5968:
4487:
5963:{\displaystyle {\begin{aligned}{\ddot {y}}&\equiv {\frac {d^{2}y}{dt^{2}}}={\frac {d}{dt}}\left({\frac {dy}{dt}}\right)={\frac {d}{dt}}{\Bigl (}{\dot {y}}{\Bigr )}={\frac {d}{dt}}{\Bigl (}f'(t){\Bigr )}=D_{t}^{2}y=f''(t)=y''_{t}\\{\overset {...}{y}}&={\dot {\ddot {y}}}\equiv {\frac {d^{3}y}{dt^{3}}}=D_{t}^{3}y=f'''(t)=y'''_{t}\\{\overset {\,4}{\dot {y}}}&={\overset {....}{y}}={\ddot {\ddot {y}}}\equiv {\frac {d^{4}y}{dt^{4}}}=D_{t}^{4}y=f^{\rm {IV}}(t)=y_{t}^{(4)}\\{\overset {\,5}{\dot {y}}}&={\ddot {\overset {...}{y}}}={\dot {\ddot {\ddot {y}}}}={\ddot {\dot {\ddot {y}}}}\equiv {\frac {d^{5}y}{dt^{5}}}=D_{t}^{5}y=f^{\rm {V}}(t)=y_{t}^{(5)}\\{\overset {\,6}{\dot {y}}}&={\overset {...}{\overset {...}{y}}}\equiv {\frac {d^{6}y}{dt^{6}}}=D_{t}^{6}y=f^{\rm {VI}}(t)=y_{t}^{(6)}\\{\overset {\,7}{\dot {y}}}&={\dot {\overset {...}{\overset {...}{y}}}}\equiv {\frac {d^{7}y}{dt^{7}}}=D_{t}^{7}y=f^{\rm {VII}}(t)=y_{t}^{(7)}\\{\overset {\,10}{\dot {y}}}&={\ddot {\ddot {\ddot {\ddot {\ddot {y}}}}}}\equiv {\frac {d^{10}y}{dt^{10}}}=D_{t}^{10}y=f^{\rm {X}}(t)=y_{t}^{(10)}\\{\overset {\,n}{\dot {y}}}&\equiv {\frac {d^{n}y}{dt^{n}}}=D_{t}^{n}y=f^{(n)}(t)=y_{t}^{(n)}\end{aligned}}}
11474:
10812:
11469:{\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {A} &=\left({\partial A_{z} \over {\partial y}}-{\partial A_{y} \over {\partial z}},{\partial A_{x} \over {\partial z}}-{\partial A_{z} \over {\partial x}},{\partial A_{y} \over {\partial x}}-{\partial A_{x} \over {\partial y}}\right)\\&=\left({\partial A_{z} \over {\partial y}}-{\partial A_{y} \over {\partial z}}\right)\mathbf {i} +\left({\partial A_{x} \over {\partial z}}-{\partial A_{z} \over {\partial x}}\right)\mathbf {j} +\left({\partial A_{y} \over {\partial x}}-{\partial A_{x} \over {\partial y}}\right)\mathbf {k} \\&={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\cfrac {\partial }{\partial x}}&{\cfrac {\partial }{\partial y}}&{\cfrac {\partial }{\partial z}}\\A_{x}&A_{y}&A_{z}\end{vmatrix}}\\&=\nabla \times \mathbf {A} \end{aligned}}}
7090:
8196:
6522:
7682:
7085:{\displaystyle {\begin{aligned}{\mathcal {X}}\ &=\ f(x,y)\,,\\\cdot {\mathcal {X}}\ &=\ x{\frac {\partial f}{\partial x}}=xf_{x}\,,\\{\mathcal {X}}\!\cdot \ &=\ y{\frac {\partial f}{\partial y}}=yf_{y}\,,\\\colon \!{\mathcal {X}}\,{\text{ or }}\,\cdot \!\left(\cdot {\mathcal {X}}\right)\ &=\ x^{2}{\frac {\partial ^{2}f}{\partial x^{2}}}=x^{2}f_{xx}\,,\\{\mathcal {X}}\colon \,{\text{ or }}\,\left({\mathcal {X}}\cdot \right)\!\cdot \ &=\ y^{2}{\frac {\partial ^{2}f}{\partial y^{2}}}=y^{2}f_{yy}\,,\\\cdot {\mathcal {X}}\!\cdot \ \ &=\ xy{\frac {\partial ^{2}f}{\partial x\,\partial y}}=xyf_{xy}\,,\end{aligned}}}
8191:{\displaystyle {\begin{aligned}{\overset {\,\prime \prime \prime }{y}}&=\Box {\overset {\,\prime \prime }{y}}\equiv \int {\overset {\,\prime \prime }{y}}\,dt=\int g(t)\,dt=D_{t}^{-3}y=G(t)+C_{3}\\{\overset {\,\prime \prime \prime \prime }{y}}&=\Box {\overset {\,\prime \prime \prime }{y}}\equiv \int {\overset {\,\prime \prime \prime }{y}}\,dt=\int G(t)\,dt=D_{t}^{-4}y=h(t)+C_{4}\\{\overset {\;n}{\overset {\,\prime }{y}}}&=\Box {\overset {\;n-1}{\overset {\,\prime }{y}}}\equiv \int {\overset {\;n-1}{\overset {\,\prime }{y}}}\,dt=\int s(t)\,dt=D_{t}^{-n}y=S(t)+C_{n}\end{aligned}}}
2995:
10158:
10468:
4116:
2645:
9956:
10243:
36:
7489:
2990:{\displaystyle {\begin{aligned}f^{\prime }&={\frac {\partial f}{\partial x}}=f_{x}\\f_{\prime }&={\frac {\partial f}{\partial y}}=f_{y}\\f^{\prime \prime }&={\frac {\partial ^{2}f}{\partial x^{2}}}=f_{xx}\\f_{\prime }^{\prime }&={\frac {\partial ^{2}f}{\partial y\partial x}}\ =f_{xy}\\f_{\prime \prime }&={\frac {\partial ^{2}f}{\partial y^{2}}}=f_{yy}\end{aligned}}}
3856:
6511:
10690:
10153:{\displaystyle {\begin{aligned}\operatorname {grad} \varphi &=\left({\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}}\right)\\&=\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\varphi \\&=\nabla \varphi \end{aligned}}}
10463:{\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &={\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}\\&=\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot \mathbf {A} \\&=\nabla \cdot \mathbf {A} \end{aligned}}}
9201:
4111:{\displaystyle {\begin{aligned}&\partial _{xx}f={\frac {\partial ^{2}f}{\partial x^{2}}},\\&\partial _{xy}f={\frac {\partial ^{2}f}{\partial y\,\partial x}},\\&\partial _{yx}f={\frac {\partial ^{2}f}{\partial x\,\partial y}},\\&\partial _{yy}f={\frac {\partial ^{2}f}{\partial y^{2}}}.\end{aligned}}}
9528:
7172:
7671:
8961:
10563:
6246:
9831:
9047:
11623:
1499:
1047:
have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the
1643:
8401:
7484:{\displaystyle {\begin{aligned}y&=\Box {\dot {y}}\equiv \int {\dot {y}}\,dt=\int f'(t)\,dt=D_{t}^{-1}(D_{t}y)=f(t)+C_{0}=y_{t}+C_{0}\\{\overset {\,\prime }{y}}&=\Box y\equiv \int y\,dt=\int f(t)\,dt=D_{t}^{-1}y=F(t)+C_{1}\end{aligned}}}
1322:
9398:
9349:
4492:
2377:
7521:
6527:
1756:
8826:
9674:
10685:{\displaystyle {\begin{aligned}\operatorname {div} \operatorname {grad} \varphi &=\nabla \cdot (\nabla \varphi )\\&=(\nabla \cdot \nabla )\varphi \\&=\nabla ^{2}\varphi \\&=\Delta \varphi \\\end{aligned}}}
2650:
6506:{\displaystyle {\frac {\dot {y}}{\dot {x}}}={\dot {y}}:{\dot {x}}\equiv {\frac {dy}{dt}}:{\frac {dx}{dt}}={\frac {\frac {dy}{dt}}{\frac {dx}{dt}}}={\frac {dy}{dx}}={\frac {d}{dx}}{\Bigl (}f(x){\Bigr )}=Dy=f'(x)=y'.}
1857:
3861:
9036:
8558:
8811:
8746:
10528:
9748:
9196:{\displaystyle {\begin{aligned}&(f_{x})_{y}=f_{xy},\\&{\frac {\partial }{\partial y}}\!\left({\frac {\partial f}{\partial x}}\right)={\frac {\partial ^{2}f}{\partial y\,\partial x}}.\end{aligned}}}
9052:
8831:
8682:
8288:
3123:
2490:
176:
10817:
10568:
10248:
9961:
7687:
7177:
1333:
4476:
1510:
10755:
10789:
10220:
3539:
1990:
8600:
9899:
9721:
9390:
8283:
11491:
3661:
D-notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be made explicit by putting its name as a subscript: if
2265:
2048:
12162:
3128:
However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of
1189:
9237:
2227:
11481:
Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in
Cartesian coordinates. For example, the single-variable
9523:{\displaystyle \partial ^{\alpha }f={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}f}
8817:. This becomes necessary in situations where the number of variables exceeds the degrees of freedom, so that one has to choose which other variables are to be kept fixed.
6207:
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3314:
3267:
3178:
2549:
1212:
6174:
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4231:
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3774:
3737:
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3220:
10548:
9944:
9919:
1917:
11681:
9262:
3700:
3651:
3619:
3587:
11661:
12449:
9257:
8643:
8623:
57:
7666:{\displaystyle {\overset {\,\prime \prime }{y}}=\Box {\overset {\,\prime }{y}}\equiv \int {\overset {\,\prime }{y}}\,dt=\int F(t)\,dt=D_{t}^{-2}y=g(t)+C_{2}}
11895:
2281:
8956:{\displaystyle {\begin{aligned}&{\frac {\partial ^{2}f}{\partial x^{2}}}=f_{xx},\\&{\frac {\partial ^{3}f}{\partial x^{3}}}=f_{xxx},\end{aligned}}}
1670:
9606:
2057:). Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the
12844:
12849:
12712:
12839:
11810:
1776:
6516:
Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by
Whiteside are below:
8972:
8494:
1762:
Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering
272:
12442:
8763:
8698:
12499:
10501:
9826:{\displaystyle \nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\!,}
44:
11921:
12612:
11629:
3835:". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function
11875:
11728:
9534:
8648:
8206:
11951:
12435:
2388:
12656:
11695: – 19th-century British group who promoted the use of Leibnizian or analytical calculus, as opposed to Newtonian calculus
12550:
12387:
12325:
12062:
11993:
9210:
is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on
1494:{\displaystyle {\frac {d^{2}y}{dx^{2}}},{\frac {d^{3}y}{dx^{3}}},{\frac {d^{4}y}{dx^{4}}},\ldots ,{\frac {d^{n}y}{dx^{n}}}.}
12834:
12296:
12025:
538:
513:
12744:
12759:
12540:
12213:
12101:
1134:
1638:{\displaystyle {\frac {d\left({\frac {dy}{dx}}\right)}{dx}}=\left({\frac {d}{dx}}\right)^{2}y={\frac {d^{2}y}{dx^{2}}}.}
12183:
11756:
Grosse, Johann; Breitkopf, Bernhard
Christoph; Martin, Johann Christian; Gleditsch, Johann Friedrich (September 1749).
2382:
to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in
1014:
577:
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3045:
95:
12545:
12509:
12417:
12242:
11786:
4431:
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251:
11973:
12607:
12519:
10723:
1878:) on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as
518:
10760:
10191:
3471:
1937:
12575:
11887:
11757:
8566:
8449:
Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operator
854:
528:
503:
185:
17:
8687:
Other notations can be found in various subfields of mathematics, physics, and engineering; see for example the
8396:{\displaystyle {\begin{aligned}f_{x}&={\frac {df}{dx}}\\f_{xx}&={\frac {d^{2}f}{dx^{2}}}.\end{aligned}}}
12961:
12956:
12890:
12489:
11618:{\displaystyle (fg)'=f'g+fg'~~~\Longrightarrow ~~~\nabla (\phi \psi )=(\nabla \phi )\psi +\phi (\nabla \psi ).}
9869:
12646:
9679:
9041:
In this last case the variables are written in inverse order between the two notations, explained as follows:
12661:
12504:
12494:
9354:
636:
583:
464:
12514:
9537:) that are tedious to write in other ways can be expressed succinctly -- some examples can be found in the
262:
373:
12749:
12619:
12580:
12137:
9592:
887:
495:
333:
305:
1875:
1154:
758:
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378:
267:
257:
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11485:
has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in
9213:
2001:
12870:
12789:
522:
12925:
12895:
10550:
is a scalar, which is symbolically expressed by the scalar multiplication of ∇ and the scalar field
8813:
is the derivative of the temperature with respect to the volume while keeping constant the pressure
2271:. The use of repeated prime marks eventually becomes unwieldy. Some authors continue by employing
1317:{\displaystyle {\frac {df}{dx}}(x){\text{ or }}{\frac {df(x)}{dx}}{\text{ or }}{\frac {d}{dx}}f(x).}
358:
12784:
657:
217:
4188:
D-notation can be used for antiderivatives in the same way that
Lagrange's notation is as follows
12794:
12764:
12754:
12651:
11831:
8645:
when all variables are allowed to vary simultaneously, whereas with a partial derivative such as
6183:
3321:
3274:
3227:
3138:
2509:
971:
763:
652:
49:
9344:{\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n}),\ \alpha _{i}\in \mathbb {Z} _{\geq 0}}
6150:
4396:
4282:
4238:
4194:
3781:
3744:
3707:
11876:
http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN308900308%7CLOG_0017&physid=PHYS_0031
11640:
9731:
3186:
1862:
Leibniz's notation for differentiation does not require assigning a meaning to symbols such as
1504:
This is a suggestive notational device that comes from formal manipulations of symbols, as in,
1040:
1007:
936:
897:
781:
717:
641:
10533:
9929:
9904:
1899:
12910:
12905:
12799:
12723:
12702:
12570:
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12560:
12555:
12479:
12458:
11867:
11666:
9922:
8259:
8202:
6222:
4129:
3431:
2236:
2132:
1883:
1109:
1028:
981:
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418:
363:
324:
230:
3675:
3626:
3594:
3562:
2198:
12804:
12728:
12697:
11737:
11646:
11635:
Further notations have been developed for more exotic types of spaces. For calculations in
9538:
9207:
6218:
4133:
3549:
2157:
986:
966:
892:
561:
480:
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368:
8:
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193:
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926:
829:
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753:
748:
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707:
588:
507:
413:
408:
212:
207:
2372:{\displaystyle f^{\mathrm {iv} }(x),f^{\mathrm {v} }(x),f^{\mathrm {vi} }(x),\ldots ,}
12393:
12383:
12238:
11989:
11947:
11792:
11782:
8688:
2230:
1115:
1000:
834:
612:
490:
443:
300:
295:
12920:
12900:
3181:
2268:
1751:{\displaystyle \left.{\frac {dy}{dx}}\right|_{x=a}{\text{ or }}{\frac {dy}{dx}}(a)}
1148:. Leibniz's notation makes this relationship explicit by writing the derivative as
844:
738:
712:
573:
485:
449:
12321:
12058:
11643:, also called the d'Alembertian, wave operator, or box operator is represented as
12880:
12809:
12421:
11888:"The D operator - Differential - Calculus - Maths Reference with Worked Examples"
11762:
11722: – Matrix of all first-order partial derivatives of a vector-valued function
11719:
11636:
9669:{\displaystyle \mathbf {A} =(\mathbf {A} _{x},\mathbf {A} _{y},\mathbf {A} _{z})}
9569:
9549:
8263:
976:
849:
803:
798:
685:
598:
543:
12885:
12292:
12021:
12875:
12682:
11872:
Nouvelle méthode pour résoudre les équations littérales par le moyen des séries
11713:
8820:
Higher-order partial derivatives with respect to one variable are expressed as
8692:
8563:
What makes this distinction important is that a non-partial derivative such as
6025:(third derivative) ← replaced by "combining diaeresis" + "combining dot above".
3424:
2140:
2136:
1053:
1049:
859:
667:
434:
12209:
8277:, we can express the derivative using subscripts of the independent variable:
1882:. Later authors have assigned them other meanings, such as infinitesimals in
12950:
12397:
11796:
10796:
9565:
7494:
To denote multiple integrals, Newton used two small vertical bars or primes (
2272:
2092:
1879:
839:
603:
353:
310:
12179:
4374:
for differentiation) places a dot over the dependent variable. That is, if
1118:
is used throughout mathematics. It is particularly common when the equation
12930:
12865:
12779:
12316:
12287:
12053:
12016:
11482:
9735:
9724:
9600:
9561:
6210:
4357:
2131:
One of the most common modern notations for differentiation is named after
593:
338:
12413:
12377:
6221:. It also appears in areas of mathematics connected with physics such as
6133:
Newton's notation is generally used when the independent variable denotes
12687:
12427:
11985:
11981:
10227:
9840:
reflects that the operator ∇ will also be treated as an ordinary vector.
2081:
2058:
956:
1852:{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}
12915:
11701:
10184:
9553:
7135:: he wrote a small vertical bar or prime above the dependent variable (
2077:
1767:
1036:
702:
626:
348:
343:
247:
9031:{\displaystyle {\frac {\partial ^{2}f}{\partial y\partial x}}=f_{xy}.}
8553:{\displaystyle {\frac {\partial f}{\partial x}}=f_{x}=\partial _{x}f.}
8258:
When more specific types of differentiation are necessary, such as in
3039:
When taking the antiderivative, Lagrange followed
Leibniz's notation:
12677:
12253:"Patterns of Mathematical Thought in the Later Seventeenth Century",
10494:
2195:
Higher derivatives are indicated using additional prime marks, as in
631:
621:
12717:
9862:
9557:
8806:{\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{\!P}}
8741:{\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{\!S}}
8205:
did not become widespread because of printing difficulties and the
7163:
7124:
6177:
697:
439:
396:
85:
8605:, depending on the context, be interpreted as a rate of change in
11707:
10523:{\displaystyle \operatorname {div} \operatorname {grad} \varphi }
8457:" symbol. For example, we can indicate the partial derivative of
6214:
4366:
35:
9568:. Several notations specific to the case of three-dimensional
2139:
and just popularized by the former. In
Lagrange's notation, a
11755:
6042:(fourth derivative) ← replaced by "combining diaeresis" twice.
4425:
Higher derivatives are represented using multiple dots, as in
3000:
12268:
Families of Curves and the
Origins of Partial Differentiation
8454:
3826:
11710: – Historical mathematical concept; form of derivative
9742:
or nabla, is symbolically defined in the form of a vector,
8677:{\displaystyle \textstyle {\frac {\partial f}{\partial x}}}
6134:
1676:
2554:
Unicode characters related to
Lagrange's notation include
9739:
8966:
and so on. Mixed partial derivatives can be expressed as
5973:
Unicode characters related to Newton's notation include:
3825:
is a function of several variables, it is common to use "
3180:
for the first integral (this is easily confused with the
2485:{\displaystyle f^{(4)}(x),f^{(5)}(x),f^{(6)}(x),\ldots .}
12056:, 1736), pp. 313-318 and p. 265 (p. 163 in original MS:
11683:
when not in conflict with the symbol for the
Laplacian.
