Differential (calculus)
Encyclopedia
In calculus
, a differential is traditionally an infinitesimal
ly small change in a variable
. For example, if x is a variable, then a change in the value of x is often denoted Δx (or δx when this change is considered to be small). The differential dx represents such a change, but is infinitely small. Although, as stated, it is not a precise mathematical concept, it is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.
The key property of the differential is that if y is a function of x, then the differential dy of y is related to dx by the formula
where dy/dx denotes the derivative
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 Δy/Δx as Δx becomes infinitesimally small.
There are several approaches for making the notion of differentials mathematically precise.
These approaches are very different from each other, but they have in common the idea to be quantitative, i.e., to say not just that a differential is infinitesimally small, but how small it is.
quantities played a significant role in the development of calculus. Archimedes
used them, even though he didn't believe that arguments involving infinitesimals were rigorous. Isaac Newton
referred to them as fluxions
. However, it was Gottfried Leibniz
who coined the term differentials for infinitesimal quantities, and introduced the notation for them which is still used today.
In Leibniz's notation, if x is a variable quantity, then dx denotes an infinitesimally small change in the variable x. Thus, if y is a function of x, then the derivative
of y with respect to x is often denoted
, which would otherwise be denoted (in the notation of Newton or Lagrange) 
or 
. The use of differentials in this form attracted much criticism, for instance in the famous pamphlet The Analyst
by Bishop Berkeley. Nevertheless the notation has remained popular because it suggests strongly the idea that the derivative of a function y(x) is its slope, which may be obtained by taking the limit
of the ratio
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
, where a differential such as dx has the same dimensions as the variable x.
Differentials are also used in the notation for integral
s because an integral can be regarded as an infinite sum of infinitesimally small quantities: the area under a graph is obtained by subdividing the graph into infinitesimally thin strips and summing their areas. In an expression such as
the integral sign (which is a modified long s
) denotes the infinite sum, f(x) denotes the 'height' of a thin strip, and the differential dx denotes its arbitrarily thin width.
as a function on the real line, the standard coordinate or identity map
which takes a real number p to itself (
): then 
denotes the composite 
of f with x, whose value at p is 
. The differential 
is then a function on the real line whose value at p (usually denoted 
) is not a number, but a linear map from 
to 
. Since a linear map from 
to 
is given by a 1×1 matrix
, it is essentially the same thing as a number, but the change in the point of view allows us to think of
as an infinitesimal and compare it with the standard infinitesimal 
which is again just the identity map from 
to 
(a 1×1 matrix
with entry 1). It may seem fanciful to regard the identity map as an infinitesimal, but it does at least have the property that if
is very small, then 
is very small. The differential 
has the same property, because it is just a multiple of 
, and this multiple is the derivative 
by definition. We therefore obtain that 
, and hence
. Thus we recover the idea that 
is the ratio of the differentials 
and 
.
This would just be a trick were it not for the fact that:
For instance if f is a function from
to 
then we say f is differentiable at 
if there is a linear map 
from 
to 
such that for any 
, there is a neighbourhood
of p such that for 
:
We can now use the same trick as in the one dimensional case, and think of the expression
as the composite of 
with the standard coordinates 
on 
(so that 
is the j-th component of 
). Then the differentials 
, 
, 
(at a point p) form a basis
for the vector space
of linear maps from
to 
and therefore, if f is differentiable at p, we can write 
as a linear combination
of these basis elements:
The coefficients
are (by definition) the partial derivative
s of f at p with respect to
, 
, …, 
. Hence, if f is differentiable on all of 
, we can write, more concisely:
In the one-dimensional case this becomes
as before.
This idea generalizes straightforwardly to functions from
to 
. 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 maps between smooth manifolds.
Aside: Note that the existence of all the partial derivative
s of
at x is a necessary condition for the existence of a differential at x. However it is not a sufficient condition. For counterexamples, see Gâteaux derivative
.
, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements. The simplest example is the ring of dual number
s R[ε], where ε2 = 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)1 (where 1 is the identity function) belongs to the ideal
Ip of functions on R which vanish at p. If the derivative f vanishes at p, then f−f(p)1 belongs to the square Ip2 of this ideal. Hence the derivative of f at p may be captured by the equivalence class [f−f(p)1] in the quotient space
Ip/Ip2, 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 Ip2. 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 Ip) but R[ε] which is the quotient space of functions on R modulo Ip2. Such a thickened point is a simple example of a scheme
.
or smooth infinitesimal analysis
. 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
with another category
of smoothly varying sets which is a topos
. 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
in this new category is not identical to the familiar logic of the category of sets: in particular, the law of the excluded middle does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are constructive (e.g., do not use proof by contradiction). Some regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.
s of infinitely large numbers. Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of real number
s, so that, for example, the sequence (1, 1/2, 1/3, ..., 1/n, ...) represents an infinitesimal. The first-order logic
of this new set of hyperreal number
s is the same as the logic for the usual real numbers, but the completeness axiom
(which involves second-order logic
) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle
.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, a differential is traditionally an infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
ly small change in a variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. For example, if x is a variable, then a change in the value of x is often denoted Δx (or δx when this change is considered to be small). The differential dx represents such a change, but is infinitely small. Although, as stated, it is not a precise mathematical concept, it is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.
The key property of the differential is that if y is a function of x, then the differential dy of y is related to dx by the formula
where dy/dx denotes the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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 Δy/Δx as Δx becomes infinitesimally small.
There are several approaches for making the notion of differentials mathematically precise.
- Differentials as linear maps. This approach underlies the definition of the derivativeTotal derivativeIn the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...
and the exterior derivativeExterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
in differential geometry. - Differentials as nilpotentNilpotentIn mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
elements of commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s. This approach is popular in algebraic geometry. - Differentials in smooth models of set theory. This approach is known as synthetic differential geometrySynthetic differential geometryIn mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
or smooth infinitesimal analysisSmooth infinitesimal analysisSmooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete...
and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced. - Differentials as infinitesimals in hyperreal numberHyperreal numberThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
systems, which are extensions of the real numbers which contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham RobinsonAbraham RobinsonAbraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
.
These approaches are very different from each other, but they have in common the idea to be quantitative, i.e., to say not just that a differential is infinitesimally small, but how small it is.
History and usage
InfinitesimalInfinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
quantities played a significant role in the development of calculus. Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
used them, even though he didn't believe that arguments involving infinitesimals were rigorous. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
referred to them as fluxions
Method of Fluxions
Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
. However, it was Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
who coined the term differentials for infinitesimal quantities, and introduced the notation for them which is still used today.
In Leibniz's notation, if x is a variable quantity, then dx denotes an infinitesimally small change in the variable x. Thus, if y is a function of x, then the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of y with respect to x is often denoted
, which would otherwise be denoted (in the notation of Newton or Lagrange) or . The use of differentials in this form attracted much criticism, for instance in the famous pamphlet The AnalystThe Analyst
The Analyst, subtitled "A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith", is a book published by...
by Bishop Berkeley. Nevertheless the notation has remained popular because it suggests strongly the idea that the derivative of a function y(x) is its slope, which may be obtained by taking the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
of the ratio
of the change in y over the change in x, as the change in x becomes arbitrarily small. Differentials are also compatible with dimensional analysisDimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
, where a differential such as dx has the same dimensions as the variable x.
Differentials are also used in the notation for integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s because an integral can be regarded as an infinite sum of infinitesimally small quantities: the area under a graph is obtained by subdividing the graph into infinitesimally thin strips and summing their areas. In an expression such as
the integral sign (which is a modified long s
Long s
The long, medial or descending s is a form of the minuscule letter s formerly used where s occurred in the middle or at the beginning of a word, for example "ſinfulneſs" . The modern letterform was called the terminal, round, or short s.-History:The long s is derived from the old Roman cursive...
) denotes the infinite sum, f(x) denotes the 'height' of a thin strip, and the differential dx denotes its arbitrarily thin width.
