Inmathematics,differential refers to several related notions^[1] derived from the early days ofcalculus, put on a rigorous footing, such asinfinitesimal differences and thederivatives of functions.^[2]

The term is used in various branches of mathematics such ascalculus,differential geometry,algebraic geometry andalgebraic topology.

The termdifferential is used nonrigorously incalculus to refer to aninfinitesimal ("infinitely small") change in somevarying quantity. For example, ifx is avariable, then a change in the value ofx is often denoted Δx (pronounceddelta x). The differentialdx represents an infinitely small change in the variablex. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise.

Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically usingderivatives. Ify is a function ofx, then the differentialdy ofy is related todx by the formula$${\displaystyle dy=\frac{dy}{dx}dx,}$$where${\displaystyle \frac{dy}{dx}}$denotes not 'dy divided by dx' as one would intuitively read, but 'thederivative ofy with respect tox '. This formula summarizes the idea that the derivative ofy with respect tox is the limit of the ratio of differences Δy/Δx as Δx approaches zero.

• Incalculus, thedifferential represents a change in thelinearization of afunction.
  • Thetotal differential is its generalization for functions of multiple variables.
• In traditional approaches to calculus, differentials (e.g.dx,dy,dt, etc.) are interpreted asinfinitesimals. There are several methods of defining infinitesimals rigorously, but it is sufficient to say that an infinitesimal number is smaller in absolute value than any positive real number, just as an infinitely large number is larger than any real number.
• Thedifferential is another name for theJacobian matrix ofpartial derivatives of a function fromRⁿ toR^m (especially when thismatrix is viewed as alinear map).
• More generally, thedifferential orpushforward refers to the derivative of a map betweensmooth manifolds and the pushforward operations it defines. The differential is also used to define the dual concept ofpullback.
• Stochastic calculus provides a notion ofstochastic differential and an associated calculus forstochastic processes.
• Theintegrator in aStieltjes integral is represented as the differential of a function. Formally, the differential appearing under the integral behaves exactly as a differential: thus, theintegration by substitution andintegration by parts formulae for Stieltjes integral correspond, respectively, to thechain rule andproduct rule for the differential.

Infinitesimal quantities played a significant role in the development of calculus.Archimedes used them, even though he did not believe that arguments involving infinitesimals were rigorous.^[3]Isaac Newton referred to them asfluxions. However, it wasGottfried Leibniz who coined the termdifferentials for infinitesimal quantities and introduced the notation for them which is still used today.

InLeibniz's notation, ifx is a variable quantity, thendx denotes an infinitesimal change in the variablex. Thus, ify is a function ofx, then thederivative ofy with respect tox is often denoteddy/dx, which would otherwise be denoted (in the notation of Newton orLagrange)ẏ ory′. The use of differentials in this form attracted much criticism, for instance in the famous pamphletThe Analyst by Bishop Berkeley. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative ofy atx is itsinstantaneous rate of change (theslope of the graph'stangent line), which may be obtained by taking thelimit of the ratio Δy/Δx as Δx becomes arbitrarily small. Differentials are also compatible withdimensional analysis, where a differential such asdx has the same dimensions as the variablex.

Calculus evolved into a distinct branch of mathematics during the 17th century CE, although there were antecedents going back to antiquity. The presentations of, e.g., Newton, Leibniz, were marked by non-rigorous definitions of terms like differential,fluent and "infinitely small". While many of the arguments inBishop Berkeley's 1734The Analyst are theological in nature, modern mathematicians acknowledge the validity of his argument against "the Ghosts of departed Quantities"; however, the modern approaches do not have the same technical issues. Despite the lack of rigor, immense progress was made in the 17th and 18th centuries. In the 19th century, Cauchy and others gradually developed theEpsilon, delta approach to continuity, limits and derivatives, giving a solid conceptual foundation for calculus.

In the 20th century, several new concepts in, e.g., multivariable calculus, differential geometry, seemed to encapsulate the intent of the old terms, especiallydifferential; both differential and infinitesimal are used with new, more rigorous, meanings.

Differentials are also used in the notation forintegrals because an integral can be regarded as an infinite sum of infinitesimal quantities: the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as$${\displaystyle \int f(x)dx,}$$the integral sign (which is a modifiedlong s) denotes the infinite sum,f(x) denotes the "height" of a thin strip, and the differentialdx denotes its infinitely thin width.

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There are several approaches for making the notion of differentials mathematically precise.

