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67-505: [REDACTED] Look up differential in Wiktionary, the free dictionary. Differential may refer to: Mathematics [ edit ] Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function Differential algebra Differential calculus Differential of

134-483: A Hilbert space , a Banach space , or more generally, a topological vector space . The case of the Real line is the easiest to explain. This type of differential is also known as a covariant vector or cotangent vector , depending on context. Suppose f ( x ) {\displaystyle f(x)} is a real-valued function on R {\displaystyle \mathbb {R} } . We can reinterpret

201-422: A linear combination of these basis elements: d f p = ∑ j = 1 n D j f ( p ) ( d x j ) p . {\displaystyle df_{p}=\sum _{j=1}^{n}D_{j}f(p)\,(dx_{j})_{p}.} The coefficients D j f ( p ) {\displaystyle D_{j}f(p)} are (by definition)

268-422: A differential motivates several concepts in differential geometry (and differential topology ). The term differential has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in a cochain complex ( C ∙ , d ∙ ) , {\displaystyle (C_{\bullet },d_{\bullet }),}

335-555: A differential such as dx has the same dimensions as the variable x . Calculus evolved into a distinct branch of mathematics during the 17th century CE, although there were antecedents going back to antiquity. The presentations of, e.g., Newton, Leibniz, were marked by non-rigorous definitions of terms like differential, fluent and "infinitely small". While many of the arguments in Bishop Berkeley 's 1734 The Analyst are theological in nature, modern mathematicians acknowledge

402-403: A function , represents a change in the linearization of a function Total differential is its generalization for functions of multiple variables Differential (infinitesimal) (e.g. dx , dy , dt etc.) are interpreted as infinitesimals Differential topology Differential (pushforward) The total derivative of a map between manifolds. Differential exponent , an exponent in

469-416: A main driver for generating the large-scale magnetic field, through magneto-hydrodynamical (dynamo) mechanisms in the outer envelopes. The interface between these two regions is where angular rotation gradients are strongest and thus where dynamo processes are expected to be most efficient. The inner differential rotation is one part of the mixing processes in stars, mixing the materials and the heat/energy of

536-422: A method of transmitting electronic signals over a pair of wires to reduce interference Differential amplifier an electronic amplifier that amplifies signals. Social sciences [ edit ] Semantic and structural differentials in psychology Quality spread differential , in finance Compensating differential , in labor economics Medicine [ edit ] Differential diagnosis ,

603-669: A number of ways to make the notion mathematically precise. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives . If y is a function of x , then the differential dy of y is related to dx by the formula d y = d y d x d x , {\displaystyle dy={\frac {dy}{dx}}\,dx,} where d y d x {\displaystyle {\frac {dy}{dx}}\,} denotes not 'dy divided by dx' as one would intuitively read, but 'the derivative of y with respect to x '. This formula summarizes

670-502: A point p {\displaystyle p} form a basis for the vector space of linear maps from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } and therefore, if f {\displaystyle f} is differentiable at p {\displaystyle p} , we can write d ⁡ f p {\displaystyle \operatorname {d} f_{p}} as

737-523: A positive thing, since it forces one to find constructive arguments wherever they are available. The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way. In the nonstandard analysis approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the reciprocals of infinitely large numbers. Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of real numbers , so that, for example,

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804-447: A smooth function f at p , denoted d f p {\displaystyle \mathrm {d} f_{p}} , is [ f − f ( p ) ] p / I p 2 {\displaystyle [f-f(p)]_{p}/{\mathcal {I}}_{p}^{2}} . A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch. Then

871-472: A systematic fashion over a number of years. On the Sun, active regions are common sources of radio flares. The researchers concluded that this effect was best explained by active regions emerging and disappearing at different latitudes, such as occurs during the solar sunspot cycle . Gradients in angular rotation caused by angular momentum redistribution within the convective layers of a star are expected to be

938-473: Is differentiable at p ∈ R n {\displaystyle p\in \mathbb {R} ^{n}} if there is a linear map d f p {\displaystyle df_{p}} from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } such that for any ε > 0 {\displaystyle \varepsilon >0} , there

1005-419: Is a necessary condition for the existence of a differential at x {\displaystyle x} . However it is not a sufficient condition . For counterexamples, see Gateaux derivative . The same procedure works on a vector space with a enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as a complete inner product space , where