2627:
When there are two independent variables for a function
12088:
th derivatives: Articles "Differential" and "Fluxion",
11733:
Pages displaying short descriptions of redirect targets
11724:
Pages displaying short descriptions of redirect targets
11379:
11367:
11346:
11334:
11313:
11301:
1664:
may be expressed in two ways using
Leibniz's notation:
12257:
Vol. 1, No. 3 (D. T. Whiteside, 1961), pp. 361-362,378
12116:
Vol. 7 1691-1695 (D. T. Whiteside, 1976), pp.88 and 17
11382:
11370:
11349:
11337:
11316:
11304:
11272:
8652:
8570:
8406:
This type of notation is especially useful for taking
7915:
7912:
7890:
7887:
7861:
7858:
7855:
7751:
7729:
7703:
7700:
7535:
2921:
2769:
2070:
Some authors and journals set the differential symbol
12140:
11669:
11649:
11494:
10815:
10763:
10726:
10566:
10536:
10504:
10246:
10194:
9959:
9932:
9907:
9872:
9751:
9682:
9609:
9401:
9357:
9265:
9245:
9216:
9050:
8975:
8829:
8766:
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8651:
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8497:
8286:
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This notation also makes it possible to describe the
2391:
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2160:
2004:
1940:
1902:
1779:
1673:
1513:
1336:
1215:
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98:
12266:
S.B. Engelsman has given more strict definitions in
11781:. Stark, Robert M., 1930-2017. Hoboken, New Jersey.
11697:
Pages displaying wikidata descriptions as a fallback
11628:
Many other rules from single variable calculus have
10795:
is a vector, which is symbolically expressed by the
10226:
is a scalar, which is symbolically expressed by the
9921:
is a vector, which is symbolically expressed by the
7095:
12279:Newton's notation for integration reproduced from:
11716: – (Mathematical) matrix of second derivatives
11632:for the gradient, divergence, curl, and Laplacian.
9239:, we define a multi-index to be an ordered list of
8684:it is explicit that only one variable should vary.
6228:When taking the derivative of a dependent variable
12293:"Newton Papers : On the Quadrature of Curves"
12156:
12022:"Newton Papers : On the Quadrature of Curves"
11704: – Instantaneous rate of change (mathematics)
11675:
11655:
11617:
11468:
10783:
10749:
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10542:
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3355:
3308:
3261:
3214:
3172:
3117:
2989:
2543:
2484:
2371:
2259:
2221:
2180:
2042:
1984:
1911:
1851:
1750:
1637:
1493:
1316:
1183:
170:
12845:List of nonlinear ordinary differential equations
11911:Weisstein, Eric W. "Differential Operator." From
9819:
9120:
8797:
8732:
6992:
6887:
6738:
6720:
6653:
6455:
6436:
4682:
4658:
4633:
4614:
4360:'s notation for differentiation (also called the
4139:
3829:", a stylized cursive lower-case d, rather than "
1133:is regarded as a functional relationship between
12948:
12850:List of nonlinear partial differential equations
11907:
11905:
11776:
7148:), or the inclosure of the term in a rectangle (
3118:{\displaystyle f(x)=\int f'(x)\,dx=\int y'\,dx.}
2147:is a function, then its derivative evaluated at
171:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
11731: – Meanings of symbols used in mathematics
8756:while keeping constant the entropy (subscript)
7515:, to denote the second time integral (absity).
4471:{\displaystyle {\ddot {y}},{\overset {...}{y}}}
12840:List of linear ordinary differential equations
12347:1st to 2nd integrals: Articles 622 and 365 in
9544:
7123:Newton developed many different notations for
3544:Higher derivatives are notated as "powers" of
3026:The single and double indefinite integrals of
2095:scientific style guide recommends this style.
12443:
12319:, 1736), pp. 265-266 (p. 163 in original MS:
12099:th derivatives: Articles 622, 580 and 579 in
11941:Weisstein, Eric W. "Repeated Integral." From
11902:
10750:{\displaystyle \mathrm {curl} \,\mathbf {A} }
7676:Higher order time integrals were as follows:
2639:), the following convention may be followed:
2126:, differentiated once in Lagrange's notation.
1008:
12364:Methodus Incrementorum Directa & Inversa
10784:{\displaystyle \mathrm {rot} \,\mathbf {A} }
10215:{\displaystyle \mathrm {div} \,\mathbf {A} }
3534:{\displaystyle (Df)(x)={\frac {df(x)}{dx}}.}
2054:
1985:{\displaystyle dy={\frac {dy}{dx}}\cdot dx,}
12237:(1929), Dover Publications, Inc. New York.
8595:{\displaystyle \textstyle {\frac {df}{dx}}}
3001:Lagrange's notation for antidifferentiation
12457:
12450:
12436:
12358:th integral notation is deducted from the
11847:
11845:
11843:
11841:
11809:: CS1 maint: location missing publisher (
8087:
8054:
8023:
2135:, even though it was actually invented by
1015:
1001:
12147:
10775:
10741:
10206:
9894:{\displaystyle \mathrm {grad\,} \varphi }
9886:
9366:
9328:
9219:
9176:
8125:
8100:
8080:
8047:
8016:
7946:
7921:
7908:
7883:
7851:
7782:
7757:
7747:
7725:
7696:
7604:
7579:
7572:
7553:
7531:
7418:
7393:
7364:
7253:
7223:
7074:
7042:
6974:
6866:
6860:
6842:
6734:
6728:
6709:
6638:
6568:
5835:
5658:
5477:
5306:
5086:
4903:
4030:
3969:
3105:
3084:
131:
12090:Dictionary of Pure and Mixed Mathematics
11978:A First Course in Differential Equations
9730:The differential operator introduced by
9716:{\displaystyle \varphi =\varphi (x,y,z)}
7500:), or a combination of previous symbols
7114:The first and second antiderivatives of
4122:
3548:(where the superscripts denote iterated
60:of all important aspects of the article.
12114:The Mathematical Papers of Isaac Newton
11861:
11838:
11830:. Boston: D. C. Heath and co. pp.
11823:
11729:List of mathematical symbols by subject
9385:{\displaystyle f:\mathbb {R} ^{n}\to X}
539:Differentiating under the integral sign
14:
12949:
12203:Weisstein, Eric W. "Double Dot." From
11852:The Differential and Integral Calculus
9533:In this way some results (such as the
8212:
4128:D-notation is useful in the study of
2098:
56:Please consider expanding the lead to
12431:
12290:, 1704), p. 7 (p. 5r in original MS:
12255:Archive for History of Exact Sciences
12019:, 1704), p. 7 (p. 5r in original MS:
8748:is the derivative of the temperature
4481:Newton extended this idea quite far:
1059:
1035:. Instead, various notations for the
12835:List of named differential equations
12414:Earliest Uses of Symbols of Calculus
12344:(James Hodgson, 1736), pp. 54 and 72
12127:(Hans Niels Jahnke, 2000), pp. 84-85
11971:
10173:The divergence of the vector field
8410:of a function of several variables.
6074:COMBINING DOUBLE VERTICAL LINE ABOVE
4348:The first and second derivatives of
4329:
2192:It first appeared in print in 1749.
1095:The first and second derivatives of
29:
12760:Method of undetermined coefficients
12541:Dependent and independent variables
12362:th derivative. It could be used in
12349:A History of Mathematical Notations
12235:A History of Mathematical Notations
12173:Weisstein, Eric W. "Overdot." From
12157:{\displaystyle {\overset {\,n}{y}}}
12102:A History of Mathematical Notations
12008:Newton's notation reproduced from:
8207:Leibniz–Newton calculus controversy
6240:), an alternative notation exists:
1135:dependent and independent variables
24:
12382:(2 ed.). New York: Springer.
12375:
11827:Differential and Integral Calculus
11670:
11603:
11582:
11561:
11451:
11384:
11372:
11351:
11339:
11318:
11306:
11238:
11223:
11208:
11193:
11163:
11148:
11133:
11118:
11088:
11073:
11058:
11043:
11011:
10996:
10981:
10966:
10951:
10936:
10921:
10906:
10891:
10876:
10861:
10846:
10771:
10768:
10765:
10737:
10734:
10731:
10728:
10672:
10650:
10630:
10624:
10602:
10593:
10483:The Laplacian of the scalar field
10445:
10413:
10409:
10395:
10391:
10377:
10373:
10347:
10332:
10317:
10302:
10287:
10272:
10202:
10199:
10196:
10140:
10113:
10109:
10095:
10091:
10077:
10073:
10042:
10034:
10019:
10011:
9996:
9988:
9883:
9880:
9877:
9874:
9805:
9801:
9787:
9783:
9769:
9765:
9752:
9489:
9472:
9438:
9421:
9403:
9177:
9170:
9156:
9136:
9128:
9111:
9107:
9000:
8994:
8980:
8911:
8897:
8853:
8839:
8783:
8775:
8718:
8710:
8664:
8656:
8535:
8509:
8501:
8081:
8048:
8017:
7909:
7884:
7852:
7748:
7726:
7697:
7573:
7554:
7532:
7365:
7043:
7036:
7022:
6987:
6932:
6918:
6874:
6852:
6800:
6786:
6749:
6723:
6684:
6676:
6648:
6613:
6605:
6581:
6532:
5779:
5602:
5599:
5596:
5421:
5418:
5250:
5030:
5027:
4085:
4071:
4049:
4031:
4024:
4010:
3988:
3970:
3963:
3949:
3927:
3902:
3888:
3866:
3411:This notation is sometimes called
2951:
2937:
2918:
2881:
2875:
2861:
2845:
2840:
2799:
2785:
2766:
2735:
2727:
2712:
2681:
2673:
2658:
2345:
2342:
2318:
2294:
2291:
1925:is assigned a meaning in terms of
1903:
1896:is left undefined or equated with
1327:Higher derivatives are written as
1114:The original notation employed by
80:Part of a series of articles about
25:
12973:
12407:
8273:of a single independent variable
7096:Newton's notation for integration
4179:and the second antiderivative of
1184:{\displaystyle {\frac {dy}{dx}}.}
27:Notation of differential calculus
12657:Carathéodory's existence theorem
11898:from the original on 2016-01-19.
11458:
11290:
11283:
11276:
11253:
11178:
11103:
10827:
10777:
10743:
10452:
10431:
10258:
10208:
9653:
9638:
9623:
9611:
9232:{\displaystyle \mathbb {R} ^{n}}
4277:for a second antiderivative, and
2503:is a variable. This is written
2043:{\displaystyle df=f'(x)\cdot dx}
1995:which may also be written, e.g.
1770:easy to remember and recognize:
34:
12369:
12328:from the original on 2017-04-06
12322:"Newton Papers : Fluxions"
12299:from the original on 2016-02-28
12273:
12260:
12247:
12233:Article 580 in Florian Cajori,
12227:
12216:from the original on 2016-03-03
12197:
12186:from the original on 2015-09-05
12167:
12134:th derivative may be omitted (
12081:(Colin MacLaurin, 1742), p. 613
12065:from the original on 2017-04-06
12059:"Newton Papers : Fluxions"
12028:from the original on 2016-02-28
12002:
11954:from the original on 2016-02-01
11924:from the original on 2016-01-21
11777:Morris, Carla C. (2015-07-28).
7118:, in one of Newton's notations.
3454:). When applied to a function
3421:Louis François Antoine Arbogast
2275:, usually in lower case, as in
1648:The value of the derivative of
1194:Furthermore, the derivative of
48:may be too short to adequately
12077:1st to 5th derivatives :
11965:
11935:
11880:
11817:
11770:
11758:"Notation for differentiation"
11749:
11609:
11600:
11588:
11579:
11573:
11564:
11549:
11505:
11495:
10633:
10621:
10608:
10599:
9710:
9692:
9663:
9618:
9376:
9304:
9272:
9072:
9058:
8266:, other notations are common.
8168:
8162:
8122:
8116:
7989:
7983:
7943:
7937:
7825:
7819:
7779:
7773:
7647:
7641:
7601:
7595:
7461:
7455:
7415:
7409:
7312:
7306:
7297:
7281:
7250:
7244:
6565:
6553:
6486:
6480:
6450:
6444:
6213:. This notation is popular in
6123:COMBINING LATIN SMALL LETTER N
5951:
5945:
5929:
5923:
5918:
5912:
5813:
5807:
5791:
5785:
5636:
5630:
5614:
5608:
5455:
5449:
5433:
5427:
5284:
5278:
5262:
5256:
5064:
5058:
5042:
5036:
4867:
4861:
4725:
4719:
4677:
4671:
4308:
4302:
4264:
4258:
4220:
4214:
4140:D-notation for antiderivatives
3514:
3508:
3493:
3487:
3484:
3475:
3419:although it was introduced by
3350:
3344:
3339:
3330:
3303:
3297:
3292:
3283:
3256:
3250:
3245:
3236:
3209:
3203:
3167:
3161:
3156:
3147:
3081:
3075:
3058:
3052:
2535:
2529:
2524:
2518:
2470:
2464:
2459:
2453:
2442:
2436:
2431:
2425:
2414:
2408:
2403:
2397:
2357:
2351:
2330:
2324:
2306:
2300:
2254:
2248:
2216:
2210:
2175:
2169:
2028:
2022:
1745:
1739:
1308:
1302:
1265:
1259:
1242:
1236:
165:
159:
150:
144:
128:
122:
58:provide an accessible overview
13:
1:
12416:, maintained by Jeff Miller (
11743:
9851:Gradient of the scalar field
6057:COMBINING VERTICAL LINE ABOVE
4370:, or sometimes, crudely, the
3776:for the third derivative, and
3621:for the third derivative, and
3402:and the second derivative of
3371:
1031:, there is no single uniform
465:Integral of inverse functions
12485:Notation for differentiation
12379:An introduction to manifolds
3665:is a function of a variable
1033:notation for differentiation
7:
12581:Exact differential equation
11824:Osborne, George A. (1908).
11686:
9593:Cartesian coordinate system
9545:Notation in vector calculus
8752:with respect to the volume
6202:{\displaystyle {\ddot {y}}}
4233:for a first antiderivative,
3356:{\displaystyle f^{(-n)}(x)}
3316:for the third integral, and
3309:{\displaystyle f^{(-3)}(x)}
3262:{\displaystyle f^{(-2)}(x)}
3173:{\displaystyle f^{(-1)}(x)}
3034:, in the Lagrange notation.
2544:{\displaystyle f^{(n)}(x).}
888:Calculus on Euclidean space
306:Logarithmic differentiation
10:
12978:
12207:--A Wolfram Web Resource.
12177:--A Wolfram Web Resource.
11945:--A Wolfram Web Resource.
11915:--A Wolfram Web Resource.
7141:), a prefixing rectangle (
6169:{\displaystyle {\dot {y}}}
6106:COMBINING ENCLOSING SQUARE
6023:COMBINING THREE DOTS ABOVE
4415:{\displaystyle {\dot {y}}}
4314:{\displaystyle D^{-n}f(x)}
4270:{\displaystyle D^{-2}f(x)}
4226:{\displaystyle D^{-1}f(x)}
4123:§ Partial derivatives
3806:{\displaystyle D_{x}^{n}f}
3769:{\displaystyle D_{x}^{3}f}
3739:for the second derivative,
3732:{\displaystyle D_{x}^{2}f}
3669:, this is done by writing
3589:for the second derivative,
2143:denotes a derivative. If
1107:
1103:, in the Leibniz notation.
12891:Józef Maria Hoene-Wroński
12871:Gottfried Wilhelm Leibniz
12858:
12827:
12737:
12670:
12662:Cauchy–Kowalevski theorem
12639:
12632:
12594:
12533:
12472:
12465:
11630:vector calculus analogues
10705:The curl of vector field
6040:COMBINING FOUR DOTS ABOVE
4382:, then the derivative of
3702:for the first derivative,
3215:{\displaystyle f^{-1}(x)}
622:Summand limit (term test)
12785:Finite difference method
12342:The Doctrine of Fluxions
12011:1st to 5th derivatives:
11972:Zill, Dennis G. (2009).
11779:Fundamentals of calculus
10543:{\displaystyle \varphi }
9939:{\displaystyle \varphi }
9914:{\displaystyle \varphi }
9539:article on multi-indices
3269:for the second integral,
1912:{\displaystyle \Delta x}
301:Implicit differentiation
291:Differentiation notation
218:Inverse function theorem
12765:Variation of parameters
12755:Separation of variables
12652:Peano existence theorem
12647:Picard–Lindelöf theorem
12534:Attributes of variables
11918:"Differential Operator"
11676:{\displaystyle \Delta }
9259:non-negative integers:
8440:differentiated against
8241:differentiated against
3452:Newton–Leibniz operator
2260:{\displaystyle f'''(x)}
764:Helmholtz decomposition
12926:Carl David Tolmé Runge
12500:Differential-algebraic
12459:Differential equations
12376:Tu, Loring W. (2011).
12311:1st to 3rd integrals:
12282:1st to 3rd integrals:
12158:
12079:A Treatise of Fluxions
11677:
11657:
11619:
11470:
10791:, of the vector field
10785:
10751:
10686:
10544:
10524:
10464:
10216:
10154:
9940:
9925:of ∇ and scalar field
9915:
9895:
9836:where the terminology
9827:
9732:William Rowan Hamilton
9717:
9670:
9524:
9386:
9351:. We then define, for
9345:
9253:
9233:
9197:
9032:
8957:
8807:
8742:
8678:
8639:
8619:
8596:
8554:
8397:
8192:
7667:
7485:
7086:
6507:
6223:differential equations
6203:
6170:
5964:
4472:
4416:
4315:
4271:
4227:
4130:differential equations
4112:
3807:
3770:
3733:
3696:
3695:{\displaystyle D_{x}f}
3647:
3646:{\displaystyle D^{n}f}
3615:
3614:{\displaystyle D^{3}f}
3583:
3582:{\displaystyle D^{2}f}
3535:
3357:
3310:
3263:
3216:
3174:
3119:
2991:
2545:
2486:
2373:
2261:
2223:
2222:{\displaystyle f''(x)}
2182:
2044:
1986:
1913:
1853:
1752:
1639:
1495:
1318:
1185:
1054:indefinite integration
898:Limit of distributions
718:Directional derivative
374:Faà di Bruno's formula
172:
12962:Mathematical notation
12957:Differential calculus
12911:Augustin-Louis Cauchy
12906:Joseph-Louis Lagrange
12800:Finite element method
12790:Crank–Nicolson method
12724:Numerical integration
12703:Exponential stability
12595:Relation to processes
12480:Differential operator
12159:
12125:A History of Analysis
12095:1st to 4th, 10th and
11678:
11658:
11656:{\displaystyle \Box }
11620:
11471:
10786:
10752:
10687:
10545:
10525:
10465:
10217:
10155:
9941:
9916:
9896:
9828:
9718:
9671:
9525:
9387:
9346:
9254:
9234:
9198:
9033:
8958:
8808:
8743:
8679:
8640:
8620:
8597:
8555:
8398:
8260:multivariate calculus
8203:mathematical notation
8193:
7668:
7486:
7087:
6508:
6204:
6171:
5965:
4473:
4417:
4316:
4272:
4228:
4113:
3808:
3771:
3734:
3697:
3648:
3616:
3584:
3536:
3432:differential operator
3430:This notation uses a
3358:
3311:
3264:
3217:
3175:
3120:
2992:
2546:
2499:th derivative, where
2487:
2374:
2262:
2224:
2183:
2181:{\displaystyle f'(x)}
2133:Joseph Louis Lagrange
2045:
1987:
1914:
1884:non-standard analysis
1854:
1766:. It also makes the
1753:
1640:
1496:
1319:
1206:is therefore written
1186:
1029:differential calculus
982:Mathematical analysis
893:Generalized functions
578:arithmetico-geometric
419:Leibniz integral rule
173:
12805:Finite volume method
12729:Dirac delta function
12698:Asymptotic stability
12640:Existence/uniqueness
12505:Integro-differential
12366:(Brook Taylor, 1715)
12138:
12092:(Peter Barlow, 1814)
11858:, 1842). pp. 267-268
11763:Nova Acta Eruditorum
11738:Operational calculus
11667:
11647:
11492:
10813:
10799:of ∇ and the vector
10761:
10724:
10564:
10534:
10530:of the scalar field
10502:
10244:
10230:of ∇ and the vector
10222:of the vector field
10192:
9957:
9930:
9905:
9901:of the scalar field
9870:
9749:
9680:
9607:
9399:
9355:
9263:
9243:
9214:
9208:multi-index notation
9048:
8973:
8827:
8764:
8699:
8649:
8629:
8609:
8567:
8495:
8284:
7683:
7522:
7173:
6523:
6247:
6219:mathematical physics
6184:
6151:
4488:
4432:
4397:
4352:, Newton's notation.
