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# differential

[dif-uh-ren-shuhl]
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.

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

In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.

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Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.

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Field of mathematics in which methods of calculus are applied to the local geometry of curves and surfaces (i.e., to a small portion of a surface or curve around a point). A simple example is finding the tangent line on a two-dimensional curve at a given point. Similar operations may be extended to calculate the curvature and length of a curve and to analogous properties of surfaces in any number of dimensions.

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In automotive mechanics, a gear arrangement that transmits power from the engine to a pair of driving wheels, dividing the force equally between them but permitting them to follow paths of different lengths, as when turning a corner or traversing an uneven road. On a straight road the wheels rotate at the same speed; when turning a corner the outside wheel has farther to go and would turn faster than the inner wheel if unrestrained. The automobile differential was invented in 1827; originally used on steam-driven vehicles, it was well known when internal-combustion engines finally appeared.

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Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. Differential equations are very common in physics, engineering, and all fields involving quantitative study of change. They are used whenever a rate of change is known but the process giving rise to it is not. The solution of a differential equation is generally a function whose derivatives satisfy the equation. Differential equations are classified into several broad categories. The most important are ordinary differential equations (ODEs), in which change depends on a single variable, and partial differential equations (PDEs), in which change depends on several variables. Seealso differentiation.

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Mathematical process of finding the derivative of a function. Defined abstractly as a process involving limits, in practice it may be done using algebraic manipulations that rely on three basic formulas and four rules of operation. The formulas are: (1) the derivative of math.xmath.n is math.nmath.xmath.n − 1, (2) the derivative of sin math.x is cos math.x, and (3) the derivative of the exponential function math.emath.x is itself. The rules are: (1) (math.amath.f + math.bmath.g)' = math.amath.f' + math.bmath.g', (2) (math.fmath.g)' = math.fmath.g' + math.gmath.f', (3) (math.f/math.g)' = (math.gmath.f' − math.fmath.g')/math.g2, and (4) (math.f(math.g))' = math.f'(math.g)math.g', where math.a and math.b are constants, math.f and math.g are functions, and a prime (') indicates the derivative. The last formula is called the chain rule. The derivation and exploration of these formulas and rules is the subject of differential calculus. Seealso integration.

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In calculus, an expression based on the derivative of a function, useful for approximating certain values of the function. The differential of an independent variable math.x, written Δmath.x, is an infinitesimal change in its value. The corresponding differential of its dependent variable math.y is given by Δmath.y = math.f(math.x + Δmath.x) − math.f(math.x). Because the derivative of the function math.f(math.x), math.f'(math.x), is equal to the ratio Δmath.yΔmath.x as Δmath.x approaches zero (see limit), for small values of Δmath.x, Δmath.y ≅ math.f'(math.xmath.x. This formula often enables a quick and fairly accurate approximation to be made for what otherwise would be a tedious calculation.

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Differential may refer to:

## Mathematics

• Differential calculus comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function

* In traditional approaches to calculus, the differential of a function represents an extremely small change in its value and employs the dx, dy, dt symbols.
* In differential geometry, differentials represent the exterior derivatives of the coordinate functions on a differentiable manifold.

## Other uses

The word differential is also used in many fields as an adjective to refer to differences, particularly those of a variable or continuous nature. Examples include differential hardening in metallurgy, differential rotation in astronomy, differential centrifugation in cell biology, differential scanning calorimetry in materials science, differential signalling in communications, differential algebra in mathematics, differential execution in computer science, and differential GPS technology.

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