1072-576: Is a neighbourhood N {\displaystyle N} of p {\displaystyle p} such that for x ∈ N {\displaystyle x\in N} , | f ( x ) − f ( p ) − d f p ( x − p ) | < ε | x − p | . {\displaystyle \left|f(x)-f(p)-df_{p}(x-p)\right|<\varepsilon \left|x-p\right|.} We can now use

1139-470: Is a Banach space and any topological vector space is complete. As a result, you can define a coordinate system from an arbitrary basis and use the same technique as for R n {\displaystyle \mathbb {R} ^{n}} . This approach works on any differentiable manifold . If then f is equivalent to g at p , denoted f ∼ p g {\displaystyle f\sim _{p}g} , if and only if there

1206-513: Is a movement of mass, due to steep temperature gradients from the core outwards. This mass carries a portion of the star's angular momentum, thus redistributing the angular velocity, possibly even far enough out for the star to lose angular velocity in stellar winds . Differential rotation thus depends on temperature differences in adjacent regions. There are many ways to measure and calculate differential rotation in stars to see if different latitudes have different angular velocities. The most obvious

1273-402: Is again just the identity map from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } (a 1 × 1 {\displaystyle 1\times 1} matrix with entry 1 {\displaystyle 1} ). The identity map has the property that if ε {\displaystyle \varepsilon }

1340-450: Is an open W ⊆ U ∩ V {\displaystyle W\subseteq U\cap V} containing p such that f ( x ) = g ( x ) {\displaystyle f(x)=g(x)} for every x in W . The germ of f at p , denoted [ f ] p {\displaystyle [f]_{p}} , is the set of all real continuous functions equivalent to f at p ; if f

1407-461: Is closely related to the algebraic-geometric approach, except that the infinitesimals are more implicit and intuitive. The main idea of this approach is to replace the category of sets with another category of smoothly varying sets which is a topos . In this category, one can define the real numbers, smooth functions, and so on, but the real numbers automatically contain nilpotent infinitesimals, so these do not need to be introduced by hand as in

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1474-415: Is given by a 1 × 1 {\displaystyle 1\times 1} matrix , it is essentially the same thing as a number, but the change in the point of view allows us to think of d f p {\displaystyle df_{p}} as an infinitesimal and compare it with the standard infinitesimal d x p {\displaystyle dx_{p}} , which

1541-430: Is smooth at p then [ f ] p {\displaystyle [f]_{p}} is a smooth germ. If then This shows that the germs at p form an algebra . Define I p {\displaystyle {\mathcal {I}}_{p}} to be the set of all smooth germs vanishing at p and I p 2 {\displaystyle {\mathcal {I}}_{p}^{2}} to be

1608-506: Is the j {\displaystyle j} -th component of p ∈ R n {\displaystyle p\in \mathbb {R} ^{n}} ). Then the differentials ( d x 1 ) p , ( d x 2 ) p , … , ( d x n ) p {\displaystyle \left(dx_{1}\right)_{p},\left(dx_{2}\right)_{p},\ldots ,\left(dx_{n}\right)_{p}} at

1675-449: Is the heliographic latitude , measured from the equator. On the Sun, the study of oscillations revealed that rotation is roughly constant within the whole radiative interior and variable with radius and latitude within the convective envelope. The Sun has an equatorial rotation speed of ~2 km/s; its differential rotation implies that the angular velocity decreases with increased latitude. The poles make one rotation every 34.3 days and

1742-468: Is the composite of f {\displaystyle f} with x {\displaystyle x} , whose value at p {\displaystyle p} is f ( x ( p ) ) = f ( p ) {\displaystyle f(x(p))=f(p)} . The differential d ⁡ f {\displaystyle \operatorname {d} f} (which of course depends on f {\displaystyle f} )

1809-412: Is the derivative f ′ ( p ) {\displaystyle f'(p)} by definition. We therefore obtain that d f p = f ′ ( p ) d x p {\displaystyle df_{p}=f'(p)\,dx_{p}} , and hence d f = f ′ d x {\displaystyle df=f'\,dx} . Thus we recover