4283:
4239:
4195:
4134:differential algebra
3857:
3782:
3745:
3708:
3676:
3627:
3595:
3563:
3472:
3423:, and it seems that
3322:
3275:
3228:
3187:
3139:
3046:
2646:
2510:
2389:
2282:
2237:
2199:
2158:
2002:
1938:
1900:
1888:exterior derivatives
1777:
1671:
1511:
1334:
1213:
1155:
1056:) are listed below.
987:Nonstandard analysis
455:Lebesgue integration
325:Rules and identities
96:
12815:Perturbation theory
12795:Runge–Kutta methods
12775:Integral transforms
12708:Rate of convergence
12604:(discrete analogue)
12284:Quadratura curvarum
12270:(2000), pp. 223-226
12048:+1)th derivatives:
12013:Quadratura curvarum
11948:"Repeated Integral"
11641:d'Alembert operator
11381:
11369:
11348:
11336:
11315:
11303:
9513:
9462:
8408:partial derivatives
8213:Partial derivatives
8152:
7973:
7809:
7631:
7445:
7280:
7129:Quadratura curvarum
6008:(double derivative)
6006:COMBINING DIAERESIS
5989:COMBINING DOT ABOVE
5955:
5900:
5817:
5766:
5640:
5583:
5459:
5405:
5288:
5237:
5068:
5014:
4885:
4846:
4743:
4704:
3799:
3762:
3725:
3465:, it is defined by
2849:
2623:(fourth derivative)
2589:(double derivative)
2099:Lagrange's notation
1931:, via the equation
1764:partial derivatives
1050:antidifferentiation
658:Cauchy condensation
460:Contour integration
186:Fundamental theorem
113:
12936:Sofya Kovalevskaya
12770:Integrating factor
12693:Lyapunov stability
12613:Stochastic partial
12420:2020-07-26 at the
12313:Method of Fluxions
12154:
12050:Method of Fluxions
11874:(1770), p. 25-26.
11856:Augustus De Morgan
11693:Analytical Society
11673:
11653:
11615:
11466:
11464:
11435:
11391:
11376:
11358:
11343:
11325:
11310:
10781:
10747:
10682:
10680:
10540:
10520:
10460:
10458:
10212:
10150:
10148:
9936:
9911:
9891:
9823:
9713:
9666:
9520:
9492:
9441:
9382:
9341:
9249:
9229:
9193:
9191:
9028:
8953:
8951:
8803:
8738:
8674:
8673:
8635:
8615:
8592:
8591:
8550:
8393:
8391:
8188:
8186:
8135:
7956:
7792:
7663:
7614:
7481:
7479:
7428:
7263:
7162:or time integral (
7082:
7080:
6503:
6199:
6166:
5960:
5958:
5935:
5886:
5797:
5752:
5620:
5569:
5439:
5391:
5268:
5223:
5048:
5000:
4873:
4832:
4731:
4690:
4468:
4412:
4325:th antiderivative.
4311:
4267:
4223:
4175:antiderivative of
4108:
4106:
3803:
3785:
3766:
3748:
3729:
3711:
3692:
3643:
3611:
3579:
3531:
3353:
3306:
3259:
3212:
3170:
3132:may be written as
3115:
2987:
2985:
2835:
2606:(third derivative)
2541:
2482:
2369:
2257:
2219:
2178:
2040:
1982:
1909:
1849:
1748:
1635:
1491:
1314:
1181:
1110:Leibniz's notation
1060:Leibniz's notation
830:Partial derivative
759:generalized Stokes
653:Alternating series
534:Reduction formulae
523:Heaviside's method
504:tangent half-angle
491:Cylindrical shells
414:Integral transform
409:Lists of integrals
213:Mean value theorem
168:
99:
12944:
12943:
12823:
12822:
12628:
12627:
12389:978-1-4419-7400-6
12351:(F .Cajori, 1929)
12152:
12105:(F .Cajori, 1929)
11995:978-0-495-10824-5
11560:
11557:
11554:
11548:
11545:
11542:
11393:
11380:
11368:
11360:
11347:
11335:
11327:
11314:
11302:
11245:
11215:
11170:
11140:
11095:
11065:
11018:
10988:
10958:
10928:
10898:
10868:
10420:
10402:
10384:
10354:
10324:
10294:
10188:: The divergence
10120:
10102:
10084:
10049:
10026:
10003:
9812:
9794:
9776:
9515:
9464:
9312:
9252:{\displaystyle n}
9184:
9143:
9118:
9007:
8925:
8867:
8790:
8725:
8689:Maxwell relations
8671:
8638:{\displaystyle x}
8618:{\displaystyle f}
8589:
8516:
8488:in several ways:
8384:
8326:
8098:
8085:
8065:
8052:
8028:
8021:
7919:
7894:
7865:
7755:
7733:
7707:
7577:
7558:
7539:
7369:
7220:
7202:
7050:
7011:
7001:
6998:
6946:
6903:
6893:
6864:
6814:
6771:
6761:
6732:
6691:
6669:
6659:
6620:
6598:
6588:
6549:
6539:
6432:
6414:
6391:
6390:
6372:
6348:
6325:
6301:
6286:
6272:
6270:
6260:
6196:
6163:
6143:is a function of
6076:(second integral)
5881:
5840:
5832:
5747:
5709:
5704:
5699:
5694:
5689:
5663:
5655:
5564:
5526:
5522:
5510:
5482:
5474:
5386:
5349:
5337:
5311:
5303:
5218:
5180:
5175:
5170:
5151:
5146:
5141:
5122:
5118:
5091:
5083:
4995:
4957:
4952:
4936:
4908:
4900:
4827:
4789:
4784:
4764:
4654:
4628:
4610:
4588:
4564:
4546:
4504:
4466:
4444:
4409:
4378:is a function of
4372:flyspeck notation
4330:Newton's notation
4183:, Euler notation.
4099:
4038:
3977:
3916:
3526:
3427:did not use it.
3406:, Euler notation.
2965:
2892:
2888:
2813:
2742:
2688:
2231:second derivative
1968:
1844:
1821:
1798:
1737:
1717:
1696:
1630:
1580:
1557:
1542:
1486:
1443:
1406:
1369:
1297:
1282:
1277:
1248:
1234:
1176:
1116:Gottfried Leibniz
1025:
1024:
905:
904:
867:
866:
835:Multiple integral
771:
770:
675:
674:
642:Direct comparison
613:Convergence tests
551:
550:
519:Partial fractions
386:
385:
296:Second derivative
75:
74:
16:(Redirected from
12969:
12921:Phyllis Nicolson
12901:Rudolf Lipschitz
12738:Solution methods
12713:Series solutions
12637:
12636:
12470:
12469:
12452:
12445:
12438:
12429:
12428:
12402:
12401:
12373:
12367:
12336:
12334:
12333:
12307:
12305:
12304:
12277:
12271:
12264:
12258:
12251:
12245:
12231:
12225:
12224:
12222:
12221:
12201:
12195:
12194:
12192:
12191:
12171:
12165:
12163:
12161:
12160:
12155:
12153:
12151:
12142:
12123:th derivatives:
12112:th derivatives:
12073:
12071:
12070:
12036:
12034:
12033:
12006:
12000:
11999:
11980:(9th ed.).
11969:
11963:
11962:
11960:
11959:
11939:
11933:
11932:
11930:
11929:
11909:
11900:
11899:
11892:www.codecogs.com
11884:
11878:
11865:
11859:
11849:
11836:
11835:
11821:
11815:
11814:
11808:
11800:
11774:
11768:
11767:
11753:
11734:
11725:
11698:
11682:
11680:
11679:
11674:
11662:
11660:
11659:
11654:
11624:
11622:
11621:
11616:
11558:
11555:
11552:
11546:
11543:
11540:
11539:
11522:
11511:
11475:
11473:
11472:
11467:
11465:
11461:
11444:
11440:
11439:
11432:
11431:
11420:
11419:
11408:
11407:
11394:
11392:
11390:
11377:
11375:
11365:
11361:
11359:
11357:
11344:
11342:
11332:
11328:
11326:
11324:
11311:
11309:
11299:
11293:
11286:
11279:
11260:
11256:
11251:
11247:
11246:
11244:
11236:
11235:
11234:
11221:
11216:
11214:
11206:
11205:
11204:
11191:
11181:
11176:
11172:
11171:
11169:
11161:
11160:
11159:
11146:
11141:
11139:
11131:
11130:
11129:
11116:
11106:
11101:
11097:
11096:
11094:
11086:
11085:
11084:
11071:
11066:
11064:
11056:
11055:
11054:
11041:
11028:
11024:
11020:
11019:
11017:
11009:
11008:
11007:
10994:
10989:
10987:
10979:
10978:
10977:
10964:
10959:
10957:
10949:
10948:
10947:
10934:
10929:
10927:
10919:
10918:
10917:
10904:
10899:
10897:
10889:
10888:
10887:
10874:
10869:
10867:
10859:
10858:
10857:
10844:
10830:
10790:
10788:
10787:
10782:
10780:
10774:
10756:
10754:
10753:
10748:
10746:
10740:
10691:
10689:
10688:
10683:
10681:
10665:
10658:
10657:
10642:
10614:
10549:
10547:
10546:
10541:
10529:
10527:
10526:
10521:
10498:: The Laplacian
10469:
10467:
10466:
10461:
10459:
10455:
10438:
10434:
10426:
10422:
10421:
10419:
10408:
10403:
10401:
10390:
10385:
10383:
10372:
10359:
10355:
10353:
10345:
10344:
10343:
10330:
10325:
10323:
10315:
10314:
10313:
10300:
10295:
10293:
10285:
10284:
10283:
10270:
10261:
10221:
10219:
10218:
10213:
10211:
10205:
10159:
10157:
10156:
10151:
10149:
10133:
10126:
10122:
10121:
10119:
10108:
10103:
10101:
10090:
10085:
10083:
10072:
10059:
10055:
10051:
10050:
10048:
10040:
10032:
10027:
10025:
10017:
10009:
10004:
10002:
9994:
9986:
9945:
9943:
9942:
9937:
9920:
9918:
9917:
9912:
9900:
9898:
9897:
9892:
9887:
9832:
9830:
9829:
9824:
9818:
9814:
9813:
9811:
9800:
9795:
9793:
9782:
9777:
9775:
9764:
9722:
9720:
9719:
9714:
9675:
9673:
9672:
9667:
9662:
9661:
9656:
9647:
9646:
9641:
9632:
9631:
9626:
9614:
9603:with components
9590:
9529:
9527:
9526:
9521:
9516:
9514:
9512:
9511:
9510:
9500:
9487:
9486:
9485:
9484:
9470:
9465:
9463:
9461:
9460:
9459:
9449:
9436:
9435:
9434:
9433:
9419:
9411:
9410:
9391:
9389:
9388:
9383:
9375:
9374:
9369:
9350:
9348:
9347:
9342:
9340:
9339:
9331:
9322:
9321:
9310:
9303:
9302:
9284:
9283:
9258:
9256:
9255:
9250:
9238:
9236:
9235:
9230:
9228:
9227:
9222:
9202:
9200:
9199:
9194:
9192:
9185:
9183:
9168:
9164:
9163:
9153:
9148:
9144:
9142:
9134:
9126:
9119:
9117:
9106:
9103:
9096:
9095:
9080:
9079:
9070:
9069:
9054:
9037:
9035:
9034:
9029:
9024:
9023:
9008:
9006:
8992:
8988:
8987:
8977:
8962:
8960:
8959:
8954:
8952:
8945:
8944:
8926:
8924:
8923:
8922:
8909:
8905:
8904:
8894:
8891:
8884:
8883:
8868:
8866:
8865:
8864:
8851:
8847:
8846:
8836:
8833:
8812:
8810:
8809:
8804:
8802:
8801:
8795:
8791:
8789:
8781:
8773:
8747:
8745:
8744:
8739:
8737:
8736:
8730:
8726:
8724:
8716:
8708:
8683:
8681:
8680:
8675:
8672:
8670:
8662:
8654:
8644:
8642:
8641:
8636:
8624:
8622:
8621:
8616:
8601:
8599:
8598:
8593:
8590:
8588:
8580:
8572:
8559:
8557:
8556:
8551:
8543:
8542:
8530:
8529:
8517:
8515:
8507:
8499:
8476:with respect to
8475:
8433:
8431:
8430:
8425:
8422:
8402:
8400:
8399:
8394:
8392:
8385:
8383:
8382:
8381:
8368:
8364:
8363:
8353:
8344:
8343:
8327:
8325:
8317:
8309:
8300:
8299:
8234:
8226:
8197:
8195:
8194:
8189:
8187:
8183:
8182:
8151:
8143:
8099:
8097:
8084:
8075:
8074:
8066:
8064:
8051:
8042:
8041:
8029:
8027:
8020:
8011:
8010:
8004:
8003:
7972:
7964:
7920:
7918:
7903:
7895:
7893:
7878:
7866:
7864:
7846:
7840:
7839:
7808:
7800:
7756:
7754:
7742:
7734:
7732:
7720:
7708:
7706:
7691:
7672:
7670:
7669:
7664:
7662:
7661:
7630:
7622:
7578:
7576:
7567:
7559:
7557:
7548:
7540:
7538:
7526:
7514:
7513:
7506:
7499:
7490:
7488:
7487:
7482:
7480:
7476:
7475:
7444:
7436:
7370:
7368:
7359:
7353:
7352:
7340:
7339:
7327:
7326:
7293:
7292:
7279:
7271:
7243:
7222:
7221:
7213:
7204:
7203:
7195:
7156:) to denote the
7155:
7154:
7147:
7140:
7111:
7106:
7091:
7089:
7088:
7083:
7081:
7073:
7072:
7051:
7049:
7034:
7030:
7029:
7019:
7009:
6999:
6996:
6991:
6990:
6973:
6972:
6960:
6959:
6947:
6945:
6944:
6943:
6930:
6926:
6925:
6915:
6913:
6912:
6901:
6891:
6886:
6882:
6878:
6877:
6865:
6862:
6856:
6855:
6841:
6840:
6828:
6827:
6815:
6813:
6812:
6811:
6798:
6794:
6793:
6783:
6781:
6780:
6769:
6759:
6758:
6754:
6753:
6752:
6733:
6730:
6727:
6726:
6708:
6707:
6692:
6690:
6682:
6674:
6667:
6657:
6652:
6651:
6637:
6636:
6621:
6619:
6611:
6603:
6596:
6586:
6585:
6584:
6547:
6537:
6536:
6535:
6512:
6510:
6509:
6504:
6499:
6479:
6459:
6458:
6440:
6439:
6433:
6431:
6420:
6415:
6413:
6405:
6397:
6392:
6389:
6381:
6373:
6371:
6363:
6355:
6354:
6349:
6347:
6339:
6331:
6326:
6324:
6316:
6308:
6303:
6302:
6294:
6288:
6287:
6279:
6273:
6271:
6263:
6261:
6253:
6251:
6208:
6206:
6205:
6200:
6198:
6197:
6189:
6175:
6173:
6172:
6167:
6165:
6164:
6156:
6142:
6124:
6121:
6120:
6116:
6114:
6107:
6104:
6103:
6099:
6097:
6090:
6087:
6084:
6082:
6075:
6072:
6071:
6067:
6065:
6058:
6055:
6054:
6050:
6048:
6041:
6038:
6037:
6033:
6031:
6024:
6021:
6020:
6016:
6014:
6007:
6004:
6003:
5999:
5997:
5990:
5987:
5986:
5982:
5980:
5969:
5967:
5966:
5961:
5959:
5954:
5943:
5922:
5921:
5899:
5894:
5882:
5880:
5879:
5878:
5865:
5861:
5860:
5850:
5841:
5839:
5833:
5825:
5823:
5816:
5805:
5784:
5783:
5782:
5765:
5760:
5748:
5746:
5745:
5744:
5731:
5727:
5726:
5716:
5711:
5710:
5705:
5700:
5695:
5690:
5682:
5680:
5678:
5676:
5674:
5664:
5662:
5656:
5648:
5646:
5639:
5628:
5607:
5606:
5605:
5582:
5577:
5565:
5563:
5562:
5561:
5548:
5544:
5543:
5533:
5528:
5527:
5521:
5509:
5495:
5494:
5493:
5483:
5481:
5475:
5467:
5465:
5458:
5447:
5426:
5425:
5424:
5404:
5399:
5387:
5385:
5384:
5383:
5370:
5366:
5365:
5355:
5350:
5348:
5336:
5322:
5321:
5312:
5310:
5304:
5296:
5294:
5287:
5276:
5255:
5254:
5253:
5236:
5231:
5219:
5217:
5216:
5215:
5202:
5198:
5197:
5187:
5182:
5181:
5176:
5171:
5163:
5161:
5159:
5153:
5152:
5147:
5142:
5134:
5132:
5130:
5124:
5123:
5117:
5103:
5102:
5092:
5090:
5084:
5076:
5074:
5067:
5056:
5035:
5034:
5033:
5013:
5008:
4996:
4994:
4993:
4992:
4979:
4975:
4974:
4964:
4959:
4958:
4953:
4945:
4943:
4937:
4935:
4918:
4909:
4907:
4901:
4893:
4891:
4881:
4860:
4845:
4840:
4828:
4826:
4825:
4824:
4811:
4807:
4806:
4796:
4791:
4790:
4785:
4777:
4775:
4765:
4763:
4749:
4739:
4718:
4703:
4698:
4686:
4685:
4670:
4662:
4661:
4655:
4653:
4642:
4637:
4636:
4630:
4629:
4621:
4618:
4617:
4611:
4609:
4598:
4593:
4589:
4587:
4579:
4571:
4565:
4563:
4552:
4547:
4545:
4544:
4543:
4530:
4526:
4525:
4515:
4506:
4505:
4497:
4477:
4475:
4474:
4469:
4467:
4465:
4451:
4446:
4445:
4437:
4421:
4419:
4418:
4413:
4411:
4410:
4402:
4386:with respect to
4345:
4340:
4320:
4318:
4317:
4312:
4298:
4297:
4276:
4274:
4273:
4268:
4254:
4253:
4232:
4230:
4229:
4224:
4210:
4209:
4158:
4157:
4117:
4115:
4114:
4109:
4107:
4100:
4098:
4097:
4096:
4083:
4079:
4078:
4068:
4060:
4059:
4046:
4039:
4037:
4022:
4018:
4017:
4007:
3999:
3998:
3985:
3978:
3976:
3961:
3957:
3956:
3946:
3938:
3937:
3924:
3917:
3915:
3914:
3913:
3900:
3896:
3895:
3885:
3877:
3876:
3863:
3849:
3834:
3812:
3810:
3809:
3804:
3798:
3793:
3775:
3773:
3772:
3767:
3761:
3756:
3738:
3736:
3735:
3730:
3724:
3719:
3701:
3699:
3698:
3693:
3688:
3687:
3652:
3650:
3649:
3644:
3639:
3638:
3620:
3618:
3617:
3612:
3607:
3606:
3588:
3586:
3585:
3580:
3575:
3574:
3540:
3538:
3537:
3532:
3527:
3525:
3517:
3500:
3464:
3449:
3439:
3417:
3416:
3415:Euler's notation
3362:
3360:
3359:
3354:
3343:
3342:
3315:
3313:
3312:
3307:
3296:
3295:
3268:
3266:
3265:
3260:
3249:
3248:
3221:
3219:
3218:
3213:
3202:
3201:
3182:inverse function
3179:
3177:
3176:
3171:
3160:
3159:
3124:
3122:
3121:
3116:
3104:
3074:
3030:with respect to
2996:
2994:
2993:
2988:
2986:
2982:
2981:
2966:
2964:
2963:
2962:
2949:
2945:
2944:
2934:
2925:
2924:
2908:
2907:
2890:
2889:
2887:
2873:
2869:
2868:
2858:
2848:
2843:
2830:
2829:
2814:
2812:
2811:
2810:
2797:
2793:
2792:
2782:
2773:
2772:
2756:
2755:
2743:
2741:
2733:
2725:
2716:
2715:
2702:
2701:
2689:
2687:
2679:
2671:
2662:
2661:
2622:
2619:
2618:
2614:
2612:
2605:
2602:
2601:
2597:
2595:
2588:
2585:
2584:
2580:
2578:
2571:
2568:
2567:
2563:
2561:
2550:
2548:
2547:
2542:
2528:
2527:
2491:
2489:
2488:
2483:
2463:
2462:
2435:
2434:
2407:
2406:
2378:
2376:
2375:
2370:
2350:
2349:
2348:
2323:
2322:
2321:
2299:
2298:
2297:
2269:third derivative
2266:
2264:
2263:
2258:
2247:
2228:
2226:
2225:
2220:
2209:
2187:
2185:
2184:
2179:
2168:
2110:
2090:
2075:
2066:
2049:
2047:
2046:
2041:
2021:
1991:
1989:
1988:
1983:
1969:
1967:
1959:
1951:
1930:
1924:
1918:
1916:
1915:
1910:
1895:
1873:
1867:
1858:
1856:
1855:
1850:
1845:
1843:
1835:
1827:
1822:
1820:
1812:
1804:
1799:
1797:
1789:
1781:
1757:
1755:
1754:
1749:
1738:
1736:
1728:
1720:
1718:
1715:
1713:
1712:
1701:
1697:
1695:
1687:
1679:
1663:
1653:
1644:
1642:
1641:
1636:
1631:
1629:
1628:
1627:
1614:
1610:
1609:
1599:
1591:
1590:
1585:
1581:
1579:
1568:
1558:
1556:
1548:
1547:
1543:
1541:
1533:
1525:
1515:
1500:
1498:
1497:
1492:
1487:
1485:
1484:
1483:
1470:
1466:
1465:
1455:
1444:
1442:
1441:
1440:
1427:
1423:
1422:
1412:
1407:
1405:
1404:
1403:
1390:
1386:
1385:
1375:
1370:
1368:
1367:
1366:
1353:
1349:
1348:
1338:
1323:
1321:
1320:
1315:
1298:
1296:
1285:
1283:
1280:
1278:
1276:
1268:
1251:
1249:
1246:
1235:
1233:
1225:
1217:
1205:
1199:
1190:
1188:
1187:
1182:
1177:
1175:
1167:
1159:
1147:
1141:
1132:
1099:with respect to
1017:
1010:
1003:
951:
916:
882:
881:
878:
845:Surface integral
788:
787:
784:
692:
691:
688:
648:Limit comparison
568:
567:
564:
450:Riemann integral
403:
402:
399:
359:L'Hôpital's rule
316:Taylor's theorem
237:
236:
233:
177:
175:
174:
169:
121:
112:
107:
77:
76:
70:
67:
61:
38:
30:
21:
12977:
12976:
12972:
12971:
12970:
12968:
12967:
12966:
12947:
12946:
12945:
12940:
12881:Jacob Bernoulli
12854:
12819:
12810:Galerkin method
12733:
12671:Solution topics
12666:
12624:
12590:
12529:
12461:
12456:
12422:Wayback Machine
12410:
12405:
12390:
12374:
12370:
12340:4th integrals:
12331:
12329:
12320:
12302:
12300:
12291:
12278:
12274:
12265:
12261:
12252:
12248:
12232:
12228:
12219:
12217:
12208:
12202:
12198:
12189:
12187:
12178:
12172:
12168:
12146:
12141:
12139:
12136:
12135:
12119:1st to 3rd and
12108:1st to 6th and
12084:1st to 4th and
12068:
12066:
12057:
12031:
12029:
12020:
12007:
12003:
11996:
11970:
11966:
11957:
11955:
11946:
11940:
11936:
11927:
11925:
11916:
11910:
11903:
11886:
11885:
11881:
11866:
11862:
11850:
11839:
11822:
11818:
11802:
11801:
11789:
11775:
11771:
11754:
11750:
11746:
11732:
11723:
11720:Jacobian matrix
11696:
11689:
11668:
11665:
11664:
11648:
11645:
11644:
11637:Minkowski space
11532:
11515:
11504:
11493:
11490:
11489:
11463:
11462:
11457:
11442:
11441:
11434:
11433:
11427:
11423:
11421:
11415:
11411:
11409:
11403:
11399:
11396:
11395:
11383:
11378:
11371:
11366:
11364:
11362:
11350:
11345:
11338:
11333:
11331:
11329:
11317:
11312:
11305:
11300:
11298:
11295:
11294:
11289:
11287:
11282:
11280:
11275:
11268:
11267:
11258:
11257:
11252:
11237:
11230:
11226:
11222:
11220:
11207:
11200:
11196:
11192:
11190:
11189:
11185:
11177:
11162:
11155:
11151:
11147:
11145:
11132:
11125:
11121:
11117:
11115:
11114:
11110:
11102:
11087:
11080:
11076:
11072:
11070:
11057:
11050:
11046:
11042:
11040:
11039:
11035:
11026:
11025:
11010:
11003:
10999:
10995:
10993:
10980:
10973:
10969:
10965:
10963:
10950:
10943:
10939:
10935:
10933:
10920:
10913:
10909:
10905:
10903:
10890:
10883:
10879:
10875:
10873:
10860:
10853:
10849:
10845:
10843:
10842:
10838:
10831:
10826:
10816:
10814:
10811:
10810:
10776:
10764:
10762:
10759:
10758:
10742:
10727:
10725:
10722:
10721:
10720:: The rotation
10712:
10711:
10710:
10703:
10679:
10678:
10663:
10662:
10653:
10649:
10640:
10639:
10612:
10611:
10586:
10567:
10565:
10562:
10561:
10535:
10532:
10531:
10503:
10500:
10499:
10490:
10489:
10488:
10481:
10457:
10456:
10451:
10436:
10435:
10430:
10412:
10407:
10394:
10389:
10376:
10371:
10370:
10366:
10357:
10356:
10346:
10339:
10335:
10331:
10329:
10316:
10309:
10305:
10301:
10299:
10286:
10279:
10275:
10271:
10269:
10262:
10257:
10247:
10245:
10242:
10241:
10207:
10195:
10193:
10190:
10189:
10180:
10179:
10178:
10171:
10147:
10146:
10131:
10130:
10112:
10107:
10094:
10089:
10076:
10071:
10070:
10066:
10057:
10056:
10041:
10033:
10031:
10018:
10010:
10008:
9995:
9987:
9985:
9984:
9980:
9973:
9960:
9958:
9955:
9954:
9931:
9928:
9927:
9906:
9903:
9902:
9873:
9871:
9868:
9867:
9866:: The gradient
9858:
9857:
9856:
9849:
9804:
9799:
9786:
9781:
9768:
9763:
9762:
9758:
9750:
9747:
9746:
9681:
9678:
9677:
9657:
9652:
9651:
9642:
9637:
9636:
9627:
9622:
9621:
9610:
9608:
9605:
9604:
9576:
9570:Euclidean space
9554:differentiation
9550:Vector calculus
9547:
9506:
9502:
9501:
9496:
9488:
9480:
9476:
9475:
9471:
9469:
9455:
9451:
9450:
9445:
9437:
9429:
9425:
9424:
9420:
9418:
9406:
9402:
9400:
9397:
9396:
9392:, the notation
9370:
9365:
9364:
9356:
9353:
9352:
9332:
9327:
9326:
9317:
9313:
9298:
9294:
9279:
9275:
9264:
9261:
9260:
9244:
9241:
9240:
9223:
9218:
9217:
9215:
9212:
9211:
9190:
9189:
9169:
9159:
9155:
9154:
9152:
9135:
9127:
9125:
9121:
9110:
9105:
9101:
9100:
9088:
9084:
9075:
9071:
9065:
9061:
9051:
9049:
9046:
9045:
9016:
9012:
8993:
8983:
8979:
8978:
8976:
8974:
8971:
8970:
8950:
8949:
8934:
8930:
8918:
8914:
8910:
8900:
8896:
8895:
8893:
8889:
8888:
8876:
8872:
8860:
8856:
8852:
8842:
8838:
8837:
8835:
8830:
8828:
8825:
8824:
8796:
8782:
8774:
8772:
8768:
8767:
8765:
8762:
8761:
8731:
8717:
8709:
8707:
8703:
8702:
8700:
8697:
8696:
8663:
8655:
8653:
8650:
8647:
8646:
8630:
8627:
8626:
8610:
8607:
8606:
8581:
8573:
8571:
8568:
8565:
8564:
8538:
8534:
8525:
8521:
8508:
8500:
8498:
8496:
8493:
8492:
8458:
8447:
8446:
8445:
8443:
8439:
8434:
8426:
8423:
8418:
8417:
8415:
8390:
8389:
8377:
8373:
8369:
8359:
8355:
8354:
8352:
8345:
8336:
8332:
8329:
8328:
8318:
8310:
8308:
8301:
8295:
8291:
8287:
8285:
8282:
8281:
8269:For a function
8264:tensor analysis
8256:
8255:
8254:
8252:
8248:
8245:, then against
8244:
8240:
8235:
8232:
8227:
8224:
8219:
8215:
8185:
8184:
8178:
8174:
8144:
8139:
8086:
8079:
8073:
8053:
8046:
8040:
8030:
8022:
8015:
8009:
8006:
8005:
7999:
7995:
7965:
7960:
7907:
7902:
7882:
7877:
7867:
7850:
7845:
7842:
7841:
7835:
7831:
7801:
7796:
7746:
7741:
7724:
7719:
7709:
7695:
7690:
7686:
7684:
7681:
7680:
7657:
7653:
7623:
7618:
7571:
7566:
7552:
7547:
7530:
7525:
7523:
7520:
7519:
7509:
7508:
7501:
7495:
7478:
7477:
7471:
7467:
7437:
7432:
7371:
7363:
7358:
7355:
7354:
7348:
7344:
7335:
7331:
7322:
7318:
7288:
7284:
7272:
7267:
7236:
7212:
7211:
7194:
7193:
7183:
7176:
7174:
7171:
7170:
7150:
7149:
7142:
7136:
7121:
7120:
7119:
7117:
7112:
7107:
7102:
7098:
7079:
7078:
7065:
7061:
7035:
7025:
7021:
7020:
7018:
7002:
6986:
6985:
6979:
6978:
6965:
6961:
6955:
6951:
6939:
6935:
6931:
6921:
6917:
6916:
6914:
6908:
6904:
6894:
6873:
6872:
6871:
6867:
6861:
6851:
6850:
6847:
6846:
6833:
6829:
6823:
6819:
6807:
6803:
6799:
6789:
6785:
6784:
6782:
6776:
6772:
6762:
6748:
6747:
6743:
6739:
6729:
6722:
6721:
6714:
6713:
6703:
6699:
6683:
6675:
6673:
6660:
6647:
6646:
6643:
6642:
6632:
6628:
6612:
6604:
6602:
6589:
6580:
6579:
6573:
6572:
6540:
6531:
6530:
6526:
6524:
6521:
6520:
6492:
6472:
6454:
6453:
6435:
6434:
6424:
6419:
6406:
6398:
6396:
6382:
6374:
6364:
6356:
6353:
6340:
6332:
6330:
6317:
6309:
6307:
6293:
6292:
6278:
6277:
6262:
6252:
6250:
6248:
6245:
6244:
6188:
6187:
6185:
6182:
6181:
6155:
6154:
6152:
6149:
6148:
6138:
6137:. If location
6122:
6118:
6117:
6112:
6111:
6105:
6101:
6100:
6095:
6094:
6089:WHITE RECTANGLE
6088:
6085:
6080:
6079:
6073:
6069:
6068:
6063:
6062:
6056:
6052:
6051:
6046:
6045:
6039:
6035:
6034:
6029:
6028:
6022:
6018:
6017:
6012:
6011:
6005:
6001:
6000:
5995:
5994:
5988:
5984:
5983:
5978:
5977:
5957:
5956:
5944:
5939:
5911:
5907:
5895:
5890:
5874:
5870:
5866:
5856:
5852:
5851:
5849:
5842:
5834:
5824:
5822:
5819:
5818:
5806:
5801:
5778:
5777:
5773:
5761:
5756:
5740:
5736:
5732:
5722:
5718:
5717:
5715:
5681:
5679:
5677:
5675:
5673:
5672:
5665:
5657:
5647:
5645:
5642:
5641:
5629:
5624:
5595:
5594:
5590:
5578:
5573:
5557:
5553:
5549:
5539:
5535:
5534:
5532:
5511:
5499:
5492:
5491:
5484:
5476:
5466:
5464:
5461:
5460:
5448:
5443:
5417:
5416:
5412:
5400:
5395:
5379:
5375:
5371:
5361:
5357:
5356:
5354:
5338:
5326:
5320:
5313:
5305:
5295:
5293:
5290:
5289:
5277:
5272:
5249:
5248:
5244:
5232:
5227:
5211:
5207:
5203:
5193:
5189:
5188:
5186:
5162:
5160:
5158:
5157:
5133:
5131:
5129:
5128:
5107:
5101:
5100:
5093:
5085:
5075:
5073:
5070:
5069:
5057:
5052:
5026:
5025:
5021:
5009:
5004:
4988:
4984:
4980:
4970:
4966:
4965:
4963:
4944:
4942:
4941:
4922:
4917:
4910:
4902:
4892:
4890:
4887:
4886:
4877:
4853:
4841:
4836:
4820:
4816:
4812:
4802:
4798:
4797:
4795:
4776:
4774:
4773:
4766:
4753:
4748:
4745:
4744:
4735:
4711:
4699:
4694:
4681:
4680:
4663:
4657:
4656:
4646:
4641:
4632:
4631:
4620:
4619:
4613:
4612:
4602:
4597:
4580:
4572:
4570:
4566:
4556:
4551:
4539:
4535:
4531:
4521:
4517:
4516:
4514:
4507:
4496:
4495:
4491:
4489:
4486:
4485:
4455:
4450:
4436:
4435:
4433:
4430:
4429:
4401:
4400:
4398:
4395:
4394:
4355:
4354:
4353:
4351:
4346:
4341:
4336:
4332:
4290:
4286:
4284:
4281:
4280:
4246:
4242:
4240:
4237:
4236:
4202:
4198:
4196:
4193:
4192:
4186:
4185:
4184:
4182:
4178:
4174:
4169:
4162:
4156:
4151:
4150:
4149:
4142:
4105:
4104:
4092:
4088:
4084:
4074:
4070:
4069:
4067:
4052:
4048:
4044:
4043:
4023:
4013:
4009:
4008:
4006:
3991:
3987:
3983:
3982:
3962:
3952:
3948:
3947:
3945:
3930:
3926:
3922:
3921:
3909:
3905:
3901:
3891:
3887:
3886:
3884:
3869:
3865:
3860:
3858:
3855:
3854:
3836:
3830:
3794:
3789:
3783:
3780:
3779:
3757:
3752:
3746:
3743:
3742:
3720:
3715:
3709:
3706:
3705:
3683:
3679:
3677:
3674:
3673:
3634:
3630:
3628:
3625:
3624:
3602:
3598:
3596:
3593:
3592:
3570:
3566:
3564:
3561:
3560:
3518:
3501:
3499:
3473:
3470:
3469:
3455:
3445:
3435:
3414:
3413:
3409:
3408:
3407:
3405:
3401:
3397:
3392:
3385:
3382:
3374:
3329:
3325:
3323:
3320:
3319:
3282:
3278:
3276:
3273:
3272:
3235:
3231:
3229:
3226:
3225:
3194:
3190:
3188:
3185:
3184:
3146:
3142:
3140:
3137:
3136:
3097:
3067:
3047:
3044:
3043:
3037:
3036:
3035:
3033:
3029:
3024:
3015:
3003:
2984:
2983:
2974:
2970:
2958:
2954:
2950:
2940:
2936:
2935:
2933:
2926:
2917:
2913:
2910:
2909:
2900:
2896:
2874:
2864:
2860:
2859:
2857:
2850:
2844:
2839:
2832:
2831:
2822:
2818:
2806:
2802:
2798:
2788:
2784:
2783:
2781:
2774:
2765:
2761:
2758:
2757:
2751:
2747:
2734:
2726:
2724:
2717:
2711:
2707:
2704:
2703:
2697:
2693:
2680:
2672:
2670:
2663:
2657:
2653:
2649:
2647:
2644:
2643:
2621:QUADRUPLE PRIME
2620:
2616:
2615:
2610:
2609:
2603:
2599:
2598:
2593:
2592:
2586:
2582:
2581:
2576:
2575:
2569:
2565:
2564:
2559:
2558:
2517:
2513:
2511:
2508:
2507:
2452:
2448:
2424:
2420:
2396:
2392:
2390:
2387:
2386:
2341:
2340:
2336:
2317:
2316:
2312:
2290:
2289:
2285:
2283:
2280:
2279:
2240:
2238:
2235:
2234:
2202:
2200:
2197:
2196:
2161:
2159:
2156:
2155:
2129:
2128:
2127:
2125:
2121:
2116:
2108:
2101:
2085:
2071:
2062:
2014:
2003:
2000:
1999:
1960:
1952:
1950:
1939:
1936:
1935:
1926:
1920:
1901:
1898:
1897:
1891:
1869:
1863:
1836:
1828:
1826:
1813:
1805:
1803:
1790:
1782:
1780:
1778:
1775:
1774:
1729:
1721:
1719:
1714:
1702:
1688:
1680:
1678:
1675:
1674:
1672:
1669:
1668:
1655:
1649:
