# differential

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## differential

**differential,**in the automobile, a set of gears used on the driving (usually rear) axle. The two wheels on the driving axle must be interconnected in order to receive their energy from the same source, the driving shaft; at the same time they must be free to revolve at different speeds when necessary (e.g., when rounding a curve, the outer wheel travels farther and thus must revolve faster than the inner wheel in order to prevent skidding). These two requirements are met by the differential gearing. Furthermore, through it the rotating motion of the driving shaft is transmitted to the axle and the wheels. The axle is in two halves; to each half is attached a wheel at one end and, at the inner end, a gear (see gear). The end of the driving shaft is also equipped with a gear. By an ingenious arrangement of these and other gears, together constituting the differential, a difference in speed of the two wheels is compensated for without a loss of tractive force. A disadvantage of the conventional differential is that when one wheel is on a dry and the other on a slippery surface, the differential causes the wheel on the slippery surface to revolve at double speed while the other wheel remains stationary. This hazard can be avoided by use of a limited slip differential, which feeds power to one wheel when the other wheel starts to slip and thus keeps the automobile moving.

### Bibliography

See R. T. Hinkle, *Kinematics of Machines* (2d ed. 1960).

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## 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 =* x_{0}, then the increment

Δ*y* = *f*(*x*_{0} + Δ*x*) - *f*(*x*_{0})

of the function *f*(*x*) may be given as

Δ*y* = *f*’(*x*_{0}) Δ*x* + R

where the term *R* is infinitely small compared to Δ*x*. The first term

*dy* = *f*’(*x*_{0}) Δ*x*

in this expansion is called the differential of the function *f*(*x*) at point *x*_{0}. From this formula it is seen that the differential *dy* depends linearly on the increment of the independent variable ΔJ, and the equality

Δ*y = 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* = {*x*_{1}, …, *x*_{n}} always has the form

L(x) = a_{1}x_{1} + … + a_{n}x_{n}

where *a*_{1} …, *a*_{n} 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

Δ*f* = *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 Δ*f* 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 + h_{2};h_{1}) - df(x;h_{1})

Further, considering *h* = *h*_{1} to be constant, we can define the differential of the differential df(x; *h*) as the principal part of the increment

*df(x + h _{2};h_{1}) - df(x;h_{1})*

where h_{2} is an increment of x independent of h_{1} The second differential, d^{2}f=d^{2}f(x;h_{1},h_{2}), obtained in such a way, is a function of three vector arguments x, h_{1} and h_{2} and is linear in each of the last two arguments. If d^{2}f depends continuously on *x*, then it is symmetric in h_{1} and h_{2}

*d ^{2}f(x;h_{1}h_{2}) = d^{2}f(x;h_{2}h_{1})*

The differential d^{n}f = d^{n}f (x;h_{1} … ,h_{n}) 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 d^{2}f of the functional f[x(t)] are called its first and second variations and are denoted by δf and δ^{2}f.

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]*x*), where

*x*is a vector, evaluated at a given vector

*c*, is the linear, real-valued function whose graph is the tangent hyperplane to the graph of ƒ(

*x*) at

*x*=

*c*; if

*x*is a real number, the usual notation is

*d*ƒ = ƒ′(

*c*)

*dx*.

## 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).

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

## differential

**1.**

*Maths*of, containing, or involving one or more derivatives or differentials

**2.**

*Physics*

*Engineering*relating to, operating on, or based on the difference between two effects, motions, forces, etc.

**3.**

*Maths*

**a.**an increment in a given function, expressed as the product of the derivative of that function and the corresponding increment in the independent variable

**b.**an increment in a given function of two or more variables, f(

*x*

_{1},

*x*

_{2}, …

*x*), expressed as the sum of the products of each partial derivative and the increment in the corresponding variable. FORMULA

_{n}**4.**

*Engineering*an epicyclic gear train that permits two shafts to rotate at different speeds while being driven by a third shaft

**5.**(in commerce) a difference in rates, esp between comparable labour services or transportation routes