Differential (infinitesimal)
   HOME

TheInfoList



Click Here for Items Related To - Differential (infinitesimal)
OR:
Differential (infinitesimal) on:   [Wikipedia]   [Google]   [Amazon]

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, differential refers to several related notions derived from the early days of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, put on a rigorous footing, such as
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
differences and the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of functions. The term is used in various branches of mathematics such as
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
,
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.


Introduction

The term differential is used nonrigorously in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
to refer to an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
("infinitely small") change in some varying quantity. For example, if ''x'' is a variable, then a change in the value of ''x'' is often denoted Δ''x'' (pronounced '' delta x''). The differential ''dx'' represents an infinitely small change in the variable ''x''. 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 using
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s. If ''y'' is a function of ''x'', then the differential ''dy'' of ''y'' is related to ''dx'' by the formula dy = \frac \,dx, where \frac \,denotes the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of ''y'' with respect to ''x''. This formula summarizes the intuitive idea that the derivative of ''y'' with respect to ''x'' is the limit of the ratio of differences Δ''y''/Δ''x'' as Δ''x'' becomes infinitesimal. There are several approaches for making the notion of differentials mathematically precise. # Differentials as
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s. This approach underlies the definition of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
and the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
. # Differentials as equivalence classes of germs of functions # Differentials as
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
elements of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s. This approach is popular in algebraic geometry.. # Differentials in smooth models of set theory. This approach is known as
synthetic differential geometry In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic ...
or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from
topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
are used to ''hide'' the mechanisms by which nilpotent infinitesimals are introduced. # Differentials as infinitesimals in
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach of
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
pioneered by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
.See and . These approaches are very different from each other, but they have in common the idea of being ''quantitative'', i.e., saying not just that a differential is infinitely small, but ''how'' small it is.


Basic notions

* In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the differential represents a change in the
linearization In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linea ...
of a function. ** The total differential is its generalization for functions of multiple variables. * In traditional approaches to calculus, the differentials (e.g. ''dx'', ''dy'', ''dt'', etc.) are interpreted as
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s. 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. * The differential is another name for the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of a function from R''n'' to R''m'' (especially when this
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
is viewed as a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
). * More generally, the differential or '' pushforward'' refers to the derivative of a map between
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s and the pushforward operations it defines. The differential is also used to define the dual concept of
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
. *
Stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
provides a notion of stochastic differential and an associated calculus for
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
es. * The
integrator An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an importan ...
in a
Stieltjes integral Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at ...
is represented as the differential of a function. Formally, the differential appearing under the integral behaves exactly as a differential: thus, the
integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and ...
and
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
formulae for Stieltjes integral correspond, respectively, to the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
and
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
for the differential.


History and usage

Infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
quantities played a significant role in the development of calculus.
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
used them, even though he didn't believe that arguments involving infinitesimals were rigorous.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
referred to them as fluxions. However, it was
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
who coined the term ''differentials'' for infinitesimal quantities and introduced the notation for them which is still used today. In
Leibniz's notation In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ...
, if ''x'' is a variable quantity, then ''dx'' denotes an infinitesimal change in the variable ''x''. Thus, if ''y'' is a function of ''x'', then the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of ''y'' with respect to ''x'' is often denoted ''dy''/''dx'', which would otherwise be denoted (in the notation of Newton or
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaThe Analyst by Bishop Berkeley. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative of ''y'' at ''x'' is its
instantaneous rate of change In physics and the philosophy of science, instant refers to an infinitesimal interval in time, whose passage is instantaneous. In ordinary speech, an instant has been defined as "a point or very short space of time," a notion deriving from its ...
(the
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the graph's
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
), which may be obtained by taking the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the ratio Δ''y''/Δ''x'' of the change in ''y'' over the change in ''x'', as the change in ''x'' becomes arbitrarily small. Differentials are also compatible with
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
, where a differential such as ''dx'' has the same dimensions as the variable ''x''. 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 in
Bishop Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immater ...
's 1734 The 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 the Epsilon, 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, especially ''differential''; both differential and infinitesimal are used with new, more rigorous, meanings. Differentials are also used in the notation for
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s 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 \int f(x) \,dx, the integral sign (which is a modified
long s The long s , also known as the medial s or initial s, is an archaic form of the lowercase letter . It replaced the single ''s'', or one or both of the letters ''s'' in a 'double ''s sequence (e.g., "ſinfulneſs" for "sinfulness" and "po ...
) denotes the infinite sum, ''f''(''x'') denotes the "height" of a thin strip, and the differential ''dx'' denotes its infinitely thin width.