1623:
1619:
1615:
1605:
1601:
1600:
1598:
1586:
1572:
1567:
1563:
1562:
1549:
1534:
1526:
1524:
1520:
1516:
1514:
1512:
1509:
1508:
1479:
1475:
1471:
1461:
1457:
1456:
1454:
1436:
1432:
1428:
1418:
1414:
1413:
1411:
1399:
1395:
1391:
1381:
1377:
1376:
1374:
1362:
1358:
1354:
1344:
1340:
1339:
1337:
1335:
1332:
1331:
1289:
1284:
1279:
1269:
1252:
1250:
1245:
1226:
1218:
1216:
1214:
1211:
1210:
1201:
1195:
1168:
1160:
1158:
1156:
1153:
1152:
1143:
1137:
1119:
1112:
1106:
1105:
1104:
1102:
1098:
1093:
1092:
1091:
1086:
1077:
1076:
1071:
1062:
1021:
992:
991:
977:Integration Bee
952:
949:
942:
941:
917:
914:
907:
906:
879:
876:
869:
868:
850:Volume integral
785:
780:
773:
772:
689:
684:
677:
676:
646:
565:
560:
553:
552:
544:Risch algorithm
514:Euler's formula
400:
395:
388:
387:
369:General Leibniz
252:generalizations
234:
229:
222:
208:Rolle's theorem
203:
178:
114:
108:
103:
97:
94:
93:
71:
65:
62:
55:
43:This article's
39:
28:
23:
22:
18:Newton notation
15:
12:
11:
5:
12975:
12965:
12964:
12959:
12942:
12941:
12939:
12938:
12933:
12928:
12923:
12918:
12913:
12908:
12903:
12898:
12896:Ernst Lindelöf
12893:
12888:
12883:
12878:
12876:Leonhard Euler
12873:
12868:
12862:
12860:
12859:Mathematicians
12856:
12855:
12853:
12852:
12847:
12842:
12837:
12831:
12829:
12825:
12824:
12821:
12820:
12818:
12817:
12812:
12807:
12802:
12797:
12792:
12787:
12782:
12777:
12772:
12767:
12762:
12757:
12752:
12747:
12741:
12739:
12735:
12734:
12732:
12731:
12726:
12721:
12715:
12710:
12705:
12700:
12695:
12690:
12685:
12683:Phase portrait
12680:
12674:
12672:
12668:
12667:
12665:
12664:
12659:
12654:
12649:
12643:
12641:
12634:
12630:
12629:
12626:
12625:
12623:
12622:
12617:
12616:
12615:
12605:
12598:
12596:
12592:
12591:
12589:
12588:
12586:On jet bundles
12583:
12578:
12573:
12568:
12563:
12558:
12553:
12551:Nonhomogeneous
12548:
12543:
12537:
12535:
12531:
12530:
12528:
12527:
12522:
12517:
12512:
12507:
12502:
12497:
12492:
12487:
12482:
12476:
12474:
12467:
12466:Classification
12463:
12462:
12455:
12454:
12447:
12440:
12432:
12426:
12425:
12409:
12408:External links
12406:
12404:
12403:
12388:
12368:
12353:
12352:
12345:
12338:
12309:
12272:
12259:
12246:
12226:
12196:
12166:
12150:
12145:
12129:
12128:
12117:
12106:
12093:
12082:
12075:
12038:
12001:
11994:
11964:
11934:
11901:
11879:
11860:
11837:
11816:
11787:
11769:
11747:
11745:
11742:
11741:
11740:
11735:
11726:
11717:
11714:Hessian matrix
11711:
11705:
11699:
11688:
11685:
11672:
11652:
11626:
11625:
11614:
11611:
11608:
11605:
11602:
11599:
11596:
11593:
11590:
11587:
11584:
11581:
11578:
11575:
11572:
11569:
11566:
11563:
11551:
11538:
11535:
11531:
11528:
11525:
11521:
11518:
11514:
11510:
11507:
11503:
11500:
11497:
11479:
11478:
11477:
11476:
11460:
11456:
11453:
11450:
11447:
11445:
11443:
11438:
11430:
11426:
11422:
11418:
11414:
11410:
11406:
11402:
11398:
11397:
11389:
11386:
11374:
11363:
11356:
11353:
11341:
11330:
11323:
11320:
11308:
11297:
11296:
11292:
11288:
11285:
11281:
11278:
11274:
11273:
11271:
11266:
11263:
11261:
11259:
11255:
11250:
11243:
11240:
11233:
11229:
11225:
11219:
11213:
11210:
11203:
11199:
11195:
11188:
11184:
11180:
11175:
11168:
11165:
11158:
11154:
11150:
11144:
11138:
11135:
11128:
11124:
11120:
11113:
11109:
11105:
11100:
11093:
11090:
11083:
11079:
11075:
11069:
11063:
11060:
11053:
11049:
11045:
11038:
11034:
11031:
11029:
11027:
11023:
11016:
11013:
11006:
11002:
10998:
10992:
10986:
10983:
10976:
10972:
10968:
10962:
10956:
10953:
10946:
10942:
10938:
10932:
10926:
10923:
10916:
10912:
10908:
10902:
10896:
10893:
10886:
10882:
10878:
10872:
10866:
10863:
10856:
10852:
10848:
10841:
10837:
10834:
10832:
10829:
10825:
10822:
10819:
10818:
10805:
10804:
10779:
10773:
10770:
10767:
10745:
10739:
10736:
10733:
10730:
10704:
10698:
10697:
10696:
10695:
10694:
10693:
10692:
10677:
10674:
10671:
10668:
10666:
10664:
10661:
10656:
10652:
10648:
10645:
10643:
10641:
10638:
10635:
10632:
10629:
10626:
10623:
10620:
10617:
10615:
10613:
10610:
10607:
10604:
10601:
10598:
10595:
10592:
10589:
10587:
10585:
10582:
10579:
10576:
10573:
10570:
10569:
10556:
10555:
10539:
10519:
10516:
10513:
10510:
10507:
10482:
10476:
10475:
10474:
10473:
10472:
10471:
10470:
10454:
10450:
10447:
10444:
10441:
10439:
10437:
10433:
10429:
10425:
10418:
10415:
10411:
10406:
10400:
10397:
10393:
10388:
10382:
10379:
10375:
10369:
10365:
10362:
10360:
10358:
10352:
10349:
10342:
10338:
10334:
10328:
10322:
10319:
10312:
10308:
10304:
10298:
10292:
10289:
10282:
10278:
10274:
10268:
10265:
10263:
10260:
10256:
10253:
10250:
10249:
10236:
10235:
10210:
10204:
10201:
10198:
10172:
10166:
10165:
10164:
10163:
10162:
10161:
10160:
10145:
10142:
10139:
10136:
10134:
10132:
10129:
10125:
10118:
10115:
10111:
10106:
10100:
10097:
10093:
10088:
10082:
10079:
10075:
10069:
10065:
10062:
10060:
10058:
10054:
10047:
10044:
10039:
10036:
10030:
10024:
10021:
10016:
10013:
10007:
10001:
9998:
9993:
9990:
9983:
9979:
9976:
9974:
9972:
9969:
9966:
9963:
9962:
9949:
9948:
9935:
9923:multiplication
9910:
9890:
9885:
9882:
9879:
9876:
9850:
9844:
9843:
9842:
9834:
9833:
9822:
9817:
9810:
9807:
9803:
9798:
9792:
9789:
9785:
9780:
9774:
9771:
9767:
9761:
9757:
9754:
9712:
9709:
9706:
9703:
9700:
9697:
9694:
9691:
9688:
9685:
9665:
9660:
9655:
9650:
9645:
9640:
9635:
9630:
9625:
9620:
9617:
9613:
9546:
9543:
9531:
9530:
9519:
9509:
9505:
9499:
9495:
9491:
9483:
9479:
9474:
9468:
9458:
9454:
9448:
9444:
9440:
9432:
9428:
9423:
9417:
9414:
9409:
9405:
9381:
9378:
9373:
9368:
9363:
9360:
9338:
9335:
9330:
9325:
9320:
9316:
9309:
9306:
9301:
9297:
9293:
9290:
9287:
9282:
9278:
9274:
9271:
9268:
9248:
9226:
9221:
9204:
9203:
9188:
9182:
9179:
9175:
9172:
9167:
9162:
9158:
9151:
9147:
9141:
9138:
9133:
9130:
9124:
9116:
9113:
9109:
9104:
9102:
9099:
9094:
9091:
9087:
9083:
9078:
9074:
9068:
9064:
9060:
9057:
9055:
9053:
9039:
9038:
9027:
9022:
9019:
9015:
9011:
9005:
9002:
8999:
8996:
8991:
8986:
8982:
8964:
8963:
8948:
8943:
8940:
8937:
8933:
8929:
8921:
8917:
8913:
8908:
8903:
8899:
8892:
8890:
8887:
8882:
8879:
8875:
8871:
8863:
8859:
8855:
8850:
8845:
8841:
8834:
8832:
8800:
8794:
8788:
8785:
8780:
8777:
8771:
8735:
8729:
8723:
8720:
8715:
8712:
8706:
8693:thermodynamics
8669:
8666:
8661:
8658:
8634:
8614:
8587:
8584:
8579:
8576:
8561:
8560:
8549:
8546:
8541:
8537:
8533:
8528:
8524:
8520:
8514:
8511:
8506:
8503:
8441:
8437:
8435:
8414:
8413:
8412:
8404:
8403:
8388:
8380:
8376:
8372:
8367:
8362:
8358:
8351:
8348:
8346:
8342:
8339:
8335:
8331:
8330:
8324:
8321:
8316:
8313:
8307:
8304:
8302:
8298:
8294:
8290:
8289:
8250:
8246:
8242:
8238:
8236:
8230:
8222:
8218:
8217:
8216:
8214:
8211:
8199:
8198:
8181:
8177:
8173:
8170:
8167:
8164:
8161:
8158:
8155:
8150:
8147:
8142:
8138:
8134:
8131:
8128:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8096:
8093:
8090:
8083:
8078:
8072:
8069:
8063:
8060:
8057:
8050:
8045:
8039:
8036:
8033:
8031:
8026:
8019:
8014:
8008:
8007:
8002:
7998:
7994:
7991:
7988:
7985:
7982:
7979:
7976:
7971:
7968:
7963:
7959:
7955:
7952:
7949:
7945:
7942:
7939:
7936:
7933:
7930:
7927:
7924:
7917:
7914:
7911:
7906:
7901:
7898:
7892:
7889:
7886:
7881:
7876:
7873:
7870:
7868:
7863:
7860:
7857:
7854:
7849:
7844:
7843:
7838:
7834:
7830:
7827:
7824:
7821:
7818:
7815:
7812:
7807:
7804:
7799:
7795:
7791:
7788:
7785:
7781:
7778:
7775:
7772:
7769:
7766:
7763:
7760:
7753:
7750:
7745:
7740:
7737:
7731:
7728:
7723:
7718:
7715:
7712:
7710:
7705:
7702:
7699:
7694:
7689:
7688:
7674:
7673:
7660:
7656:
7652:
7649:
7646:
7643:
7640:
7637:
7634:
7629:
7626:
7621:
7617:
7613:
7610:
7607:
7603:
7600:
7597:
7594:
7591:
7588:
7585:
7582:
7575:
7570:
7565:
7562:
7556:
7551:
7546:
7543:
7537:
7534:
7529:
7492:
7491:
7474:
7470:
7466:
7463:
7460:
7457:
7454:
7451:
7448:
7443:
7440:
7435:
7431:
7427:
7424:
7421:
7417:
7414:
7411:
7408:
7405:
7402:
7399:
7396:
7392:
7389:
7386:
7383:
7380:
7377:
7374:
7372:
7367:
7362:
7357:
7356:
7351:
7347:
7343:
7338:
7334:
7330:
7325:
7321:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7296:
7291:
7287:
7283:
7278:
7275:
7270:
7266:
7262:
7259:
7256:
7252:
7249:
7246:
7242:
7239:
7235:
7232:
7229:
7226:
7219:
7216:
7210:
7207:
7201:
7198:
7192:
7189:
7186:
7184:
7182:
7179:
7178:
7115:
7113:
7101:
7100:
7099:
7097:
7094:
7093:
7092:
7077:
7071:
7068:
7064:
7060:
7057:
7054:
7048:
7045:
7041:
7038:
7033:
7028:
7024:
7017:
7014:
7008:
7005:
7003:
6995:
6989:
6984:
6981:
6980:
6977:
6971:
6968:
6964:
6958:
6954:
6950:
6942:
6938:
6934:
6929:
6924:
6920:
6911:
6907:
6900:
6897:
6895:
6890:
6885:
6881:
6876:
6870:
6863: or
6859:
6854:
6849:
6848:
6845:
6839:
6836:
6832:
6826:
6822:
6818:
6810:
6806:
6802:
6797:
6792:
6788:
6779:
6775:
6768:
6765:
6763:
6757:
6751:
6746:
6742:
6737:
6731: or
6725:
6719:
6716:
6715:
6712:
6706:
6702:
6698:
6695:
6689:
6686:
6681:
6678:
6672:
6666:
6663:
6661:
6656:
6650:
6645:
6644:
6641:
6635:
6631:
6627:
6624:
6618:
6615:
6610:
6607:
6601:
6595:
6592:
6590:
6583:
6578:
6575:
6574:
6571:
6567:
6564:
6561:
6558:
6555:
6552:
6546:
6543:
6541:
6534:
6529:
6528:
6514:
6513:
6502:
6498:
6495:
6491:
6488:
6485:
6482:
6478:
6475:
6471:
6468:
6465:
6462:
6457:
6452:
6449:
6446:
6443:
6438:
6430:
6427:
6423:
6418:
6412:
6409:
6404:
6401:
6395:
6388:
6385:
6380:
6377:
6370:
6367:
6362:
6359:
6352:
6346:
6343:
6338:
6335:
6329:
6323:
6320:
6315:
6312:
6306:
6300:
6297:
6291:
6285:
6282:
6276:
6269:
6266:
6259:
6256:
6195:
6192:
6162:
6159:
6131:
6130:
6129:th derivative)
6109:
6092:
6077:
6060:
6043:
6026:
6009:
5992:
5971:
5970:
5953:
5950:
5947:
5942:
5938:
5934:
5931:
5928:
5925:
5920:
5917:
5914:
5910:
5906:
5903:
5898:
5893:
5889:
5885:
5877:
5873:
5869:
5864:
5859:
5855:
5848:
5845:
5843:
5838:
5831:
5828:
5821:
5820:
5815:
5812:
5809:
5804:
5800:
5796:
5793:
5790:
5787:
5781:
5776:
5772:
5769:
5764:
5759:
5755:
5751:
5743:
5739:
5735:
5730:
5725:
5721:
5714:
5708:
5703:
5698:
5693:
5688:
5685:
5671:
5668:
5666:
5661:
5654:
5651:
5644:
5643:
5638:
5635:
5632:
5627:
5623:
5619:
5616:
5613:
5610:
5604:
5601:
5598:
5593:
5589:
5586:
5581:
5576:
5572:
5568:
5560:
5556:
5552:
5547:
5542:
5538:
5531:
5525:
5520:
5517:
5514:
5508:
5505:
5502:
5498:
5490:
5487:
5485:
5480:
5473:
5470:
5463:
5462:
5457:
5454:
5451:
5446:
5442:
5438:
5435:
5432:
5429:
5423:
5420:
5415:
5411:
5408:
5403:
5398:
5394:
5390:
5382:
5378:
5374:
5369:
5364:
5360:
5353:
5347:
5344:
5341:
5335:
5332:
5329:
5325:
5319:
5316:
5314:
5309:
5302:
5299:
5292:
5291:
5286:
5283:
5280:
5275:
5271:
5267:
5264:
5261:
5258:
5252:
5247:
5243:
5240:
5235:
5230:
5226:
5222:
5214:
5210:
5206:
5201:
5196:
5192:
5185:
5179:
5174:
5169:
5166:
5156:
5150:
5145:
5140:
5137:
5127:
5121:
5116:
5113:
5110:
5106:
5099:
5096:
5094:
5089:
5082:
5079:
5072:
5071:
5066:
5063:
5060:
5055:
5051:
5047:
5044:
5041:
5038:
5032:
5029:
5024:
5020:
5017:
5012:
5007:
5003:
4999:
4991:
4987:
4983:
4978:
4973:
4969:
4962:
4956:
4951:
4948:
4940:
4934:
4931:
4928:
4925:
4921:
4916:
4913:
4911:
4906:
4899:
4896:
4889:
4888:
4884:
4880:
4876:
4872:
4869:
4866:
4863:
4859:
4856:
4852:
4849:
4844:
4839:
4835:
4831:
4823:
4819:
4815:
4810:
4805:
4801:
4794:
4788:
4783:
4780:
4772:
4769:
4767:
4762:
4759:
4756:
4752:
4747:
4746:
4742:
4738:
4734:
4730:
4727:
4724:
4721:
4717:
4714:
4710:
4707:
4702:
4697:
4693:
4689:
4684:
4679:
4676:
4673:
4669:
4666:
4660:
4652:
4649:
4645:
4640:
4635:
4627:
4624:
4616:
4608:
4605:
4601:
4596:
4592:
4586:
4583:
4578:
4575:
4569:
4562:
4559:
4555:
4550:
4542:
4538:
4534:
4529:
4524:
4520:
4513:
4510:
4508:
4503:
4500:
4494:
4493:
4479:
4478:
4464:
4461:
4458:
4454:
4449:
4443:
4440:
4423:
4422:
4408:
4405:
4349:
4347:
4335:
4334:
4333:
4331:
4328:
4327:
4326:
4310:
4307:
4304:
4301:
4296:
4293:
4289:
4278:
4266:
4263:
4260:
4257:
4252:
4249:
4245:
4234:
4222:
4219:
4216:
4213:
4208:
4205:
4201:
4180:
4176:
4172:
4170:
4152:
4145:
4144:
4143:
4141:
4138:
4119:
4118:
4103:
4095:
4091:
4087:
4082:
4077:
4073:
4066:
4063:
4058:
4055:
4051:
4047:
4045:
4042:
4036:
4033:
4029:
4026:
4021:
4016:
4012:
4005:
4002:
3997:
3994:
3990:
3986:
3984:
3981:
3975:
3972:
3968:
3965:
3960:
3955:
3951:
3944:
3941:
3936:
3933:
3929:
3925:
3923:
3920:
3912:
3908:
3904:
3899:
3894:
3890:
3883:
3880:
3875:
3872:
3868:
3864:
3862:
3819:
3818:
3817:th derivative.
3802:
3797:
3792:
3788:
3777:
3765:
3760:
3755:
3751:
3740:
3728:
3723:
3718:
3714:
3703:
3691:
3686:
3682:
3659:
3658:
3657:th derivative.
