differential
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differential
Bibliography
See R. T. Hinkle, Kinematics of Machines (2d ed. 1960).
Differential
in mathematics, the principal linear part of the increment of a function. If a function y = f(x) of one variable x has a derivative at x = x0, then the increment
Δy = f(x0 + Δx) - f(x0)
of the function f(x) may be given as
Δy = f’(x0) Δx + R
where the term R is infinitely small compared to Δx. The first term
dy = f’(x0) Δx
in this expansion is called the differential of the function f(x) at point x0. From this formula it is seen that the differential dy depends linearly on the increment of the independent variable ΔJ, and the equality
Δy = dy + R
shows in what sense the differential dy is the principal part of the increment Δy.
Generalization of the concept. The generalization of the concept of differential to vector functions (work which was begun in the early 20th century by the French mathematicians R. Gateaux and M. Fréchet) enables us to better understand the meaning of the concept “differential” for functions with several variables and, as applied to functionals, leads to the concept of variation, which is the basis of the calculus of variations.
The concept of a linear function (linear transformation) plays an important part in this generalization. The function L(x) of the vector argument x is called linear if it is continuous and satisfies the equality
L(x’+x”) = L(x’) + L(x”)
for any x’ and x” in the domain of definition. A linear function of an n-dimensional argument x = {x1, …, xn} always has the form
L(x) = a1x1 + … + anxn
where a1 …, an are constants. The increment
ΔL = L(x+h) - L(x)
of the linear function L(x) has the form
ΔL = L(h)
that is, it depends linearly on the vector increment h alone. The function f(x) is said to be differentiable at x if its increment Δf = f(x+h)-f(h), viewed as a function of h, has the principal linear part L(h), that is, is expressible as
Δf = L(h) + R(h)
where the remainder R(h) as h→0 is infinitely small compared to h. The principal linear part L(h) of the increment Δf is called the differential df of the function f at the point x. Depending on the sense assigned to the term “R(h) is infinitely small compared to h,” we distinguish between the weak differential (Gateaux differential) and the strong differential (Fréchet differential). If there exists a strong differential, then there also exists a weak differential equal to the strong one. The weak differential can exist when there is no strong differential.
For the case f(x) ≡ x, it follows from the general definition that df = h, that is, the increment h can be considered the differential of the argument x and denoted by dx.
If we now vary the point at which the differential df is defined, then df will be a function of two variables:
df(x + h2;h1) - df(x;h1)
Further, considering h = h1 to be constant, we can define the differential of the differential df(x; h) as the principal part of the increment
df(x + h2;h1) - df(x;h1)
where h2 is an increment of x independent of h1 The second differential, d2f=d2f(x;h1,h2), obtained in such a way, is a function of three vector arguments x, h1 and h2 and is linear in each of the last two arguments. If d2f depends continuously on x, then it is symmetric in h1 and h2
d2f(x;h1h2) = d2f(x;h2h1)
The differential dnf = dnf (x;h1 … ,hn) of any order n is defined in an analogous manner.
In the calculus of variations the vector argument x itself is a function x(t), and the differentials df and d2f of the functional f[x(t)] are called its first and second variations and are denoted by δf and δ2f.
The foregoing discussion has dealt with the generalization of the concept of differential to numerical functions of a vector variable. There also exists a generalization of the concept of differential to vector functions that assume values in Banach spaces.
REFERENCES
Il’in, B. A., and E. G. Pozniak. Osnovy matematicheskogo analiza, 2nd ed. Moscow, 1967.Kolmogorov, A. N., and S. V. Fomin. Elementy teorii funktsii i funktsial’nogo analiza, 2nd ed. Moscow, 1968.
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 7th ed., vol. 1. Moscow, 1969.
Kudriavtsev, L. D. Matematicheskii analiz, vol. 1. Moscow, 1970.
Rudin, W. Osnovy matematicheskogo analiza. Moscow, 1966. (Translated from English.)
Dieudonné, J. Osnovy sovremennogo analiza. Moscow, 1964. (Translated from English.)
A. N. KOLMOGOROV
differential
[‚dif·ə′ren·chəl]Differential
A mechanism which permits a rear axle to turn corners with one wheel rolling faster than the other. An automobile differential is located in the case carrying the rearaxle drive gear (see illustration).
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The differential gears consist of the two side gears carrying the inner ends of the axle shafts, meshing with two pinions mounted on a common pin located in the differential case. The case carries a ring gear driven by a pinion at the end of the drive shaft. This arrangement permits the drive to be carried to both wheels, but at the same time as the outer wheel on a turn overruns the differential case, the inner wheel lags by a like amount.
Special differentials permit one wheel to drive the car by a predetermined amount even though the opposite wheel is on slippery pavement; they have been used on racing cars for years and are now used by a number of car manufacturers. See Automotive transmission