1. Differentials aslinear maps. This approach underlies the definition of thederivative and theexterior derivative indifferential geometry.^[4]
2. Differentials asnilpotent elements ofcommutative rings. This approach is popular in algebraic geometry.^[5]
3. Differentials in smooth models of set theory. This approach is known assynthetic differential geometry orsmooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas fromtopos theory are used tohide the mechanisms by which nilpotent infinitesimals are introduced.^[6]
4. Differentials as infinitesimals inhyperreal number systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach ofnonstandard analysis pioneered byAbraham Robinson.^[7]

These approaches are very different from each other, but they have in common the idea of beingquantitative, i.e., saying not just that a differential is infinitely small, buthow small it is.

There is a simple way to make precise sense of differentials, first used on the Real line by regarding them aslinear maps. It can be used on${\displaystyle \mathbb{R}}$,${\displaystyle {\mathbb{R}}^n}$, aHilbert space, aBanach space, or more generally, atopological vector space. The case of the Real line is the easiest to explain. This type of differential is also known as acovariant vector orcotangent vector, depending on context.

Suppose${\displaystyle f(x)}$ is a real-valued function on${\displaystyle \mathbb{R}}$. We can reinterpret the variable${\displaystyle x}$ in${\displaystyle f(x)}$ as being a function rather than a number, namely theidentity map on the real line, which takes a real number${\displaystyle p}$ to itself:${\displaystyle x(p)=p}$. Then${\displaystyle f(x)}$ is the composite of${\displaystyle f}$ with${\displaystyle x}$, whose value at${\displaystyle p}$ is${\displaystyle f(x(p))=f(p)}$. The differential${\displaystyle \mathrm{d}f}$ (which of course depends on${\displaystyle f}$) is then a function whose value at${\displaystyle p}$ (usually denoted${\displaystyle d{f}_p}$) is not a number, but a linear map from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$. Since a linear map from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$ is given by a${\displaystyle 1\times 1}$matrix, it is essentially the same thing as a number, but the change in the point of view allows us to think of${\displaystyle d{f}_p}$ as an infinitesimal andcompare it with thestandard infinitesimal${\displaystyle d{x}_p}$, which is again just the identity map from${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$ (a${\displaystyle 1\times 1}$matrix with entry${\displaystyle 1}$). The identity map has the property that if${\displaystyle \epsilon }$ is very small, then${\displaystyle d{x}_p(\epsilon )}$ is very small, which enables us to regard it as infinitesimal. The differential${\displaystyle d{f}_p}$ has the same property, because it is just a multiple of${\displaystyle d{x}_p}$, and this multiple is the derivative${\displaystyle f^{\prime }(p)}$ by definition. We therefore obtain that${\displaystyle d{f}_p=f^{\prime }(p)d{x}_p}$, and hence${\displaystyle df=f^{\prime }dx}$. Thus we recover the idea that${\displaystyle f^{\prime }}$ is the ratio of the differentials${\displaystyle df}$ and${\displaystyle dx}$.

This would just be a trick were it not for the fact that:

1. it captures the idea of the derivative of${\displaystyle f}$ at${\displaystyle p}$ as thebest linear approximation to${\displaystyle f}$ at${\displaystyle p}$;
2. it has many generalizations.

If${\displaystyle f}$ is a function from${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle \mathbb{R}}$, then we say that${\displaystyle f}$ isdifferentiable^[8] at${\displaystyle p\in {\mathbb{R}}^n}$ if there is a linear map${\displaystyle d{f}_p}$ from${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle \mathbb{R}}$ such that for any${\displaystyle \epsilon >0}$, there is aneighbourhood${\displaystyle N}$ of${\displaystyle p}$ such that for${\displaystyle x\in N}$,

We can now use the same trick as in the one-dimensional case and think of the expression${\displaystyle f(x_1,x_2,\dots ,x_n)}$ as the composite of${\displaystyle f}$ with the standard coordinates${\displaystyle x_1,x_2,\dots ,x_n}$ on${\displaystyle {\mathbb{R}}^n}$ (so that${\displaystyle x_j(p)}$ is the${\displaystyle j}$-th component of${\displaystyle p\in {\mathbb{R}}^n}$). Then the differentials${\displaystyle {\left(d{x}_1\right)}_p,{\left(d{x}_2\right)}_p,\dots ,{\left(d{x}_n\right)}_p}$ at a point${\displaystyle p}$ form abasis for thevector space of linear maps from${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle \mathbb{R}}$ and therefore, if${\displaystyle f}$ is differentiable at${\displaystyle p}$, we can writedf_p as alinear combination of these basis elements:$${\displaystyle d{f}_p=\sum _{j=1}^n{D}_{j}f(p)(d{x}_j{)}_p.}$$

The coefficients${\displaystyle D_{j}f(p)}$ are (by definition) thepartial derivatives of${\displaystyle f}$ at${\displaystyle p}$ with respect to${\displaystyle x_1,x_2,\dots ,x_n}$. Hence, if${\displaystyle f}$ is differentiable on all of${\displaystyle {\mathbb{R}}^n}$, we can write, more concisely:$${\displaystyle df=\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_1}d{x}_1+\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_2}d{x}_2+\cdots +\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_n}d{x}_n.}$$

In the one-dimensional case this becomes$${\displaystyle df=\frac{df}{dx}dx}$$as before.

