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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 the
principal part In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_ ...
of the change in a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
''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 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 ''f'' with respect to ''x'', and ''dx'' is an additional real variable (so that ''dy'' is a function of ''x'' and ''dx''). The notation is such that the equation :dy = \frac\, dx holds, where the derivative is represented in the
Leibniz 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 ...
''dy''/''dx'', and this is consistent with regarding the derivative as the quotient of the differentials. One also writes :df(x) = f'(x)\,dx. The precise meaning of the variables ''dy'' and ''dx'' depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular
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 ...
, or analytical significance if the differential is regarded as a
linear approximation In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving o ...
to the increment of a function. Traditionally, the variables ''dx'' and ''dy'' are considered to be very small (
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 ...
), and this interpretation is made rigorous in
non-standard 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 epsilon–delta ...
.


History and usage

The differential was first introduced via an intuitive or heuristic definition by
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 ...
and furthered by
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 thought of the differential ''dy'' as an infinitely small (or
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 ...
) change in the value ''y'' of the function, corresponding to an infinitely small change ''dx'' in the function's argument ''x''. For that reason, the instantaneous rate of change of ''y'' with respect to ''x'', which is the value 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. ...
of the function, is denoted by the fraction : \frac in what is called the
Leibniz 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 ...
for derivatives. The quotient ''dy''/''dx'' is not infinitely small; rather it is a
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 ...
. The use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley.
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
(
1823 Events January–March * January 22 – By secret treaty signed at the Congress of Verona, the Quintuple Alliance gives France a mandate to invade Spain for the purpose of restoring Ferdinand VII (who has been captured by armed revolutio ...
) defined the differential without appeal to the atomism of Leibniz's infinitesimals. Instead, Cauchy, following
d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopé ...
, inverted the logical order of Leibniz and his successors: the derivative itself became the fundamental object, defined as a
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 difference quotients, and the differentials were then defined in terms of it. That is, one was free to ''define'' the differential ''dy'' by an expression :dy = f'(x)\,dx in which ''dy'' and ''dx'' are simply new variables taking finite real values, not fixed infinitesimals as they had been for Leibniz. According to , Cauchy's approach was a significant logical improvement over the infinitesimal approach of Leibniz because, instead of invoking the metaphysical notion of infinitesimals, the quantities ''dy'' and ''dx'' could now be manipulated in exactly the same manner as any other real quantities in a meaningful way. Cauchy's overall conceptual approach to differentials remains the standard one in modern analytical treatments, although the final word on rigor, a fully modern notion of the limit, was ultimately due to
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
. In physical treatments, such as those applied to the theory of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, the infinitesimal view still prevails. reconcile the physical use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the particular purpose for which they are intended. Thus "physical infinitesimals" need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense. Following twentieth-century developments in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
and
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 ...
, it became clear that the notion of the differential of a function could be extended in a variety of ways. In
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, it is more desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
of an increment Δ''x''. This approach allows the differential (as a linear map) to be developed for a variety of more sophisticated spaces, ultimately giving rise to such notions as the Fréchet or
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 ...
. Likewise, 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 ...
, the differential of a function at a point is a linear function of a
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
(an "infinitely small displacement"), which exhibits it as a kind of one-form: 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 ...
of the function. In non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing (see
differential (infinitesimal) In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathema ...
).