3642:
3637:
3633:
3622:
3610:
3605:
3601:
3590:
3578:
3573:
3569:
3542:
3541:
3530:
3524:
3521:
3516:
3513:
3510:
3507:
3504:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3425:Leonhard Euler
3403:
3399:
3398:derivative of
3395:
3393:
3380:
3377:
3376:
3375:
3373:
3370:
3369:
3368:
3352:
3349:
3346:
3341:
3338:
3335:
3332:
3328:
3317:
3305:
3302:
3299:
3294:
3291:
3288:
3285:
3281:
3270:
3258:
3255:
3252:
3247:
3244:
3241:
3238:
3234:
3223:
3211:
3208:
3205:
3200:
3197:
3193:
3169:
3166:
3163:
3158:
3155:
3152:
3149:
3145:
3126:
3125:
3114:
3111:
3108:
3103:
3100:
3096:
3093:
3090:
3087:
3083:
3080:
3077:
3073:
3070:
3066:
3063:
3060:
3057:
3054:
3051:
3031:
3027:
3025:
3006:
3005:
3004:
3002:
2999:
2998:
2997:
2980:
2977:
2973:
2969:
2961:
2957:
2953:
2948:
2943:
2939:
2932:
2929:
2927:
2923:
2920:
2916:
2912:
2911:
2906:
2903:
2899:
2895:
2886:
2883:
2880:
2877:
2872:
2867:
2863:
2856:
2853:
2851:
2847:
2842:
2838:
2834:
2833:
2828:
2825:
2821:
2817:
2809:
2805:
2801:
2796:
2791:
2787:
2780:
2777:
2775:
2771:
2768:
2764:
2760:
2759:
2754:
2750:
2746:
2740:
2737:
2732:
2729:
2723:
2720:
2718:
2714:
2710:
2706:
2705:
2700:
2696:
2692:
2686:
2683:
2678:
2675:
2669:
2666:
2664:
2660:
2656:
2652:
2651:
2625:
2624:
2607:
2590:
2573:
2552:
2551:
2540:
2537:
2534:
2531:
2526:
2523:
2520:
2516:
2493:
2492:
2481:
2478:
2475:
2472:
2469:
2466:
2461:
2458:
2455:
2451:
2447:
2444:
2441:
2438:
2433:
2430:
2427:
2423:
2419:
2416:
2413:
2410:
2405:
2402:
2399:
2395:
2380:
2379:
2368:
2365:
2362:
2359:
2356:
2353:
2347:
2344:
2339:
2335:
2332:
2329:
2326:
2320:
2315:
2311:
2308:
2305:
2302:
2296:
2293:
2288:
2273:Roman numerals
2256:
2253:
2250:
2246:
2243:
2218:
2215:
2212:
2208:
2205:
2190:
2189:
2177:
2174:
2171:
2167:
2164:
2123:
2119:
2117:
2104:
2103:
2102:
2100:
2097:
2051:
2050:
2039:
2036:
2033:
2030:
2027:
2024:
2020:
2017:
2013:
2010:
2007:
1993:
1992:
1981:
1978:
1975:
1972:
1966:
1963:
1958:
1955:
1949:
1946:
1943:
1908:
1905:
1880:infinitesimals
1860:
1859:
1848:
1842:
1839:
1834:
1831:
1825:
1819:
1816:
1811:
1808:
1802:
1796:
1793:
1788:
1785:
1760:
1759:
1747:
1744:
1741:
1735:
1732:
1727:
1724:
1716: or
1711:
1708:
1705:
1700:
1694:
1691:
1686:
1683:
1677:
1646:
1645:
1634:
1626:
1622:
1618:
1613:
1608:
1604:
1597:
1594:
1589:
1584:
1578:
1575:
1571:
1566:
1561:
1555:
1552:
1546:
1540:
1537:
1532:
1529:
1523:
1519:
1502:
1501:
1490:
1482:
1478:
1474:
1469:
1464:
1460:
1453:
1450:
1447:
1439:
1435:
1431:
1426:
1421:
1417:
1410:
1402:
1398:
1394:
1389:
1384:
1380:
1373:
1365:
1361:
1357:
1352:
1347:
1343:
1325:
1324:
1313:
1310:
1307:
1304:
1301:
1295:
1292:
1288:
1281: or
1275:
1272:
1267:
1264:
1261:
1258:
1255:
1247: or
1244:
1241:
1238:
1232:
1229:
1224:
1221:
1192:
1191:
1180:
1174:
1171:
1166:
1163:
1108:Main article:
1100:
1096:
1094:
1087:
1079:
1078:
1072:
1067:
1066:
1065:
1064:
1063:
1061:
1058:
1023:
1022:
1020:
1019:
1012:
1005:
997:
994:
993:
990:
989:
984:
979:
974:
972:List of topics
969:
964:
959:
953:
948:
947:
944:
943:
940:
939:
934:
929:
924:
918:
913:
912:
909:
908:
903:
902:
901:
900:
895:
890:
880:
875:
874:
871:
870:
865:
864:
863:
862:
857:
852:
847:
842:
837:
832:
824:
823:
819:
818:
817:
816:
811:
806:
801:
793:
792:
786:
779:
778:
775:
774:
769:
768:
767:
766:
761:
756:
751:
746:
741:
733:
732:
728:
727:
726:
725:
720:
715:
710:
705:
700:
690:
683:
682:
679:
678:
673:
672:
671:
670:
665:
660:
655:
650:
644:
639:
634:
629:
624:
616:
615:
609:
608:
607:
606:
601:
596:
591:
586:
581:
566:
559:
558:
555:
554:
549:
548:
547:
546:
541:
536:
531:
529:Changing order
526:
516:
511:
493:
488:
483:
475:
474:
473:Integration by
470:
469:
468:
467:
462:
457:
452:
447:
437:
435:Antiderivative
429:
428:
424:
423:
422:
421:
416:
411:
401:
394:
393:
390:
389:
384:
383:
382:
381:
376:
371:
366:
361:
356:
351:
346:
341:
336:
328:
327:
321:
320:
319:
318:
313:
308:
303:
298:
293:
285:
284:
280:
279:
278:
277:
276:
275:
270:
265:
255:
242:
241:
235:
228:
227:
224:
223:
221:
220:
215:
210:
204:
202:
201:
196:
190:
189:
188:
180:
179:
167:
164:
161:
158:
155:
152:
149:
146:
143:
140:
137:
134:
130:
127:
124:
120:
117:
111:
106:
102:
92:
89:
88:
82:
81:
73:
72:
52:the key points
42:
40:
33:
26:
9:
6:
4:
3:
2:
12974:
12963:
12960:
12958:
12955:
12954:
12952:
12937:
12934:
12932:
12929:
12927:
12924:
12922:
12919:
12917:
12914:
12912:
12909:
12907:
12904:
12902:
12899:
12897:
12894:
12892:
12889:
12887:
12884:
12882:
12879:
12877:
12874:
12872:
12869:
12867:
12864:
12863:
12861:
12857:
12851:
12848:
12846:
12843:
12841:
12838:
12836:
12833:
12832:
12830:
12826:
12816:
12813:
12811:
12808:
12806:
12803:
12801:
12798:
12796:
12793:
12791:
12788:
12786:
12783:
12781:
12778:
12776:
12773:
12771:
12768:
12766:
12763:
12761:
12758:
12756:
12753:
12751:
12748:
12746:
12743:
12742:
12740:
12736:
12730:
12727:
12725:
12722:
12719:
12716:
12714:
12711:
12709:
12706:
12704:
12701:
12699:
12696:
12694:
12691:
12689:
12686:
12684:
12681:
12679:
12676:
12675:
12673:
12669:
12663:
12660:
12658:
12655:
12653:
12650:
12648:
12645:
12644:
12642:
12638:
12635:
12631:
12621:
12618:
12614:
12611:
12610:
12609:
12606:
12603:
12600:
12599:
12597:
12593:
12587:
12584:
12582:
12579:
12577:
12574:
12572:
12569:
12567:
12564:
12562:
12559:
12557:
12554:
12552:
12549:
12547:
12544:
12542:
12539:
12538:
12536:
12532:
12526:
12523:
12521:
12518:
12516:
12513:
12511:
12508:
12506:
12503:
12501:
12498:
12496:
12493:
12491:
12488:
12486:
12483:
12481:
12478:
12477:
12475:
12471:
12468:
12464:
12460:
12453:
12448:
12446:
12441:
12439:
12434:
12433:
12430:
12423:
12419:
12415:
12412:
12411:
12399:
12395:
12391:
12385:
12381:
12380:
12372:
12365:
12361:
12357:
12350:
12346:
12343:
12339:
12327:
12323:
12318:
12314:
12310:
12298:
12294:
12289:
12285:
12281:
12280:
12276:
12269:
12263:
12256:
12250:
12244:
12243:0-486-67766-4
12240:
12236:
12230:
12215:
12211:
12206:
12200:
12185:
12181:
12176:
12170:
12148:
12143:
12133:
12126:
12122:
12118:
12115:
12111:
12107:
12104:
12103:
12098:
12094:
12091:
12087:
12083:
12080:
12076:
12064:
12060:
12055:
12051:
12047:
12043:
12039:
12027:
12023:
12018:
12014:
12010:
12009:
12005:
11997:
11991:
11988:. p. 3.
11987:
11983:
11979:
11975:
11968:
11953:
11949:
11944:
11938:
11923:
11919:
11914:
11908:
11906:
11897:
11893:
11889:
11883:
11877:
11873:
11869:
11864:
11857:
11853:
11848:
11846:
11844:
11842:
11833:
11829:
11828:
11820:
11812:
11806:
11798:
11794:
11790:
11788:9781119015314
11784:
11780:
11773:
11765:
11764:
11759:
11752:
11748:
11739:
11736:
11730:
11727:
11721:
11718:
11715:
11712:
11709:
11706:
11703:
11700:
11694:
11691:
11690:
11684:
11650:
11642:
11638:
11633:
11631:
11612:
11606:
11597:
11594:
11591:
11585:
11576:
11570:
11567:
11536:
11533:
11529:
11526:
11523:
11519:
11516:
11512:
11508:
11501:
11498:
11488:
11487:
11486:
11484:
11454:
11448:
11446:
11436:
11428:
11424:
11416:
11412:
11404:
11400:
11387:
11354:
11321:
11269:
11264:
11262:
11248:
11241:
11231:
11227:
11217:
11211:
11201:
11197:
11186:
11182:
11173:
11166:
11156:
11152:
11142:
11136:
11126:
11122:
11111:
11107:
11098:
11091:
11081:
11077:
11067:
11061:
11051:
11047:
11036:
11032:
11030:
11021:
11014:
11004:
11000:
10990:
10984:
10974:
10970:
10960:
10954:
10944:
10940:
10930:
10924:
10914:
10910:
10900:
10894:
10884:
10880:
10870:
10864:
10854:
10850:
10839:
10835:
10833:
10823:
10820:
10809:
10808:
10807:
10806:
10802:
10798:
10797:cross product
10794:
10719:
10718:
10714:
10713:
10708:
10702:
10675:
10669:
10667:
10659:
10654:
10646:
10644:
10636:
10627:
10618:
10616:
10605:
10596:
10590:
10588:
10583:
10580:
10577:
10574:
10571:
10560:
10559:
10558:
10557:
10553:
10537:
10517:
10514:
10511:
10508:
10505:
10497:
10496:
10492:
10491:
10486:
10480:
10448:
10442:
10440:
10427:
10423:
10416:
10404:
10398:
10386:
10380:
10367:
10363:
10361:
10350:
10340:
10336:
10326:
10320:
10310:
10306:
10296:
10290:
10280:
10276:
10266:
10264:
10254:
10251:
10240:
10239:
10238:
10237:
10233:
10229:
10225:
10187:
10186:
10182:
10181:
10176:
10170:
10143:
10137:
10135:
10127:
10123:
10116:
10104:
10098:
10086:
10080:
10067:
10063:
10061:
10052:
10045:
10037:
10028:
10022:
10014:
10005:
9999:
9991:
9981:
9977:
9975:
9970:
9967:
9964:
9953:
9952:
9951:
9950:
9946:
9933:
9924:
9908:
9888:
9865:
9864:
9860:
9859:
9854:
9848:
9841:
9839:
9820:
9815:
9808:
9796:
9790:
9778:
9772:
9759:
9755:
9745:
9744:
9743:
9741:
9737:
9733:
9728:
9726:
9707:
9704:
9701:
9698:
9695:
9689:
9686:
9683:
9658:
9648:
9643:
9633:
9628:
9615:
9602:
9598:
9594:
9588:
9584:
9580:
9573:
9571:
9567:
9566:scalar fields
9563:
9559:
9555:
9551:
9542:
9540:
9536:
9517:
9507:
9503:
9497:
9493:
9481:
9477:
9466:
9456:
9452:
9446:
9442:
9430:
9426:
9415:
9412:
9407:
9395:
9394:
9393:
9379:
9371:
9361:
9358:
9336:
9333:
9323:
9318:
9314:
9307:
9299:
9295:
9291:
9288:
9285:
9280:
9276:
9269:
9266:
9246:
9224:
9209:
9186:
9180:
9173:
9165:
9160:
9149:
9145:
9139:
9131:
9122:
9114:
9097:
9092:
9089:
9085:
9081:
9076:
9066:
9062:
9056:
9044:
9043:
9042:
9025:
9020:
9017:
9013:
9009:
9003:
8997:
8989:
8984:
8969:
8968:
8967:
8946:
8941:
8938:
8935:
8931:
8927:
8919:
8915:
8906:
8901:
8885:
8880:
8877:
8873:
8869:
8861:
8857:
8848:
8843:
8823:
8822:
8821:
8818:
8816:
8798:
8792:
8786:
8778:
8769:
8759:
8755:
8751:
8733:
8727:
8721:
8713:
8704:
8695:. The symbol
8694:
8690:
8685:
8667:
8659:
8632:
8612:
8604:
8585:
8582:
8577:
8574:
8547:
8544:
8539:
8531:
8526:
8522:
8518:
8512:
8504:
8491:
8490:
8489:
8487:
8483:
8480:, but not to
8479:
8473:
8469:
8465:
8461:
8456:
8452:
8429:
8421:
8411:
8409:
8386:
8378:
8374:
8370:
8365:
8360:
8356:
8349:
8347:
8340:
8337:
8333:
8322:
8319:
8314:
8311:
8305:
8303:
8296:
8292:
8280:
8279:
8278:
8276:
8272:
8267:
8265:
8261:
8233:
8225:
8210:
8208:
8204:
8179:
8175:
8171:
8165:
8159:
8156:
8153:
8148:
8145:
8140:
8136:
8132:
8129:
8126:
8119:
8113:
8110:
8107:
8104:
8101:
8094:
8091:
8088:
8076:
8070:
8067:
8061:
8058:
8055:
8043:
8037:
8034:
8032:
8024:
8012:
8000:
7996:
7992:
7986:
7980:
7977:
7974:
7969:
7966:
7961:
7957:
7953:
7950:
7947:
7940:
7934:
7931:
7928:
7925:
7922:
7904:
7899:
7896:
7879:
7874:
7871:
7869:
7847:
7836:
7832:
7828:
7822:
7816:
7813:
7810:
7805:
7802:
7797:
7793:
7789:
7786:
7783:
7776:
7770:
7767:
7764:
7761:
7758:
7743:
7738:
7735:
7721:
7716:
7713:
7711:
7692:
7679:
7678:
7677:
7658:
7654:
7650:
7644:
7638:
7635:
7632:
7627:
7624:
7619:
7615:
7611:
7608:
7605:
7598:
7592:
7589:
7586:
7583:
7580:
7568:
7563:
7560:
7549:
7544:
7541:
7527:
7518:
7517:
7516:
7512:
7505:
7498:
7472:
7468:
7464:
7458:
7452:
7449:
7446:
7441:
7438:
7433:
7429:
7425:
7422:
7419:
7412:
7406:
7403:
7400:
7397:
7394:
7390:
7387:
7384:
7381:
7378:
7375:
7373:
7360:
7349:
7345:
7341:
7336:
7332:
7328:
7323:
7319:
7315:
7309:
7303:
7300:
7294:
7289:
7285:
7276:
7273:
7268:
7264:
7260:
7257:
7254:
7247:
7240:
7237:
7233:
7230:
7227:
7224:
7217:
7214:
7208:
7205:
7199:
7196:
7190:
7187:
7185:
7180:
7169:
7168:
7167:
7165:
7161:
7160:
7153:
7146:
7139:
7134:
7130:
7126:
7110:
7105:
7075:
7069:
7066:
7062:
7058:
7055:
7052:
7046:
7039:
7031:
7026:
7015:
7012:
7006:
7004:
6993:
6982:
6975:
6969:
6966:
6962:
6956:
6952:
6948:
6940:
6936:
6927:
6922:
6909:
6905:
6898:
6896:
6888:
6883:
6879:
6868:
6857:
6843:
6837:
6834:
6830:
6824:
6820:
6816:
6808:
6804:
6795:
6790:
6777:
6773:
6766:
6764:
6755:
6744:
6740:
6735:
6717:
6710:
6704:
6700:
6696:
6693:
6687:
6679:
6670:
6664:
6662:
6654:
6639:
6633:
6629:
6625:
6622:
6616:
6608:
6599:
6593:
6591:
6576:
6569:
6562:
6559:
6556:
6550:
6544:
6542:
6519:
6518:
6517:
6500:
6496:
6493:
6489:
6483:
6476:
6473:
6469:
6466:
6463:
6460:
6447:
6441:
6428:
6425:
6421:
6416:
6410:
6407:
6402:
6399:
6393:
6386:
6383:
6378:
6375:
6368:
6365:
6360:
6357:
6350:
6344:
6341:
6336:
6333:
6327:
6321:
6318:
6313:
6310:
6304:
6298:
6295:
6289:
6283:
6280:
6274:
6267:
6264:
6257:
6254:
6243:
6242:
6241:
6239:
6235:
6231:
6226:
6224:
6220:
6216:
6212:
6193:
6190:
6179:
6160:
6157:
6146:
6141:
6136:
6128:
6110:
6093:
6078:
6061:
6044:
6027:
6010:
5993:
5976:
5975:
5974:
5948:
5940:
5936:
5932:
5926:
5915:
5908:
5904:
5901:
5896:
5891:
5887:
5883:
5875:
5871:
5867:
5862:
5857:
5853:
5846:
5844:
5836:
5829:
5826:
5810:
5802:
5798:
5794:
5788:
5774:
5770:
5767:
5762:
5757:
5753:
5749:
5741:
5737:
5733:
5728:
5723:
5719:
5712:
5706:
5701:
5696:
5691:
5686:
5683:
5669:
5667:
5659:
5652:
5649:
5633:
5625:
5621:
5617:
5611:
5591:
5587:
5584:
5579:
5574:
5570:
5566:
5558:
5554:
5550:
5545:
5540:
5536:
5529:
5523:
5518:
5515:
5512:
5506:
5503:
5500:
5496:
5488:
5486:
5478:
5471:
5468:
5452:
5444:
5440:
5436:
5430:
5413:
5409:
5406:
5401:
5396:
5392:
5388:
5380:
5376:
5372:
5367:
5362:
5358:
5351:
5345:
5342:
5339:
5333:
5330:
5327:
5323:
5317:
5315:
5307:
5300:
5297:
5281:
5273:
5269:
5265:
5259:
5245:
5241:
5238:
5233:
5228:
5224:
5220:
5212:
5208:
5204:
5199:
5194:
5190:
5183:
5177:
5172:
5167:
5164:
5154:
5148:
5143:
5138:
5135:
5125:
5119:
5114:
5111:
5108:
5104:
5097:
5095:
5087:
5080:
5077:
5061:
5053:
5049:
5045:
5039:
5022:
5018:
5015:
5010:
5005:
5001:
4997:
4989:
4985:
4981:
4976:
4971:
4967:
4960:
4954:
4949:
4946:
4938:
4932:
4929:
4926:
4923:
4919:
4914:
4912:
4904:
4897:
4894:
4882:
4878:
4874:
4870:
4864:
4857:
4854:
4850:
4847:
4842:
4837:
4833:
4829:
4821:
4817:
4813:
4808:
4803:
4799:
4792:
4786:
4781:
4778:
4770:
4768:
4760:
4757:
4754:
4750:
4740:
4736:
4732:
4728:
4722:
4715:
4712:
4708:
4705:
4700:
4695:
4691:
4687:
4674:
4667:
4664:
4650:
4647:
4643:
4638:
4625:
4622:
4606:
4603:
4599:
4594:
4590:
4584:
4581:
4576:
4573:
4567:
4560:
4557:
4553:
4548:
4540:
4536:
4532:
4527:
4522:
4518:
4511:
4509:
4501:
4498:
4484:
4483:
4482:
4462:
4459:
4456:
4452:
4447:
4441:
4438:
4428:
4427:
4426:
4406:
4403:
4393:
4392:
4391:
4389:
4385:
4381:
4377:
4373:
4369:
4368:
4363:
4359:
4344:
4339:
4324:
4305:
4299:
4294:
4291:
4287:
4279:
4261:
4255:
4250:
4247:
4243:
4235:
4217:
4211:
4206:
4203:
4199:
4191:
4190:
4189:
4168:
4165:
4161:
4155:
4148:
4137:
4135:
4131:
4126:
4124:
4101:
4093:
4089:
4080:
4075:
4064:
4061:
4056:
4053:
4040:
4034:
4027:
4019:
4014:
4003:
4000:
3995:
3992:
3979:
3973:
3966:
3958:
3953:
3942:
3939:
3934:
3931:
3918:
3910:
3906:
3897:
3892:
3881:
3878:
3873:
3870:
3853:
3852:
3851:
3847:
3843:
3839:
3833:
3828:
3824:
3816:
3800:
3795:
3790:
3786:
3778:
3763:
3758:
3753:
3749:
3741:
3726:
3721:
3716:
3712:
3704:
3689:
3684:
3680:
3672:
3671:
3670:
3668:
3664:
3656:
3640:
3635:
3631:
3623:
3608:
3603:
3599:
3591:
3576:
3571:
3567:
3559:
3558:
3557:
3555:
3551:
3547:
3528:
3522:
3519:
3511:
3505:
3502:
3496:
3490:
3481:
3478:
3468:
3467:
3466:
3462:
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3426:
3422:
3418:
3391:
3388:
3384:
3366:
3347:
3336:
3333:
3326:
3318:
3300:
3289:
3286:
3279:
3271:
3253:
3242:
3239:
3232:
3224:
3206:
3198:
3195:
3191:
3183:
3164:
3153:
3150:
3143:
3135:
3134:
3133:
3131:
3112:
3109:
3106:
3101:
3098:
3094:
3091:
3088:
3085:
3078:
3071:
3068:
3064:
3061:
3055:
3049:
3042:
3041:
3040:
3022:
3018:
3013:
3009:
2978:
2975:
2971:
2967:
2959:
2955:
2946:
2941:
2930:
2928:
2914:
2904:
2901:
2897:
2893:
2884:
2878:
2870:
2865:
2854:
2852:
2836:
2826:
2823:
2819:
2815:
2807:
2803:
2794:
2789:
2778:
2776:
2762:
2752:
2748:
2744:
2738:
2730:
2721:
2719:
2708:
2698:
2694:
2690:
2684:
2676:
2667:
2665:
2654:
2642:
2641:
2640:
2638:
2634:
2630:
2608:
2591:
2574:
2557:
2556:
2555:
2538:
2532:
2521:
2514:
2506:
2505:
2504:
2502:
2498:
2479:
2476:
2473:
2467:
2456:
2449:
2445:
2439:
2428:
2421:
2417:
2411:
2400:
2393:
2385:
2384:
2383:
2366:
2363:
2360:
2354:
2337:
2333:
2327:
2313:
2309:
2303:
2286:
2278:
2277:
2276:
2274:
2270:
2251:
2244:
2241:
2232:
2213:
2206:
2203:
2193:
2172:
2165:
2162:
2154:
2153:
2152:
2150:
2146:
2142:
2138:
2134:
2114:
2107:
2096:
2094:
2093:ISO/IEC 80000
2089:
2083:
2079:
2074:
2068:
2065:
2060:
2056:
2037:
2034:
2031:
2025:
2018:
2015:
2011:
2008:
2005:
1998:
1997:
1996:
1979:
1976:
1973:
1970:
1964:
1961:
1956:
1953:
1947:
1944:
1941:
1934:
1933:
1932:
1929:
1923:
1906:
1894:
1889:
1885:
1881:
1877:
1876:differentials
1872:
1866:
1846:
1840:
1837:
1832:
1829:
1823:
1817:
1814:
1809:
1806:
1800:
1794:
1791:
1786:
1783:
1773:
1772:
1771:
1769:
1765:
1742:
1733:
1730:
1725:
1722:
1709:
1706:
1703:
1698:
1692:
1689:
1684:
1681:
1667:
1666:
1665:
1662:
1658:
1652:
1632:
1624:
1620:
1616:
1611:
1606:
1602:
1595:
1592:
1587:
1582:
1576:
1573:
1569:
1564:
1559:
1553:
1550:
1544:
1538:
1535:
1530:
1527:
1521:
1517:
1507:
1506:
1505:
1488:
1480:
1476:
1472:
1467:
1462:
1458:
1451:
1448:
1445:
1437:
1433:
1429:
1424:
1419:
1415:
1408:
1400:
1396:
1392:
1387:
1382:
1378:
1371:
1363:
1359:
1355:
1350:
1345:
1341:
1330:
1329:
1328:
1311:
1305:
1299:
1293:
1290:
1286:
1273:
1270:
1262:
1256:
1253:
1239:
1230:
1227:
1222:
1219:
1209:
1208:
1207:
1204:
1198:
1178:
1172:
1169:
1164:
1161:
1151:
1150:
1149:
1146:
1140:
1136:
1130:
1126:
1122:
1117:
1111:
1090:
1085:
1082:
1075:
1070:
1057:
1055:
1051:
1046:
1042:
1038:
1034:
1030:
1018:
1013:
1011:
1006:
1004:
999:
998:
996:
995:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
954:
946:
945:
938:
935:
933:
930:
928:
925:
923:
920:
919:
911:
910:
899:
896:
894:
891:
889:
886:
885:
884:
883:
873:
872:
861:
858:
856:
853:
851:
848:
846:
843:
841:
840:Line integral
838:
836:
833:
831:
828:
827:
826:
825:
821:
820:
815:
812:
810:
807:
805:
802:
800:
797:
796:
795:
794:
790:
789:
783:
782:Multivariable
777:
776:
765:
762:
760:
757:
755:
752:
750:
747:
745:
742:
740:
737:
736:
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734:
730:
729:
724:
721:
719:
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649:
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635:
633:
630:
628:
625:
623:
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619:
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614:
611:
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605:
602:
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590:
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585:
582:
579:
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572:
571:
570:
569:
563:
557:
556:
545:
542:
540:
537:
535:
532:
530:
527:
524:
520:
517:
515:
512:
509:
505:
501:
500:trigonometric
497:
494:
492:
489:
487:
484:
482:
479:
478:
477:
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471:
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463:
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430:
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425:
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412:
410:
407:
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404:
398:
392:
391:
380:
377:
375:
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367:
365:
362:
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350:
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345:
342:
340:
337:
335:
332:
331:
330:
329:
326:
323:
322:
317:
314:
312:
311:Related rates
309:
307:
304:
302:
299:
297:
294:
292:
289:
288:
287:
286:
282:
281:
274:
271:
269:
268:of a function
266:
264:
263:infinitesimal
261:
260:
259:
256:
253:
249:
246:
245:
244:
243:
239:
238:
232:
226:
225:
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162:
156:
153:
147:
141:
138:
135:
132:
125:
118:
115:
109:
104:
100:
91:
90:
87:
84:
83:
79:
78:
69:
59:
53:
51:
46:
41:
37:
32:
31:
19:
12931:Martin Kutta
12886:Émile Picard
12866:Isaac Newton
12780:Euler method
12750:Substitution
12484:
12378:
12371:
12363:
12359:
12355:
12348:
12341:
12330:. Retrieved
12312:
12301:. Retrieved
12283:
12275:
12267:
12262:
12254:
12249:
12234:
12229:
12218:. Retrieved
12210:"Double Dot"
12204:
12199:
12188:. Retrieved
12174:
12169:
12131:
12130:The dot for
12124:
12120:
12113:
12109:
12100:
12096:
12089:
12085:
12078:
12067:. Retrieved
12049:
12045:
12041:
12040:1st to 7th,
12030:. Retrieved
12012:
12004:
11977:
11967:
11956:. Retrieved
11942:
11937:
11926:. Retrieved
11912:
11891:
11882:
11871:
11863:
11851:
11826:
11819:
11778:
11772:
11761:
11751:
11634:
11627:
11483:product rule
11480:
10800:
10792:
10715:
10706:
10700:
10551:
10493:
10484:
10478:
10231:
10223:
10183:
10174:
10168:
9926:
9861:
9852:
9846:
9838:symbolically
9837:
9835:
9729:
9725:scalar field
9601:vector field
9596:
9586:
9582:
9578:
9575:Assume that
9574:
9572:are common.
9548:
9535:Leibniz rule
9532:
9205:
9040:
8965:
8819:
8814:
8757:
8753:
8749:
8686:
8625:relative to
8602:
8562:
8485:
8481:
8477:
8471:
8467:
8463:
8459:
8450:
8448:
8427:
8419:
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8270:
8268:
8257:
8228:
8220:
8200:
7675:
7510:
7503:
7496:
7493:
7157:
7151:
7144:
7137:
7128:
7122:
7108:
7103:
6515:
6237:
6233:
6229:
6227:
6211:acceleration
6144:
6139:
6132:
6126:
5991:(derivative)
5972:
4480:
4424:
4387:
4383:
4379:
4375:
4371:
4365:
4362:dot notation
4361:
4358:Isaac Newton
4356:
4342:
4337:
4322:
4187:
4166:
4163:
4159:
4153:
4146:
4127:
4120:
3845:
3841:
3837:
3831:
3822:
3820:
3814:
3666:
3662:
3660:
3654:
3553:
3545:
3543:
3460:
3456:
3451:
3446:
3441:
3436:
3429:
3412:
3410:
3389:
3386:
3378:
3367:th integral.
3364:
3129:
3127:
3038:
3020:
3016:
3011:
3007:
2636:
2632:
2628:
2626:
2604:TRIPLE PRIME
2587:DOUBLE PRIME
2572:(derivative)
2553:
2500:
2496:
2494:
2381:
2194:
2191:
2148:
2144:
2130:
2112:
2105:
2087:
2072:
2069:
2063:
2052:
1994:
1927:
1921:
1892:
1890:. Commonly,
1870:
1864:
1861:
1761:
1660:
1656:
1650:
1647:
1503:
1326:
1202:
1196:
1193:
1144:
1138:
1128:
1124:
1120:
1113:
1088:
1083:
1080:
1073:
1068:
1032:
1026:
496:Substitution
290:
258:Differential
231:Differential
63:
47:
45:lead section
12688:Phase space
12546:Homogeneous
11986:Brooks/Cole
11982:Belmont, CA
10228:dot product
9738:and called
9676:, and that
9591:is a given
9558:integration
8436:A function
8237:A function
7133:later works
7131:(1704) and
7125:integration
3550:composition
3434:denoted as
2151:is written
2118:A function
2080:instead of
2059:coefficient
1654:at a point
957:Precalculus
950:Miscellanea
915:Specialized
822:Definitions
589:Alternating
427:Definitions
240:Definitions
12951:Categories
12916:John Crank
12745:Inspection
12608:Stochastic
12602:Difference
12576:Autonomous
12520:Non-linear
12510:Fractional
12473:Operations
12332:2016-02-05
12303:2016-02-05
12220:2016-02-05
12190:2016-02-05
12069:2016-02-05
12032:2016-02-05
11958:2016-02-07
11928:2016-02-07
11744:References
11702:Derivative
10185:Divergence
9734:, written
9206:So-called
6108:(integral)
6091:(integral)
6059:(integral)
3442:D operator
3372:D-notation
2141:prime mark
2078:roman type
1874:(known as
1768:chain rule
1037:derivative
937:Variations
932:Stochastic
922:Fractional
791:Formalisms
754:Divergence
723:Identities
703:Divergence
248:Derivative
199:Continuity
66:March 2023
12720:solutions
12678:Wronskian
12633:Solutions
12561:Decoupled
12525:Holonomic
12398:682907530
12205:MathWorld
12180:"Overdot"
12175:MathWorld
11943:MathWorld
11913:MathWorld
11805:cite book
11797:893974565
11671:Δ
11651:◻
11607:ψ
11604:∇
11598:ϕ
11592:ψ
11586:ϕ
11583:∇
11571:ψ
11568:ϕ
11562:∇
11550:⟹
11455:×
11452:∇
11385:∂
11373:∂
11352:∂
11340:∂
11319:∂
11307:∂
11239:∂
11224:∂
11218:−
11209:∂
11194:∂
11164:∂
11149:∂
11143:−
11134:∂
11119:∂
11089:∂
11074:∂
11068:−
11059:∂
11044:∂
11012:∂
10997:∂
10991:−
10982:∂
10967:∂
10952:∂
10937:∂
10931:−
10922:∂
10907:∂
10892:∂
10877:∂
10871:−
10862:∂
10847:∂
10824:
10676:φ
10673:Δ
10660:φ
10651:∇
10637:φ
10631:∇
10628:⋅
10625:∇
10606:φ
10603:∇
10597:⋅
10594:∇
10584:φ
10581:
10575:
10538:φ
10518:φ
10515:
10509:
10495:Laplacian
10449:⋅
10446:∇
10428:⋅
10414:∂
10410:∂
10396:∂
10392:∂
10378:∂
10374:∂
10348:∂
10333:∂
10318:∂
10303:∂
10288:∂
10273:∂
10255:
10144:φ
10141:∇
10128:φ
10114:∂
10110:∂
10096:∂
10092:∂
10078:∂
10074:∂
10043:∂
10038:φ
10035:∂
10020:∂
10015:φ
10012:∂
9997:∂
9992:φ
9989:∂
9971:φ
9968:
9934:φ
9909:φ
9889:φ
9806:∂
9802:∂
9788:∂
9784:∂
9770:∂
9766:∂
9753:∇
9690:φ
9684:φ
9552:concerns
9504:α
9490:∂
9478:α
9473:∂
9467:⋯
9453:α
9439:∂
9427:α
9422:∂
9408:α
9404:∂
9377:→
9334:≥
9324:∈
9315:α
9296:α
9289:…
9277:α
9267:α
9178:∂
9171:∂
9157:∂
9137:∂
9129:∂
9112:∂
9108:∂
9001:∂
8995:∂
8981:∂
8912:∂
8898:∂
8854:∂
8840:∂
8784:∂
8776:∂
8719:∂
8711:∂
8665:∂
8657:∂
8536:∂
8510:∂
8502:∂
8146:−
8111:∫
8092:−
8082:′
8071:∫
8068:≡
8059:−
8049:′
8038:◻
8018:′
7967:−
7932:∫
7916:′
7913:′
7910:′
7900:∫
7897:≡
7891:′
7888:′
7885:′
7875:◻
7862:′
7859:′
7856:′
7853:′
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7768:∫
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7749:′
7739:∫
7736:≡
7730:′
7727:′
7717:◻
7704:′
7701:′
7698:′
7625:−
7590:∫
7574:′
7564:∫
7561:≡
7555:′
7545:◻
7536:′
7533:′
7439:−
7404:∫
7388:∫
7385:≡
7379:◻
7366:′
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7234:∫
7218:˙
7209:∫
7206:≡
7200:˙
7191:◻
7044:∂
7037:∂
7023:∂
6994:⋅
6983:⋅
6933:∂
6919:∂
6889:⋅
6880:⋅
6858::
6801:∂
6787:∂
6745:⋅
6736:⋅
6718::
6685:∂
6677:∂
6655:⋅
6614:∂
6606:∂
6577:⋅
6305:≡
6299:˙
6284:˙
6268:˙
6258:˙
6194:¨
6161:˙
6119:◌ᷠ
6102:◌⃞
6036:◌⃜
6019:◌⃛
5847:≡
5830:˙
5713:≡
5707:¨
5702:¨
5697:¨
5692:¨
5687:¨
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5530:≡
5524:˙
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5352:≡
5301:˙
5184:≡
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5168:¨
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5139:¨
5120:¨
5081:˙
4961:≡
4955:¨
4950:¨
4898:˙
4793:≡
4787:˙
4782:¨
4626:˙
4512:≡
4502:¨
4442:¨
4407:˙
4292:−
4248:−
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4086:∂
4072:∂
4050:∂
4032:∂
4025:∂
4011:∂
3989:∂
3971:∂
3964:∂
3950:∂
3928:∂
3903:∂
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3867:∂
3556:), as in
3334:−
3287:−
3240:−
3196:−
3151:−
3095:∫
3065:∫
2952:∂
2938:∂
2922:′
2919:′
2882:∂
2876:∂
2862:∂
2846:′
2841:′
2800:∂
2786:∂
2770:′
2767:′
2736:∂
2728:∂
2713:′
2682:∂
2674:∂
2659:′
2617:◌⁗
2600:◌‴
2583:◌″
2566:◌′
2477:…
2364:…
2032:⋅
1971:⋅
1919:, while
1904:Δ
1824:⋅
1449:…
927:Malliavin
814:Geometric
713:Laplacian
663:Dirichlet
574:Geometric
154:−
101:∫
50:summarize
12828:Examples
12718:Integral
12490:Ordinary
12418:Archived
12326:Archived
12297:Archived
12214:Archived
12184:Archived
12063:Archived
12044:th and (
12026:Archived
11952:Archived
11922:Archived
11896:Archived
11868:Lagrange
11687:See also
11663:, or as
11537:′
11520:′
11509:′
10717:Rotation
9863:Gradient
8760:, while
8470:,
8466:,
8453:with a "
7241:′
7164:absement
6497:′
6477:′
6209:denotes
6178:velocity
6176:denotes
6086:▭
6070:◌̎
6053:◌̍
6002:◌̈
5985:◌̇
4883:‴
4858:‴
4741:″
4716:″
4668:′
4367:fluxions
3813:for the
3653:for the
3363:for the
3102:′
3072:′
2635:,
2267:for the
2245:‴
2229:for the
2207:″
2166:′
2019:′
1045:variable
1041:function
967:Glossary
877:Advanced
855:Jacobian
809:Exterior
739:Gradient
731:Theorems
698:Gradient
637:Integral
599:Binomial
584:Harmonic
444:improper
440:Integral
397:Integral
379:Reynolds
354:Quotient
283:Concepts
119:′
86:Calculus
12556:Coupled
12495:Partial
11708:Fluxion
9595:, that
8432:
8416:
7507:
7127:in his
6215:physics
6147:, then
4321:for an
4132:and in
962:History
860:Hessian
749:Stokes'
744:Green's
576: (
498: (
442: (
364:Inverse
339:Product
250: (
12571:Degree
12515:Linear
12396:
12386:
12317:Newton
12288:Newton
12241:
12054:Newton
12017:Newton
11992:
11795:
11785:
11766:: 512.
11639:, the
11559:
11556:
11553:
11547:
11544:
11541:
9562:vector
9311:
7159:fluent
7010:
7000:
6997:
6902:
6892:
6770:
6760:
6668:
6658:
6597:
6587:
6548:
6538:
6115:
6113:U+1DE0
6098:
6096:U+20DE
6083:
6081:U+25AD
6066:
6064:U+030E
6049:
6047:U+030D
6032:
6030:U+20DC
6015:
6013:U+20DB
5998:
5996:U+0308
5981:
5979:U+0307
2891:
2613:
2611:U+2057
2596:
2594:U+2034
2579:
2577:U+2033
2562:
2560:U+2032
2091:. The
2082:italic
804:Tensor
799:Matrix
686:Vector
604:Taylor
562:Series
194:Limits
12620:Delay
12566:Order
11974:"1.1"
10757:, or
9723:is a
9599:is a
8201:This
3850:are:
3821:When
3444:) or
2570:PRIME
2137:Euler
2055:below
2053:(see
1886:, or
1039:of a
627:Ratio
594:Power
508:Euler
486:Discs
481:Parts
349:Power
344:Chain
273:total
12394:OCLC
12384:ISBN
12354:The
12239:ISBN
11990:ISBN
11834:-65.