1876-438: Is the ring of dual numbers R [ ε ], where ε = 0. This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p . For this, note first that f  − f ( p ) belongs to the ideal I p of functions on R which vanish at p . If the derivative f vanishes at p , then f  − f ( p ) belongs to the square I p of this ideal. Hence

1943-472: Is then a function whose value at p {\displaystyle p} (usually denoted d f p {\displaystyle df_{p}} ) is not a number, but a linear map from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } . Since a linear map from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} }

2010-427: Is tracking spots on the stellar surface. By doing helioseismological measurements of solar "p-modes" it is possible to deduce the differential rotation. The Sun has very many acoustic modes that oscillate in the interior simultaneously, and the inversion of their frequencies can yield the rotation of the solar interior. This varies with both depth and (especially) latitude. The broadened shapes of absorption lines in

2077-429: Is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity . For example, if x is a variable , then a change in the value of x is often denoted Δ x (pronounced delta x ). The differential dx represents an infinitely small change in the variable x . The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are

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2144-418: Is very small, then d x p ( ε ) {\displaystyle dx_{p}(\varepsilon )} is very small, which enables us to regard it as infinitesimal. The differential d f p {\displaystyle df_{p}} has the same property, because it is just a multiple of d x p {\displaystyle dx_{p}} , and this multiple

2211-1018: The partial derivatives of f {\displaystyle f} at p {\displaystyle p} with respect to x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} . Hence, if f {\displaystyle f} is differentiable on all of R n {\displaystyle \mathbb {R} ^{n}} , we can write, more concisely: d ⁡ f = ∂ f ∂ x 1 d x 1 + ∂ f ∂ x 2 d x 2 + ⋯ + ∂ f ∂ x n d x n . {\displaystyle \operatorname {d} f={\frac {\partial f}{\partial x_{1}}}\,dx_{1}+{\frac {\partial f}{\partial x_{2}}}\,dx_{2}+\cdots +{\frac {\partial f}{\partial x_{n}}}\,dx_{n}.} In

2278-400: The product of ideals I p I p {\displaystyle {\mathcal {I}}_{p}{\mathcal {I}}_{p}} . Then a differential at p (cotangent vector at p ) is an element of I p / I p 2 {\displaystyle {\mathcal {I}}_{p}/{\mathcal {I}}_{p}^{2}} . The differential of

2345-399: The 20th century, several new concepts in, e.g., multivariable calculus, differential geometry, seemed to encapsulate the intent of the old terms, especially differential ; both differential and infinitesimal are used with new, more rigorous, meanings. Differentials are also used in the notation for integrals because an integral can be regarded as an infinite sum of infinitesimal quantities:

2412-436: The Sun had different rotational periods at the poles and at the equator, in good agreement with modern values. Stars and planets rotate in the first place because conservation of angular momentum turns random drifting of parts of the molecular cloud that they form from into rotating motion as they coalesce. Given this average rotation of the whole body, internal differential rotation is caused by convection in stars which

2479-418: The algebraic geometric approach. However the logic in this new category is not identical to the familiar logic of the category of sets: in particular, the law of the excluded middle does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are constructive (e.g., do not use proof by contradiction ). Constuctivists regard this disadvantage as

2546-465: The area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as ∫ f ( x ) d x , {\displaystyle \int f(x)\,dx,} the integral sign (which is a modified long s ) denotes the infinite sum, f ( x ) denotes the "height" of a thin strip, and the differential dx denotes its infinitely thin width. There are several approaches for making

2613-539: The body and/or in time. This indicates that the object is not rigid . In fluid objects, such as accretion disks , this leads to shearing . Galaxies and protostars usually show differential rotation; examples in the Solar System include the Sun , Jupiter and Saturn . Around the year 1610, Galileo Galilei observed sunspots and calculated the rotation of the Sun . In 1630, Christoph Scheiner reported that

2680-431: The calculation used in producing golf handicaps See also [ edit ] All pages with titles beginning with Differential All pages with titles containing Differential Different (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Differential . If an internal link led you here, you may wish to change

2747-407: The center. For observed sunspots, the differential rotation can be calculated as: Ω = Ω 0 − Δ Ω sin 2 ⁡ Ψ {\displaystyle \Omega =\Omega _{0}-\Delta \Omega \sin ^{2}\Psi } where Ω 0 {\displaystyle \Omega _{0}} is the rotation rate at

