 lecture
1110:Marquis de l'Hôpital
993:variational calculus
947:Nonstandard analysis
415:Lebesgue integration
285:Rules and identities
56:
18:Variational calculus
22241:Continuous function
22194:Functional analysis
21841:Jensen's inequality
21711:Lagrange multiplier
21701:Convex optimization
21696:Convex metric space
21539:Functional calculus
21529:Covariance operator
21450:Gelfand–Pettis/Weak
21412:measurable function
21327:Hadamard derivative
21154:. Example problems.
21101:Weinstock, Robert:
21091:, pp. 551–588.
21045:Applied Mathematics
20790:"Riemann biography"
20732:1985ArRMA..90..325B
20655:Weisstein, Eric W.
20465:1954PNAS...40..231B
20197:bilinear functional
19154:elementary calculus
19029:Functional analysis
18531:and is denoted by,
18150:and is denoted by,
17510:tautochrone problem
16806:
16111:
15822:
15444:
15051:
15014:
14953:
14916:
14440:
14377:
14349:
14235:
14150:
14120:
14098:
14052:
13959:(the derivative of
13743:
12243:
12167:{\displaystyle Q/R}
11863:
11782:{\displaystyle Q/R}
11493:
11353:
11204:will be denoted by
10909:{\displaystyle Q/R}
10778:
10516:
10443:eigenvalue problems
10437:Eigenvalue problems
9879:{\displaystyle v=0}
9435:will be denoted by
8875:
8830:Dirichlet principle
8505:is arclength along
8463:is the boundary of
7837:
7794:
7633:{\displaystyle x,y}
7107:
6601:
6162:Beltrami's identity
5690:which implies that
4854:Note that assuming
4647:
4074:
3742:
3653:
3621:
3525:
3410:
3328:
2700:
2540:{\displaystyle y=f}
2024:{\displaystyle y'.}
1692:
1610:necessary condition
1255:dynamic programming
618:Cauchy condensation
420:Contour integration
146:Fundamental theorem
73:
22273:Mathematics portal
22156:Lists of integrals
21969:(cs, bcs)-complete
21940:Algebraic interior
21658:Convex combination
21486:Nash–Moser theorem
21363:Canonical Gaussian
21310:Gateaux derivative
21293:Fréchet derivative
20967:Dacorogna, Bernard
20942:Cassel, Kevin W.:
20740:10.1007/BF00276295
20173:
20172:
20146:
20095:
20075:
20055:
20035:
20000:
19956:
19946:are functions and
19936:
19903:
19825: and
19815:
19761:
19724:
19683:
19651:
19634:{\displaystyle h.}
19631:
19608:
19588:
19507:
19483:
19467:and is denoted by
19457:
19437:
19394:{\displaystyle x,}
19391:
19364:
19335:
19266:
19246:
19204:
19009:Fermat's principle
18967:
18929:
18890:
18855:
18817:
18782:
18740:
18714:
18694:
18626:
18585:
18521:
18492:
18456:
18424:
18398:
18362:
18326:
18221:
18190:
18140:
18111:
18082:
18050:
18027:{\displaystyle h,}
18024:
18001:
17975:
17946:
17877:
17846:
17778:{\displaystyle y,}
17775:
17752:
17717:
17688:
17668:
17633:
17558:Hamiltonian optics
17467:
17375:{\displaystyle X.}
17372:
17349:
17314:
17215:
17195:
17175:
17143:
17102:
17044:
16989:
16966:
16879:
16847:
16778:
16747:
16723:
16703:
16671:{\displaystyle L.}
16668:
16648:{\displaystyle S,}
16645:
16603:
16583:
16561:
16480:
16430:
16377:
16357:
16321:
16284:
16236:
16214:
16153:
16097:
16083:{\displaystyle X.}
16080:
16057:
16037:
15975:Hamiltonian optics
15959:
15936:
15916:
15906:then the integral
15899:{\displaystyle P,}
15896:
15873:
15851:
15794:
15760:
15709:
15687:
15602:
15526:{\displaystyle C.}
15523:
15498:
15416:
15384:
15349:{\displaystyle C,}
15346:
15323:
15294:
15274:
15194:
15174:
15141:
15121:
15086:
15039:
15002:
14941:
14904:
14809:{\displaystyle f'}
14806:
14781:
14762:
14733:
14704:
14678:
14645:
14612:
14607:
14483:Hamiltonian optics
14465:
14428:
14358:
14337:
14268:
14223:
14138:
14108:
14086:
14024:
13969:
13949:
13925:
13849:
13814:
13715:
13683:
13648:
13636:Fermat's principle
13614:{\displaystyle B.}
13611:
13588:
13512:
13492:
13434:
13336:
13319:{\displaystyle B.}
13316:
13293:
13244:
13224:
13127:
12967:
12947:
12915:
12831:
12721:
12701:
12681:
12646:
12571:
12551:
12531:
12215:
12164:
12118:
11835:
11779:
11751:
11716:
11689:
11662:
11465:
11433:
11410:
11325:
11308:
11284:
11260:
11221:
11194:
11174:
11154:
11096:
11076:
11004:
10984:
10955:
10926:
10906:
10878:
10849:
10820:
10750:
10718:
10698:
10627:
10607:
10488:
10464:eigenvalue problem
10422:
10349:
10333:{\displaystyle c,}
10330:
10307:
10209:
10189:
10163:{\displaystyle C.}
10160:
10137:
10110:
10093:{\displaystyle C.}
10090:
10067:
10006:
9986:
9969:{\displaystyle D.}
9966:
9943:
9902:{\displaystyle C,}
9899:
9876:
9850:
9698:
9569:
9534:{\displaystyle v.}
9531:
9508:
9488:
9468:
9451:{\displaystyle u.}
9448:
9424:{\displaystyle C.}
9421:
9398:
9372:{\displaystyle C,}
9369:
9346:
9327:an external force
9320:{\displaystyle D,}
9317:
9294:
9259:
9048:
9022:
9003:
8968:
8927:
8907:
8858:
8817:
8798:{\displaystyle D.}
8795:
8772:
8740:{\displaystyle D.}
8737:
8714:
8694:
8632:
8612:
8595:{\displaystyle C.}
8592:
8569:
8549:
8515:
8495:
8479:{\displaystyle D,}
8476:
8453:
8433:
8265:
8142:
8107:{\displaystyle D,}
8104:
8081:
8061:
8044:{\displaystyle D.}
8041:
8018:
7998:
7978:
7886:
7823:
7780:
7737:
7714:
7630:
7604:
7584:
7537:
7496:
7463:
7430:
7394:
7358:
7319:
7270:
7155:
7093:
7044:
7024:
6959:
6935:
6908:
6712:
6692:
6587:
6564:
6532:
6500:
6457:
6435:
6398:
6374:
6354:
6285:
6265:
6231:
6202:
6148:
6110:
6081:
6052:
6023:
5820:
5770:
5720:
5680:
5598:
5559:
5454:{\displaystyle c.}
5451:
5438:for some constant
5428:
5338:
5229:
5209:
5142:
5116:{\displaystyle L,}
5113:
5090:
5068:
5001:
4919:{\displaystyle A.}
4916:
4884:
4844:
4699:
4619:
4583:
4530:
4480:
4433:{\displaystyle J.}
4430:
4390:
4352:
4306:
4269:
4177:
4046:
4029:
4002:
3975:
3945:
3919:
3833:
3831:
3714:
3625:
3559:
3497:
3382:
3300:
3277:
3176:
3120:
3077:
2947:{\displaystyle x,}
2944:
2921:
2901:
2851:
2816:
2757:
2672:
2592:
2566:
2537:
2521:has a minimum for
2511:
2482:
2424:
2404:{\displaystyle J,}
2401:
2379:in the functional
2369:
2349:
2318:
2302:and is denoted by
2292:
2268:
2243:
2186:
2166:
2136:
2109:
2083:{\displaystyle f,}
2080:
2053:
2034:If the functional
2021:
1993:
1964:
1895:
1833:
1798:
1755:
1664:
1582:
1556:{\displaystyle f,}
1553:
1530:
1501:
1468:
1451:{\displaystyle f.}
1448:
1425:
1401:
1348:
1328:
1291:
1215:Gilbert Ames Bliss
1100:problem raised by
1046:Fermat's principle
1021:definite integrals
790:Partial derivative
719:generalized Stokes
613:Alternating series
494:Reduction formulae
483:Heaviside's method
464:tangent half-angle
451:Cylindrical shells
374:Integral transform
369:Lists of integrals
173:Mean value theorem
128:
59:
22280:
22279:
22246:Special functions
22209:Harmonic analysis
22047:
22046:
21606:
21605:
21503:Sazonov's theorem
21389:Projection-valued
21089:978-3-527-41188-7
21043:Logan, J. David:
21025:978-3-662-06201-2
21017:978-3-662-03278-7
20985:978-1-78326-551-0
20937:978-1-4181-8201-4
20685:978-1-4704-1495-5
20435:Dimitri Bertsekas
20397:978-0-387-40247-5
20125:
20098:{\displaystyle h}
20078:{\displaystyle b}
20058:{\displaystyle a}
19966:is a real number.
19654:{\displaystyle y}
19611:{\displaystyle y}
19460:{\displaystyle J}
19269:{\displaystyle h}
19207:{\displaystyle f}
19181:Harold J. Kushner
19119:Stampacchia Medal
19084:Chaplygin problem
19069:Noether's theorem
18961:
18887:
18814:
18792:has a minimum at
18785:{\displaystyle J}
18717:{\displaystyle h}
18638:strongly positive
18524:{\displaystyle J}
18224:{\displaystyle J}
18143:{\displaystyle J}
17880:{\displaystyle J}
17691:{\displaystyle y}
17636:{\displaystyle J}
17533:Plateau's problem
17448:
17411:
17299:
17272:
17218:{\displaystyle H}
17198:{\displaystyle L}
17172:
17137:
17090:
17075:
17039:
17035:
16992:{\displaystyle P}
16961:
16938:
16934:
16909:
16828:
16750:{\displaystyle U}
16726:{\displaystyle T}
16606:{\displaystyle A}
16556:
16552:
16549:
16534:
16517:
16469:
16412:
16380:{\displaystyle P}
16060:{\displaystyle c}
15971:Lagrangian optics
15919:{\displaystyle A}
15838:
15712:{\displaystyle P}
15682:
15681:
15678:
15663:
15649:
15590:
15587:
15572:
15553:
15486:
15483:
15468:
15372:
15297:{\displaystyle t}
15197:{\displaystyle x}
15144:{\displaystyle x}
15076:
15075:
14978:
14977:
14784:{\displaystyle f}
14590:
14550:
14479:Lagrangian optics
14457:
14379:
14378:
14303:
14252:
14168:
14167:
13972:{\displaystyle A}
13952:{\displaystyle A}
13803:
13651:{\displaystyle x}
13559:
13515:{\displaystyle u}
13487:
13339:{\displaystyle u}
13247:{\displaystyle u}
12970:{\displaystyle B}
12950:{\displaystyle D}
12830:
12724:{\displaystyle v}
12704:{\displaystyle u}
12645:
12574:{\displaystyle v}
12554:{\displaystyle v}
12200:
11828:
11436:{\displaystyle y}
11311:{\displaystyle Q}
11287:{\displaystyle u}
11177:{\displaystyle u}
11149:
11007:{\displaystyle u}
10987:{\displaystyle R}
10958:{\displaystyle Q}
10929:{\displaystyle y}
10721:{\displaystyle R}
10630:{\displaystyle y}
10352:{\displaystyle V}
10212:{\displaystyle c}
10113:{\displaystyle u}
10041:
10009:{\displaystyle u}
9989:{\displaystyle v}
9976:Then if we allow
9815:
9572:{\displaystyle V}
9511:{\displaystyle u}
9491:{\displaystyle g}
9471:{\displaystyle f}
9202:
9115:
9025:{\displaystyle W}
8820:{\displaystyle u}
8717:{\displaystyle v}
8635:{\displaystyle C}
8615:{\displaystyle v}
8572:{\displaystyle u}
8518:{\displaystyle C}
8498:{\displaystyle s}
8456:{\displaystyle C}
8421:
8177:
8145:{\displaystyle V}
8084:{\displaystyle v}
8064:{\displaystyle u}
8001:{\displaystyle V}
7934:
7740:{\displaystyle D}
7725:Plateau's problem
7695:
7607:{\displaystyle D}
7316:
7245:
7047:{\displaystyle f}
7013:
6962:{\displaystyle L}
6938:{\displaystyle J}
6896:
6859:
6798:
6769:
6751:
6715:{\displaystyle y}
6535:{\displaystyle S}
6512:Noether's theorem
6492:
6460:{\displaystyle x}
6401:{\displaystyle L}
6377:{\displaystyle C}
6342:
6297:Beltrami identity
6288:{\displaystyle x}
6191:
6113:{\displaystyle A}
6084:{\displaystyle A}
6021:
5940:
5934:
5875:
5872:
5868:
5678:
5540:
5416:
5415:
5327:
5323:
5322:
5263:
5259:
5232:{\displaystyle L}
5219:Substituting for
5197:
5172:
5093:{\displaystyle f}
5062:
4993:
4968:
4950:
4858:is a function of
4807:
4804:
4765:
4762:
4754:
4686:
4309:{\displaystyle J}
4261:
4236:
4218:
4189:According to the
4153:
4128:
4110:
3955:and we have used
3815:
3790:
3766:
3695:
3670:
3587:
3544:
3465:
3434:
3351:
3264:
3233:
3210:
3160:
3112:
3075:
3050:
3022:
3002:
2979:
2723:
2651:
2576:has a minimum at
2514:{\displaystyle J}
2372:{\displaystyle y}
2295:{\displaystyle f}
2056:{\displaystyle J}
1890:
1504:{\displaystyle J}
1471:{\displaystyle f}
1428:{\displaystyle y}
1351:{\displaystyle f}
1331:{\displaystyle J}
1294:{\displaystyle y}
1247:Ralph Rockafellar
1145:Vincenzo Brunacci
1141:Gottfried Leibniz
1077:Plateau's problem
985:
984:
865:
864:
827:
826:
795:Multiple integral
731:
730:
635:
634:
602:Direct comparison
573:Convergence tests
511:
510:
479:Partial fractions
346:
345:
256:Second derivative
16:(Redirected from
22305:
22199:Fourier analysis
22179:Complex analysis
22080:Major topics in
22074:
22067:
22060:
22051:
22050:
21965:(cs, lcs)-closed
21911:Effective domain
21866:Robinson–Ursescu
21742:Convex conjugate
21633:
21626:
21619:
21610:
21609:
21598:Hilbert manifold
21593:Fréchet manifold
21377: like
21337:Quasi-derivative
21203:
21196:
21189:
21180:
21179:
21070:
21065:. Archived from
20902:
20896:
20890:
20884:
20878:
20872:
20866:
20865:, pp. 97–98
20860:
20854:
20853:, pp. 11–12
20848:
20842:
20836:
20830:
20824:
20818:
20812:
20806:
20800:
20794:
20793:
20785:
20779:
20778:
20758:
20752:
20751:
20715:
20709:
20708:
20696:
20690:
20689:
20671:
20665:
20664:
20652:
20646:
20645:
20632:
20626:
20625:
20600:
20594:
20593:, pp. 14–15
20588:
20582:
20576:
20570:
20569:, pp. 12–13
20564:
20558:
20557:
20532:
20526:
20525:
20523:
20522:
20503:
20497:
20496:
20486:
20476:
20444:
20438:
20432:
20426:
20425:
20423:
20411:
20402:
20401:
20383:
20372:
20371:
20348:
20342:
20341:
20319:
20310:
20309:
20282:
20276:
20270:
20252:
20249:
20243:
20239:
20233:
20225:
20219:
20206:
20200:
20189:
20183:
20182:
20180:
20179:
20174:
20168:
20151:
20145:
20104:
20102:
20101:
20096:
20084:
20082:
20081:
20076:
20064:
20062:
20061:
20056:
20044:
20042:
20041:
20036:
20009:
20007:
20006:
20001:
19973:
19967:
19965:
19963:
19962:
19957:
19945:
19943:
19942:
19937:
19935:
19934:
19912:
19910:
19909:
19904:
19899:
19895:
19894:
19860:
19856:
19855:
19854:
19824:
19822:
19821:
19816:
19770:
19768:
19767:
19762:
19740:
19734:
19733:
19731:
19730:
19725:
19692:
19690:
19689:
19684:
19660:
19658:
19657:
19652:
19640:
19638:
19637:
19632:
19617:
19615:
19614:
19609:
19597:
19595:
19594:
19589:
19564:
19558:
19555:
19549:
19546:
19540:
19537:
19531:
19524:
19518:
19516:
19514:
19513:
19508:
19492:
19490:
19489:
19484:
19466:
19464:
19463:
19458:
19446:
19444:
19443:
19438:
19427:
19408:
19402:
19400:
19398:
19397:
19392:
19373:
19371:
19370:
19365:
19344:
19342:
19341:
19336:
19314:
19308:
19305:
19299:
19292:
19286:
19283:
19277:
19275:
19273:
19272:
19267:
19255:
19253:
19252:
19247:
19239:
19225:
19213:
19211:
19210:
19205:
19192:
19186:
19176:
19170:
19167:
19161:
19150:
19044:Obstacle problem
18976:
18974:
18973:
18968:
18963:
18962:
18954:
18938:
18936:
18935:
18930:
18916:
18915:
18899:
18897:
18896:
18891:
18889:
18888:
18880:
18864:
18862:
18861:
18856:
18826:
18824:
18823:
18818:
18816:
18815:
18807:
18791:
18789:
18788:
18783:
18749:
18747:
18746:
18741:
18723:
18721:
18720:
18715:
18703:
18701:
18700:
18695:
18690:
18689:
18656:
18655:
18635:
18633:
18632:
18627:
18613:
18612:
18594:
18592:
18591:
18586:
18572:
18571:
18547:
18546:
18530:
18528:
18527:
18522:
18501:
18499:
18498:
18493:
18482:
18481:
18465:
18463:
18462:
18457:
18433:
18431:
18430:
18425:
18407:
18405:
18404:
18399:
18388:
18387:
18371:
18369:
18368:
18363:
18352:
18351:
18335:
18333:
18332:
18327:
18322:
18321:
18291:
18290:
18269:
18268:
18230:
18228:
18227:
18222:
18199:
18197:
18196:
18191:
18149:
18147:
18146:
18141:
18120:
18118:
18117:
18112:
18091:
18089:
18088:
18083:
18059:
18057:
18056:
18051:
18033:
18031:
18030:
18025:
18010:
18008:
18007:
18002:
17984:
17982:
17981:
17976:
17955:
17953:
17952:
17947:
17886:
17884:
17883:
17878:
17855:
17853:
17852:
17847:
17784:
17782:
17781:
17776:
17761:
17759:
17758:
17753:
17726:
17724:
17723:
17718:
17697:
17695:
17694:
17689:
17677:
17675:
17674:
17669:
17642:
17640:
17639:
17634:
17614:For example, if
17609:second variation
17592:image processing
17529:minimal surfaces
17508:Solution to the
17501:Solution to the
17476:
17474:
17473:
17468:
17460:
17456:
17449:
17447:
17439:
17431:
17412:
17410:
17402:
17394:
17381:
17379:
17378:
17373:
17358:
17356:
17355:
17350:
17323:
17321:
17320:
17315:
17301:
17300:
17292:
17274:
17273:
17265:
17224:
17222:
17221:
17216:
17204:
17202:
17201:
17196:
17184:
17182:
17181:
17176:
17174:
17173:
17165:
17152:
17150:
17149:
17144:
17139:
17138:
17130:
17111:
17109:
17108:
17103:
17098:
17097:
17092:
17091:
17083:
17076:
17068:
17054:For example, if
17053:
17051:
17050:
17045:
17040:
17038:
17037:
17036:
17028:
17021:
17013:
16998:
16996:
16995:
16990:
16975:
16973:
16972:
16967:
16962:
16960:
16952:
16944:
16939:
16937:
16936:
16935:
16927:
16920:
16912:
16910:
16908:
16897:
16888:
16886:
16885:
16880:
16856:
16854:
16853:
16848:
16830:
16829:
16821:
16805:
16804:
16803:
16793:
16792:
16791:
16759:potential energy
16756:
16754:
16753:
16748:
16732:
16730:
16729:
16724:
16712:
16710:
16709:
16704:
16677:
16675:
16674:
16669:
16654:
16652:
16651:
16646:
16628:Action (physics)
16612:
16610:
16609:
16604:
16592:
16590:
16589:
16584:
16570:
16568:
16567:
16562:
16557:
16551:
16550:
16542:
16536:
16535:
16527:
16524:
16523:
16518:
16516:
16508:
16500:
16489:
16487:
16486:
16481:
16470:
16468:
16460:
16452:
16439:
16437:
16436:
16431:
16413:
16411:
16403:
16395:
16386:
16384:
16383:
16378:
16366:
16364:
16363:
16358:
16330:
16328:
16327:
16322:
16314:
16293:
16291:
16290:
16285:
16280:
16279:
16245:
16243:
16242:
16237:
16223:
16221:
16220:
16215:
16162:
16160:
16159:
16154:
16133:
16132:
16110:
16105:
16089:
16087:
16086:
16081:
16066:
16064:
16063:
16058:
16046:
16044:
16043:
16038:
16021:
16020:
16008:
16007:
15968:
15966:
15965:
15960:
15945:
15943:
15942:
15937:
15925:
15923:
15922:
15917:
15905:
15903:
15902:
15897:
15882:
15880:
15879:
15874:
15860:
15858:
15857:
15852:
15840:
15839:
15831:
15821:
15820:
15819:
15809:
15808:
15807:
15769:
15767:
15766:
15761:
15756:
15755:
15718:
15716:
15715:
15710:
15696:
15694:
15693:
15688:
15683:
15680:
15679:
15671:
15665:
15664:
15656:
15653:
15652:
15651:
15650:
15642:
15626:
15611:
15609:
15608:
15603:
15591:
15589:
15588:
15580:
15574:
15573:
15565:
15562:
15554:
15552:
15541:
15532:
15530:
15529:
15524:
15507:
15505:
15504:
15499:
15487:
15485:
15484:
15476:
15470:
15469:
15461:
15458:
15443:
15442:
15441:
15431:
15430:
15429:
15393:
15391:
15390:
15385:
15374:
15373:
15365:
15355:
15353:
15352:
15347:
15332:
15330:
15329:
15324:
15303:
15301:
15300:
15295:
15283:
15281:
15280:
15275:
15267:
15266:
15254:
15253:
15241:
15240:
15203:
15201:
15200:
15195:
15183:
15181:
15180:
15175:
15173:
15172:
15150:
15148:
15147:
15142:
15130:
15128:
15127:
15122:
15120:
15119:
15095:
15093:
15092:
15087:
15082:
15078:
15077:
15074:
15073:
15064:
15063:
15047:
15032:
15031:
15027:
15026:
15010:
15000:
14998:
14997:
14979:
14976:
14975:
14966:
14965:
14949:
14934:
14933:
14929:
14928:
14912:
14902:
14900:
14899:
14870:
14869:
14854:
14853:
14841:
14840:
14815:
14813:
14812:
14807:
14805:
14790:
14788:
14787:
14782:
14771:
14769:
14768:
14763:
14742:
14740:
14739:
14734:
14713:
14711:
14710:
14705:
14687:
14685:
14684:
14679:
14677:
14676:
14654:
14652:
14651:
14646:
14644:
14643:
14621:
14619:
14618:
14613:
14611:
14610:
14591:
14588:
14584:
14583:
14551:
14548:
14544:
14543:
14474:
14472:
14471:
14466:
14458:
14456:
14455:
14436:
14421:
14416:
14415:
14397:
14396:
14384:
14380:
14376:
14375:
14366:
14351:
14350:
14345:
14333:
14332:
14310:
14304:
14302:
14291:
14277:
14275:
14274:
14269:
14258:
14254:
14253:
14251:
14250:
14231:
14216:
14214:
14213:
14201:
14200:
14182:
14181:
14169:
14166:
14165:
14146:
14131:
14130:
14116:
14094:
14082:
14081:
14059:
14051:
14050:
14049:
14039:
14038:
14037:
14017:
14016:
14004:
14003:
13978:
13976:
13975:
13970:
13958:
13956:
13955:
13950:
13934:
13932:
13931:
13926:
13915:
13914:
13890:
13889:
13858:
13856:
13855:
13850:
13823:
13821:
13820:
13815:
13804:
13802:
13801:
13786:
13772:
13742:
13741:
13740:
13730:
13729:
13728:
13692:
13690:
13689:
13684:
13657:
13655:
13654:
13649:
13622:Hilbert (1953).
13620:
13618:
13617:
13612:
13598:on the boundary
13597:
13595:
13594:
13589:
13560:
13558:
13550:
13542:
13521:
13519:
13518:
13513:
13501:
13499:
13498:
13493:
13488:
13486:
13472:
13458:
13443:
13441:
13440:
13435:
13345:
13343:
13342:
13337:
13325:
13323:
13322:
13317:
13302:
13300:
13299:
13294:
13277:
13253:
13251:
13250:
13245:
13233:
13231:
13230:
13225:
13199:
13198:
13168:
13167:
13136:
13134:
13133:
13128:
13116:
13115:
13094:
13093:
13060:
13059:
13008:
13007:
12976:
12974:
12973:
12968:
12956:
12954:
12953:
12948:
12924:
12922:
12921:
12916:
12905:
12904:
12889:
12888:
12873:
12872:
12860:
12849:
12848:
12832:
12828:
12812:
12811:
12796:
12795:
12780:
12779:
12767:
12756:
12755:
12730:
12728:
12727:
12722:
12710:
12708:
12707:
12702:
12690:
12688:
12687:
12682:
12677:
12676:
12658:
12657:
12647:
12643:
12612:
12604:
12580:
12578:
12577:
12572:
12560:
12558:
12557:
12552:
12540:
12538:
12537:
12532:
12521:
12520:
12505:
12504:
12489:
12488:
12476:
12465:
12464:
12443:
12442:
12418:
12417:
12402:
12401:
12386:
12385:
12373:
12362:
12361:
12337:
12336:
12311:
12307:
12285:
12277:
12242:
12241:
12240:
12230:
12229:
12228:
12211:
12210:
12201:
12196:
12182:
12173:
12171:
12170:
12165:
12151:
12127:
12125:
12124:
12119:
12114:
12110:
12106:
12105:
12087:
12086:
12071:
12070:
12055:
12054:
12036:
12035:
12020:
12019:
12000:
11996:
11905:
11888:
11862:
11861:
11860:
11850:
11849:
11848:
11829:
11827:
11810:
11805:
11804:
11788:
11786:
11785:
11780:
11775:
11760:
11758:
11757:
11752:
11725:
11723:
11722:
11717:
11715:
11714:
11698:
11696:
11695:
11690:
11688:
11687:
11671:
11669:
11668:
11663:
11658:
11657:
11648:
11647:
11632:
11631:
11619:
11618:
11609:
11608:
11593:
11592:
11573:
11569:
11568:
11567:
11534:
11533:
11518:
11492:
11491:
11490:
11480:
11479:
11478:
11442:
11440:
11439:
11434:
11419:
11417:
11416:
11411:
11375:
11374:
11352:
11351:
11350:
11340:
11339:
11338:
11317:
11315:
11314:
11309:
11293:
11291:
11290:
11285:
11269:
11267:
11266:
11261:
11247:
11246:
11230:
11228:
11227:
11222:
11220:
11219:
11203:
11201:
11200:
11195:
11183:
11181:
11180:
11175:
11163:
11161:
11160:
11155:
11150:
11148:
11134:
11120:
11106:is the quotient
11105:
11103:
11102:
11097:
11085:
11083:
11082:
11077:
11045:
11037:
11013:
11011:
11010:
11005:
10993:
10991:
10990:
10985:
10964:
10962:
10961:
10956:
10935:
10933:
10932:
10927:
10915:
10913:
10912:
10907:
10902:
10887:
10885:
10884:
10879:
10858:
10856:
10855:
10850:
10829:
10827:
10826:
10821:
10809:
10808:
10777:
10776:
10775:
10765:
10764:
10763:
10727:
10725:
10724:
10719:
10707:
10705:
10704:
10699:
10688:
10687:
10659:
10658:
10636:
10634:
10633:
10628:
10616:
10614:
10613:
10608:
10596:
10592:
10591:
10590:
10557:
10556:
10541:
10515:
10514:
10513:
10503:
10502:
10501:
10431:
10429:
10428:
10423:
10405:
10404:
10375:
10374:
10358:
10356:
10355:
10350:
10339:
10337:
10336:
10331:
10316:
10314:
10313:
10308:
10303:
10299:
10288:
10287:
10258:
10257:
10218:
10216:
10215:
10210:
10198:
10196:
10195:
10190:
10169:
10167:
10166:
10161:
10146:
10144:
10143:
10138:
10119:
10117:
10116:
10111:
10099:
10097:
10096:
10091:
10076:
10074:
10073:
10068:
10042:
10040:
10032:
10024:
10015:
10013:
10012:
10007:
9995:
9993:
9992:
9987:
9975:
9973:
9972:
9967:
9952:
9950:
9949:
9944:
9908:
9906:
9905:
9900:
9885:
9883:
9882:
9877:
9860:If we first set
9859:
9857:
9856:
9851:
9836:
9832:
9816:
9814:
9806:
9798:
9788:
9787:
9761:
9757:
9724:
9723:
9707:
9705:
9704:
9699:
9684:
9680:
9656:
9655:
9629:
9625:
9595:
9594:
9578:
9576:
9575:
9570:
9540:
9538:
9537:
9532:
9517:
9515:
9514:
9509:
9497:
9495:
9494:
9489:
9477:
9475:
9474:
9469:
9457:
9455:
9454:
9449:
9430:
9428:
9427:
9422:
9407:
9405:
9404:
9399:
9378:
9376:
9375:
9370:
9356:on the boundary
9355:
9353:
9352:
9347:
9326:
9324:
9323:
9318:
9303:
9301:
9300:
9295:
9268:
9266:
9265:
9260:
9248:
9244:
9225:
9224:
9203:
9195:
9188:
9187:
9160:
9156:
9116:
9108:
9101:
9100:
9057:
9055:
9054:
9049:
9031:
9029:
9028:
9023:
9012:
9010:
9009:
9004:
8977:
8975:
8974:
8969:
8936:
8934:
8933:
8928:
8916:
8914:
8913:
8908:
8899:
8898:
8889:
8874:
8869:
8826:
8824:
8823:
8818:
8804:
8802:
8801:
8796:
8781:
8779:
8778:
8773:
8746:
8744:
8743:
8738:
8723:
8721:
8720:
8715:
8703:
8701:
8700:
8695:
8658:
8657:
8641:
8639:
8638:
8633:
8621:
8619:
8618:
8613:
8601:
8599:
8598:
8593:
8578:
8576:
8575:
8570:
8558:
8556:
8555:
8550:
8542:
8524:
8522:
8521:
8516:
8504:
8502:
8501:
8496:
8485:
8483:
8482:
8477:
8462:
8460:
8459:
8454:
8442:
8440:
8439:
8434:
8422:
8420:
8412:
8404:
8399:
8398:
8339:
8338:
8291:
8290:
8274:
8272:
8271:
8266:
8229:
8228:
8216:
8215:
8204:
8200:
8178:
8176:
8165:
8151:
8149:
8148:
8143:
8113:
8111:
8110:
8105:
8090:
8088:
8087:
8082:
8070:
8068:
8067:
8062:
8050:
8048:
8047:
8042:
8027:
8025:
8024:
8019:
8007:
8005:
8004:
7999:
7987:
7985:
7984:
7979:
7945:
7944:
7935:
7927:
7895:
7893:
7892:
7887:
7879:
7878:
7866:
7865:
7856:
7855:
7836:
7831:
7813:
7812:
7793:
7788:
7770:
7769:
7749:minimal surfaces
7746:
7744:
7743:
7738:
7723:
7721:
7720:
7715:
7696:
7673:
7671:
7670:
7639:
7637:
7636:
7631:
7613:
7611:
7610:
7605:
7593:
7591:
7590:
7585:
7559:For example, if
7546:
7544:
7543:
7538:
7505:
7503:
7502:
7497:
7495:
7494:
7472:
7470:
7469:
7464:
7462:
7461:
7439:
7437:
7436:
7431:
7426:
7425:
7403:
7401:
7400:
7395:
7393:
7392:
7367:
7365:
7364:
7359:
7357:
7356:
7328:
7326:
7325:
7320:
7318:
7317:
7309:
7279:
7277:
7276:
7271:
7243:
7203:
7202:
7175:
7164:
7162:
7161:
7156:
7151:
7150:
7149:
7136:
7135:
7120:
7119:
7106:
7101:
7053:
7051:
7050:
7045:
7033:
7031:
7030:
7025:
7014:
7012:
7011:
7010:
7009:
6992:
6988:
6987:
6977:
6968:
6966:
6965:
6960:
6944:
6942:
6941:
6936:
6917:
6915:
6914:
6909:
6901:
6897:
6895:
6894:
6893:
6874:
6866:
6860:
6858:
6857:
6856:
6843:
6842:
6833:
6831:
6830:
6803:
6799:
6797:
6796:
6784:
6776:
6770:
6768:
6757:
6752:
6750:
6742:
6734:
6721:
6719:
6718:
6713:
6701:
6699:
6698:
6693:
6670:
6669:
6636:
6600:
6595:
6573:
6571:
6570:
6565:
6541:
6539:
6538:
6533:
6509:
6507:
6506:
6501:
6493:
6491:
6483:
6475:
6466:
6464:
6463:
6458:
6444:
6442:
6441:
6436:
6422:
6408:with respect to
6407:
6405:
6404:
6399:
6383:
6381:
6380:
6375:
6363:
6361:
6360:
6355:
6343:
6341:
6340:
6328:
6320:
6318:
6294:
6292:
6291:
6286:
6274:
6272:
6271:
6266:
6255:
6240:
6238:
6237:
6232:
6211:
6209:
6208:
6203:
6192:
6190:
6182:
6174:
6157:
6155:
6154:
6149:
6119:
6117:
6116:
6111:
6090:
6088:
6087:
6082:
6061:
6059:
6058:
6053:
6032:
6030:
6029:
6024:
6022:
6020:
6019:
6018:
6006:
6005:
5995:
5994:
5993:
5984:
5983:
5971:
5970:
5961:
5960:
5950:
5941:
5938:
5935:
5933:
5932:
5931:
5919:
5918:
5908:
5907:
5906:
5894:
5893:
5883:
5873:
5870:
5869:
5866:
5829:
5827:
5826:
5821:
5819:
5815:
5814:
5813:
5801:
5800:
5779:
5777:
5776:
5771:
5769:
5765:
5764:
5763:
5751:
5750:
5729:
5727:
5726:
5721:
5704:
5689:
5687:
5686:
5681:
5679:
5677:
5676:
5675:
5659:
5658:
5649:
5644:
5643:
5625:
5608:Solving, we get
5607:
5605:
5604:
5599:
5591:
5590:
5568:
5566:
5565:
5560:
5554:
5553:
5541:
5539:
5538:
5537:
5519:
5501:
5500:
5499:
5481:
5469:
5460:
5458:
5457:
5452:
5437:
5435:
5434:
5429:
5417:
5414:
5413:
5395:
5378:
5377:
5367:
5358:
5347:
5345:
5344:
5339:
5325:
5324:
5321:
5320:
5302:
5285:
5284:
5274:
5265:
5261:
5260:
5258:
5247:
5238:
5236:
5235:
5230:
5218:
5216:
5215:
5210:
5198:
5196:
5195:
5183:
5175:
5173:
5171:
5160:
5151:
5149:
5148:
5143:
5122:
5120:
5119:
5114:
5099:
5097:
5096:
5091:
5077:
5075:
5074:
5069:
5063:
5061:
5060:
5042:
5025:
5010:
5008:
5007:
5002:
4994:
4992:
4991:
4979:
4971:
4969:
4967:
4956:
4951:
4949:
4941:
4933:
4925:
4923:
4922:
4917:
4893:
4891:
4890:
4885:
4861:
4857:
4853:
4851:
4850:
4845:
4836:
4835:
4817:
4816:
4805:
4802:
4794:
4793:
4775:
4774:
4763:
4760:
4755:
4753:
4745:
4737:
4723:
4708:
4706:
4705:
4700:
4687:
4685:
4684:
4666:
4649:
4646:
4645:
4644:
4634:
4633:
4632:
4592:
4590:
4589:
4584:
4579:
4575:
4574:
4573:
4561:
4560:
4539:
4537:
4536:
4531:
4529:
4525:
4524:
4523:
4511:
4510:
4489:
4487:
4486:
4481:
4439:
4437:
4436:
4431:
4399:
4397:
4396:
4391:
4361:
4359:
4358:
4353:
4333:
4315:
4313:
4312:
4307:
4278:
4276:
4275:
4270:
4262:
4260:
4259:
4247:
4239:
4237:
4235:
4224:
4219:
4217:
4209:
4201:
4186:
4184:
4183:
4178:
4159:
4155:
4154:
4152:
4151:
4139:
4131:
4129:
4127:
4116:
4111:
4109:
4101:
4093:
4073:
4072:
4071:
4061:
4060:
4059:
4038:
4036:
4035:
4030:
4028:
4027:
4011:
4009:
4008:
4003:
4001:
4000:
3984:
3982:
3981:
3976:
3954:
3952:
3951:
3946:
3928:
3926:
3925:
3920:
3918:
3914:
3913:
3882:
3878:
3877:
3842:
3840:
3839:
3834:
3832:
3821:
3817:
3816:
3814:
3813:
3801:
3793:
3791:
3789:
3778:
3767:
3765:
3757:
3749:
3741:
3740:
3739:
3729:
3728:
3727:
3707:
3696:
3694:
3693:
3681:
3673:
3671:
3669:
3658:
3652:
3651:
3650:
3640:
3639:
3638:
3620:
3619:
3618:
3608:
3607:
3606:
3596:
3592:
3588:
3586:
3585:
3573:
3565:
3545:
3543:
3535:
3527:
3524:
3523:
3522:
3512:
3511:
3510:
3490:
3479:
3475:
3474:
3466:
3464:
3463:
3451:
3443:
3435:
3433:
3425:
3417:
3409:
3408:
3407:
3397:
3396:
3395:
3368:
3367:
3356:
3352:
3350:
3342:
3334:
3327:
3326:
3325:
3315:
3314:
3313:
3286:
3284:
3283:
3278:
3273:
3265:
3263:
3262:
3250:
3242:
3234:
3232:
3224:
3216:
3211:
3209:
3201:
3193:
3185:
3183:
3182:
3177:
3172:
3161:
3159:
3151:
3150:
3138:
3129:
3127:
3126:
3121:
3113:
3111:
3103:
3095:
3086:
3084:
3083:
3078:
3076:
3074:
3066:
3065:
3053:
3051:
3049:
3048:
3036:
3028:
3023:
3021:
3013:
3005:
3003:
3001:
2993:
2985:
2980:
2978:
2970:
2962:
2953:
2951:
2950:
2945:
2930:
2928:
2927:
2922:
2910:
2908:
2907:
2902:
2900:
2886:
2875:
2860:
2858:
2857:
2852:
2825:
2823:
2822:
2817:
2812:
2808:
2807:
2771:total derivative
2766:
2764:
2763:
2758:
2740:
2739:
2728:
2724:
2722:
2714:
2706:
2699:
2698:
2697:
2687:
2686:
2685:
2668:
2667:
2656:
2652:
2650:
2642:
2634:
2616:
2601:
2599:
2598:
2593:
2575:
2573:
2572:
2567:
2546:
2544:
2543:
2538:
2520:
2518:
2517:
2512:
2491:
2489:
2488:
2483:
2433:
2431:
2430:
2425:
2410:
2408:
2407:
2402:
2378:
2376:
2375:
2370:
2358:
2356:
2355:
2350:
2327:
2325:
2324:
2319:
2301:
2299:
2298:
2293:
2282:of the function
2277:
2275:
2274:
2269:
2252:
2250:
2249:
2244:
2195:
2193:
2192:
2187:
2175:
2173:
2172:
2167:
2162:
2161:
2145:
2143:
2142:
2137:
2135:
2134:
2118:
2116:
2115:
2110:
2089:
2087:
2086:
2081:
2062:
2060:
2059:
2054:
2030:
2028:
2027:
2022:
2017:
2002:
2000:
1999:
1994:
1973:
1971:
1970:
1965:
1963:
1959:
1949:
1904:
1902:
1901:
1896:
1891:
1889:
1881:
1873:
1859:
1842:
1840:
1839:
1834:
1807:
1805:
1804:
1799:
1797:
1796:
1784:
1783:
1764:
1762:
1761:
1756:
1743:
1739:
1729:
1691:
1690:
1689:
1679:
1678:
1677:
1591:
1589:
1588:
1583:
1562:
1560:
1559:
1554:
1539:
1537:
1536:
1531:
1510:
1508:
1507:
1502:
1477:
1475:
1474:
1469:
1457:
1455:
1454:
1449:
1434:
1432:
1431:
1426:
1410:
1408:
1407:
1402:
1357:
1355:
1354:
1349:
1337:
1335:
1334:
1329:
1300:
1298:
1297:
1292:
1231:Jacques Hadamard
1102:Johann Bernoulli
1069:Laplace equation
995:) is a field of
977:
970:
963:
911:
876:
842:
841:
838:
805:Surface integral
748:
747:
744:
652:
651:
648:
608:Limit comparison
528:
527:
524:
410:Riemann integral
363:
362:
359:
319:L'Hôpital's rule
276:Taylor's theorem
197:
196:
193:
137:
135:
134:
129:
81:
72:
67:
37:
36:
21:
22313:
22312:
22308:
22307:
22306:
22304:
22303:
22302:
22283:
22282:
22281:
22276:
22265:
22214:P-adic analysis
22165:
22151:Matrix calculus
22146:Tensor calculus
22141:Vector calculus
22104:Differentiation
22084:
22078:
22048:
22043:
22027:
21994:
21949:
21880:
21806:
21797:Semi-continuity
21782:Convex function
21763:Logarithmically
21730:
21691:Convex geometry
21672:
21663:Convex function
21646:
21640:Convex analysis
21637:
21607:
21602:
21573:Banach manifold
21561:
21533:
21512:
21464:
21440:Direct integral
21421:
21341:
21269:
21265:Vector calculus
21260:Matrix calculus
21216:
21207:
21112:
21077:". Chap.17 in:
21073:Roubicek, T.: "
21051:Pike, Ralph W.
20997:Forsyth, A.R.:
20990:Elsgolc, L.E.:
20910:
20908:Further reading
20905:
20897:
20893:
20885:
20881:
20873:
20869:
20861:
20857:
20849:
20845:
20837:
20833:
20825:
20821:
20813:
20809:
20801:
20797:
20786:
20782:
20759:
20755:
20716:
20712:
20697:
20693:
20686:
20672:
20668:
20653:
20649:
20636:Kelland, Philip
20633:
20629:
20622:
20601:
20597:
20589:
20585:
20577:
20573:
20565:
20561:
20554:
20533:
20529:
20520:
20518:
20505:
20504:
20500:
20445:
20441:
20433:
20429:
20412:
20405:
20398:
20384:
20375:
20368:
20349:
20345:
20338:
20320:
20313:
20306:
20283:
20279:
20271:
20264:
20260:
20255:
20250:
20246:
20237:
20231:
20223:
20217:
20207:
20203:
20190:
20186:
20164:
20147:
20129:
20110:
20107:
20106:
20090:
20087:
20086:
20070:
20067:
20066:
20050:
20047:
20046:
20015:
20012:
20011:
19980:
19977:
19976:
19975:For a function
19974:
19970:
19951:
19948:
19947:
19930:
19926:
19918:
19915:
19914:
19890:
19886:
19882:
19850:
19846:
19839:
19835:
19830:
19827:
19826:
19780:
19777:
19776:
19747:
19744:
19743:
19741:
19737:
19698:
19695:
19694:
19666:
19663:
19662:
19646:
19643:
19642:
19623:
19620:
19619:
19603:
19600:
19599:
19571:
19568:
19567:
19565:
19561:
19556:
19552:
19547:
19543:
19538:
19534:
19525:
19521:
19502:
19499:
19498:
19495:first variation
19472:
19469:
19468:
19452:
19449:
19448:
19420:
19415:
19412:
19411:
19409:
19405:
19383:
19380:
19379:
19350:
19347:
19346:
19321:
19318:
19317:
19315:
19311:
19306:
19302:
19293:
19289:
19284:
19280:
19261:
19258:
19257:
19235:
19221:
19219:
19216:
19215:
19199:
19196:
19195:
19193:
19189:
19177:
19173:
19168:
19164:
19158:infinitesimally
19151:
19147:
19143:
19138:
19089:Nehari manifold
19059:Optimal control
18989:First variation
18984:
18979:
18977:
18953:
18952:
18944:
18941:
18940:
18911:
18907:
18905:
18902:
18901:
18879:
18878:
18870:
18867:
18866:
18832:
18829:
18828:
18806:
18805:
18797:
18794:
18793:
18768:
18765:
18764:
18763:The functional
18729:
18726:
18725:
18709:
18706:
18705:
18685:
18681:
18651:
18647:
18645:
18642:
18641:
18608:
18604:
18602:
18599:
18598:
18567:
18563:
18542:
18538:
18536:
18533:
18532:
18507:
18504:
18503:
18477:
18473:
18471:
18468:
18467:
18439:
18436:
18435:
18413:
18410:
18409:
18383:
18379:
18377:
18374:
18373:
18347:
18343:
18341:
18338:
18337:
18317:
18313:
18286:
18282:
18264:
18260:
18240:
18237:
18236:
18207:
18204:
18203:
18202:The functional
18155:
18152:
18151:
18126:
18123:
18122:
18097:
18094:
18093:
18065:
18062:
18061:
18039:
18036:
18035:
18016:
18013:
18012:
18011:is the norm of
17990:
17987:
17986:
17961:
17958:
17957:
17896:
17893:
17892:
17863:
17860:
17859:
17858:The functional
17790:
17787:
17786:
17767:
17764:
17763:
17732:
17729:
17728:
17703:
17700:
17699:
17683:
17680:
17679:
17648:
17645:
17644:
17619:
17616:
17615:
17605:first variation
17601:
17538:Optimal control
17503:brachistochrone
17482:
17440:
17432:
17430:
17423:
17419:
17403:
17395:
17393:
17391:
17388:
17387:
17364:
17361:
17360:
17329:
17326:
17325:
17291:
17290:
17264:
17263:
17230:
17227:
17226:
17210:
17207:
17206:
17190:
17187:
17186:
17164:
17163:
17161:
17158:
17157:
17129:
17128:
17117:
17114:
17113:
17093:
17082:
17081:
17080:
17067:
17059:
17056:
17055:
17027:
17026:
17022:
17014:
17012:
17004:
17001:
17000:
16999:are defined by
16984:
16981:
16980:
16953:
16945:
16943:
16926:
16925:
16921:
16913:
16911:
16901:
16896:
16894:
16891:
16890:
16862:
16859:
16858:
16820:
16819:
16799:
16795:
16794:
16787:
16783:
16782:
16770:
16767:
16766:
16742:
16739:
16738:
16718:
16715:
16714:
16683:
16680:
16679:
16660:
16657:
16656:
16637:
16634:
16633:
16630:
16624:
16598:
16595:
16594:
16578:
16575:
16574:
16541:
16540:
16526:
16525:
16522:
16509:
16501:
16499:
16497:
16494:
16493:
16461:
16453:
16451:
16449:
16446:
16445:
16404:
16396:
16394:
16392:
16389:
16388:
16372:
16369:
16368:
16340:
16337:
16336:
16310:
16299:
16296:
16295:
16275:
16271:
16251:
16248:
16247:
16231:
16228:
16227:
16170:
16167:
16166:
16128:
16124:
16106:
16101:
16095:
16092:
16091:
16072:
16069:
16068:
16052:
16049:
16048:
16016:
16012:
16000:
15996:
15994:
15991:
15990:
15983:
15951:
15948:
15947:
15931:
15928:
15927:
15911:
15908:
15907:
15888:
15885:
15884:
15868:
15865:
15864:
15830:
15829:
15815:
15811:
15810:
15803:
15799:
15798:
15777:
15774:
15773:
15751:
15747:
15724:
15721:
15720:
15704:
15701:
15700:
15670:
15669:
15655:
15654:
15641:
15640:
15627:
15625:
15617:
15614:
15613:
15579:
15578:
15564:
15563:
15561:
15545:
15540:
15538:
15535:
15534:
15515:
15512:
15511:
15475:
15474:
15460:
15459:
15457:
15437:
15433:
15432:
15425:
15421:
15420:
15399:
15396:
15395:
15364:
15363:
15361:
15358:
15357:
15338:
15335:
15334:
15309:
15306:
15305:
15289:
15286:
15285:
15262:
15258:
15249:
15245:
15236:
15232:
15221:
15218:
15217:
15214:
15189:
15186:
15185:
15162:
15158:
15156:
15153:
15152:
15136:
15133:
15132:
15109:
15105:
15103:
15100:
15099:
15069:
15065:
15059:
15055:
15043:
15022:
15018:
15006:
15001:
14999:
14987:
14983:
14971:
14967:
14961:
14957:
14945:
14924:
14920:
14908:
14903:
14901:
14889:
14885:
14884:
14880:
14865:
14861:
14849:
14845:
14836:
14832:
14821:
14818:
14817:
14798:
14796:
14793:
14792:
14776:
14773:
14772:
14748:
14745:
14744:
14719:
14716:
14715:
14693:
14690:
14689:
14666:
14662:
14660:
14657:
14656:
14633:
14629:
14627:
14624:
14623:
14606:
14605:
14587:
14585:
14573:
14569:
14566:
14565:
14547:
14545:
14533:
14529:
14522:
14521:
14498:
14495:
14494:
14491:
14451:
14447:
14432:
14420:
14411:
14407:
14392:
14388:
14371:
14367:
14362:
14341:
14328:
14324:
14311:
14309:
14305:
14295:
14290:
14285:
14282:
14281:
14246:
14242:
14227:
14215:
14209:
14205:
14196:
14192:
14177:
14173:
14161:
14157:
14142:
14112:
14090:
14077:
14073:
14060:
14058:
14057:
14053:
14045:
14041:
14040:
14033:
14029:
14028:
14012:
14008:
13999:
13995:
13984:
13981:
13980:
13964:
13961:
13960:
13944:
13941:
13940:
13937:first variation
13910:
13906:
13885:
13881:
13864:
13861:
13860:
13829:
13826:
13825:
13797:
13793:
13779:
13771:
13736:
13732:
13731:
13724:
13720:
13719:
13698:
13695:
13694:
13663:
13660:
13659:
13643:
13640:
13639:
13633:
13628:
13603:
13600:
13599:
13551:
13543:
13541:
13527:
13524:
13523:
13507:
13504:
13503:
13502:The minimizing
13473:
13459:
13457:
13449:
13446:
13445:
13351:
13348:
13347:
13331:
13328:
13327:
13308:
13305:
13304:
13273:
13259:
13256:
13255:
13239:
13236:
13235:
13194:
13190:
13163:
13159:
13142:
13139:
13138:
13111:
13107:
13089:
13085:
13055:
13051:
13003:
12999:
12982:
12979:
12978:
12962:
12959:
12958:
12942:
12939:
12938:
12935:
12900:
12896:
12884:
12880:
12868:
12864:
12853:
12844:
12840:
12826:
12807:
12803:
12791:
12787:
12775:
12771:
12760:
12751:
12747:
12736:
12733:
12732:
12716:
12713:
12712:
12696:
12693:
12692:
12672:
12668:
12653:
12649:
12641:
12605:
12597:
12586:
12583:
12582:
12566:
12563:
12562:
12546:
12543:
12542:
12516:
12512:
12500:
12496:
12484:
12480:
12469:
12460:
12456:
12438:
12434:
12413:
12409:
12397:
12393:
12381:
12377:
12366:
12357:
12353:
12332:
12328:
12278:
12270:
12260:
12256:
12236:
12232:
12231:
12224:
12220:
12219:
12206:
12202:
12183:
12181:
12179:
12176:
12175:
12147:
12133:
12130:
12129:
12101:
12097:
12082:
12078:
12066:
12062:
12050:
12046:
12031:
12027:
12015:
12011:
11898:
11881:
11868:
11864:
11856:
11852:
11851:
11844:
11840:
11839:
11834:
11830:
11814:
11809:
11800:
11796:
11794:
11791:
11790:
11771:
11766:
11763:
11762:
11731:
11728:
11727:
11710:
11706:
11704:
11701:
11700:
11683:
11679:
11677:
11674:
11673:
11653:
11649:
11643:
11639:
11627:
11623:
11614:
11610:
11604:
11600:
11588:
11584:
11563:
11559:
11529:
11525:
11511:
11498:
11494:
11486:
11482:
11481:
11474:
11470:
11469:
11448:
11445:
11444:
11428:
11425:
11424:
11370:
11366:
11346:
11342:
11341:
11334:
11330:
11329:
11323:
11320:
11319:
11303:
11300:
11299:
11279:
11276:
11275:
11242:
11238:
11236:
11233:
11232:
11215:
11211:
11209:
11206:
11205:
11189:
11186:
11185:
11169:
11166:
11165:
11135:
11121:
11119:
11111:
11108:
11107:
11091:
11088:
11087:
11038:
11030:
11019:
11016:
11015:
10999:
10996:
10995:
10970:
10967:
10966:
10941:
10938:
10937:
10921:
10918:
10917:
10898:
10893:
10890:
10889:
10864:
10861:
10860:
10835:
10832:
10831:
10804:
10800:
10771:
10767:
10766:
10759:
10755:
10754:
10733:
10730:
10729:
10713:
10710:
10709:
10683:
10679:
10654:
10650:
10642:
10639:
10638:
10622:
10619:
10618:
10586:
10582:
10552:
10548:
10534:
10521:
10517:
10509:
10505:
10504:
10497:
10493:
10492:
10471:
10468:
10467:
10460:
10454:
10448:
10439:
10400:
10396:
10370:
10366:
10364:
10361:
10360:
10344:
10341:
10340:
10322:
10319:
10318:
10283:
10279:
10253:
10249:
10248:
10244:
10224:
10221:
10220:
10204:
10201:
10200:
10175:
10172:
10171:
10152:
10149:
10148:
10132:
10129:
10128:
10105:
10102:
10101:
10082:
10079:
10078:
10033:
10025:
10023:
10021:
10018:
10017:
10001:
9998:
9997:
9981:
9978:
9977:
9958:
9955:
9954:
9914:
9911:
9910:
9891:
9888:
9887:
9865:
9862:
9861:
9807:
9799:
9797:
9796:
9792:
9783:
9779:
9729:
9725:
9719:
9715:
9713:
9710:
9709:
9661:
9657:
9651:
9647:
9600:
9596:
9590:
9586:
9584:
9581:
9580:
9546:
9543:
9542:
9523:
9520:
9519:
9503:
9500:
9499:
9483:
9480:
9479:
9463:
9460:
9459:
9440:
9437:
9436:
9413:
9410:
9409:
9384:
9381:
9380:
9361:
9358:
9357:
9332:
9329:
9328:
9309:
9306:
9305:
9274:
9271:
9270:
9220:
9216:
9194:
9193:
9189:
9183:
9179:
9107:
9106:
9102:
9096:
9092:
9075:
9072:
9071:
9068:
9037:
9034:
9033:
9017:
9014:
9013:
8983:
8980:
8979:
8942:
8939:
8938:
8922:
8919:
8918:
8894:
8890:
8882:
8870:
8862:
8841:
8838:
8837:
8812:
8809:
8808:
8787:
8784:
8783:
8752:
8749:
8748:
8729:
8726:
8725:
8709:
8706:
8705:
8653:
8649:
8647:
8644:
8643:
8627:
8624:
8623:
8607:
8604:
8603:
8584:
8581:
8580:
8564:
8561:
8560:
8538:
8530:
8527:
8526:
8510:
8507:
8506:
8490:
8487:
8486:
8468:
8465:
8464:
8448:
8445:
8444:
8413:
8405:
8403:
8394:
8390:
8334:
8330:
8286:
8282:
8280:
8277:
8276:
8224:
8220:
8205:
8169:
8164:
8163:
8160:
8159:
8157:
8154:
8153:
8119:
8116:
8115:
8096:
8093:
8092:
8076:
8073:
8072:
8056:
8053:
8052:
8033:
8030:
8029:
8013:
8010:
8009:
7993:
7990:
7989:
7988:The functional
7940:
7936:
7926:
7909:
7906:
7905:
7902:
7871:
7867:
7861:
7857:
7851:
7847:
7832:
7827:
7805:
7801:
7789:
7784:
7762:
7758:
7756:
7753:
7752:
7732:
7729:
7728:
7672:
7666:
7662:
7645:
7642:
7641:
7619:
7616:
7615:
7599:
7596:
7595:
7564:
7561:
7560:
7557:
7511:
7508:
7507:
7484:
7480:
7478:
7475:
7474:
7451:
7447:
7445:
7442:
7441:
7415:
7411:
7409:
7406:
7405:
7382:
7378:
7376:
7373:
7372:
7346:
7342:
7334:
7331:
7330:
7308:
7304:
7287:
7284:
7283:
7192:
7188:
7171:
7169:
7166:
7165:
7145:
7141:
7137:
7131:
7127:
7115:
7111:
7102:
7097:
7076:
7073:
7072:
7060:
7039:
7036:
7035:
7005:
7001:
6997:
6993:
6983:
6979:
6978:
6976:
6974:
6971:
6970:
6954:
6951:
6950:
6930:
6927:
6926:
6923:
6883:
6879:
6875:
6867:
6865:
6861:
6852:
6848:
6844:
6838:
6834:
6832:
6826:
6822:
6789:
6785:
6777:
6775:
6771:
6761:
6756:
6743:
6735:
6733:
6731:
6728:
6727:
6707:
6704:
6703:
6659:
6655:
6629:
6596:
6591:
6579:
6576:
6575:
6547:
6544:
6543:
6527:
6524:
6523:
6520:
6484:
6476:
6474:
6472:
6469:
6468:
6452:
6449:
6448:
6415:
6413:
6410:
6409:
6393:
6390:
6389:
6369:
6366:
6365:
6333:
6329:
6321:
6319:
6311:
6303:
6300:
6299:
6280:
6277:
6276:
6248:
6246:
6243:
6242:
6217:
6214:
6213:
6183:
6175:
6173:
6171:
6168:
6167:
6164:
6125:
6122:
6121:
6096:
6093:
6092:
6067:
6064:
6063:
6038:
6035:
6034:
6014:
6010:
6001:
5997:
5996:
5989:
5985:
5979:
5975:
5966:
5962:
5956:
5952:
5951:
5949:
5937:
5927:
5923:
5914:
5910:
5909:
5902:
5898:
5889:
5885:
5884:
5882:
5865:
5835:
5832:
5831:
5809:
5805:
5796:
5792:
5791:
5787:
5785:
5782:
5781:
5759:
5755:
5746:
5742:
5741:
5737:
5735:
5732:
5731:
5697:
5695:
5692:
5691:
5671:
5667:
5660:
5654:
5650:
5648:
5639:
5635:
5618:
5613:
5610:
5609:
5586:
5582:
5574:
5571:
5570:
5549:
5545:
5533:
5529:
5512:
5502:
5495:
5491:
5474:
5470:
5468:
5466:
5463:
5462:
5443:
5440:
5439:
5409:
5405:
5388:
5360:
5359:
5357:
5355:
5352:
5351:
5316:
5312:
5295:
5267:
5266:
5264:
5251:
5246:
5244:
5241:
5240:
5224:
5221:
5220:
5188:
5184:
5176:
5174:
5164:
5159:
5157:
5154:
5153:
5128:
5125:
5124:
5105:
5102:
5101:
5085:
5082:
5081:
5056:
5052:
5035:
5024:
5016:
5013:
5012:
4984:
4980:
4972:
4970:
4960:
4955:
4942:
4934:
4932:
4930:
4927:
4926:
4899:
4896:
4895:
4870:
4867:
4866:
4859:
4855:
4831:
4827:
4812:
4808:
4789:
4785:
4770:
4766:
4746:
4738:
4736:
4716:
4714:
4711:
4710:
4680:
4676:
4659:
4648:
4640:
4636:
4635:
4628:
4624:
4623:
4602:
4599:
4598:
4569:
4565:
4556:
4552:
4551:
4547:
4545:
4542:
4541:
4519:
4515:
4506:
4502:
4501:
4497:
4495:
4492:
4491:
4457:
4454:
4453:
4450:
4413:
4410:
4409:
4373:
4370:
4369:
4329:
4321:
4318:
4317:
4316:and is denoted
4292:
4289:
4288:
4252:
4248:
4240:
4238:
4228:
4223:
4210:
4202:
4200:
4198:
4195:
4194:
4144:
4140:
4132:
4130:
4120:
4115:
4102:
4094:
4092:
4091:
4087:
4067:
4063:
4062:
4055:
4051:
4050:
4044:
4041:
4040:
4023:
4019:
4017:
4014:
4013:
3996:
3992:
3990:
3987:
3986:
3964:
3961:
3960:
3934:
3931:
3930:
3906:
3893:
3889:
3870:
3857:
3853:
3848:
3845:
3844:
3830:
3829:
3806:
3802:
3794:
3792:
3782:
3777:
3758:
3750:
3748:
3747:
3743:
3735:
3731:
3730:
3723:
3719:
3718:
3705:
3704:
3686:
3682:
3674:
3672:
3662:
3657:
3646:
3642:
3641:
3634:
3630:
3629:
3614:
3610:
3609:
3602:
3598:
3597:
3578:
3574:
3566:
3564:
3563:
3560:
3536:
3528:
3526:
3518:
3514:
3513:
3506:
3502:
3501:
3488:
3487:
3467:
3456:
3452:
3444:
3442:
3426:
3418:
3416:
3415:
3411:
3403:
3399:
3398:
3391:
3387:
3386:
3375:
3357:
3343:
3335:
3333:
3330:
3329:
3321:
3317:
3316:
3309:
3305:
3304:
3296:
3294:
3291:
3290:
3266:
3255:
3251:
3243:
3241:
3225:
3217:
3215:
3202:
3194:
3192:
3190:
3187:
3186:
3165:
3152:
3143:
3139:
3137:
3135:
3132:
3131:
3104:
3096:
3094:
3092:
3089:
3088:
3067:
3058:
3054:
3052:
3041:
3037:
3029:
3027:
3014:
3006:
3004:
2994:
2986:
2984:
2971:
2963:
2961:
2959:
2956:
2955:
2936:
2933:
2932:
2916:
2913:
2912:
2893:
2879:
2868:
2866:
2863:
2862:
2831:
2828:
2827:
2800:
2787:
2783:
2778:
2775:
2774:
2729:
2715:
2707:
2705:
2702:
2701:
2693:
2689:
2688:
2681:
2677:
2676:
2657:
2643:
2635:
2633:
2630:
2629:
2609:
2607:
2604:
2603:
2581:
2578:
2577:
2552:
2549:
2548:
2526:
2523:
2522:
2497:
2494:
2493:
2440:
2437:
2436:
2416:
2413:
2412:
2384:
2381:
2380:
2364:
2361:
2360:
2335:
2332:
2331:
2307:
2304:
2303:
2287:
2284:
2283:
2260:
2257:
2256:
2201:
2198:
2197:
2181:
2178:
2177:
2157:
2153:
2151:
2148:
2147:
2130:
2126:
2124:
2121:
2120:
2095:
2092:
2091:
2072:
2069:
2068:
2039:
2036:
2035:
2010:
2008:
2005:
2004:
1979:
1976:
1975:
1942:
1920:
1916:
1911:
1908:
1907:
1882:
1874:
1872:
1852:
1850:
1847:
1846:
1819:
1816:
1815:
1792:
1788:
1779:
1775:
1773:
1770:
1769:
1722:
1700:
1696:
1685:
1681:
1680:
1673:
1669:
1668:
1647:
1644:
1643:
1628:
1622:
1568:
1565:
1564:
1545:
1542:
1541:
1516:
1513:
1512:
1487:
1484:
1483:
1463:
1460:
1459:
1440:
1437:
1436:
1420:
1417:
1416:
1363:
1360:
1359:
1343:
1340:
1339:
1314:
1311:
1310:
1309:. A functional
1286:
1283:
1282:
1267:
1259:Richard Bellman
1223:Leonida Tonelli
1200:Hilbert problem
1106:Jacob Bernoulli
1090:
981:
952:
951:
937:Integration Bee
912:
909:
902:
901:
877:
874:
867:
866:
839:
836:
829:
828:
810:Volume integral
745:
740:
733:
732:
649:
644:
637:
636:
606:
525:
520:
513:
512:
504:Risch algorithm
474:Euler's formula
360:
355:
348:
347:
329:General Leibniz
212:generalizations
194:
189:
182:
168:Rolle's theorem
163:
138:
74:
68:
63:
57:
54:
53:
35:
28:
23:
22:
15:
12:
11:
5:
22311:
22301:
22300:
22295:
22278:
22277:
22270:
22267:
22266:
22264:
22263:
22258:
22253:
22248:
22243:
22238:
22232:
22231:
22226:
22224:Measure theory
22221:
22218:P-adic numbers
22211:
22206:
22201:
22196:
22191:
22181:
22176:
22170:
22167:
22166:
22164:
22163:
22158:
22153:
22148:
22143:
22138:
22133:
22128:
22127:
22126:
22121:
22116:
22106:
22101:
22089:
22086:
22085:
22077:
22076:
22069:
22062:
22054:
22045:
22044:
22042:
22041:
22035:
22033:
22029:
22028:
22026:
22025:
22020:
22018:Strong duality
22015:
22010:
22004:
22002:
21996:
21995:
21993:
21992:
21957:
21955:
21951:
21950:
21948:
21947:
21942:
21933:
21928:
21926:John ellipsoid
21923:
21918:
21913:
21908:
21894:
21888:
21886:
21882:
21881:
21879:
21878:
21873:
21868:
21863:
21858:
21853:
21848:
21843:
21838:
21833:
21828:
21823:
21817:
21815:
21813:results (list)
21808:
21807:
21805:
21804:
21799:
21794:
21789:
21787:Invex function
21784:
21775:
21770:
21765:
21760:
21755:
21749:
21744:
21738:
21736:
21732:
21731:
21729:
21728:
21723:
21718:
21713:
21708:
21703:
21698:
21693:
21688:
21686:Choquet theory
21682:
21680:
21674:
21673:
21671:
21670:
21665:
21660:
21654:
21652:
21651:Basic concepts
21648:
21647:
21636:
21635:
21628:
21621:
21613:
21604:
21603:
21601:
21600:
21595:
21590:
21588:Choquet theory
21585:
21580:
21569:
21567:
21563:
21562:
21560:
21559:
21554:
21549:
21543:
21541:
21535:
21534:
21532:
21531:
21526:
21520:
21518:
21514:
21513:
21511:
21510:
21505:
21500:
21495:
21490:
21489:
21488:
21478:
21472:
21470:
21466:
21465:
21463:
21462:
21457:
21452:
21447:
21442:
21437:
21431:
21429:
21423:
21422:
21420:
21419:
21414:
21398:
21397:
21396:
21391:
21386:
21372:
21371:
21370:
21365:
21355:
21349:
21347:
21343:
21342:
21340:
21339:
21334:
21329:
21324:
21319:
21318:
21317:
21307:
21302:
21301:
21300:
21290:
21285:
21279:
21277:
21271:
21270:
21268:
21267:
21262:
21257:
21252:
21247:
21242:
21237:
21236:
21235:
21224:
21222:
21221:Basic concepts
21218:
21217:
21206:
21205:
21198:
21191:
21183:
21177:
21176:
21165:
21160:. Lectures on
21155:
21149:
21137:
21125:
21111:
21110:External links
21108:
21107:
21106:
21099:
21098:, Dover, 1992.
21092:
21071:
21069:on 2007-07-05.
21048:
21041:
21034:
21027:
21009:
21004:Fox, Charles:
21002:
21001:, Dover, 1960.
20995:
20988:
20964:
20954:
20947:
20940:
20922:
20909:
20906:
20904:
20903:
20891:
20879:
20867:
20855:
20843:
20831:
20819:
20807:
20795:
20780:
20769:(4): 378–388.
20753:
20726:(4): 325–388.
20710:
20691:
20684:
20666:
20647:
20627:
20621:978-0471504474
20620:
20595:
20583:
20571:
20559:
20553:978-0471504474
20552:
20527:
20498:
20459:(4): 231–235.
20439:
20427:
20403:
20396:
20373:
20366:
20343:
20336:
20311:
20305:978-0486414485
20304:
20286:Gelfand, I. M.