Differentials as linear maps
There is a simple way to make precise sense of differentials by regarding them as linear maps. One way to explain this point of view is to regard the variable x in an expression such as as a function on the real line, the standard coordinate or identity mapMap (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
which takes a real number p to itself (
): then denotes the composite of f with x, whose value at p is . The differential is then a function on the real line whose value at p (usually denoted ) is not a number, but a linear map from to . Since a linear map from to is given by a 1×1 matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, it is essentially the same thing as a number, but the change in the point of view allows us to think of
as an infinitesimal and compare it with the standard infinitesimal which is again just the identity map from to (a 1×1 matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
with entry 1). It may seem fanciful to regard the identity map as an infinitesimal, but it does at least have the property that if
is very small, then is very small. The differential has the same property, because it is just a multiple of , and this multiple is the derivative by definition. We therefore obtain that , and hence. Thus we recover the idea that is the ratio of the differentials and .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.
For instance if f is a function from
to then we say f is differentiable at if there is a linear map from to such that for any , there is a neighbourhoodNeighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of p such that for :We can now use the same trick as in the one dimensional case, and think of the expression
as the composite of with the standard coordinates on (so that is the j-th component of ). Then the differentials , , (at a point p) form a basisBasis
Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis of futures, the value differential between a future and the spot price...
for the vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of linear maps from
to and therefore, if f is differentiable at p, we can write as a linear combinationLinear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of these basis elements:
The coefficients
are (by definition) the partial derivativePartial 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...
s of f at p with respect to
, , …, . Hence, if f is differentiable on all of , we can write, more concisely:In the one-dimensional case this becomes
as before.
This idea generalizes straightforwardly to functions from
to . Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariantInvariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under changes of coordinates. This means that the same idea can be used to define the differential of smooth maps between smooth manifolds.
Aside: Note that the existence of all the partial derivative
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...
s of
at x is a necessary condition for the existence of a differential at x. However it is not a sufficient condition. For counterexamples, see Gâteaux derivativeGâteaux derivative
In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector...
.
Algebraic geometry
In algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements. The simplest example is the ring of dual number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...
s R[ε], where ε2 = 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)1 (where 1 is the identity function) belongs to the ideal
Ideal
-In philosophy:* Ideal , values that one actively pursues as goals* Platonic ideal, a philosophical idea of trueness of form, associated with Plato-In mathematics:* Ideal , special subsets of a ring considered in abstract algebra...
Ip of functions on R which vanish at p. If the derivative f vanishes at p, then f−f(p)1 belongs to the square Ip2 of this ideal. Hence the derivative of f at p may be captured by the equivalence class [f−f(p)1] in the quotient space
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
Ip/Ip2, and
the 1-jet
Jet (mathematics)
In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions modulo Ip2. 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 Ip) but R[ε] which is the quotient space of functions on R modulo Ip2. Such a thickened point is a simple example of a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
.
Synthetic differential geometry
A third approach to infinitesimals is the method of synthetic differential geometrySynthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
or smooth infinitesimal analysis
Smooth infinitesimal analysis
Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete...
. 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
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 all functions from A to B...
with another category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of smoothly varying sets which is a topos
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
. 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
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
in this new category is not identical to the familiar logic of the category of sets: in particular, the law of the excluded middle does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are constructive (e.g., do not use proof by contradiction). 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 approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the reciprocalMultiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
s of infinitely large numbers. Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, so that, for example, the sequence (1, 1/2, 1/3, ..., 1/n, ...) represents an infinitesimal. The first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
of this new set of hyperreal number
Hyperreal number
The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s is the same as the logic for the usual real numbers, but the completeness axiom
Completeness axiom
In mathematics the completeness axiom, also called Dedekind completeness of the real numbers, is a fundamental property of the set R of real numbers...
(which involves second-order logic
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....
) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle
Transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...
.
See also
- Infinitesimal calculusInfinitesimal calculusInfinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
- Differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
- Differential formDifferential formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
- Differential of a functionDifferential of a functionIn calculus, the differential represents the principal part of the change in a function y = ƒ with respect to changes in the independent variable. The differential dy is defined bydy = f'\,dx,...