Differential

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The main linear part of increment of a function.

1) A real-valued function of a real variable is said to be differentiable at a point if it is defined in some neighbourhood of this point and if there exists a number such that the increment

may be written (if the point lies in this neighbourhood) in the form

where as . Here is usually denoted by and is called the differential of at . For a given the differential is proportional to , i.e. is a linear function of . By definition, as the additional term is infinitely small of a higher order than (and also than if ). This is why the differential is said to be the main part of increment of the function.

For a function which is differentiable at a point , if , i.e. a function which is differentiable at a point is continuous at that point. A function is differentiable at a point if and only if it has, at that point, a finite derivative

moreover,

There exist continuous functions which are not differentiable.

The designation may be used instead of , and the above equation assumes the form

The increment of the argument is then usually denoted by , and is said to be the differential of the independent variable. Accordingly, one may write

Hence , i.e. the derivative is equal to the ratio of the differentials and . If , then as , i.e. if , then and are infinitesimals of the same order as ; this fact, along with the simple structure of the differential (i.e. linearity with respect to ), is often used in approximate computations, by assuming that for small . E.g., if it is desired to compute from a known when is small, it is assumed that

Obviously, such reasoning is useful only if it is possible to estimate the magnitude of the error involved.

Geometric interpretation of the differential. The equation of the tangent to the graph of a function at a point is of the form . If one puts , then . The right-hand side represents the value of the differential of the function at the point corresponding to the value of being considered. Thus, the differential is identical with the corresponding increment of the ordinate of the tangent to the curve (cf. the segment in Fig. a). Here , i.e. the value of coincides with the length of the segment .

Figure: d031810a

2) The definitions of differentiability and differential are readily extended to real-valued functions of real variables. Thus, in the case a real-valued function is said to be differentiable at a point with respect to both variables and if it is defined in some neighbourhood of this point and if its total increment

may be written as

where and are real numbers, if , ; it is assumed that the point belongs to the neighbourhood mentioned above (Fig. b).

Figure: d031810b

One introduces the notation

is said to be the total differential, or simply the differential, of the function at the point (the phrase "with respect to both variables x and y" is sometimes added). For a given point the differential is a linear function of and ; the difference is infinitely small of a higher order than . In this sense is the main linear part of the increment .

If is differentiable at the point , then it is continuous at this point and has finite partial derivatives (cf. Derivative)

at this point. Thus

The increments and of the independent variables are usually denoted by and , as in the case of a single variable. One may write, accordingly,

The existence of finite partial derivatives does not, in general, entail the differentiability of the function (even if it is assumed to be continuous).

If a function has a partial derivative with respect to at a point , the product is said to be its partial differential with respect to ; in the same manner, is the partial differential with respect to . If the function is differentiable, its total differential is equal to the sum of the partial differentials. Geometrically, the total differential is the increment in the -direction in the tangent plane to the surface at the point , where (Fig. c).

Figure: d031810c

The following is a sufficient criterion for the differentiability of a function: If in a certain neighbourhood of a point a function has a partial derivative which is continuous at and, in addition, has a partial derivative at that point, then is differentiable at that point.

If a function is differentiable at all points of an open domain , then at any point of the domain

where , . If, in addition, there exist continuous partial derivatives and in , then, everywhere in ,

This proves, in particular, that not every expression

with continuous and (in a domain ) is the total differential of some function of two variables. This is a difference from functions of one variable, where any expression with a continuous function in some interval is the differential of some function.

The expression is the total differential of some function in a simply-connected open domain if and are continuous in this domain, meet the condition and, in addition: a) and are continuous or b) and are everywhere differentiable in with respect to both variables and [7], [8].

See also Differential calculus for differentials of real-valued functions of one or more real variables and for differentials of higher orders.

3) Let a function be defined on some set of real numbers, let be a limit point of this set, let , , , where if ; then the function is called differentiable with respect to the set at , while is called its differential with respect to the set at . This is a generalization of the differential of a real-valued function of one real variable. Special kinds of this generalization include differentials at the end points of the interval within which the function is defined, and the approximate differential (cf. Approximate differentiability).

Differentials with respect to a set for real-valued functions of several real variables are introduced in a similar manner.

4) All definitions of differentiability and a differential given above can be extended, almost unchanged, to complex-valued functions of one or more real variables; to real-valued and complex-valued vector-functions of one or more real variables; and to complex functions and vector-functions of one or more complex variables. In functional analysis they are extended to functions of the points of an abstract space. One may speak of differentiability and of the differential of a set function with respect to some measure.

References

 [1] G.P. Tolstov, "Elements of mathematical analysis" , 1–2 , Moscow (1974) (In Russian) [2] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964) [3] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) [4] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) [5] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) [6] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) [7] G.P. Tolstov, "On curvilinear and iterated integrals" Trudy Mat. Inst. Steklov. , 35 (1950) (In Russian) [8] G.P. Tolstov, "On the total differential" Uspekhi Mat. Nauk , 3 : 5 (1948) pp. 167–170