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2814-685: The characterization of the underlying cause of pathological states based on specific tests White blood cell differential , the enumeration of each type of white blood cell either manually or using automated analyzers Other [ edit ] Differential hardening , in metallurgy Differential rotation , in astronomy Differential centrifugation , in cell biology Differential scanning calorimetry , in materials science Differential signalling , in communications Differential GPS , in satellite navigation technology Differential interferometry in radar Differential , an extended play by The Sixth Lie Handicap differential , part of

2881-406: The decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. This means that the same idea can be used to define the differential of smooth maps between smooth manifolds . Aside: Note that the existence of all the partial derivatives of f ( x ) {\displaystyle f(x)} at x {\displaystyle x}

2948-408: The derivative of f at p may be captured by the equivalence class [ f  − f ( p )] in the quotient space I p / I p , and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions modulo I p . Algebraic geometers regard this equivalence class as the restriction of f to a thickened version of

3015-681: The difference, usually computed by XOR, between two plaintexts, and the difference of the corresponding ciphertexts Science and technology [ edit ] Differential (mechanical device) , as part of a motor vehicle drivetrain, the device that allows driving wheels or axles on opposite sides to rotate at different speeds Limited-slip differential Differential steering , the steering method used by tanks and similar tracked vehicles Electronic differential , an electric motor controller which substitutes its mechanical counterpart with significant advantages in electric vehicle application Differential signaling , in electronics, applies to

3082-420: The differential of f at p is the set of all functions differentially equivalent to f − f ( p ) {\displaystyle f-f(p)} at p . In algebraic geometry , differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements . The simplest example

3149-451: The equator every 25.05 days, as measured relative to distant stars (sidereal rotation). The highly turbulent nature of solar convection and anisotropies induced by rotation complicate the dynamics of modeling. Molecular dissipation scales on the Sun are at least six orders of magnitude smaller than the depth of the convective envelope. A direct numerical simulation of solar convection would have to resolve this entire range of scales in each of

3216-394: The equator, and Δ Ω = ( Ω 0 − Ω p o l e ) {\displaystyle \Delta \Omega =(\Omega _{0}-\Omega _{\mathrm {pole} })} is the difference in angular velocity between pole and equator, called the strength of the rotational shear. Ψ {\displaystyle \Psi }

3283-406: The factorisation of the different ideal Differential geometry , exterior differential, or exterior derivative , is a generalization to differential forms of the notion of differential of a function on a differentiable manifold Differential (coboundary) , in homological algebra and algebraic topology, one of the maps of a cochain complex Differential cryptanalysis , a pair consisting of

3350-427: The famous pamphlet The Analyst by Bishop Berkeley. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative of y at x is its instantaneous rate of change (the slope of the graph's tangent line ), which may be obtained by taking the limit of the ratio Δ y /Δ x as Δ x becomes arbitrarily small. Differentials are also compatible with dimensional analysis , where

3417-547: The idea that f ′ {\displaystyle f'} is the ratio of the differentials d f {\displaystyle df} and d x {\displaystyle dx} . This would just be a trick were it not for the fact that: If f {\displaystyle f} is a function from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } , then we say that f {\displaystyle f}

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3484-423: The idea that the derivative of y with respect to x is the limit of the ratio of differences Δ y /Δ x as Δ x approaches zero. Infinitesimal quantities played a significant role in the development of calculus. Archimedes used them, even though he did not believe that arguments involving infinitesimals were rigorous. Isaac Newton referred to them as fluxions . However, it was Gottfried Leibniz who coined

3551-424: The inner product and its associated norm define a suitable concept of distance. The same procedure works for a Banach space, also known as a complete Normed vector space . However, for a more general topological vector space, some of the details are more abstract because there is no concept of distance. For the important case of a finite dimension, any inner product space is a Hilbert space, any normed vector space

3618-549: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Differential&oldid=1209919598 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Differential (mathematics) The term is used in various branches of mathematics such as calculus , differential geometry , algebraic geometry and algebraic topology . The term differential