Approaches


Naive approach

Some texts for primary and undergraduate students use the old naïve approach and nomenclature rather than giving rigorous axioms, definitions and basic results. This approach to
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
uses the term differential to refer to an "
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
" ("infinitely small") change in some varying quantity. For example, if ''x'' is a variable, then a change in the value of ''x'' is often denoted Δ''x'' (pronounced '' delta x''). The differential ''dx'' represents an infinitely small change in the variable ''x''. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, except when it confuses students who notice the inconsistencies. 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 using
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s. If ''y'' is a function of ''x'', then the differential ''dy'' of ''y'' is related to ''dx'' by the formula dy = \frac \,dx, where \frac \,denotes the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of ''y'' with respect to ''x''. This formula summarizes the intuitive idea that the derivative of ''y'' with respect to ''x'' is the ratio of differences \Delta y/\Delta x when \Delta x is infinitesimal. In a more rigorous approach it is the limit of the ratio of differences \Delta y/\Delta x as \Delta x approaches 0. There are several approaches for making the notion of differentials mathematically precise. # Differentials as
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s. This approach underlies the definition of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
and the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
. # Differentials as
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
elements of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s. This approach is popular in algebraic geometry.. # Differentials in smooth models of set theory. This approach is known as
synthetic differential geometry In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic ...
or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from
topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
are used to ''hide'' the mechanisms by which nilpotent infinitesimals are introduced. # Differentials as infinitesimals in
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach of
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
pioneered by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
.See and . These approaches are very different from each other, but they have in common the idea of being ''quantitative'', i.e., saying not just that a differential is infinitely small, but ''how'' small it is.


Differentials as linear maps

There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s. It can be used on \mathbb, \mathbb^n, a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, or more generally, a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. The case of the Real line is the easiest to explain. This type of differential is also known as a covariant vector or
cotangent vector In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
, depending on context.


Differentials as linear maps on R

Suppose f(x) is a real-valued function on \mathbb. We can reinterpret the variable x in f(x) as being a function rather than a number, namely the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on the real line, which takes a real number p to itself: x(p)=p. Then f(x) is the composite of f with x, whose value at p is f(x(p))=f(p). The differential \operatornamef (which of course depends on f) is then a function whose value at p (usually denoted df_p) is not a number, but a linear map from \mathbb to \mathbb. Since a linear map from \mathbb to \mathbb is given by a 1\times 1
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, it is essentially the same thing as a number, but the change in the point of view allows us to think of df_p as an infinitesimal and ''compare'' it with the ''standard infinitesimal'' dx_p, which is again just the identity map from \mathbb to \mathbb (a 1\times 1
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
with entry 1). The identity map has the property that if \varepsilon is very small, then dx_p(\varepsilon) is very small, which enables us to regard it as infinitesimal. The differential df_p has the same property, because it is just a multiple of dx_p, and this multiple is the derivative f'(p) by definition. We therefore obtain that df_p=f'(p)\,dx_p, and hence df=f'\,dx. Thus we recover the idea that f' is the ratio of the differentials df and dx. This would just be a trick were it not for the fact that: # it captures the idea of the derivative of f at p as the ''best linear approximation'' to f at p; # it has many generalizations.


Differentials as linear maps on Rn

If f is a function from \mathbb^n to \mathbb, then we say that f is ''differentiable'' at p\in\mathbb^n if there is a linear map df_p from \mathbb^n to \mathbb such that for any \varepsilon>0, there is a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
N of p such that for x\in N, \left, f(x) - f(p) - df_p(x-p)\ < \varepsilon \left, x-p\ . We can now use the same trick as in the one-dimensional case and think of the expression f(x_1, x_2, \ldots, x_n) as the composite of f with the standard coordinates x_1, x_2, \ldots, x_n on \mathbb^n (so that x_j(p) is the j-th component of p\in\mathbb^n). Then the differentials \left(dx_1\right)_p, \left(dx_2\right)_p, \ldots, \left(dx_n\right)_p at a point p form a basis for the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of linear maps from \mathbb^n to \mathbb and therefore, if f is differentiable at p, we can write ''\operatornamef_p'' as a linear combination of these basis elements: df_p = \sum_^n D_j f(p) \,(dx_j)_p. The coefficients D_j f(p) are (by definition) the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of f at p with respect to x_1, x_2, \ldots, x_n. Hence, if f is differentiable on all of \mathbb^n, we can write, more concisely: \operatornamef = \frac \,dx_1 + \frac \,dx_2 + \cdots +\frac \,dx_n. In the one-dimensional case this becomes df = \fracdx as before. This idea generalizes straightforwardly to functions from \mathbb^n to \mathbb^m. Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. This means that the same idea can be used to define the differential of
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s between
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. Aside: Note that the existence of all the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of f(x) at x is a
necessary condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
for the existence of a differential at x. However it is not a
sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
. For counterexamples, see
Gateaux derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
.


Differentials as linear maps on a vector space

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 a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, where the inner product and its associated norm define a suitable concept of distance. The same procedure works for a Banach space, also known as a complete
Normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
. 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 \mathbb^n.