20277:
20261:
20259:
20256:
20254:
20253:
20244:
20242:
20241:
20227:
20201:
20184:
20171:
20167:
20163:
20160:
20157:
20154:
20150:
20144:
20141:
20138:
20135:
20132:
20128:
20123:
20120:
20117:
20114:
20094:
20074:
20054:
20034:
20031:
20028:
20025:
20022:
20019:
19999:
19996:
19993:
19990:
19987:
19984:
19968:
19955:
19933:
19929:
19925:
19922:
19902:
19898:
19893:
19889:
19885:
19881:
19878:
19875:
19872:
19869:
19866:
19863:
19859:
19853:
19849:
19845:
19842:
19838:
19834:
19814:
19811:
19808:
19805:
19802:
19799:
19796:
19793:
19790:
19787:
19784:
19771:is said to be
19760:
19757:
19754:
19751:
19735:
19723:
19720:
19717:
19714:
19711:
19708:
19705:
19702:
19682:
19679:
19676:
19673:
19670:
19650:
19630:
19627:
19607:
19587:
19584:
19581:
19578:
19575:
19559:
19550:
19541:
19532:
19519:
19506:
19482:
19479:
19476:
19456:
19436:
19433:
19430:
19426:
19423:
19419:
19403:
19390:
19387:
19377:
19363:
19360:
19357:
19354:
19334:
19331:
19328:
19325:
19309:
19300:
19287:
19278:
19265:
19245:
19242:
19238:
19234:
19231:
19228:
19224:
19203:
19187:
19171:
19162:
19144:
19142:
19139:
19137:
19136:
19131:
19126:
19121:
19116:
19111:
19106:
19101:
19096:
19091:
19086:
19081:
19076:
19071:
19066:
19061:
19056:
19051:
19046:
19041:
19036:
19031:
19026:
19021:
19016:
19011:
19006:
19001:
18996:
18991:
18985:
18983:
18980:
18966:
18960:
18957:
18951:
18948:
18928:
18925:
18922:
18919:
18914:
18910:
18886:
18883:
18877:
18874:
18854:
18851:
18848:
18845:
18842:
18839:
18836:
18813:
18810:
18804:
18801:
18781:
18778:
18775:
18772:
18762:
18755:
18739:
18736:
18733:
18713:
18693:
18688:
18684:
18680:
18677:
18674:
18671:
18668:
18665:
18662:
18659:
18654:
18650:
18636:is said to be
18625:
18622:
18619:
18616:
18611:
18607:
18584:
18581:
18578:
18575:
18570:
18566:
18562:
18559:
18556:
18553:
18550:
18545:
18541:
18520:
18517:
18514:
18511:
18491:
18488:
18485:
18480:
18476:
18455:
18452:
18449:
18446:
18443:
18423:
18420:
18417:
18397:
18394:
18391:
18386:
18382:
18361:
18358:
18355:
18350:
18346:
18325:
18320:
18316:
18312:
18309:
18306:
18303:
18300:
18297:
18294:
18289:
18285:
18281:
18278:
18275:
18272:
18267:
18263:
18259:
18256:
18253:
18250:
18247:
18244:
18231:is said to be
18220:
18217:
18214:
18211:
18189:
18186:
18183:
18180:
18177:
18174:
18171:
18168:
18165:
18162:
18159:
18139:
18136:
18133:
18130:
18110:
18107:
18104:
18101:
18081:
18078:
18075:
18072:
18069:
18049:
18046:
18043:
18023:
18020:
18000:
17997:
17994:
17974:
17971:
17968:
17965:
17945:
17942:
17939:
17936:
17933:
17930:
17927:
17924:
17921:
17918:
17915:
17912:
17909:
17906:
17903:
17900:
17889:differentiable
17887:is said to be
17876:
17873:
17870:
17867:
17845:
17842:
17839:
17836:
17833:
17830:
17827:
17824:
17821:
17818:
17815:
17812:
17809:
17806:
17803:
17800:
17797:
17794:
17774:
17771:
17751:
17748:
17745:
17742:
17739:
17736:
17716:
17713:
17710:
17707:
17687:
17667:
17664:
17661:
17658:
17655:
17652:
17632:
17629:
17626:
17623:
17600:
17597:
17596:
17595:
17585:
17579:
17573:
17567:
17561:
17554:
17540:
17535:
17525:
17519:
17512:
17506:
17499:
17493:
17481:
17478:
17466:
17463:
17459:
17455:
17452:
17446:
17443:
17438:
17435:
17429:
17426:
17422:
17418:
17415:
17409:
17406:
17401:
17398:
17371:
17368:
17348:
17345:
17342:
17339:
17336:
17333:
17313:
17310:
17307:
17304:
17298:
17295:
17289:
17286:
17283:
17280:
17277:
17271:
17268:
17261:
17258:
17255:
17252:
17249:
17246:
17243:
17240:
17237:
17234:
17214:
17194:
17171:
17168:
17142:
17136:
17133:
17127:
17124:
17121:
17101:
17096:
17089:
17086:
17079:
17074:
17071:
17066:
17063:
17043:
17034:
17031:
17025:
17020:
17017:
17011:
17008:
16988:
16965:
16959:
16956:
16951:
16948:
16942:
16933:
16930:
16924:
16919:
16916:
16907:
16904:
16900:
16878:
16875:
16872:
16869:
16866:
16846:
16843:
16839:
16836:
16833:
16827:
16824:
16818:
16815:
16812:
16809:
16802:
16798:
16790:
16786:
16781:
16777:
16774:
16746:
16735:kinetic energy
16722:
16702:
16699:
16696:
16693:
16690:
16687:
16667:
16664:
16644:
16641:
16626:Main article:
16623:
16620:
16602:
16582:
16560:
16555:
16548:
16545:
16539:
16533:
16530:
16521:
16515:
16512:
16507:
16504:
16479:
16476:
16473:
16467:
16464:
16459:
16456:
16442:the light rays
16429:
16426:
16423:
16419:
16416:
16410:
16407:
16402:
16399:
16376:
16356:
16353:
16350:
16347:
16344:
16320:
16317:
16313:
16309:
16306:
16303:
16283:
16278:
16274:
16270:
16267:
16264:
16261:
16258:
16255:
16235:
16226:In that case,
16213:
16210:
16207:
16204:
16201:
16198:
16195:
16192:
16189:
16186:
16183:
16180:
16177:
16174:
16152:
16149:
16146:
16143:
16140:
16137:
16131:
16127:
16123:
16120:
16117:
16114:
16109:
16104:
16100:
16079:
16076:
16056:
16036:
16033:
16030:
16027:
16024:
16019:
16015:
16011:
16006:
16003:
15999:
15982:
15979:
15958:
15955:
15935:
15915:
15895:
15892:
15872:
15850:
15847:
15844:
15837:
15834:
15828:
15825:
15818:
15814:
15806:
15802:
15797:
15793:
15790:
15787:
15784:
15781:
15759:
15754:
15750:
15746:
15743:
15740:
15737:
15734:
15731:
15728:
15708:
15686:
15677:
15674:
15668:
15662:
15659:
15648:
15645:
15639:
15636:
15633:
15630:
15624:
15621:
15601:
15598:
15595:
15586:
15583:
15577:
15571:
15568:
15560:
15557:
15551:
15548:
15544:
15522:
15519:
15497:
15494:
15491:
15482:
15479:
15473:
15467:
15464:
15456:
15453:
15450:
15447:
15440:
15436:
15428:
15424:
15419:
15415:
15412:
15409:
15406:
15403:
15383:
15380:
15377:
15371:
15368:
15345:
15342:
15322:
15319:
15316:
15313:
15293:
15273:
15270:
15265:
15261:
15257:
15252:
15248:
15244:
15239:
15235:
15231:
15228:
15225:
15213:
15210:
15193:
15171:
15168:
15165:
15161:
15140:
15118:
15115:
15112:
15108:
15085:
15081:
15072:
15068:
15062:
15058:
15054:
15050:
15046:
15042:
15038:
15035:
15030:
15025:
15021:
15017:
15013:
15009:
15005:
14996:
14993:
14990:
14986:
14982:
14974:
14970:
14964:
14960:
14956:
14952:
14948:
14944:
14940:
14937:
14932:
14927:
14923:
14919:
14915:
14911:
14907:
14898:
14895:
14892:
14888:
14883:
14879:
14876:
14873:
14868:
14864:
14860:
14857:
14852:
14848:
14844:
14839:
14835:
14831:
14828:
14825:
14804:
14801:
14780:
14761:
14758:
14755:
14752:
14732:
14729:
14726:
14723:
14703:
14700:
14697:
14675:
14672:
14669:
14665:
14642:
14639:
14636:
14632:
14609:
14604:
14601:
14598:
14595:
14586:
14582:
14579:
14576:
14572:
14568:
14567:
14564:
14561:
14558:
14555:
14546:
14542:
14539:
14536:
14532:
14528:
14527:
14525:
14520:
14517:
14514:
14511:
14508:
14505:
14502:
14490:
14487:
14464:
14461:
14454:
14450:
14446:
14443:
14439:
14435:
14431:
14427:
14424:
14419:
14414:
14410:
14406:
14403:
14400:
14395:
14391:
14387:
14383:
14374:
14370:
14365:
14361:
14357:
14354:
14348:
14344:
14340:
14336:
14331:
14327:
14323:
14320:
14317:
14314:
14308:
14301:
14298:
14294:
14289:
14267:
14264:
14261:
14257:
14249:
14245:
14241:
14238:
14234:
14230:
14226:
14222:
14219:
14212:
14208:
14204:
14199:
14195:
14191:
14188:
14185:
14180:
14176:
14172:
14164:
14160:
14156:
14153:
14149:
14145:
14141:
14137:
14134:
14129:
14126:
14123:
14119:
14115:
14111:
14107:
14104:
14101:
14097:
14093:
14089:
14085:
14080:
14076:
14072:
14069:
14066:
14063:
14056:
14048:
14044:
14036:
14032:
14027:
14023:
14020:
14015:
14011:
14007:
14002:
13998:
13994:
13991:
13988:
13968:
13948:
13924:
13921:
13918:
13913:
13909:
13905:
13902:
13899:
13896:
13893:
13888:
13884:
13880:
13877:
13874:
13871:
13868:
13848:
13845:
13842:
13839:
13836:
13833:
13813:
13810:
13807:
13800:
13796:
13792:
13789:
13785:
13782:
13778:
13775:
13770:
13767:
13764:
13761:
13758:
13755:
13752:
13749:
13746:
13739:
13735:
13727:
13723:
13718:
13714:
13711:
13708:
13705:
13702:
13682:
13679:
13676:
13673:
13670:
13667:
13647:
13632:
13629:
13627:
13624:
13610:
13607:
13587:
13584:
13581:
13578:
13575:
13572:
13569:
13566:
13563:
13557:
13554:
13549:
13546:
13540:
13537:
13534:
13531:
13511:
13491:
13485:
13482:
13479:
13476:
13471:
13468:
13465:
13462:
13456:
13453:
13433:
13430:
13427:
13424:
13421:
13418:
13415:
13412:
13409:
13406:
13403:
13400:
13397:
13394:
13391:
13388:
13385:
13382:
13379:
13376:
13373:
13370:
13367:
13364:
13361:
13358:
13355:
13335:
13315:
13312:
13292:
13289:
13286:
13283:
13280:
13276:
13272:
13269:
13266:
13263:
13243:
13223:
13220:
13217:
13213:
13210:
13206:
13203:
13197:
13193:
13189:
13186:
13183:
13180:
13177:
13174:
13171:
13166:
13162:
13158:
13155:
13152:
13149:
13146:
13126:
13123:
13120:
13114:
13110:
13106:
13103:
13100:
13097:
13092:
13088:
13084:
13081:
13078:
13074:
13071:
13067:
13064:
13058:
13054:
13050:
13047:
13044:
13041:
13038:
13035:
13032:
13029:
13026:
13023:
13020:
13017:
13014:
13011:
13006:
13002:
12998:
12995:
12992:
12989:
12986:
12966:
12957:with boundary
12946:
12934:
12931:
12914:
12911:
12908:
12903:
12899:
12895:
12892:
12887:
12883:
12879:
12876:
12871:
12867:
12863:
12859:
12856:
12852:
12847:
12843:
12839:
12836:
12824:
12821:
12818:
12815:
12810:
12806:
12802:
12799:
12794:
12790:
12786:
12783:
12778:
12774:
12770:
12766:
12763:
12759:
12754:
12750:
12746:
12743:
12740:
12720:
12700:
12680:
12675:
12671:
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12664:
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10819:
10816:
10813:
10807:
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10438:
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10247:
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3414:
3406:
3402:
3394:
3390:
3385:
3381:
3378:
3376:
3374:
3371:
3366:
3363:
3360:
3355:
3349:
3346:
3341:
3338:
3332:
3324:
3320:
3312:
3308:
3303:
3299:
3298:
3276:
3272:
3269:
3261:
3258:
3254:
3249:
3246:
3240:
3237:
3231:
3228:
3223:
3220:
3214:
3208:
3205:
3200:
3197:
3175:
3171:
3168:
3164:
3158:
3155:
3149:
3146:
3142:
3119:
3116:
3110:
3107:
3102:
3099:
3073:
3070:
3064:
3061:
3057:
3047:
3044:
3040:
3035:
3032:
3026:
3020:
3017:
3012:
3009:
3000:
2997:
2992:
2989:
2983:
2977:
2974:
2969:
2966:
2943:
2940:
2920:
2899:
2896:
2892:
2889:
2885:
2882:
2878:
2874:
2871:
2850:
2847:
2844:
2841:
2838:
2835:
2815:
2811:
2806:
2803:
2799:
2796:
2793:
2790:
2786:
2782:
2756:
2752:
2749:
2746:
2743:
2738:
2735:
2732:
2727:
2721:
2718:
2713:
2710:
2704:
2696:
2692:
2684:
2680:
2675:
2671:
2666:
2663:
2660:
2655:
2649:
2646:
2641:
2638:
2632:
2628:
2625:
2622:
2619:
2615:
2612:
2591:
2588:
2585:
2565:
2562:
2559:
2556:
2536:
2533:
2530:
2510:
2507:
2504:
2501:
2481:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2423:
2420:
2400:
2397:
2394:
2391:
2388:
2368:
2348:
2345:
2342:
2339:
2317:
2314:
2311:
2291:
2278:is called the
2267:
2264:
2242:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2185:
2165:
2160:
2156:
2133:
2129:
2108:
2105:
2102:
2099:
2079:
2076:
2052:
2049:
2046:
2043:
2032:
2031:
2020:
2016:
2013:
1992:
1989:
1986:
1983:
1962:
1958:
1955:
1952:
1948:
1945:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1919:
1915:
1905:
1894:
1888:
1885:
1880:
1877:
1871:
1868:
1865:
1862:
1858:
1855:
1844:
1832:
1829:
1826:
1823:
1813:
1795:
1791:
1787:
1782:
1778:
1754:
1750:
1747:
1742:
1738:
1735:
1732:
1728:
1725:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1699:
1695:
1688:
1684:
1676:
1672:
1667:
1663:
1660:
1657:
1654:
1651:
1624:Main article:
1621:
1618:
1594:strong extrema
1581:
1578:
1575:
1572:
1552:
1549:
1529:
1526:
1523:
1520:
1500:
1497:
1494:
1491:
1467:
1447:
1444:
1424:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1347:
1327:
1324:
1321:
1318:
1303:function space
1290:
1266:
1263:
1243:Lev Pontryagin
1227:Henri Lebesgue
1185:Alfred Clebsch
1153:Siméon Poisson
1114:Leonhard Euler
1089:
1086:
1050:optical length
1011:from a set of
983:
982:
980:
979:
972:
965:
957:
954:
953:
950:
949:
944:
939:
934:
932:List of topics
929:
924:
919:
913:
908:
907:
904:
903:
900:
899:
894:
889:
884:
878:
873:
872:
869:
868:
863:
862:
861:
860:
855:
850:
840:
835:
834:
831:
830:
825:
824:
823:
822:
817:
812:
807:
802:
797:
792:
784:
783:
779:
778:
777:
776:
771:
766:
761:
753:
752:
746:
739:
738:
735:
734:
729:
728:
727:
726:
721:
716:
711:
706:
701:
693:
692:
688:
687:
686:
685:
680:
675:
670:
665:
660:
650:
643:
642:
639:
638:
633:
632:
631:
630:
625:
620:
615:
610:
604:
599:
594:
589:
584:
576:
575:
569:
568:
567:
566:
561:
556:
551:
546:
541:
526:
519:
518:
515:
514:
509:
508:
507:
506:
501:
496:
491:
489:Changing order
486:
476:
471:
453:
448:
443:
435:
434:
433:Integration by
430:
429:
428:
427:
422:
417:
412:
407:
397:
395:Antiderivative
389:
388:
384:
383:
382:
381:
376:
371:
361:
354:
353:
350:
349:
344:
343:
342:
341:
336:
331:
326:
321:
316:
311:
306:
301:
296:
288:
287:
281:
280:
279:
278:
273:
268:
263:
258:
253:
245:
244:
240:
239:
238:
237:
236:
235:
230:
225:
215:
202:
201:
195:
188:
187:
184:
183:
181:
180:
175:
170:
164:
162:
161:
156:
150:
149:
148:
140:
139:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
90:
87:
84:
80:
77:
71:
66:
62:
52:
49:
48:
42:
41:
26:
9:
6:
4:
3:
2:
22310:
22299:
22296:
22294:
22291:
22290:
22288:
22275:
22274:
22268:
22262:
22259:
22257:
22254:
22252:
22249:
22247:
22244:
22242:
22239:
22237:
22234:
22233:
22230:
22227:
22225:
22222:
22219:
22215:
22212:
22210:
22207:
22205:
22202:
22200:
22197:
22195:
22192:
22189:
22185:
22182:
22180:
22177:
22175:
22174:Real analysis
22172:
22171:
22168:
22162:
22159:
22157:
22154:
22152:
22149:
22147:
22144:
22142:
22139:
22137:
22134:
22132:
22129:
22125:
22122:
22120:
22117:
22115:
22112:
22111:
22110:
22107:
22105:
22102:
22100:
22096:
22095:
22091:
22090:
22087:
22083:
22075:
22070:
22068:
22063:
22061:
22056:
22055:
22052:
22040:
22037:
22036:
22034:
22030:
22024:
22021:
22019:
22016:
22014:
22011:
22009:
22006:
22005:
22003:
22001:
21997:
21990:
21988:
21982:
21980:
21974:
21970:
21966:
21962:
21959:
21958:
21956:
21952:
21946:
21943:
21941:
21937:
21934:
21932:
21929:
21927:
21924:
21922:
21919:
21917:
21914:
21912:
21909:
21907:
21903:
21899:
21895:
21893:
21890:
21889:
21887:
21883:
21877:
21874:
21872:
21869:
21867:
21864:
21862:
21859:
21857:
21856:Mazur's lemma
21854:
21852:
21849:
21847:
21844:
21842:
21839:
21837:
21834:
21832:
21829:
21827:
21824:
21822:
21819:
21818:
21816:
21814:
21809:
21803:
21802:Subderivative
21800:
21798:
21795:
21793:
21790:
21788:
21785:
21783:
21779:
21776:
21774:
21771:
21769:
21766:
21764:
21761:
21759:
21756:
21754:
21750:
21748:
21745:
21743:
21740:
21739:
21737:
21733:
21727:
21724:
21722:
21719:
21717:
21714:
21712:
21709:
21707:
21704:
21702:
21699:
21697:
21694:
21692:
21689:
21687:
21684:
21683:
21681:
21679:
21678:Topics (list)
21675:
21669:
21666:
21664:
21661:
21659:
21656:
21655:
21653:
21649:
21645:
21641:
21634:
21629:
21627:
21622:
21620:
21615:
21614:
21611:
21599:
21596:
21594:
21591:
21589:
21586:
21584:
21581:
21578:
21574:
21571:
21570:
21568:
21564:
21558:
21555:
21553:
21550:
21548:
21545:
21544:
21542:
21540:
21536:
21530:
21527:
21525:
21522:
21521:
21519:
21515:
21509:
21506:
21504:
21501:
21499:
21496:
21494:
21491:
21487:
21484:
21483:
21482:
21479:
21477:
21474:
21473:
21471:
21467:
21461:
21458:
21456:
21453:
21451:
21448:
21446:
21443:
21441:
21438:
21436:
21433:
21432:
21430:
21428:
21424:
21418:
21415:
21413:
21410:
21406:
21402:
21399:
21395:
21392:
21390:
21387:
21385:
21382:
21381:
21380:
21379:set functions
21376:
21373:
21369:
21366:
21364:
21361:
21360:
21359:
21356:
21354:
21353:Besov measure
21351:
21350:
21348:
21346:Measurability
21344:
21338:
21335:
21333:
21330:
21328:
21325:
21323:
21320:
21316:
21313:
21312:
21311:
21308:
21306:
21303:
21299:
21296:
21295:
21294:
21291:
21289:
21286:
21284:
21281:
21280:
21278:
21276:
21272:
21266:
21263:
21261:
21258:
21256:
21253:
21251:
21248:
21246:
21245:Convex series
21243:
21241:
21240:Bochner space
21238:
21234:
21231:
21230:
21229:
21226:
21225:
21223:
21219:
21215:
21211:
21204:
21199:
21197:
21192:
21190:
21185:
21184:
21181:
21174:
21170:
21166:
21163:
21159:
21156:
21153:
21150:
21147:
21146:
21141:
21138:
21135:
21134:
21129:
21126:
21123:
21122:
21117:
21114:
21113:
21104:
21100:
21097:
21094:Sagan, Hans:
21093:
21090:
21086:
21082:
21081:
21076:
21072:
21068:
21064:
21060:
21059:
21054:
21049:
21046:
21042:
21039:
21035:
21032:
21028:
21026:
21022:
21018:
21014:
21010:
21007:
21003:
21000:
20996:
20993:
20989:
20986:
20982:
20978:
20977:
20972:
20968:
20965:
20962:
20958:
20955:
20952:
20949:Clegg, J.C.:
20948:
20945:
20941:
20938:
20934:
20930:
20926:
20923:
20920:
20916:
20912:
20911:
20900:
20895:
20889:, p. 100
20888:
20883:
20876:
20871:
20864:
20859:
20852:
20847:
20840:
20835:
20828:
20823:
20816:
20811:
20804:
20799:
20791:
20784:
20776:
20772:
20768:
20764:
20757:
20749:
20745:
20741:
20737:
20733:
20729:
20725:
20721:
20714:
20706:
20702:
20695:
20687:
20681:
20677:
20670:
20662:
20658:
20651:
20643:
20642:
20637:
20631:
20623:
20617:
20613:
20609:
20605:
20599:
20592:
20587:
20580:
20575:
20568:
20563:
20555:
20549:
20545:
20541:
20537:
20531:
20517:on 2018-10-01
20516:
20512:
20508:
20502:
20494:
20490:
20485:
20480:
20475:
20470:
20466:
20462:
20458:
20454:
20450:
20443:
20436:
20431:
20422:
20417:
20410:
20408:
20399:
20393:
20389:
20382:
20380:
20378:
20369:
20367:9781461381068
20363:
20359:
20358:
20353:
20347:
20339:
20337:9780080471297
20333:
20329:
20325:
20318:
20316:
20307:
20301:
20297:
20296:
20291:
20287:
20281:
20275:, p. 184
20274:
20269:
20267:
20262:
20248:
20235:
20228:
20221:
20214:
20213:
20211:
20205:
20198:
20194:
20188:
20169:
20158:
20152:
20142:
20139:
20136:
20133:
20130:
20121:
20115:
20092:
20072:
20052:
20032:
20029:
20026:
20023:
20020:
20017:
19994:
19988:
19985:
19982:
19972:
19953:
19931:
19927:
19923:
19920:
19900:
19896:
19891:
19887:
19883:
19879:
19876:
19870:
19864:
19861:
19857:
19851:
19847:
19843:
19840:
19836:
19832:
19809:
19803:
19800:
19797:
19791:
19788:
19782:
19774:
19755:
19749:
19742:A functional
19739:
19721:
19715:
19712:
19709:
19703:
19677:
19671:
19648:
19641:The argument
19628:
19625:
19605:
19582:
19576:
19563:
19554:
19545:
19536:
19529:
19523:
19504:
19496:
19480:
19477:
19474:
19454:
19431:
19424:
19417:
19407:
19388:
19385:
19375:
19358:
19352:
19329:
19323:
19313:
19304:
19297:
19291:
19282:
19263:
19243:
19240:
19232:
19229:
19226:
19201:
19191:
19184:
19182:
19175:
19166:
19159:
19155:
19149:
19145:
19135:
19132:
19130:
19127:
19125:
19122:
19120:
19117:
19115:
19112:
19110:
19107:
19105:
19102:
19100:
19097:
19095:
19092:
19090:
19087:
19085:
19082:
19080:
19077:
19075:
19072:
19070:
19067:
19065:
19062:
19060:
19057:
19055:
19054:Young measure
19052:
19050:
19047:
19045:
19042:
19040:
19037:
19035:
19032:
19030:
19027:
19025:
19022:
19020:
19017:
19015:
19012:
19010:
19007:
19005:
19002:
19000:
18997:
18995:
18992:
18990:
18987:
18986:
18964:
18955:
18949:
18946:
18923:
18917:
18912:
18908:
18881:
18875:
18872:
18852:
18849:
18843:
18837:
18834:
18808:
18802:
18799:
18776:
18770:
18761:
18760:
18754:
18751:
18737:
18734:
18731:
18711:
18691:
18686:
18678:
18672:
18669:
18663:
18657:
18652:
18648:
18639:
18620:
18614:
18609:
18605:
18595:
18582:
18576:
18568:
18564:
18560:
18554:
18548:
18543:
18539:
18515:
18509:
18486:
18478:
18474:
18453:
18444:
18421:
18415:
18392:
18384:
18380:
18356:
18348:
18344:
18323:
18318:
18310:
18304:
18301:
18295:
18287:
18283:
18279:
18273:
18265:
18261:
18257:
18251:
18245:
18234:
18215:
18209:
18200:
18187:
18181:
18175:
18172:
18166:
18160:
18157:
18134:
18128:
18105:
18099:
18079:
18070:
18047:
18041:
18021:
18018:
17995:
17969:
17963:
17943:
17937:
17931:
17928:
17922:
17916:
17913:
17907:
17901:
17890:
17871:
17865:
17856:
17843:
17837:
17831:
17828:
17822:
17819:
17816:
17810:
17807:
17801:
17795:
17772:
17769:
17746:
17740:
17737:
17734:
17714:
17711:
17708:
17705:
17685:
17662:
17656:
17653:
17650:
17627:
17621:
17612:
17610:
17606:
17593:
17589:
17586:
17583:
17580:
17577:
17574:
17571:
17568:
17565:
17562:
17559:
17555:
17552:
17548:
17544:
17541:
17539:
17536:
17534:
17530:
17526:
17524:
17520:
17517:
17516:isoperimetric
17513:
17511:
17507:
17504:
17500:
17498:
17494:
17491:
17487:
17486:
17485:
17477:
17464:
17461:
17457:
17453:
17450:
17444:
17436:
17427:
17424:
17420:
17416:
17413:
17407:
17399:
17385:
17369:
17366:
17346:
17343:
17340:
17337:
17334:
17331:
17311:
17305:
17302:
17296:
17293:
17287:
17284:
17278:
17275:
17269:
17266:
17259:
17256:
17250:
17247:
17244:
17241:
17238:
17232:
17212:
17192:
17169:
17166:
17155:
17140:
17134:
17131:
17125:
17122:
17119:
17099:
17094:
17087:
17084:
17077:
17072:
17069:
17064:
17061:
17041:
17032:
17029:
17018:
17009:
17006:
16986:
16977:
16963:
16957:
16949:
16940:
16931:
16928:
16917:
16905:
16902:
16898:
16876:
16870:
16864:
16844:
16841:
16834:
16831:
16825:
16822:
16816:
16813:
16807:
16800:
16796:
16788:
16784:
16779:
16775:
16772:
16764:
16760:
16744:
16736:
16720:
16700:
16697:
16694:
16691:
16688:
16685:
16665:
16662:
16642:
16639:
16629:
16619:
16617:
16600:
16580:
16571:
16558:
16553:
16546:
16543:
16537:
16531:
16528:
16519:
16513:
16510:
16505:
16502:
16490:
16477:
16474:
16471:
16465:
16462:
16457:
16454:
16443:
16427:
16424:
16417:
16414:
16408:
16405:
16400:
16397:
16374:
16354:
16351:
16345:
16342:
16334:
16318:
16315:
16311:
16307:
16304:
16301:
16281:
16276:
16272:
16268:
16265:
16259:
16256:
16233:
16224:
16211:
16205:
16199:
16196:
16193:
16190:
16184:
16181:
16178:
16172:
16163:
16150:
16147:
16141:
16138:
16129:
16121:
16115:
16112:
16107:
16102:
16098:
16077:
16074:
16054:
16034:
16031:
16025:
16017:
16013:
16009:
16004:
16001:
15997:
15988:
15987:wave equation
15978:
15976:
15972:
15956:
15953:
15933:
15913:
15893:
15890:
15870:
15861:
15848:
15845:
15842:
15835:
15832:
15826:
15823:
15816:
15812:
15804:
15800:
15795:
15791:
15785:
15779:
15770:
15757:
15752:
15744:
15738:
15735:
15732:
15729:
15726:
15706:
15697:
15684:
15675:
15672:
15666:
15660:
15657:
15646:
15643:
15634:
15628:
15622:
15619:
15599:
15596:
15584:
15581:
15575:
15569:
15566:
15558:
15555:
15549:
15546:
15542:
15520:
15517:
15508:
15495:
15492:
15489:
15480:
15477:
15471:
15465:
15462:
15451:
15445:
15438:
15434:
15426:
15422:
15417:
15413:
15407:
15401:
15378:
15369:
15366:
15343:
15340:
15317:
15311:
15291:
15271:
15263:
15259:
15255:
15250:
15246:
15242:
15237:
15233:
15226:
15223:
15209:
15207:
15191:
15166:
15159:
15138:
15113:
15106:
15096:
15083:
15079:
15070:
15060:
15056:
15048:
15044:
15040:
15036:
15033:
15023:
15019:
15011:
15007:
15003:
14991:
14984:
14980:
14972:
14962:
14958:
14950:
14946:
14942:
14938:
14935:
14925:
14921:
14913:
14909:
14905:
14893:
14886:
14881:
14874:
14866:
14862:
14858:
14850:
14846:
14842:
14837:
14833:
14826:
14823:
14802:
14799:
14778:
14759:
14756:
14753:
14750:
14730:
14727:
14724:
14721:
14701:
14698:
14695:
14670:
14663:
14637:
14630:
14602:
14599:
14596:
14593:
14577:
14570:
14562:
14559:
14556:
14553:
14537:
14530:
14523:
14518:
14512:
14509:
14506:
14500:
14486:
14484:
14480:
14475:
14462:
14459:
14452:
14444:
14437:
14433:
14429:
14425:
14422:
14412:
14408:
14404:
14401:
14393:
14389:
14385:
14381:
14372:
14368:
14363:
14359:
14355:
14352:
14346:
14342:
14338:
14329:
14325:
14321:
14318:
14312:
14306:
14299:
14296:
14292:
14287:
14278:
14265:
14262:
14259:
14255:
14247:
14239:
14232:
14228:
14224:
14220:
14217:
14210:
14206:
14197:
14193:
14189:
14186:
14178:
14174:
14170:
14162:
14154:
14147:
14143:
14139:
14135:
14132:
14124:
14117:
14113:
14109:
14102:
14095:
14091:
14087:
14078:
14074:
14070:
14067:
14061:
14054:
14046:
14042:
14034:
14030:
14025:
14021:
14013:
14009:
14005:
14000:
13996:
13989:
13986:
13966:
13946:
13938:
13919:
13911:
13907:
13903:
13900:
13894:
13886:
13882:
13878:
13872:
13866:
13843:
13840:
13837:
13831:
13811:
13808:
13805:
13798:
13790:
13783:
13780:
13776:
13773:
13762:
13756:
13753:
13750:
13744:
13737:
13733:
13725:
13721:
13716:
13712:
13706:
13700:
13677:
13671:
13668:
13665:
13645:
13637:
13623:
13608:
13605:
13585:
13582:
13579:
13576:
13570:
13564:
13561:
13555:
13547:
13535:
13529:
13509:
13489:
13480:
13474:
13466:
13460:
13454:
13451:
13431:
13428:
13425:
13422:
13416:
13410:
13407:
13404:
13401:
13395:
13389:
13386:
13380:
13371:
13365:
13359:
13353:
13333:
13313:
13310:
13290:
13284:
13278:
13274:
13267:
13261:
13241:
13221:
13218:
13215:
13211:
13208:
13204:
13201:
13195:
13187:
13181:
13175:
13169:
13164:
13160:
13156:
13150:
13144:
13124:
13121:
13118:
13112:
13108:
13101:
13095:
13090:
13086:
13082:
13079:
13076:
13072:
13069:
13065:
13062:
13056:
13052:
13045:
13039:
13036:
13033:
13027:
13024:
13015:
13009:
13004:
13000:
12996:
12990:
12984:
12964:
12944:
12930:
12928:
12912:
12909:
12901:
12897:
12890:
12885:
12881:
12877:
12869:
12865:
12857:
12854:
12845:
12841:
12834:
12822:
12819:
12816:
12808:
12804:
12797:
12792:
12788:
12784:
12776:
12772:
12764:
12761:
12752:
12748:
12741:
12738:
12718:
12698:
12678:
12673:
12669:
12665:
12662:
12659:
12654:
12650:
12637:
12634:
12631:
12628:
12625:
12622:
12619:
12616:
12613:
12609:
12601:
12598:
12594:
12588:
12568:
12548:
12528:
12517:
12513:
12506:
12501:
12497:
12493:
12485:
12481:
12473:
12470:
12461:
12457:
12450:
12439:
12435:
12428:
12425:
12414:
12410:
12403:
12398:
12394:
12390:
12382:
12378:
12370:
12367:
12358:
12354:
12347:
12344:
12333:
12329:
12322:
12319:
12316:
12313:
12308:
12304:
12301:
12298:
12295:
12292:
12289:
12286:
12282:
12274:
12271:
12267:
12261:
12257:
12250:
12244:
12237:
12233:
12225:
12221:
12216:
12212:
12207:
12203:
12197:
12190:
12184:
12158:
12152:
12148:
12141:
12135:
12115:
12111:
12102:
12098:
12091:
12083:
12079:
12072:
12067:
12063:
12059:
12051:
12047:
12040:
12032:
12028:
12021:
12016:
12012:
12008:
12005:
12002:
11997:
11990:
11984:
11978:
11972:
11966:
11960:
11957:
11954:
11948:
11942:
11936:
11930:
11924:
11918:
11915:
11909:
11902:
11899:
11892:
11885:
11882:
11875:
11869:
11865:
11857:
11853:
11845:
11841:
11836:
11831:
11821:
11815:
11811:
11806:
11801:
11797:
11776:
11772:
11768:
11748:
11745:
11742:
11739:
11736:
11733:
11711:
11707:
11684:
11680:
11659:
11654:
11644:
11640:
11633:
11628:
11624:
11620:
11615:
11605:
11601:
11594:
11589:
11585:
11581:
11578:
11575:
11570:
11564:
11556:
11550:
11544:
11538:
11535:
11530:
11522:
11515:
11512:
11505:
11499:
11495:
11487:
11483:
11475:
11471:
11466:
11462:
11456:
11450:
11430:
11421:
11407:
11404:
11401:
11398:
11391:
11385:
11379:
11371:
11367:
11360:
11354:
11347:
11343:
11335:
11331:
11326:
11305:
11296:
11281:
11273:
11257:
11251:
11243:
11239:
11216:
11212:
11191:
11171:
11151:
11142:
11136:
11128:
11122:
11116:
11113:
11093:
11073:
11070:
11067:
11064:
11061:
11058:
11055:
11052:
11049:
11046:
11042:
11034:
11031:
11027:
11021:
11001:
10978:
10972:
10949:
10943:
10923:
10903:
10899:
10895:
10872:
10866:
10843:
10837:
10817:
10814:
10811:
10805:
10797:
10791:
10785:
10779:
10772:
10768:
10760:
10756:
10751:
10747:
10741:
10735:
10715:
10695:
10692:
10684:
10680:
10673:
10669:
10666:
10663:
10655:
10651:
10644:
10624:
10604:
10601:
10598:
10593:
10587:
10579:
10573:
10567:
10561:
10558:
10553:
10545:
10538:
10535:
10528:
10522:
10518:
10510:
10506:
10498:
10494:
10489:
10485:
10479:
10473:
10465:
10459:
10449:
10446:
10444:
10434:
10419:
10416:
10413:
10410:
10406:
10401:
10397:
10393:
10390:
10387:
10383:
10380:
10376:
10371:
10367:
10346:
10327:
10324:
10304:
10300:
10296:
10293:
10289:
10284:
10280:
10276:
10273:
10270:
10266:
10263:
10259:
10254:
10250:
10245:
10241:
10238:
10232:
10226:
10206:
10186:
10183:
10180:
10177:
10157:
10154:
10134:
10125:
10123:
10107:
10087:
10084:
10064:
10061:
10058:
10055:
10052:
10049:
10046:
10043:
10037:
10029:
10003:
9983:
9963:
9960:
9940:
9937:
9934:
9931:
9928:
9922:
9916:
9896:
9893:
9873:
9870:
9867:
9847:
9844:
9841:
9838:
9833:
9829:
9826:
9823:
9820:
9817:
9811:
9803:
9793:
9789:
9784:
9780:
9776:
9773:
9770:
9766:
9763:
9758:
9754:
9751:
9748:
9745:
9739:
9733:
9730:
9726:
9720:
9716:
9695:
9692:
9689:
9686:
9681:
9677:
9674:
9671:
9668:
9665:
9662:
9658:
9652:
9648:
9644:
9641:
9638:
9634:
9631:
9626:
9622:
9619:
9616:
9613:
9607:
9604:
9597:
9591:
9587:
9563:
9560:
9557:
9554:
9548:
9528:
9525:
9505:
9485:
9465:
9445:
9442:
9434:
9418:
9415:
9392:
9386:
9366:
9363:
9340:
9334:
9314:
9311:
9288:
9285:
9282:
9276:
9256:
9253:
9250:
9245:
9241:
9235:
9229:
9226:
9221:
9217:
9210:
9204:
9199:
9196:
9190:
9184:
9180:
9176:
9172:
9169:
9165:
9162:
9157:
9153:
9147:
9144:
9141:
9135:
9132:
9129:
9123:
9120:
9112:
9109:
9103:
9097:
9093:
9089:
9083:
9077:
9063:
9061:
9045:
9042:
9039:
9019:
9000:
8997:
8991:
8985:
8965:
8962:
8959:
8953:
8950:
8944:
8937:that satisfy
8924:
8904:
8901:
8895:
8886:
8883:
8879:
8871:
8866:
8863:
8859:
8855:
8849:
8843:
8835:
8831:
8814:
8805:
8792:
8789:
8769:
8766:
8763:
8757:
8734:
8731:
8711:
8691:
8688:
8685:
8682:
8678:
8675:
8671:
8665:
8659:
8654:
8650:
8629:
8609:
8589:
8586:
8566:
8546:
8539:
8535:
8512:
8492:
8473:
8470:
8450:
8430:
8427:
8424:
8417:
8409:
8400:
8395:
8391:
8387:
8384:
8381:
8377:
8374:
8370:
8364:
8358:
8355:
8352:
8346:
8343:
8335:
8331:
8327:
8324:
8321:
8317:
8314:
8307:
8301:
8295:
8287:
8283:
8262:
8259:
8256:
8253:
8249:
8246:
8242:
8236:
8233:
8225:
8221:
8217:
8212:
8209:
8206:
8201:
8194:
8191:
8188:
8185:
8179:
8173:
8170:
8166:
8152:must vanish:
8136:
8133:
8130:
8127:
8121:
8101:
8098:
8078:
8058:
8038:
8035:
8015:
7995:
7975:
7972:
7969:
7965:
7962:
7958:
7952:
7949:
7941:
7937:
7931:
7928:
7923:
7917:
7911:
7897:
7883:
7880:
7875:
7872:
7868:
7862:
7858:
7852:
7848:
7844:
7841:
7833:
7828:
7824:
7820:
7817:
7809:
7806:
7802:
7798:
7790:
7785:
7781:
7777:
7774:
7766:
7763:
7759:
7750:
7734:
7726:
7711:
7708:
7705:
7701:
7698:
7692:
7686:
7683:
7677:
7674:
7667:
7663:
7659:
7653:
7647:
7627:
7624:
7621:
7601:
7578:
7575:
7572:
7566:
7552:
7549:
7534:
7528:
7525:
7522:
7519:
7516:
7513:
7491:
7488:
7485:
7481:
7458:
7455:
7452:
7448:
7427:
7419:
7416:
7412:
7389:
7386:
7383:
7379:
7369:
7350:
7347:
7343:
7339:
7336:
7313:
7310:
7305:
7301:
7295:
7289:
7280:
7267:
7261:
7258:
7252:
7246:
7240:
7237:
7234:
7228:
7222:
7219:
7213:
7210:
7207:
7199:
7196:
7193:
7189:
7185:
7182:
7176:
7172:
7152:
7146:
7142:
7138:
7132:
7124:
7121:
7116:
7112:
7103:
7098:
7094:
7090:
7084:
7078:
7069:
7064:
7055:
7041:
7021:
7018:
7015:
7006:
7002:
6998:
6989:
6984:
6956:
6948:
6932:
6918:
6905:
6902:
6898:
6887:
6880:
6871:
6862:
6853:
6849:
6845:
6839:
6835:
6827:
6819:
6816:
6810:
6807:
6804:
6800:
6793:
6790:
6781:
6772:
6765:
6762:
6758:
6753:
6747:
6739:
6725:
6709:
6689:
6686:
6683:
6674:
6663:
6656:
6652:
6649:
6646:
6640:
6633:
6630:
6626:
6620:
6614:
6611:
6608:
6602:
6597:
6592:
6588:
6584:
6581:
6561:
6555:
6549:
6529:
6515:
6513:
6497:
6494:
6488:
6480:
6454:
6445:
6432:
6426:
6419:
6416:
6395:
6387:
6371:
6351:
6347:
6344:
6337:
6334:
6325:
6315:
6312:
6308:
6305:
6298:
6282:
6259:
6252:
6249:
6225:
6219:
6199:
6196:
6193:
6187:
6179:
6159:
6145:
6139:
6133:
6130:
6127:
6104:
6098:
6075:
6069:
6046:
6040:
6015:
6011:
6007:
6002:
5998:
5990:
5986:
5980:
5976:
5972:
5967:
5963:
5957:
5953:
5946:
5943:
5928:
5924:
5920:
5915:
5911:
5903:
5899:
5895:
5890:
5886:
5879:
5876:
5861:
5858:
5855:
5852:
5849:
5843:
5837:
5816:
5810:
5806:
5802:
5797:
5793:
5788:
5766:
5760:
5756:
5752:
5747:
5743:
5738:
5717:
5714:
5708:
5701:
5698:
5672:
5668:
5664:
5661:
5655:
5651:
5645:
5640:
5629:
5622:
5619:
5595:
5592:
5587:
5583:
5579:
5576:
5556:
5550:
5546:
5542:
5534:
5523:
5516:
5513:
5506:
5503:
5496:
5485:
5478:
5475:
5448:
5445:
5425:
5421:
5418:
5410:
5399:
5392:
5389:
5382:
5379:
5371:
5364:
5361:
5348:
5335:
5331:
5328:
5317:
5306:
5299:
5296:
5289:
5286:
5278:
5271:
5268:
5255:
5252:
5248:
5226:
5206:
5202:
5199:
5192:
5189:
5180:
5168:
5165:
5161:
5136:
5130:
5110:
5107:
5087:
5078:
5065:
5057:
5046:
5039:
5036:
5029:
5026:
5021:
5018:
4998:
4995:
4988:
4985:
4976:
4964:
4961:
4957:
4952:
4946:
4938:
4913:
4907:
4901:
4878:
4872:
4863:
4841:
4832:
4828:
4821:
4818:
4813:
4809:
4799:
4790:
4786:
4779:
4776:
4771:
4767:
4757:
4750:
4747:
4742:
4739:
4733:
4727:
4720:
4717:
4696:
4692:
4689:
4681:
4670:
4663:
4660:
4653:
4650:
4641:
4637:
4629:
4625:
4620:
4616:
4610:
4604:
4596:
4580:
4576:
4570:
4566:
4562:
4557:
4553:
4548:
4526:
4520:
4516:
4512:
4507:
4503:
4498:
4477:
4471:
4465:
4462:
4459:
4445:
4443:
4427:
4421:
4415:
4407:
4403:
4387:
4381:
4375:
4367:
4362:
4349:
4343:
4337:
4334:
4330:
4326:
4323:
4300:
4294:
4286:
4282:
4266:
4263:
4256:
4253:
4244:
4232:
4229:
4225:
4220:
4214:
4206:
4192:
4187:
4174:
4170:
4167:
4164:
4161:
4156:
4148:
4145:
4136:
4124:
4121:
4117:
4112:
4106:
4098:
4088:
4081:
4075:
4068:
4064:
4056:
4052:
4047:
4024:
4020:
3997:
3993:
3972:
3969:
3966:
3958:
3942:
3939:
3936:
3915:
3910:
3907:
3903:
3900:
3897:
3894:
3890:
3886:
3879:
3874:
3871:
3867:
3864:
3861:
3858:
3854:
3850:
3826:
3823:
3818:
3810:
3807:
3798:
3786:
3783:
3779:
3774:
3771:
3768:
3762:
3754:
3744:
3736:
3732:
3724:
3720:
3715:
3711:
3709:
3701:
3698:
3690:
3687:
3678:
3666:
3663:
3659:
3654:
3647:
3643:
3635:
3631:
3626:
3622:
3615:
3611:
3603:
3599:
3593:
3589:
3582:
3579:
3570:
3556:
3553:
3550:
3546:
3540:
3532:
3519:
3515:
3507:
3503:
3498:
3494:
3492:
3484:
3481:
3476:
3471:
3468:
3460:
3457:
3448:
3439:
3436:
3430:
3422:
3412:
3404:
3400:
3392:
3388:
3383:
3379:
3377:
3372:
3369:
3364:
3361:
3358:
3353:
3347:
3344:
3339:
3336:
3322:
3318:
3310:
3306:
3301:
3287:
3274:
3270:
3267:
3259:
3256:
3247:
3238:
3235:
3229:
3221:
3212:
3206:
3203:
3198:
3195:
3173:
3169:
3166:
3162:
3156:
3153:
3147:
3144:
3140:
3117:
3114:
3108:
3105:
3100:
3097:
3071:
3068:
3062:
3059:
3055:
3045:
3042:
3033:
3024:
3018:
3015:
3010:
3007:
2998:
2990:
2981:
2975:
2972:
2967:
2964:
2941:
2938:
2918:
2897:
2894:
2890:
2887:
2883:
2880:
2876:
2872:
2869:
2848:
2845:
2842:
2839:
2836:
2833:
2813:
2809:
2804:
2801:
2797:
2794:
2791:
2788:
2784:
2780:
2772:
2767:
2754:
2750:
2747:
2744:
2741:
2736:
2733:
2730:
2725:
2719:
2716:
2711:
2708:
2694:
2690:
2682:
2678:
2673:
2669:
2664:
2661:
2658:
2653:
2647:
2644:
2636:
2626:
2620:
2613:
2589:
2586:
2583:
2560:
2547:the function
2534:
2531:
2528:
2505:
2499:
2479:
2472:
2469:
2466:
2463:
2457:
2454:
2448:
2434:
2421:
2418:
2398:
2392:
2386:
2366:
2346:
2343:
2340:
2337:
2330:Substituting
2328:
2315:
2312:
2309:
2289:
2281:
2265:
2262:
2253:
2240:
2233:
2230:
2227:
2224:
2218:
2215:
2209:
2203:
2183:
2163:
2158:
2154:
2131:
2127:
2103:
2097:
2077:
2074:
2066:
2065:local minimum
2047:
2041:
2018:
2014:
2011:
1990:
1987:
1984:
1981:
1960:
1953:
1946:
1943:
1939:
1933:
1927:
1924:
1921:
1917:
1913:
1906:
1892:
1886:
1883:
1878:
1875:
1869:
1863:
1856:
1853:
1845:
1827:
1821:
1814:
1811:
1793:
1789:
1785:
1780:
1776:
1768:
1767:
1766:
1752:
1748:
1745:
1740:
1733:
1726:
1723:
1719:
1713:
1707:
1704:
1701:
1697:
1693:
1686:
1682:
1674:
1670:
1665:
1661:
1655:
1649:
1640:
1638:
1634:
1627:
1617:
1615:
1611:
1607:
1601:
1599:
1595:
1579:
1576:
1573:
1550:
1547:
1527:
1524:
1521:
1495:
1489:
1481:
1478:is called an
1465:
1458:The function
1445:
1442:
1422:
1414:
1411:has the same
1395:
1389:
1386:
1380:
1374:
1371:
1368:
1345:
1322:
1316:
1308:
1304:
1288:
1280:
1276:
1272:
1262:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1235:Marston Morse
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1207:David Hilbert
1203:
1201:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1128:
1124:
1119:
1115:
1111:
1107:
1103:
1099:
1095:
1085:
1083:
1078:
1074:
1070:
1066:
1061:
1059:
1055:
1051:
1047:
1043:
1042:
1037:
1036:straight line
1032:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
978:
973:
971:
966:
964:
959:
958:
956:
955:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
914:
906:
905:
898:
895:
893:
890:
888:
885:
883:
880:
879:
871:
870:
859:
856:
854:
851:
849:
846:
845:
844:
843:
833:
832:
821:
818:
816:
813:
811:
808:
806:
803:
801:
800:Line integral
798:
796:
793:
791:
788:
787:
786:
785:
781:
780:
775:
772:
770:
767:
765:
762:
760:
757:
756:
755:
754:
750:
749:
743:
742:Multivariable
737:
736:
725:
722:
720:
717:
715:
712:
710:
707:
705:
702:
700:
697:
696:
695:
694:
690:
689:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
655:
654:
653:
647:
641:
640:
629:
626:
624:
621:
619:
616:
614:
611:
609:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
579:
578:
577:
574:
571:
570:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
539:
535:
532:
531:
530:
529:
523:
517:
516:
505:
502:
500:
497:
495:
492:
490:
487:
484:
480:
477:
475:
472:
469:
465:
461:
460:trigonometric
457:
454:
452:
449:
447:
444:
442:
439:
438:
437:
436:
432:
431:
426:
423:
421:
418:
416:
413:
411:
408:
405:
401:
398:
396:
393:
392:
391:
390:
386:
385:
380:
377:
375:
372:
370:
367:
366:
365:
364:
358:
352:
351:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
291:
290:
289:
286:
283:
282:
277:
274:
272:
271:Related rates
269:
267:
264:
262:
259:
257:
254:
252:
249:
248:
247:
246:
242:
241:
234:
231:
229:
228:of a function
226:
224:
223:infinitesimal
221:
220:
219:
216:
213:
209:
206:
205:
204:
203:
199:
198:
192:
186:
185:
179:
176:
174:
171:
169:
166:
165:
160:
157:
155:
152:
151:
147:
144:
143:
142:
141:
122:
116:
113:
107:
101:
98:
95:
92:
85:
78:
75:
69:
64:
60:
51:
50:
47:
44:
43:
39:
38:
33:
19:
22271:
22135:
22092:
22023:Weak duality
21986:
21978:
21898:Orthogonally
21566:Applications
21524:Crinkled arc
21460:Paley–Wiener
21143:
21131:
21119:
21078:
21067:the original
21057:
20974:
20971:Introduction
20918:
20894:
20882:
20877:, p. 99
20870:
20858:
20846:
20834:
20822:
20810:
20798:
20783:
20766:
20762:
20756:
20723:
20719:
20713:
20704:
20700:
20694:
20675:
20669:
20660:
20650:
20640:
20630:
20611:
20598:
20586:
20581:, p. 13
20574:
20562:
20543:
20530:
20519:. Retrieved
20515:the original
20510:
20501:
20456:
20452:
20442:
20430:
20421:math/0402357
20390:. Springer.
20387:
20356:
20346:
20327:
20294:
20290:Fomin, S. V.
20280:
20247:
20229:
20215:
20204:
20196:
20192:
20187:
19971:
19772:
19738:
19562:
19553:
19544:
19535:
19522:
19410:The product
19406:
19312:
19303:
19290:
19281:
19190:
19179:
19174:
19165:
19148:
19124:Fermat Prize
18758:
18757:
18752:
18637:
18596:
18232:
18201:
17888:
17857:
17613:
17608:
17604:
17602:
17531:and solving
17521:Calculating
17514:Solution to
17495:Solution to
17483:
16978:
16631:
16572:
16491:
16441:
16225:
16164:
15984:
15862:
15771:
15698:
15509:
15215:
15097:
14492:
14476:
14279:
13634:
13626:Applications
12936:
12926:
11422:
11297:
10461:
10447:
10442:
10440:
10126:
10121:
9579:is given by
9432:
9069:
8806:
8622:vanishes on
7903:
7748:
7558:
7550:
7370:
7281:
7065:
7061:
6946:
6924:
6574:that is, if
6521:
6446:
6165:
5349:
5079:
4864:
4451:
4363:
4280:
4188:
3288:
3087:and because
2931:rather than
2768:
2435:
2329:
2279:
2254:
2196:close to 0,
2033:
1641:
1629:
1602:
1598:weak extrema
1597:
1593:
1479:
1270:
1268:
1239:Morse theory
1219:Emmy Noether
1204:
1159:(1834), and
1137:Isaac Newton
1131:
1126:
1122:
1091:
1071:satisfy the
1062:
1039:
1033:
1017:real numbers
992:
988:
986:
896:
456:Substitution
218:Differential
191:Differential
22099:Integration
22013:Duality gap
22008:Dual system
21892:Convex hull
21332:Holomorphic
21315:Directional
21275:Derivatives
20957:Courant, R.
20919:SIAM Review
20841:, p. 6
20829:, p. 8
20608:Hilbert, D.
20604:Courant, R.