3685-509: The maps (or coboundary operators ) d i are often called differentials. Dually, the boundary operators in a chain complex are sometimes called codifferentials . The properties of the differential also motivate the algebraic notions of a derivation and a differential algebra . Differential rotation Differential rotation is seen when different parts of a rotating object move with different angular velocities (or rates of rotation ) at different latitudes and/or depths of

3752-537: The notion of differentials mathematically precise. These approaches are very different from each other, but they have in common the idea of being quantitative , i.e., saying not just that a differential is infinitely small, but how small it is. There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as linear maps . It can be used on R {\displaystyle \mathbb {R} } , R n {\displaystyle \mathbb {R} ^{n}} ,

3819-409: The one-dimensional case this becomes d f = d f d x d x {\displaystyle df={\frac {df}{dx}}dx} as before. This idea generalizes straightforwardly to functions from R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} . Furthermore, it has

3886-421: The optical spectrum depend on v rot sin(i), where i is the angle between the line of sight and the rotation axis, permitting the study of the rotational velocity's line-of-sight component v rot . This is calculated from Fourier transforms of the line shapes, using equation (2) below for v rot at the equator and poles. See also plot 2. Solar differential rotation is also seen in magnetograms, images showing

3953-486: The point p whose coordinate ring is not R (which is the quotient space of functions on R modulo I p ) but R [ ε ] which is the quotient space of functions on R modulo I p . Such a thickened point is a simple example of a scheme . Differentials are also important in algebraic geometry , and there are several important notions. A fifth approach to infinitesimals is the method of synthetic differential geometry or smooth infinitesimal analysis . This

4020-644: The same trick as in the one-dimensional case and think of the expression f ( x 1 , x 2 , … , x n ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n})} as the composite of f {\displaystyle f} with the standard coordinates x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} on R n {\displaystyle \mathbb {R} ^{n}} (so that x j ( p ) {\displaystyle x_{j}(p)}

4087-453: The sequence (1, 1/2, 1/3, ..., 1/ n , ...) represents an infinitesimal. The first-order logic of this new set of hyperreal numbers is the same as the logic for the usual real numbers, but the completeness axiom (which involves second-order logic ) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle . The notion of

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4154-413: The stars. Differential rotation affects stellar optical absorption-line spectra through line broadening caused by lines being differently Doppler-shifted across the stellar surface. Solar differential rotation causes shear at the so-called tachocline. This is a region where rotation changes from differential in the convection zone to nearly solid-body rotation in the interior, at 0.71 solar radii from

4221-462: The strength and location of solar magnetic fields. It may be possible to measure the differential of stars that regularly emit flares of radio emission. Using 7 years of observations of the M9 ultracool dwarf TVLM 513-46546, astronomers were able to measure subtle changes in the arrival times of the radio waves. These measurements demonstrate that the radio waves can arrive 1–2 seconds sooner or later in

4288-520: The term differentials for infinitesimal quantities and introduced the notation for them which is still used today. In Leibniz's notation , if x is a variable quantity, then dx denotes an infinitesimal change in the variable x . Thus, if y is a function of x , then the derivative of y with respect to x is often denoted dy / dx , which would otherwise be denoted (in the notation of Newton or Lagrange ) ẏ or y ′ . The use of differentials in this form attracted much criticism, for instance in

4355-436: The three dimensions. Consequently, all solar differential rotation models must involve some approximations regarding momentum and heat transport by turbulent motions that are not explicitly computed. Thus, modeling approaches can be classified as either mean-field models or large-eddy simulations according to the approximations. Disk galaxies do not rotate like solid bodies, but rather rotate differentially. The rotation speed as

4422-521: The validity of his argument against " the Ghosts of departed Quantities "; however, the modern approaches do not have the same technical issues. Despite the lack of rigor, immense progress was made in the 17th and 18th centuries. In the 19th century, Cauchy and others gradually developed the Epsilon, delta approach to continuity, limits and derivatives, giving a solid conceptual foundation for calculus. In

4489-433: The variable x {\displaystyle x} in f ( x ) {\displaystyle f(x)} as being a function rather than a number, namely the identity map on the real line, which takes a real number p {\displaystyle p} to itself: x ( p ) = p {\displaystyle x(p)=p} . Then f ( x ) {\displaystyle f(x)}

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