This idea generalizes straightforwardly to functions from${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle {\mathbb{R}}^m}$. Furthermore, it has the decisive advantage over other definitions of the derivative that it isinvariant under changes of coordinates. This means that the same idea can be used to define thedifferential ofsmooth maps betweensmooth manifolds.

Aside: Note that the existence of all thepartial derivatives of${\displaystyle f(x)}$ at${\displaystyle x}$ is anecessary condition for the existence of a differential at${\displaystyle x}$. However it is not asufficient condition. For counterexamples, seeGateaux derivative.

The same procedure works on a vector space with a enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as acompleteinner product space, where the inner product and its associatednorm define a suitable concept of distance. The same procedure works for a Banach space, also known as a completeNormed vector space. However, for a more general topological vector space, some of the details are more abstract because there is no concept of distance.

For the important case of a finite dimension, any inner product space is a Hilbert space, any normed vector space is a Banach space and any topological vector space is complete. As a result, you can define a coordinate system from an arbitrary basis and use the same technique as for${\displaystyle {\mathbb{R}}^n}$.

This approach works on anydifferentiable manifold. If

1. U andV are open sets containingp
2. ${\displaystyle f:U\to \mathbb{R}}$ is continuous
3. ${\displaystyle g:V\to \mathbb{R}}$ is continuous

thenf is equivalent tog atp, denoted${\displaystyle f{\sim }_{p}g}$, if and only ifthere is an open${\displaystyle W\subseteq U\cap V}$ containingp such that${\displaystyle f(x)=g(x)}$ for everyx inW.The germ off atp, denoted${\displaystyle [f{]}_p}$, is the set of all real continuous functions equivalent tof atp; iff is smooth atp then${\displaystyle [f{]}_p}$ is a smooth germ.If

1. ${\displaystyle U_1}$,${\displaystyle U_2}$${\displaystyle V_1}$ and${\displaystyle V_2}$ are open sets containingp
2. ${\displaystyle f_1:U_1\to \mathbb{R}}$,${\displaystyle f_2:U_2\to \mathbb{R}}$,${\displaystyle g_1:V_1\to \mathbb{R}}$ and${\displaystyle g_2:V_2\to \mathbb{R}}$ are smooth functions
3. ${\displaystyle f_1{\sim }_p{g}_1}$
4. ${\displaystyle f_2{\sim }_p{g}_2}$
5. r is a real number

then

1. ${\displaystyle r\ast f_1{\sim }_{p}r\ast g_1}$
2. ${\displaystyle f_1+f_2:U_1\cap U_2\to \mathbb{R}{\sim }_p{g}_1+g_2:V_1\cap V_2\to \mathbb{R}}$
3. ${\displaystyle f_1\ast f_2:U_1\cap U_2\to \mathbb{R}{\sim }_p{g}_1\ast g_2:V_1\cap V_2\to \mathbb{R}}$

This shows that the germs at p form analgebra.

Define${\displaystyle {\mathcal{I}}_p}$ to be the set of all smooth germs vanishing atp and${\displaystyle {\mathcal{I}}_p^2}$ to be theproduct ofideals${\displaystyle {\mathcal{I}}_p{\mathcal{I}}_p}$. Then a differential atp (cotangent vector atp) is an element of${\displaystyle {\mathcal{I}}_p/{\mathcal{I}}_p^2}$. The differential of a smooth functionf atp, denoted${\displaystyle \mathrm{d}{f}_p}$, is${\displaystyle [f-f(p){]}_p/{\mathcal{I}}_p^2}$.

A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch.Then the differential off atp is the set of all functions differentially equivalent to${\displaystyle f-f(p)}$ atp.

Inalgebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that thecoordinate ring orstructure sheaf of a space may containnilpotent elements. The simplest example is the ring ofdual numbersR[ε], whereε² = 0.