11811:link
11793:OCLC
11783:ISBN
10821:curl
10578:grad
10512:grad
9965:grad
9556:and
8249:and
6217:and
6180:and
6135:time
4171:The
4121:See
3394:The
2233:and
1142:and
708:Curl
668:Abel
632:Root
10572:div
10506:div
10252:div
9740:del
9564:or
9560:of
8691:of
8603:may
8484:or
8262:or
7166:).
4390:is
3552:of
2122:of
2076:in
2067:).
2061:of
1868:or
1200:at
1052:or
1043:or
1027:In
334:Sum
12953::
12424:).
12392:.
12324:.
12295:.
12212:.
12182:.
12061:.
12037:).
12024:.
11984::
11976:.
11950:.
11920:.
11904:^
11894:.
11890:.
11870:,
11840:^
11832:63
11807:}}
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11791:.
11760:.
10699:∇×
10167:∇∙
9727:.
9585:,
9581:,
9541:.
8428:∂x
8420:∂f
8231:xy
8209:.
7511:y̍
7504:y̍
7497:y̎
7138:y̍
7109:x̎
7104:x̍
6232:=
6225:.
5811:10
5763:10
5742:10
5724:10
5660:10
4364:,
4136:.
4125:.
3844:,
3447:D̃
3222:),
2084::
2064:dx
1928:dx
1922:dy
1893:dx
1871:dy
1865:dx
1659:=
1123:=
1089:dx
1074:dx
1069:dy
506:,
502:,
12451:e
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12437:v
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12360:n
12356:n
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12315:(
12308:)
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12223:.
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12046:n
12042:n
12035:.
12015:(
11998:.
11961:.
11931:.
11854:(
11813:)
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11613:.
11610:)
11601:(
11595:+
11589:)
11580:(
11577:=
11574:)
11565:(
11534:g
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11506:)
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11499:f
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11459:A
11449:=
11437:|
11429:z
11425:A
11417:y
11413:A
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11401:A
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11277:i
11270:|
11265:=
11254:k
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11108:+
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11092:z
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11062:y
11052:z
11048:A
11037:(
11033:=
11022:)
11015:y
11005:x
11001:A
10985:x
10975:y
10971:A
10961:,
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10941:A
10925:z
10915:x
10911:A
10901:,
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10865:y
10855:z
10851:A
10840:(
10836:=
10828:A
10803:,
10801:A
10793:A
10778:A
10772:t
10769:o
10766:r
10744:A
10738:l
10735:r
10732:u
10729:c
10709:.
10707:A
10701:A
10670:=
10655:2
10647:=
10634:)
10622:(
10619:=
10609:)
10600:(
10591:=
10554:,
10552:φ
10487:.
10485:φ
10479:φ
10477:∇
10453:A
10443:=
10432:A
10424:)
10417:z
10405:,
10399:y
10387:,
10381:x
10368:(
10364:=
10351:z
10341:z
10337:A
10327:+
10321:y
10311:y
10307:A
10297:+
10291:x
10281:x
10277:A
10267:=
10259:A
10234:,
10232:A
10224:A
10209:A
10203:v
10200:i
10197:d
10177:.
10175:A
10169:A
10138:=
10124:)
10117:z
10105:,
10099:y
10087:,
10081:x
10068:(
10064:=
10053:)
10046:z
10029:,
10023:y
10006:,
10000:x
9982:(
9978:=
9947:,
9884:d
9881:a
9878:r
9875:g
9855:.
9853:φ
9847:φ
9845:∇
9821:,
9816:)
9809:z
9797:,
9791:y
9779:,
9773:x
9760:(
9756:=
9736:∇
9711:)
9708:z
9705:,
9702:y
9699:,
9696:x
9693:(
9687:=
9664:)
9659:z
9654:A
9649:,
9644:y
9639:A
9634:,
9629:x
9624:A
9619:(
9616:=
9612:A
9597:A
9589:)
9587:z
9583:y
9579:x
9577:(
9518:f
9508:n
9498:n
9494:x
9482:n
9457:1
9447:1
9443:x
9431:1
9416:=
9413:f
9380:X
9372:n
9367:R
9362::
9359:f
9337:0
9329:Z
9319:i
9308:,
9305:)
9300:n
9292:,
9286:,
9281:1
9273:(
9270:=
9247:n
9225:n
9220:R
9187:.
9181:x
9174:y
9166:f
9161:2
9150:=
9146:)
9140:x
9132:f
9123:(
9115:y
9098:,
9093:y
9090:x
9086:f
9082:=
9077:y
9073:)
9067:x
9063:f
9059:(
9026:.
9021:y
9018:x
9014:f
9010:=
9004:x
8998:y
8990:f
8985:2
8947:,
8942:x
8939:x
8936:x
8932:f
8928:=
8920:3
8916:x
8907:f
8902:3
8886:,
8881:x
8878:x
8874:f
8870:=
8862:2
8858:x
8849:f
8844:2
8815:P
8799:P
8793:)
8787:V
8779:T
8770:(
8758:S
8754:V
8750:T
8734:S
8728:)
8722:V
8714:T
8705:(
8668:x
8660:f
8633:x
8613:f
8586:x
8583:d
8578:f
8575:d
8548:.
8545:f
8540:x
8532:=
8527:x
8523:f
8519:=
8513:x
8505:f
8486:z
8482:y
8478:x
8474:)
8472:z
8468:y
8464:x
8462:(
8460:f
8455:∂
8451:d
8444:.
8442:x
8438:f
8424:/
8387:.
8379:2
8375:x
8371:d
8366:f
8361:2
8357:d
8350:=
8341:x
8338:x
8334:f
8323:x
8320:d
8315:f
8312:d
8306:=
8297:x
8293:f
8275:x
8271:f
8253:.
8251:y
8247:x
8243:x
8239:f
8229:f
8223:x
8221:f
8180:n
8176:C
8172:+
8169:)
8166:t
8163:(
8160:S
8157:=
8154:y
8149:n
8141:t
8137:D
8133:=
8130:t
8127:d
8123:)
8120:t
8117:(
8114:s
8108:=
8105:t
8102:d
8095:1
8089:n
8077:y
8062:1
8056:n
8044:y
8035:=
8025:n
8013:y
8001:4
7997:C
7993:+
7990:)
7987:t
7984:(
7981:h
7978:=
7975:y
7970:4
7962:t
7958:D
7954:=
7951:t
7948:d
7944:)
7941:t
7938:(
7935:G
7929:=
7926:t
7923:d
7905:y
7880:y
7872:=
7848:y
7837:3
7833:C
7829:+
7826:)
7823:t
7820:(
7817:G
7814:=
7811:y
7806:3
7798:t
7794:D
7790:=
7787:t
7784:d
7780:)
7777:t
7774:(
7771:g
7765:=
7762:t
7759:d
7744:y
7722:y
7714:=
7693:y
7659:2
7655:C
7651:+
7648:)
7645:t
7642:(
7639:g
7636:=
7633:y
7628:2
7620:t
7616:D
7612:=
7609:t
7606:d
7602:)
7599:t
7596:(
7593:F
7587:=
7584:t
7581:d
7569:y
7550:y
7542:=
7528:y
7502:▭
7473:1
7469:C
7465:+
7462:)
7459:t
7456:(
7453:F
7450:=
7447:y
7442:1
7434:t
7430:D
7426:=
7423:t
7420:d
7416:)
7413:t
7410:(
7407:f
7401:=
7398:t
7395:d
7391:y
7382:y
7376:=
7361:y
7350:0
7346:C
7342:+
7337:t
7333:y
7329:=
7324:0
7320:C
7316:+
7313:)
7310:t
7307:(
7304:f
7301:=
7298:)
7295:y
7290:t
7286:D
7282:(
7277:1
7269:t
7265:D
7261:=
7258:t
7255:d
7251:)
7248:t
7245:(
7238:f
7231:=
7228:t
7225:d
7215:y
7197:y
7188:=
7181:y
7152:y
7145:y
7143:▭
7116:x
7076:,
7070:y
7067:x
7063:f
7059:y
7056:x
7053:=
7047:y
7040:x
7032:f
7027:2
7016:y
7013:x
7007:=
6988:X
6976:,
6970:y
6967:y
6963:f
6957:2
6953:y
6949:=
6941:2
6937:y
6928:f
6923:2
6910:2
6906:y
6899:=
6884:)
6875:X
6869:(
6853:X
6844:,
6838:x
6835:x
6831:f
6825:2
6821:x
6817:=
6809:2
6805:x
6796:f
6791:2
6778:2
6774:x
6767:=
6756:)
6750:X
6741:(
6724:X
6711:,
6705:y
6701:f
6697:y
6694:=
6688:y
6680:f
6671:y
6665:=
6649:X
6640:,
6634:x
6630:f
6626:x
6623:=
6617:x
6609:f
6600:x
6594:=
6582:X
6570:,
6566:)
6563:y
6560:,
6557:x
6554:(
6551:f
6545:=
6533:X
6501:.
6494:y
6490:=
6487:)
6484:x
6481:(
6474:f
6470:=
6467:y
6464:D
6461:=
6456:)
6451:)
6448:x
6445:(
6442:f
6437:(
6429:x
6426:d
6422:d
6417:=
6411:x
6408:d
6403:y
6400:d
6394:=
6387:t
6384:d
6379:x
6376:d
6369:t
6366:d
6361:y
6358:d
6351:=
6345:t
6342:d
6337:x
6334:d
6328::
6322:t
6319:d
6314:y
6311:d
6296:x
6290::
6281:y
6275:=
6265:x
6255:y
6238:x
6236:(
6234:f
6230:y
6191:y
6158:y
6145:t
6140:y
6127:n
6125:(
5952:)
5949:n
5946:(
5941:t
5937:y
5933:=
5930:)
5927:t
5924:(
5919:)
5916:n
5913:(
5909:f
5905:=
5902:y
5897:n
5892:t
5888:D
5884:=
5876:n
5872:t
5868:d
5863:y
5858:n
5854:d
5837:n
5827:y
5814:)
5808:(
5803:t
5799:y
5795:=
5792:)
5789:t
5786:(
5780:X
5775:f
5771:=
5768:y
5758:t
5754:D
5750:=
5738:t
5734:d
5729:y
5720:d
5684:y
5670:=
5650:y
5637:)
5634:7
5631:(
5626:t
5622:y
5618:=
5615:)
5612:t
5609:(
5603:I
5600:I
5597:V
5592:f
5588:=
5585:y
5580:7
5575:t
5571:D
5567:=
5559:7
5555:t
5551:d
5546:y
5541:7
5537:d
5519:.
5516:.
5513:.
5507:.
5504:.
5501:.
5497:y
5489:=
5479:7
5469:y
5456:)
5453:6
5450:(
5445:t
5441:y
5437:=
5434:)
5431:t
5428:(
5422:I
5419:V
5414:f
5410:=
5407:y
5402:6
5397:t
5393:D
5389:=
5381:6
5377:t
5373:d
5368:y
5363:6
5359:d
5346:.
5343:.
5340:.
5334:.
5331:.
5328:.
5324:y
5318:=
5308:6
5298:y
5285:)
5282:5
5279:(
5274:t
5270:y
5266:=
5263:)
5260:t
5257:(
5251:V
5246:f
5242:=
5239:y
5234:5
5229:t
5225:D
5221:=
5213:5
5209:t
5205:d
5200:y
5195:5
5191:d
5165:y
5155:=
5136:y
5126:=
5115:.
5112:.
5109:.
5105:y
5098:=
5088:5
5078:y
5065:)
5062:4
5059:(
5054:t
5050:y
5046:=
5043:)
5040:t
5037:(
5031:V
5028:I
5023:f
5019:=
5016:y
5011:4
5006:t
5002:D
4998:=
4990:4
4986:t
4982:d
4977:y
4972:4
4968:d
4947:y
4939:=
4933:.
4930:.
4927:.
4924:.
4920:y
4915:=
4905:4
4895:y
4879:t
4875:y
4871:=
4868:)
4865:t
4862:(
4855:f
4851:=
4848:y
4843:3
4838:t
4834:D
4830:=
4822:3
4818:t
4814:d
4809:y
4804:3
4800:d
4779:y
4771:=
4761:.
4758:.
4755:.
4751:y
4737:t
4733:y
4729:=
4726:)
4723:t
4720:(
4713:f
4709:=
4706:y
4701:2
4696:t
4692:D
4688:=
4683:)
4678:)
4675:t
4672:(
4665:f
4659:(
4651:t
4648:d
4644:d
4639:=
4634:)
4623:y
4615:(
4607:t
4604:d
4600:d
4595:=
4591:)
4585:t
4582:d
4577:y
4574:d
4568:(
4561:t
4558:d
4554:d
4549:=
4541:2
4537:t
4533:d
4528:y
4523:2
4519:d
4499:y
4463:.
4460:.
4457:.
4453:y
4448:,
4439:y
4404:y
4388:t
4384:y
4380:t
4376:y
4350:x
4343:ẍ
4338:ẋ
4323:n
4309:)
4306:x
4303:(
4300:f
4295:n
4288:D
4265:)
4262:x
4259:(
4256:f
4251:2
4244:D
4221:)
4218:x
4215:(
4212:f
4207:1
4200:D
4181:f
4177:y
4173:x
4167:f
4164:D
4160:y
4154:x
4147:D
4102:.
4094:2
4090:y
4081:f
4076:2
4065:=
4062:f
4057:y
4054:y
4041:,
4035:y
4028:x
4020:f
4015:2
4004:=
4001:f
3996:x
3993:y
3980:,
3974:x
3967:y
3959:f
3954:2
3943:=
3940:f
3935:y
3932:x
3919:,
3911:2
3907:x
3898:f
3893:2
3882:=
3879:f
3874:x
3871:x
3848:)
3846:y
3842:x
3840:(
3838:f
3832:D
3827:∂
3823:f
3815:n
3801:f
3796:n
3791:x
3787:D
3764:f
3759:3
3754:x
3750:D
3727:f
3722:2
3717:x
3713:D
3690:f
3685:x
3681:D
3667:x
3663:f
3655:n
3641:f
3636:n
3632:D
3609:f
3604:3
3600:D
3577:f
3572:2
3568:D
3554:D
3546:D
3529:.
3523:x
3520:d
3515:)
3512:x
3509:(
3506:f
3503:d
3497:=
3494:)
3491:x
3488:(
3485:)
3482:f
3479:D
3476:(
3463:)
3461:x
3459:(
3457:f
3450:(
3440:(
3437:D
3404:f
3400:y
3396:x
3390:f
3387:D
3383:y
3381:x
3379:D
3365:n
3351:)
3348:x
3345:(
3340:)
3337:n
3331:(
3327:f
3304:)
3301:x
3298:(
3293:)
3290:3
3284:(
3280:f
3257:)
3254:x
3251:(
3246:)
3243:2
3237:(
3233:f
3210:)
3207:x
3204:(
3199:1
3192:f
3168:)
3165:x
3162:(
3157:)
3154:1
3148:(
3144:f
3130:f
3113:.
3110:x
3107:d
3099:y
3092:=
3089:x
3086:d
3082:)
3079:x
3076:(
3069:f
3062:=
3059:)
3056:x
3053:(
3050:f
3032:x
3028:f
3023:)
3021:x
3019:(
3017:f
3014:)
3012:x
3010:(
3008:f
2979:y
2976:y
2972:f
2968:=
2960:2
2956:y
2947:f
2942:2
2931:=
2915:f
2905:y
2902:x
2898:f
2894:=
2885:x
2879:y
2871:f
2866:2
2855:=
2837:f
2827:x
2824:x
2820:f
2816:=
2808:2
2804:x
2795:f
2790:2
2779:=
2763:f
2753:y
2749:f
2745:=
2739:y
2731:f
2722:=
2709:f
2699:x
2695:f
2691:=
2685:x
2677:f
2668:=
2655:f
2637:y
2633:x
2631:(
2629:f
2539:.
2536:)
2533:x
2530:(
2525:)
2522:n
2519:(
2515:f
2501:n
2497:n
2480:.
2474:,
2471:)
2468:x
2465:(
2460:)
2457:6
2454:(
2450:f
2446:,
2443:)
2440:x
2437:(
2432:)
2429:5
2426:(
2422:f
2418:,
2415:)
2412:x
2409:(
2404:)
2401:4
2398:(
2394:f
2367:,
2361:,
2358:)
2355:x
2352:(
2346:i
2343:v
2338:f
2334:,
2331:)
2328:x
2325:(
2319:v
2314:f
2310:,
2307:)
2304:x
2301:(
2295:v
2292:i
2287:f
2255:)
2252:x
2249:(
2242:f
2217:)
2214:x
2211:(
2204:f
2188:.
2176:)
2173:x
2170:(
2163:f
2149:x
2145:f
2124:x
2120:f
2115:)
2113:x
2111:(
2109:′
2106:f
2088:x
2086:d
2073:d
2038:x
2035:d
2029:)
2026:x
2023:(
2016:f
2012:=
2009:f
2006:d
1980:,
1977:x
1974:d
1965:x
1962:d
1957:y
1954:d
1948:=
1945:y
1942:d
1907:x
1847:.
1841:x
1838:d
1833:u
1830:d
1818:u
1815:d
1810:y
1807:d
1801:=
1795:x
1792:d
1787:y
1784:d
1758:.
1746:)
1743:a
1740:(
1734:x
1731:d
1726:y
1723:d
1710:a
1707:=
1704:x
1699:|
1693:x
1690:d
1685:y
1682:d
1661:a
1657:x
1651:y
1633:.
1625:2
1621:x
1617:d
1612:y
1607:2
1603:d
1596:=
1593:y
1588:2
1583:)
1577:x
1574:d
1570:d
1565:(
1560:=
1554:x
1551:d
1545:)
1539:x
1536:d
1531:y
1528:d
1522:(
1518:d
1489:.
1481:n
1477:x
1473:d
1468:y
1463:n
1459:d
1452:,
1446:,
1438:4
1434:x
1430:d
1425:y
1420:4
1416:d
1409:,
1401:3
1397:x
1393:d
1388:y
1383:3
1379:d
1372:,
1364:2
1360:x
1356:d
1351:y
1346:2
1342:d
1312:.
1309:)
1306:x
1303:(
1300:f
1294:x
1291:d
1287:d
1274:x
1271:d
1266:)
1263:x
1260:(
1257:f
1254:d
1243:)
1240:x
1237:(
1231:x
1228:d
1223:f
1220:d
1203:x
1197:f
1179:.
1173:x
1170:d
1165:y
1162:d
1145:x
1139:y
1131:)
1129:x
1127:(
1125:f
1121:y
1101:x
1097:y
1084:y
1081:d
1016:e
1009:t
1002:v
580:)
525:)
521:(
510:)
446:)
254:)
166:)
163:a
160:(
157:f
151:)
148:b
145:(
142:f
139:=
136:t
133:d
129:)
126:t
123:(
116:f
110:b
105:a
68:)
64:(
54:.
20:)
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