17225:defined by
15206:Snell's law
14489:Snell's law
3289:Therefore,
2769:Taking the
1301:of a given
1211:Oskar Bolza
1189:Weierstrass
1161:Carl Jacobi
1025:derivatives
1005:functionals
917:Precalculus
910:Miscellanea
875:Specialized
782:Definitions
549:Alternating
387:Definitions
200:Definitions
22287:Categories
22124:stochastic
21936:Radial set
21906:Convex set
21668:Convex set
21133:PlanetMath
20788:Turnbull.
20707:: 147–153.
20540:Hilbert, D
20536:Courant, R
20521:2013-07-28
20258:References
19566:Note that
19528:Archimedes
19378:values of
19316:Note that
17547:Lagrangian
16387:satisfies
16246:satisfies
15719:satisfies
10916:among all
10456:See also:
9408:acting on
7068:Lavrentiev
6726:equation,
5152:and thus,
4595:arc length
4406:sufficient
4404:, but not
2602:and thus,
2063:attains a
1181:Otto Hesse
897:Variations
892:Stochastic
882:Fractional
751:Formalisms
714:Divergence
683:Identities
663:Divergence
208:Derivative
159:Continuity
22236:Functions
21921:Hypograph
21455:Regulated
21427:Integrals
21145:MathWorld
20925:Bolza, O.
20193:quadratic
20140:≤
20134:≤
20119:‖
20113:‖
20027:≤
20021:≤
19954:α
19880:φ
19865:φ
19833:φ
19804:φ
19801:α
19789:α
19783:φ
19750:φ
19701:Δ
19669:Δ
19574:Δ
19505:ε
19475:δ
19422:Φ
19418:ε
19324:η
19230:−
19156:is about
18959:^
18909:δ
18885:^
18835:δ
18812:^
18683:‖
18676:‖
18670:≥
18649:δ
18606:δ
18565:φ
18540:δ
18475:φ
18451:→
18448:‖
18442:‖
18419:→
18416:ε
18381:φ
18345:φ
18315:‖
18308:‖
18305:ε
18284:φ
18262:φ
18243:Δ
18176:φ
18158:δ
18100:φ
18077:→
18074:‖
18068:‖
18045:→
18042:ε
17999:‖
17993:‖
17964:φ
17941:‖
17935:‖
17932:ε
17917:φ
17899:Δ
17829:−
17793:Δ
17523:geodesics
17442:∂
17437:ψ
17434:∂
17405:∂
17400:ψ
17397:∂
17297:˙
17276:−
17270:˙
17170:˙
17135:˙
17088:˙
17033:˙
17024:∂
17016:∂
16955:∂
16947:∂
16932:˙
16923:∂
16915:∂
16826:˙
16780:∫
16695:−
16622:Mechanics
16581:ψ
16547:˙
16538:⋅
16532:˙
16422:∇
16352:ψ
16349:∇
16266:ψ
16263:∇
16260:⋅
16257:ψ
16254:∇
16234:ψ
16200:ψ
16197:−
16173:φ
16148:φ
16145:∇
16142:⋅
16139:φ
16136:∇
16099:φ
16029:∇
16026:⋅
16023:∇
15954:ψ
15934:ψ
15871:ψ
15836:˙
15827:⋅
15796:∫
15730:⋅
15676:˙
15667:⋅
15661:˙
15647:˙
15594:∇
15585:˙
15576:⋅
15570:˙
15481:˙
15472:⋅
15466:˙
15418:∫
15370:˙
15114:−
14981:−
14963:−
14926:−
14894:−
14824:δ
14638:−
14538:−
14288:−
14026:∫
13987:δ
13935:then the
13904:ε
13717:∫
13565:σ
13553:∂
13545:∂
13452:λ
13408:λ
13405:−
13378:∇
13360:⋅
13357:∇
13354:−
13285:φ
13268:φ
13182:φ
13161:∭
13151:φ
13109:φ
13096:σ
13087:∬
13053:φ
13034:φ
13031:∇
13028:⋅
13025:φ
13022:∇
13001:∭
12991:φ
12739:−
12626:λ
12623:−
12589:−
12345:−
12299:λ
12296:−
12262:−
12217:∫
11958:λ
11955:−
11837:∫
11746:ε
11467:∫
11327:∫
11213:λ
11192:λ
11114:λ
11094:λ
11059:λ
11056:−
11022:−
10752:∫
10490:∫
10398:∫
10368:∬
10281:∫
10251:∬
10181:≡
10178:φ
10135:σ
10047:σ
10035:∂
10027:∂
9926:∇
9923:⋅
9920:∇
9917:−
9821:σ
9809:∂
9801:∂
9781:∫
9743:∇
9740:⋅
9737:∇
9731:−
9717:∬
9663:σ
9649:∫
9611:∇
9608:⋅
9602:∇
9588:∬
9561:ε
9387:σ
9242:φ
9218:φ
9205:σ
9181:∫
9154:φ
9130:φ
9127:∇
9124:⋅
9121:φ
9118:∇
9094:∬
9084:φ
8986:φ
8963:−
8951:−
8945:φ
8925:φ
8884:φ
8864:−
8860:∫
8850:φ
8761:∇
8758:⋅
8755:∇
8669:∇
8666:⋅
8663:∇
8651:∬
8544:∂
8533:∂
8415:∂
8407:∂
8392:∫
8368:∇
8365:⋅
8362:∇
8350:∇
8347:⋅
8341:∇
8332:∬
8305:∇
8296:⋅
8293:∇
8284:∬
8240:∇
8237:⋅
8231:∇
8222:∬
8207:ε
8192:ε
8174:ε
8134:ε
8016:φ
7959:φ
7956:∇
7953:⋅
7950:φ
7947:∇
7938:∬
7918:φ
7869:φ
7859:φ
7849:φ
7842:−
7825:φ
7803:φ
7782:φ
7760:φ
7693:φ
7690:∇
7687:⋅
7684:φ
7681:∇
7664:∬
7654:φ
7567:φ
7532:∞
7517:≤
7423:∞
7354:∞
7340:∈
7282:Clearly,
7186:∈
7122:−
7095:∫
7016:≠
6995:∂
6981:∂
6947:weak form
6877:∂
6869:∂
6817:−
6808:⋯
6787:∂
6779:∂
6754:−
6745:∂
6737:∂
6650:…
6589:∫
6486:∂
6478:∂
6331:∂
6323:∂
6309:−
6185:∂
6177:∂
6008:−
5973:−
5921:−
5896:−
5665:−
5580:≤
5186:∂
5178:∂
4982:∂
4974:∂
4953:−
4944:∂
4936:∂
4621:∫
4402:necessary
4335:δ
4324:δ
4250:∂
4242:∂
4221:−
4212:∂
4204:∂
4142:∂
4134:∂
4113:−
4104:∂
4096:∂
4076:η
4048:∫
3967:η
3937:ε
3884:→
3804:∂
3796:∂
3775:η
3772:−
3769:η
3760:∂
3752:∂
3716:∫
3684:∂
3676:∂
3655:η
3627:∫
3623:−
3590:η
3576:∂
3568:∂
3547:η
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3530:∂
3499:∫
3469:η
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3446:∂
3437:η
3428:∂
3420:∂
3384:∫
3359:ε
3348:ε
3302:∫
3268:η
3253:∂
3245:∂
3236:η
3227:∂
3219:∂
3207:ε
3167:η
3157:ε
3118:η
3109:ε
3072:ε
3039:∂
3031:∂
3019:ε
2996:∂
2988:∂
2976:ε
2919:ε
2895:η
2891:ε
2849:η
2846:ε
2731:ε
2720:ε
2674:∫
2659:ε
2648:ε
2640:Φ
2627:≡
2611:Φ
2584:ε
2561:ε
2555:Φ
2473:η
2470:ε
2449:ε
2443:Φ
2419:ε
2347:η
2344:ε
2310:δ
2280:variation
2266:η
2263:ε
2255:The term
2234:η
2231:ε
2216:≤
2184:ε
2098:η
1810:constants
1666:∫
1577:≥
1571:Δ
1525:≤
1519:Δ
1387:−
1366:Δ
1275:functions
1054:mechanics
1041:geodesics
1013:functions
1001:functions
887:Malliavin
774:Geometric
673:Laplacian
623:Dirichlet
534:Geometric
114:−
61:∫
22261:Infinity
22114:ordinary
22094:Calculus
21945:Zonotope
21916:Epigraph
21409:Strongly
21210:Analysis
20748:55005550
20638:(1843).
20610:(1953).
20542:(1953).
20493:16589462
20354:(2012).
19425:′
19152:Whereas
18982:See also
18704:for all
17527:Finding
17518:problems
17490:catenary
15356:and let
15049:′
15012:′
14951:′
14914:′
14803:′
14438:′
14369:′
14347:′
14233:′
14148:′
14118:′
14096:′
13784:′
12858:′
12765:′
12731:only if
12610:′
12602:′
12581:only if
12474:′
12371:′
12283:′
12275:′
11903:′
11886:′
11516:′
11043:′
11035:′
10539:′
8887:′
7143:′
7066:However
7003:′
6794:′
6634:′
6420:′
6338:′
6316:′
6253:′
6091:so that
5702:′
5623:′
5517:′
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5300:′
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2884:′
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2805:′
2614:′
2015:′
1947:′
1857:′
1727:′
1606:converse
1480:extremal
1415:for all
1195:and the
1183:(1857),
1179:(1850),
1175:(1849),
1155:(1831),
1151:(1829),
1147:(1810),
1133:Legendre
1118:Lagrange
1108:and the
1082:topology
1067:for the
1009:mappings
927:Glossary
837:Advanced
815:Jacobian
769:Exterior
699:Gradient
691:Theorems
658:Gradient
597:Integral
559:Binomial
544:Harmonic
404:improper
400:Integral
357:Integral
339:Reynolds
314:Quotient
243:Concepts
79:′
46:Calculus
22119:partial
22000:Duality
21902:Pseudo-
21876:Ursescu
21773:Pseudo-
21747:Concave
21726:Simplex
21706:Duality
21575: (
21517:Related
21469:Results
21445:Dunford
21435:Bochner
21401:Bochner
21375:Measure
21173:Part II
21162:YouTube
20728:Bibcode
20461:Bibcode
20230:Chapter
20216:Chapter
19517:factor.
17505:problem
16733:is the
15204:axis.
7614:in the
6724:Poisson
4448:Example
2954:yields
1279:scalars
1271:extrema
1265:Extrema
1177:Jellett
1173:Strauch
1112:, but
1088:History
1056:is the
1015:to the
922:History
820:Hessian
709:Stokes'
704:Green's
536: (
458: (
402: (
324:Inverse
299:Product
210: (
22256:Series
21983:, and
21954:Series
21871:Simons
21778:Quasi-
21768:Proper
21753:Closed
21577:bundle
21405:Weakly
21394:Vector
21169:Part I
21087:
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20232:
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20045:where
19913:where
19773:linear
19183:(2004)
18336:where
17956:where
17727:where
16713:where
16294:where
16047:where
15612:where
14622:where
13631:Optics
13444:where
11672:where
11086:where
10617:where
10199:where
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8443:where
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6364:where
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5569:where
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5080:Since
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3843:where
2826:where
1765:where
1307:domain
1253:. The
1169:Cauchy
1165:Sarrus
764:Tensor
759:Matrix
646:Vector
564:Taylor
522:Series
154:Limits
22251:Limit
21811:Main
21298:Total
20744:S2CID
20416:arXiv
19141:Notes
17590:, an
17492:shape
17112:then
16367:then
16335:, if
7034:then
6702:then
5461:Then
5350:Thus
5011:with
4709:with
3929:when
587:Ratio
554:Power
468:Euler
446:Discs
441:Parts
309:Power
304:Chain
233:total
21931:Lens
21885:Sets
21735:Maps
21642:and
21085:ISBN
21021:ISBN
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20981:ISBN
20933:ISBN
20680:ISBN
20616:ISBN
20548:ISBN
20489:PMID
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20332:ISBN
20300:ISBN
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20226:116.
20065:and
19618:and
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18034:and
17549:and
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13137:and
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9478:and
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8525:and
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7506:for
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6241:and
5867:with
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2003:and
1808:are
1413:sign
1229:and
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1193:20th
1139:and
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987:The
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628:Abel
592:Root
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294:Sum
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4856:y
4842:.
4838:)
4833:2
4829:x
4825:(
4822:f
4819:=
4814:2
4810:y
4800:,
4796:)
4791:1
4787:x
4783:(
4780:f
4777:=
4772:1
4768:y
4758:,
4751:x
4748:d
4743:y
4740:d
4734:=
4731:)
4728:x
4725:(
4718:y
4697:,
4693:x
4690:d
4682:2
4678:]
4674:)
4671:x
4668:(
4661:y
4657:[
4654:+
4651:1
4642:2
4638:x
4630:1
4626:x
4617:=
4614:]
4611:y
4608:[
4605:A
4581:.
4577:)
4571:2
4567:y
4563:,
4558:2
4554:x
4549:(
4527:)
4521:1
4517:y
4513:,
4508:1
4504:x
4499:(
4478:,
4475:)
4472:x
4469:(
4466:f
4463:=
4460:y
4428:.
4425:]
4422:f
4419:[
4416:J
4388:.
4385:)
4382:x
4379:(
4376:f
4350:.
4347:)
4344:x
4341:(
4338:f
4331:/
4327:J
4304:]
4301:f
4298:[
4295:J
4267:0
4264:=
4254:f
4245:L
4233:x
4230:d
4226:d
4215:f
4207:L
4175:.
4171:0
4168:=
4165:x
4162:d
4157:)
4146:f
4137:L
4125:x
4122:d
4118:d
4107:f
4099:L
4089:(
4085:)
4082:x
4079:(
4069:2
4065:x
4057:1
4053:x
4025:2
4021:x
3998:1
3994:x
3973:0
3970:=
3943:0
3940:=
3916:]
3908:f
3904:,
3901:f
3898:,
3895:x
3891:[
3887:L
3880:]
3872:y
3868:,
3865:y
3862:,
3859:x
3855:[
3851:L
3827:x
3824:d
3819:)
3808:f
3799:L
3787:x
3784:d
3780:d
3763:f
3755:L
3745:(
3737:2
3733:x
3725:1
3721:x
3712:=
3702:x
3699:d
3688:f
3679:L
3667:x
3664:d
3660:d
3648:2
3644:x
3636:1
3632:x
3616:2
3612:x
3604:1
3600:x
3594:|
3580:f
3571:L
3557:+
3554:x
3551:d
3541:f
3533:L
3520:2
3516:x
3508:1
3504:x
3495:=
3485:x
3482:d
3477:)
3458:f
3449:L
3440:+
3431:f
3423:L
3413:(
3405:2
3401:x
3393:1
3389:x
3380:=
3373:x
3370:d
3365:0
3362:=
3354:|
3345:d
3340:L
3337:d
3323:2
3319:x
3311:1
3307:x
3275:.
3257:y
3248:L
3239:+
3230:y
3222:L
3213:=
3204:d
3199:L
3196:d
3174:,
3163:=
3154:d
3145:y
3141:d
3115:=
3106:d
3101:y
3098:d
3069:d
3060:y
3056:d
3043:y
3034:L
3025:+
3016:d
3011:y
3008:d
2999:y
2991:L
2982:=
2973:d
2968:L
2965:d
2942:,
2939:x
2888:+
2881:f
2877:=
2870:y
2843:+
2840:f
2837:=
2834:y
2814:,
2810:]
2802:y
2798:,
2795:y
2792:,
2789:x
2785:[
2781:L
2755:.
2751:0
2748:=
2745:x
2742:d
2737:0
2734:=
2726:|
2717:d
2712:L
2709:d
2695:2
2691:x
2683:1
2679:x
2670:=
2665:0
2662:=
2654:|
2645:d
2637:d
2624:)
2621:0
2618:(
2590:0
2587:=
2564:)
2558:(
2535:f
2532:=
2529:y
2509:]
2506:y
2503:[
2500:J
2480:.
2476:]
2467:+
2464:f
2461:[
2458:J
2455:=
2452:)
2446:(
2422:,
2399:,
2396:]
2393:y
2390:[
2387:J
2367:y
2341:+
2338:f
2316:.
2313:f
2290:f
2241:.
2237:]
2228:+
2225:f
2222:[
2219:J
2213:]
2210:f
2207:[
2204:J
2164:,
2159:2
2155:x
2132:1
2128:x
2107:)
2104:x
2101:(
2078:,
2075:f
2051:]
2048:y
2045:[
2042:J
2019:.
2012:y
1991:,
1988:y
1985:,
1982:x
1961:)
1957:)
1954:x
1951:(
1944:y
1940:,
1937:)
1934:x
1931:(
1928:y
1925:,
1922:x
1918:(
1914:L
1893:,
1887:x
1884:d
1879:y
1876:d
1870:=
1867:)
1864:x
1861:(
1854:y
1831:)
1828:x
1825:(
1822:y
1812:,
1794:2
1790:x
1786:,
1781:1
1777:x
1753:.
1749:x
1746:d
1741:)
1737:)
1734:x
1731:(
1724:y
1720:,
1717:)
1714:x
1711:(
1708:y
1705:,
1702:x
1698:(
1694:L
1687:2
1683:x
1675:1
1671:x
1662:=
1659:]
1656:y
1653:[
1650:J
1580:0
1574:J
1551:,
1548:f
1528:0
1522:J
1499:]
1496:f
1493:[
1490:J
1466:f
1446:.
1443:f
1423:y
1399:]
1396:f
1393:[
1390:J
1384:]
1381:y
1378:[
1375:J
1372:=
1369:J
1346:f
1326:]
1323:y
1320:[
1317:J
1289:y
976:e
969:t
962:v
540:)
485:)
481:(
470:)
406:)
214:)
126:)
123:a
120:(
117:f
111:)
108:b
105:(
102:f
99:=
96:t
93:d
89:)
86:t
83:(
76:f
70:b
65:a
34:.
20:)
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