This can be motivated by the algebro-geometric point of view on the derivative of a functionf fromR toR at a pointp. For this, note first thatf −f(p) belongs to theidealI_p of functions onR which vanish atp. If the derivativef vanishes atp, thenf −f(p) belongs to the squareI_p² of this ideal. Hence the derivative off atp may be captured by the equivalence class [f −f(p)] in thequotient spaceI_p/I_p², and the1-jet off (which encodes its value and its first derivative) is the equivalence class off in the space of all functions moduloI_p². Algebraic geometers regard this equivalence class as therestriction off to athickened version of the pointp whose coordinate ring is notR (which is the quotient space of functions onR moduloI_p) butR[ε] which is the quotient space of functions onR moduloI_p². Such a thickened point is a simple example of ascheme.^[5]

Differentials are also important inalgebraic geometry, and there are several important notions.

• Abelian differentials usually mean differential one-forms on analgebraic curve orRiemann surface.
• Quadratic differentials (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
• Kähler differentials provide a general notion of differential in algebraic geometry.

A fifth approach to infinitesimals is the method ofsynthetic differential geometry^[9] orsmooth infinitesimal analysis.^[10] This is closely related to the algebraic-geometric approach, except that the infinitesimals are more implicit and intuitive. The main idea of this approach is to replace thecategory of sets with anothercategory ofsmoothly varying sets which is atopos. In this category, one can define the real numbers, smooth functions, and so on, but the real numbersautomatically contain nilpotent infinitesimals, so these do not need to be introduced by hand as in the algebraic geometric approach. However thelogic in this new category is not identical to the familiar logic of the category of sets: in particular, thelaw of the excluded middle does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they areconstructive (e.g., do not useproof by contradiction).Constructivists regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.

The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way. In thenonstandard analysis approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as thereciprocals of infinitely large numbers.^[7] Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences ofreal numbers, so that, for example, the sequence (1, 1/2, 1/3, ..., 1/n, ...) represents an infinitesimal. Thefirst-order logic of this new set ofhyperreal numbers is the same as the logic for the usual real numbers, but thecompleteness axiom (which involvessecond-order logic) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, seetransfer principle.

The notion of a differential motivates several concepts indifferential geometry (anddifferential topology).

• Thedifferential (Pushforward) of a map between manifolds.
• Differential forms provide a framework which accommodates multiplication and differentiation of differentials.
• Theexterior derivative is a notion of differentiation of differential forms which generalizes thedifferential of a function (which is adifferential 1-form).
• Pullback is, in particular, a geometric name for thechain rule for composing a map between manifolds with a differential form on the target manifold.
• Covariant derivatives or differentials provide a general notion for differentiating ofvector fields andtensor fields on a manifold, or, more generally, sections of avector bundle: seeConnection (vector bundle). This ultimately leads to the general concept of aconnection.

The termdifferential has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in acochain complex${\displaystyle (C_{\bullet },d_{\bullet }),}$ the maps (orcoboundary operators)d_i are often called differentials. Dually, the boundary operators in a chain complex are sometimes calledcodifferentials.

The properties of the differential also motivate the algebraic notions of aderivation and adifferential algebra.

• Differential equation
• Differential form
• Differential of a function

• Apostol, Tom M. (1967),Calculus (2nd ed.), Wiley,ISBN 978-0-471-00005-1.
• Bell, John L. (1998),Invitation to Smooth Infinitesimal Analysis(PDF).
• Boyer, Carl B. (1991), "Archimedes of Syracuse",A History of Mathematics (2nd ed.), John Wiley & Sons, Inc.,ISBN 978-0-471-54397-8.
• Darling, R. W. R. (1994),Differential forms and connections, Cambridge, UK: Cambridge University Press,Bibcode:1994dfc..book.....D,ISBN 978-0-521-46800-8.
• Eisenbud, David;Harris, Joe (1998),The Geometry of Schemes, Springer-Verlag,ISBN 978-0-387-98637-1
• Keisler, H. Jerome (1986),Elementary Calculus: An Infinitesimal Approach (2nd ed.).
• Kock, Anders (2006),Synthetic Differential Geometry(PDF) (2nd ed.), Cambridge University Press.
• Lawvere, F.W. (1968),Outline of synthetic differential geometry(PDF) (published 1998).
• Moerdijk, I.; Reyes, Gonzalo E. (1991),Models for Smooth Infinitesimal Analysis, Springer-Verlag,ISBN 978-1-441-93095-8.
• Robinson, Abraham (1996),Non-standard analysis,Princeton University Press,ISBN 978-0-691-04490-3.
• Weisstein, Eric W."Differentials".MathWorld.

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