Differentials as germs of functions

This approach works on any
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. If # and are open sets containing # f\colon U\to \mathbb is continuous # g\colon V\to \mathbb is continuous then is equivalent to at , denoted f \sim_p g, if and only if there is an open W \subseteq U \cap V containing such that f(x) = g(x) for every in . The germ of at , denoted p, is the set of all real continuous functions equivalent to at ; if is smooth at then p is a smooth germ. If #U_1, U_2 V_1 and V_2 are open sets containing #f_1\colon U_1\to \mathbb, f_2\colon U_2\to \mathbb, g_1\colon V_1\to \mathbb and g_2\colon V_2\to \mathbb are smooth functions #f_1 \sim_p g_1 #f_2 \sim_p g_2 # is a real number then #r*f_1 \sim_p r*g_1 #f_1+f_2\colon U_1 \cap U_2\to \mathbb \sim_p g_1+g_2\colon V_1 \cap V_2\to \mathbb #f_1*f_2\colon U_1 \cap U_2\to \mathbb \sim_p g_1*g_2\colon V_1 \cap V_2\to \mathbb This shows that the germs at p form an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. Define \mathcal_p to be the set of all smooth germs vanishing at and \mathcal_p^2 to be the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of ideals \mathcal_p \mathcal_p. Then a differential at (cotangent vector at ) is an element of \mathcal_p/\mathcal_p^2. The differential of a smooth function at , denoted \mathrm d f_p, is -f(p)p/\mathcal_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 of at is the set of all functions differentially equivalent to f-f(p) at .


Algebraic geometry

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
or
structure sheaf In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
of a space may contain
nilpotent element In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s. The simplest example is the ring of
dual number In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Du ...
s R 'ε'' where ''ε''2 = 0. This can be motivated by the algebro-geometric point of view on the derivative of a function ''f'' from R to R at a point ''p''. For this, note first that ''f'' − ''f''(''p'') belongs to the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
''I''''p'' of functions on R which vanish at ''p''. If the derivative ''f'' vanishes at ''p'', then ''f'' − ''f''(''p'') belongs to the square ''I''''p''2 of this ideal. Hence the derivative of ''f'' at ''p'' may be captured by the equivalence class 'f'' − ''f''(''p'')in the quotient space ''I''''p''/''I''''p''2, and the 1-jet of ''f'' (which encodes its value and its first derivative) is the equivalence class of ''f'' in the space of all functions modulo ''I''''p''2. Algebraic geometers regard this equivalence class as the ''restriction'' of ''f'' to a ''thickened'' version of the point ''p'' whose coordinate ring is not R (which is the quotient space of functions on R modulo ''I''''p'') but R 'ε''which is the quotient space of functions on R modulo ''I''''p''2. Such a thickened point is a simple example of a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
.


Algebraic geometry notions

Differentials are also important in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, and there are several important notions. *
Abelian differential In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential ...
s usually mean differential one-forms on an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
or
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
. * Quadratic differentials (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces. *
Kähler differential In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic ...
s provide a general notion of differential in algebraic geometry.


Synthetic differential geometry

A fifth approach to infinitesimals is the method of
synthetic differential geometry In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic ...
or smooth infinitesimal analysis.See and . 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 the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
with another
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of ''smoothly varying sets'' which is a
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
. In this category, one can define the real numbers, smooth functions, and so on, but the real numbers ''automatically'' contain nilpotent infinitesimals, so these do not need to be introduced by hand as in the algebraic geometric approach. However the
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
in this new category is not identical to the familiar logic of the category of sets: in particular, the
law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are ''
constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...
'' (e.g., do not use
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...
). Some regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.


Nonstandard analysis

The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way. In the
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
s of infinitely large numbers. Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, so that, for example, the sequence (1, 1/2, 1/3, ..., 1/''n'', ...) represents an infinitesimal. The
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
of this new set of
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s is the same as the logic for the usual real numbers, but the
completeness axiom Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number l ...
(which involves
second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle.


Differential geometry

The notion of a differential motivates several concepts in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
(and
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
). *The differential (Pushforward) of a map between manifolds. *
Differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s provide a framework which accommodates multiplication and differentiation of differentials. *The
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
is a notion of differentiation of differential forms which generalizes the differential of a function (which is a
differential 1-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
). *
Pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
is, in particular, a geometric name for the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
for composing a map between manifolds with a differential form on the target manifold. * Covariant derivatives or differentials provide a general notion for differentiating of vector fields and
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
s on a manifold, or, more generally, sections of a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
: see Connection (vector bundle). This ultimately leads to the general concept of a connection.


Other meanings

The term ''differential'' has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in a
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
(C_\bullet, d_\bullet), the maps (or ''coboundary operators'') ''di'' are often called differentials. Dually, the boundary operators in a chain complex are sometimes called ''codifferentials''. The properties of the differential also motivate the algebraic notions of a ''
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
'' and a ''
differential algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
''.


See also

*
Differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
*
Differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
*
Differential of a function In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is th ...


Notes


Citations


References

* . * . * . * . * * . * . * . * . * . * Mathematical terminology Differential calculus {{Infinitesimals Calculus