Definition

The differential is defined in modern treatments of differential calculus as follows. The differential of a function ''f''(''x'') of a single real variable ''x'' is the function ''df'' of two independent real variables ''x'' and Δ''x'' given by :df(x, \Delta x) \stackrel f'(x)\,\Delta x. One or both of the arguments may be suppressed, i.e., one may see ''df''(''x'') or simply ''df''. If ''y'' = ''f''(''x''), the differential may also be written as ''dy''. Since ''dx''(''x'', Δ''x'') = Δ''x'', it is conventional to write ''dx'' = Δ''x'' so that the following equality holds: :df(x) = f'(x) \, dx This notion of differential is broadly applicable when a
linear approximation In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving o ...
to a function is sought, in which the value of the increment Δ''x'' is small enough. More precisely, if ''f'' is a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
at ''x'', then the difference in ''y''-values :\Delta y \stackrel f(x+\Delta x) - f(x) satisfies :\Delta y = f'(x)\,\Delta x + \varepsilon = df(x) + \varepsilon\, where the error ε in the approximation satisfies ''ε''/Δ''x'' → 0 as Δ''x'' → 0. In other words, one has the approximate identity :\Delta y \approx dy in which the error can be made as small as desired relative to Δ''x'' by constraining Δ''x'' to be sufficiently small; that is to say, :\frac\to 0 as Δ''x'' → 0. For this reason, the differential of a function is known as the principal (linear) part in the increment of a function: the differential is a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
of the increment Δ''x'', and although the error ε may be nonlinear, it tends to zero rapidly as Δ''x'' tends to zero.


Differentials in several variables

Following , for functions of more than one independent variable, : y = f(x_1,\dots,x_n), the partial differential of ''y'' with respect to any one of the variables ''x''1 is the principal part of the change in ''y'' resulting from a change ''dx''1 in that one variable. The partial differential is therefore : \frac dx_1 involving 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 ...
of ''y'' with respect to ''x''1. The sum of the partial differentials with respect to all of the independent variables is the total differential : dy = \frac dx_1 + \cdots + \frac dx_n, which is the principal part of the change in ''y'' resulting from changes in the independent variables ''x''''i''. More precisely, in the context of multivariable calculus, following , if ''f'' is a differentiable function, then by the definition of differentiability, the increment :\begin \Delta y &\stackrel f(x_1+\Delta x_1, \dots, x_n+\Delta x_n) - f(x_1,\dots,x_n)\\ &= \frac \Delta x_1 + \cdots + \frac \Delta x_n + \varepsilon_1\Delta x_1 +\cdots+\varepsilon_n\Delta x_n \end where the error terms ε ''i'' tend to zero as the increments Δ''x''''i'' jointly tend to zero. The total differential is then rigorously defined as :dy = \frac \Delta x_1 + \cdots + \frac \Delta x_n. Since, with this definition, :dx_i(\Delta x_1,\dots,\Delta x_n) = \Delta x_i, one has :dy = \frac\,d x_1 + \cdots + \frac\,d x_n. As in the case of one variable, the approximate identity holds :dy \approx \Delta y in which the total error can be made as small as desired relative to \sqrt by confining attention to sufficiently small increments.


Application of the total differential to error estimation

In measurement, the total differential is used in estimating the error Δ''f'' of a function ''f'' based on the errors Δ''x'', Δ''y'', ... of the parameters ''x'', ''y'', …. Assuming that the interval is short enough for the change to be approximately linear: :Δ''f''(''x'') = ''f(''x'') × Δ''x'' and that all variables are independent, then for all variables, :\Delta f = f_x \Delta x + f_y \Delta y + \cdots This is because the derivative ''f''x with respect to the particular parameter ''x'' gives the sensitivity of the function ''f'' to a change in ''x'', in particular the error Δ''x''. As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. From this principle the error rules of summation, multiplication etc. are derived, e.g.: :Let f(''a'', ''b'') = ''a'' × ''b''; :Δ''f'' = ''f''''a''Δ''a'' + ''f''''b''Δ''b''; evaluating the derivatives :Δ''f'' = ''b''Δ''a'' + ''a''Δ''b''; dividing by ''f'', which is ''a'' × ''b'' :Δ''f''/''f'' = Δ''a''/''a'' + Δ''b''/''b'' That is to say, in multiplication, the total
relative error The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
is the sum of the relative errors of the parameters. To illustrate how this depends on the function considered, consider the case where the function is instead. Then, it can be computed that the error estimate is :Δ''f''/''f'' = Δ''a''/''a'' + Δ''b''/(''b'' ln ''b'') with an extra '' factor not found in the case of a simple product. This additional factor tends to make the error smaller, as is not as large as a bare ''b''.


