Differential (mathematics)
In mathematics, differential refers to several related notions[1] derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.[2]
The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.
Introduction
The term differential is used nonrigorously in
Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula
Basic notions
- In calculus, the differential represents a change in the linearization of a function.
- The total differentialis its generalization for functions of multiple variables.
- The
- In traditional approaches to calculus, the differentials (e.g. dx, dy, dt, etc.) are interpreted as infinitesimals. 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 Jacobian matrix of partial derivatives of a function from Rn to Rm (especially when this matrix is viewed as a linear map).
- More generally, the smooth manifolds and the pushforward operations it defines. The differential is also used to define the dual concept of pullback.
- stochastic differential and an associated calculus for stochastic processes.
- The Stieltjes integral 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 and integration by parts formulae for Stieltjes integral correspond, respectively, to the chain rule and product rulefor the differential.
History and usage
In
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,
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 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. In an expression such as
Approaches
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Calculus |
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There are several approaches for making the notion of differentials mathematically precise.
- Differentials as linear maps. This approach underlies the definition of the derivative and the exterior derivative in differential geometry.[4]
- Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry.[5]
- Differentials in smooth models of set theory. This approach is known as topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced.[6]
- Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson.[7]
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 maps. It can be used on , , a
Differentials as linear maps on R
Suppose is a real-valued function on . We can reinterpret the variable in as being a function rather than a number, namely the
This would just be a trick were it not for the fact that:
- it captures the idea of the derivative of at as the best linear approximation to at ;
- it has many generalizations.
Differentials as linear maps on Rn
If is a function from to , then we say that is differentiable[8] at if there is a linear map from to such that for any , there is a neighbourhood of 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 -th component of ). Then the differentials at a point form a basis for the vector space of linear maps from to and therefore, if is differentiable at , we can write as a linear combination of these basis elements:
The coefficients are (by definition) the partial derivatives of at with respect to . Hence, if is differentiable on all of , we can write, more concisely:
In the one-dimensional case this becomes
This idea generalizes straightforwardly to functions from to . Furthermore, it has the decisive advantage over other definitions of the derivative that it is
Aside: Note that the existence of all the partial derivatives of at is a
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 inner product space, 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. 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 .
Differentials as germs of functions
This approach works on any differentiable manifold. If
- U and V are open sets containing p
- is continuous
- is continuous
then f is equivalent to g at p, denoted , if and only if there is an open containing p such that for every x in W. The germ of f at p, denoted , is the set of all real continuous functions equivalent to f at p; if f is smooth at p then is a smooth germ. If
- , and are open sets containing p
- , , and are smooth functions
- r is a real number
then
This shows that the germs at p form an algebra.
Define to be the set of all smooth germs vanishing at p and to be the product of ideals . Then a differential at p (cotangent vector at p) is an element of . The differential of a smooth function f at p, denoted , is .
A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch. Then the differential of f at p is the set of all functions differentially equivalent to at p.
Algebraic geometry
In
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. Hence the derivative of f at p may be captured by the equivalence class [f − f(p)] 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.[5]
Algebraic geometry notions
Differentials are also important in algebraic geometry, and there are several important notions.
- Abelian differentials usually mean differential one-forms on an algebraic curve or Riemann surface.
- Quadratic differentials (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
- Kähler differentials provide a general notion of differential in algebraic geometry.
Synthetic differential geometry
A fifth approach to infinitesimals is the method of
Nonstandard analysis
The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way. In the
Differential geometry
The notion of a differential motivates several concepts in differential geometry (and differential topology).
- The differential (Pushforward) of a map between manifolds.
- Differential forms provide a framework which accommodates multiplication and differentiation of differentials.
- The differential 1-form).
- Pullback is, in particular, a geometric name for the chain rule for composing a map between manifolds with a differential form on the target manifold.
- Covariant derivatives or differentials provide a general notion for differentiating of vector fields and tensor fields on a manifold, or, more generally, sections of a vector bundle: 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
The properties of the differential also motivate the algebraic notions of a
See also
Notes
Citations
- ^ "Differential". Wolfram MathWorld. Retrieved February 24, 2022.
The word differential has several related meaning in mathematics. In the most common context, it means "related to derivatives." So, for example, the portion of calculus dealing with taking derivatives (i.e., differentiation), is known as differential calculus.
The word "differential" also has a more technical meaning in the theory of differential k-forms as a so-called one-form. - ^ "differential - Definition of differential in US English by Oxford Dictionaries". Oxford Dictionaries - English. Archived from the original on January 3, 2014. Retrieved 13 April 2018.
- ^ Boyer 1991.
- ^ Darling 1994.
- ^ a b Eisenbud & Harris 1998.
- ^ See Kock 2006 and Moerdijk & Reyes 1991.
- ^ a b See Robinson 1996 and Keisler 1986.
- ^ See, for instance, Apostol 1967.
- ^ See Kock 2006 and Lawvere 1968.
- ^ See Moerdijk & Reyes 1991 and Bell 1998.
References
- ISBN 978-0-471-00005-1.
- Bell, John L. (1998), Invitation to Smooth Infinitesimal Analysis (PDF).
- ISBN 978-0-471-54397-8.
- Darling, R. W. R. (1994), Differential forms and connections, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-46800-8.
- ISBN 978-0-387-98637-1
- Keisler, H. Jerome (1986), Elementary Calculus: An Infinitesimal Approach (2nd ed.).
- Kock, Anders (2006), Synthetic Differential Geometry (PDF) (2nd ed.), Cambridge University Press.
- Lawvere, F.W. (1968), Outline of synthetic differential geometry (PDF) (published 1998).
- Moerdijk, I.; Reyes, G.E. (1991), Models for Smooth Infinitesimal Analysis, Springer-Verlag.
- ISBN 978-0-691-04490-3.
- Weisstein, Eric W. "Differentials". MathWorld.