# differential

## English

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### Etymology

Morphologically different +‎ -ial.

### Pronunciation

• IPA(key): [dɪfəˈɹənʃəɫ]
•  Audio (US): (file)

differential (comparative more differential, superlative most differential)

1. Of or pertaining to a difference.
differential characteristics
• 1856, John Lothrop Motley, The Rise of the Dutch Republic: A History, volume 1:
[Caspar Schetz, Baron of Grobbendonck] was regularly in the pay of Sir Thomas Gresham, to whom he produced differential favours, and by whose government he was rewarded by gold chains and presents of hard cash, bestowed as secretly as the equivalent was conveyed adroitly.
2. Dependent on, or making a difference; distinctive.
3. Having differences in speed or direction of motion.
4. Of or pertaining to differentiation or the differential calculus.

### Noun

differential (plural differentials)

1. The differential gear in an automobile, etc.
2. A qualitative or quantitative difference between similar or comparable things.
3. One of two coils of conducting wire so related to one another or to a magnet or armature common to both, that one coil produces polar action contrary to that of the other.
4. A form of conductor used for dividing and distributing the current to a series of electric lamps so as to maintain equal action in all.[1]
5. (calculus) A quantity representing an infinitesimal change in a variable, now only used as a heuristic aid except in nonstandard analysis but considered rigorous until the 20th century; a fluxion in Newtonian calculus, now usually written in Leibniz's notation as ${\displaystyle \operatorname {d} \!x}$.
6. (calculus, of a univariate differentiable function ${\displaystyle f(x)}$) A function giving the change in the linear approximation of ${\displaystyle f}$ at a point ${\displaystyle x}$ over a small interval ${\displaystyle \Delta x}$ or ${\displaystyle \operatorname {d} \!x}$, the function being called the differential of ${\displaystyle f}$ and denoted ${\displaystyle \operatorname {d} \!f(x,\Delta x)}$, ${\displaystyle \operatorname {d} \!f(x)}$, or simply ${\displaystyle \operatorname {d} \!f}$.
If ${\displaystyle f(x)=x^{2}}$, the differential of ${\displaystyle f}$ is the function ${\displaystyle \operatorname {d} \!f(x,\Delta x)=f'(x)\Delta x=2x\Delta x}$.
1. Any of several generalizations of this concept to functions of several variables or to higher orders: the partial differential, total differential, Gateaux differential, etc.
7. The Jacobian matrix of a function of several variables.
8. (differential geometry, of a smooth map ${\displaystyle \phi }$ between smooth manifolds) The pushforward or total derivative of ${\displaystyle \phi }$: a linear map from the tangent space at a point ${\displaystyle x}$ in ${\displaystyle \phi }$'s domain to the tangent space at ${\displaystyle \phi (x)}$ which is, in a technical sense, the best linear approximation of ${\displaystyle \phi }$ at ${\displaystyle x}$; denoted ${\displaystyle \operatorname {d} \!\phi _{x}}$.
9. Any of several generalizations of the concept(s) above: e.g. the Kähler differential in the setting of schemes, the quadratic differential in the theory of Riemann surfaces, etc.

### References

1. ^ Edward H[enry] Knight (1877) “Differential”, in Knight’s American Mechanical Dictionary. [], volumes I (A–GAS), New York, N.Y.: Hurd and Houghton [], →OCLC.

## Swedish

### Noun

differential c

1. a differential gear
2. an infinitesimal change
3. the differential operator

#### Declension

Declension of differential
Singular Plural
Indefinite Definite Indefinite Definite
Nominative differential differentialen differentialer differentialerna
Genitive differentials differentialens differentialers differentialernas