Higher-order differentials

Higher-order differentials of a function ''y'' = ''f''(''x'') of a single variable ''x'' can be defined via: :d^2y = d(dy) = d(f'(x)dx) = (df'(x))dx = f''(x)\,(dx)^2, and, in general, :d^ny = f^(x)\,(dx)^n. Informally, this motivates Leibniz's notation for higher-order derivatives :f^(x) = \frac. When the independent variable ''x'' itself is permitted to depend on other variables, then the expression becomes more complicated, as it must include also higher order differentials in ''x'' itself. Thus, for instance, : \begin d^2 y &= f''(x)\,(dx)^2 + f'(x)d^2x\\ d^3 y &= f(x)\, (dx)^3 + 3f''(x)dx\,d^2x + f'(x)d^3x \end and so forth. Similar considerations apply to defining higher order differentials of functions of several variables. For example, if ''f'' is a function of two variables ''x'' and ''y'', then :d^nf = \sum_^n \binom\frac(dx)^k(dy)^, where \binom is a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. In more variables, an analogous expression holds, but with an appropriate multinomial expansion rather than binomial expansion. Higher order differentials in several variables also become more complicated when the independent variables are themselves allowed to depend on other variables. For instance, for a function ''f'' of ''x'' and ''y'' which are allowed to depend on auxiliary variables, one has :d^2f = \left(\frac(dx)^2+2\fracdx\,dy + \frac(dy)^2\right) + \fracd^2x + \fracd^2y. Because of this notational infelicity, the use of higher order differentials was roundly criticized by , who concluded: :Enfin, que signifie ou que représente l'égalité ::d^2z = r\,dx^2 + 2s\,dx\,dy + t\,dy^2\,? :A mon avis, rien du tout. That is: ''Finally, what is meant, or represented, by the equality .. In my opinion, nothing at all.'' In spite of this skepticism, higher order differentials did emerge as an important tool in analysis. In these contexts, the ''n''th order differential of the function ''f'' applied to an increment Δ''x'' is defined by :d^nf(x,\Delta x) = \left.\frac f(x+t\Delta x)\_ or an equivalent expression, such as :\lim_\frac where \Delta^n_ f is an ''n''th forward difference with increment ''t''Δ''x''. This definition makes sense as well if ''f'' is a function of several variables (for simplicity taken here as a vector argument). Then the ''n''th differential defined in this way is a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
of degree ''n'' in the vector increment Δ''x''. Furthermore, the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of ''f'' at the point ''x'' is given by :f(x+\Delta x)\sim f(x) + df(x,\Delta x) + \fracd^2f(x,\Delta x) + \cdots + \fracd^nf(x,\Delta x) + \cdots The higher order
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 ...
generalizes these considerations to infinite dimensional spaces.


Properties

A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: *
Linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
: For constants ''a'' and ''b'' and differentiable functions ''f'' and ''g'', ::d(af+bg) = a\,df + b\,dg. *
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 two differentiable functions ''f'' and ''g'', ::d(fg) = f\,dg+g\,df. An operation ''d'' with these two properties is known in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
as 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 ...
. They imply the Power rule :: d( f^n ) = n f^ df In addition, various forms of 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 , ...
hold, in increasing level of generality: * If ''y'' = ''f''(''u'') is a differentiable function of the variable ''u'' and ''u'' = ''g''(''x'') is a differentiable function of ''x'', then ::dy = f'(u)\,du = f'(g(x))g'(x)\,dx. * If and all of the variables ''x''1, ..., ''x''''n'' depend on another variable ''t'', then by the chain rule for partial derivatives, one has :: \begin dy &= \fracdt \\ &= \frac dx_1 + \cdots + \frac dx_n\\ &= \frac \frac\,dt + \cdots + \frac \frac\,dt. \end :Heuristically, the chain rule for several variables can itself be understood by dividing through both sides of this equation by the infinitely small quantity ''dt''. * More general analogous expressions hold, in which the intermediate variables ''x''''i'' depend on more than one variable.


