The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity.
The differential dx represents an infinitely small change in the variable x.
Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives.
denotes not 'dy divided by dx' as one would intuitively read, but 'the derivative of y with respect to x '.
This formula summarizes the idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx approaches zero.
Infinitesimal quantities played a significant role in the development of calculus.
However, it was Gottfried Leibniz who coined the term differentials for infinitesimal quantities and introduced the notation for them which is still used today.
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 y at x is its instantaneous rate of change (the slope of the graph's tangent line), which may be obtained by taking the limit of the ratio Δy/Δx as Δx becomes arbitrarily small.
Differentials are also compatible with dimensional analysis, 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'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.
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.
Differentials are also used in the notation for integrals 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.
There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as linear maps.
matrix, it is essentially the same thing as a number, but the change in the point of view allows us to think of
The same procedure works on a vector space with a enough additional structure to reasonably talk about continuity.
However, for a more general topological vector space, some of the details are more abstract because there is no concept of distance.
As a result, you can define a coordinate system from an arbitrary basis and use the same technique as for
A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch.
at p. In algebraic geometry, 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.
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 Ip of functions on R which vanish at p. If the derivative f vanishes at p, then f − f(p) belongs to the square Ip2 of this ideal.
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.
[10] This is closely related to the algebraic-geometric approach, except that the infinitesimals are more implicit and intuitive.
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).
Constuctivists regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.
The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way.
In the nonstandard analysis approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the reciprocals of infinitely large numbers.
Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle.
Dually, the boundary operators in a chain complex are sometimes called codifferentials.