General formulation

A consistent notion of differential can be developed for a function between two
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
s. Let x,Δx ∈ R''n'' be a pair of
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
s. The increment in the function ''f'' is :\Delta f = f(\mathbf+\Delta\mathbf) - f(\mathbf). If there exists an ''m'' × ''n''
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 ...
''A'' such that :\Delta f = A\Delta\mathbf + \, \Delta\mathbf\, \boldsymbol in which the vector ''ε'' → 0 as Δx → 0, then ''f'' is by definition differentiable at the point x. The matrix ''A'' is sometimes known as 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 ...
, and the
linear transformation 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 pre ...
that associates to the increment Δx ∈ R''n'' the vector ''A''Δx ∈ R''m'' is, in this general setting, known as the differential ''df''(''x'') of ''f'' at the point ''x''. This is precisely the
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued ...
, and the same construction can be made to work for a function between any
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 ...
s. Another fruitful point of view is to define the differential directly as a kind of
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
: :df(\mathbf,\mathbf) = \lim_\frac = \left.\fracf(\mathbf+t\mathbf)\_, which is the approach already taken for defining higher order differentials (and is most nearly the definition set forth by Cauchy). If ''t'' represents time and x position, then h represents a velocity instead of a displacement as we have heretofore regarded it. This yields yet another refinement of the notion of differential: that it should be a linear function of a kinematic velocity. The set of all velocities through a given point of space is known as the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
, and so ''df'' gives a linear function on the tangent space: a
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 ...
. With this interpretation, the differential of ''f'' is known as 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 ...
, and has broad application 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 ...
because the notion of velocities and the tangent space makes sense 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, in addition, the output value of ''f'' also represents a position (in a Euclidean space), then a dimensional analysis confirms that the output value of ''df'' must be a velocity. If one treats the differential in this manner, then it is known as the pushforward since it "pushes" velocities from a source space into velocities in a target space.


Other approaches

Although the notion of having an infinitesimal increment ''dx'' is not well-defined in modern
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, a variety of techniques exist for defining the infinitesimal differential so that the differential of a function can be handled in a manner that does not clash with the
Leibniz 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 ...
. These include: * Defining the differential as a kind of
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 ...
, specifically 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 ...
of a function. The infinitesimal increments are then identified with vectors in the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at a point. This approach is popular 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 related fields, because it readily generalizes to mappings between
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 ...
s. * 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 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 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.See and . * 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 which 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 .


Examples and applications

Differentials may be effectively used in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
to study the propagation of experimental errors in a calculation, and thus the overall
numerical stability In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algori ...
of a problem . Suppose that the variable ''x'' represents the outcome of an experiment and ''y'' is the result of a numerical computation applied to ''x''. The question is to what extent errors in the measurement of ''x'' influence the outcome of the computation of ''y''. If the ''x'' is known to within Δ''x'' of its true value, then
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is th ...
gives the following estimate on the error Δ''y'' in the computation of ''y'': :\Delta y = f'(x)\Delta x + \fracf''(\xi) where for some . If Δ''x'' is small, then the second order term is negligible, so that Δ''y'' is, for practical purposes, well-approximated by . The differential is often useful to rewrite a
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, ...
: \frac = g(x) in the form : dy = g(x)\,dx, in particular when one wants to separate the variables.


Notes


See also

* Notation for differentiation


References

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External links


Differential Of A Function
at Wolfram Demonstrations Project {{DEFAULTSORT:Differential Of A Function Differential calculus Generalizations of the derivative Linear operators in calculus