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67-499: [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

134-479: 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)

268-399: A cell have different sizes and densities, each fragment will settle into a pellet with different minimum centrifugal forces. Thus, separation of the sample into different layers can be done by first centrifuging the original lysate under weak forces, removing the pellet, then exposing the subsequent supernatants to sequentially greater centrifugal fields. Each time a portion of different density

335-405: A centrifuge is used, Stokes' law must be modified to account for the variation in g-force with distance from the center of rotation. where Differential centrifugation can be used with intact particles (e.g. biological cells, microparticles, nanoparticles), or used to separate the component parts of a given particle. Using the example of a separation of eukaryotic organelles from intact cells,

402-422: 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 }),}

469-548: 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

536-403: 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

603-422: 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 ,

670-669: 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

737-502: 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

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804-523: 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,

871-552: A refrigerated, low-pressure chamber containing a rotor which is driven by an electrical motor capable of high speed rotation. Samples are placed in tubes within or attached to the rotor. Rotational speed may reach up to 100,000 rpm for floor model, 150,000 rpm for bench-top model (Beckman Optima Max-XP or Sorvall MTX150 or himac CS150NX), creating centrifugal speed forces of 800,000g to 1,000,000g. This force causes sedimentation of macromolecules, and can even cause non-uniform distributions of small molecules. Since different fragments of

938-447: 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

1005-473: 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

1072-419: 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

1139-573: 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

1206-470: 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

1273-400: Is a common procedure used to separate organelles and other sub-cellular particles based on their sedimentation rate . Although often applied in biological analysis, differential centrifugation is a general technique also suitable for crude purification of non-living suspended particles (e.g. nanoparticles , colloidal particles, viruses ). In a typical case where differential centrifugation

1340-402: 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 }

1407-450: 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

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1474-461: 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

1541-415: 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

1608-445: Is sedimented to the bottom of the container and extracted, and repeated application produces a rank of layers which includes different parts of the original sample. Additional steps can be taken to further refine each of the obtained pellets. Sedimentation depends on mass, shape, and partial specific volume of a macromolecule, as well as solvent density, rotor size and rate of rotation. The sedimentation velocity can be monitored during

1675-430: 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

1742-450: Is that the latter method uses solutions of different densities (e.g. sucrose , Ficoll , Percoll ) or gels through which the sample passes. This separates the sample into layers by relative density, based on the principle that molecules settle down under a centrifugal force until they reach a medium with the density the same as theirs. The degree of separation or number of layers depends on the solution or gel. Differential centrifugation, on

1809-506: 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

1876-468: 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} )

1943-412: 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

2010-438: 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

2077-410: Is the successive pelleting of particles from the previous supernatant, using increasingly higher centrifugation forces. Cellular organelles separated by differential centrifugation maintain a relatively high degree of normal functioning, as long as they are not subject to denaturing conditions during isolation. In a viscous fluid, the rate of sedimentation of a given suspended particle (as long as

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2144-472: 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} }

2211-429: 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

2278-435: Is used to analyze cell-biological phenomena (e.g. organelle distribution), a tissue sample is first lysed to break the cell membranes and release the organelles and cytosol . The lysate is then subjected to repeated centrifugations , where particles that sediment sufficiently quickly at a given centrifugal force for a given time form a compact "pellet" at the bottom of the centrifugation tube. After each centrifugation,

2345-418: 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

2412-1018: 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

2479-400: 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

2546-405: The supernatant (non-pelleted solution) is removed from the tube and re-centrifuged at an increased centrifugal force and/or time. Differential centrifugation is suitable for crude separations on the basis of sedimentation rate, but more fine grained purifications may be done on the basis of density through equilibrium density-gradient centrifugation . Thus, the differential centrifugation method

2613-399: 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:

2680-418: 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

2747-465: 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

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2814-407: The basis of density, but also of particle size and shape. In contrast, a more specialized equilibrium density-gradient centrifugation produces a separation profile dependent on particle-density alone, and therefore is suitable for more fine-grained separations. High g-force makes sedimentation of small particles much faster than Brownian diffusion , even for very small (nanoscale) particles. When

2881-431: 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

2948-445: The cell must first be lysed and homogenized (ideally by a gentle technique, such as Dounce homogenization ; harsher techniques or over homogenization will lead to a lower proportion of intact organelles). Once the crude organelle extract is obtained, it may be subjected to a varying centrifugation speeds to separate the organelles: The lysed sample is now ready for centrifugation in an ultracentrifuge . An ultracentrifuge consists of

3015-685: 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

3082-406: 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}

3149-408: 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

3216-681: 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

3283-420: 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

3350-418: The experiment to calculate molecular weight . Values of sedimentation coefficient (S) can be calculated. Large values of S (faster sedimentation rate) correspond to larger molecular weight. Dense particle sediments more rapidly. Elongated proteins have larger frictional coefficients, and sediment more slowly to ensure accuracy. The difference between differential and density gradient centrifugation techniques

3417-406: 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

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3484-427: 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

3551-547: 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}

3618-423: 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

3685-424: 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

3752-549: 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

3819-454: 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 centrifugation In biochemistry and cell biology , differential centrifugation (also known as differential velocity centrifugation )

3886-537: 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}} ,

3953-409: 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

4020-401: The other hand, does not utilize a density gradient, and the centrifugation is taken in increasing speeds. The different centrifugation speeds often create separation into not more than two fractions, so the supernatant can be separated further in additional centrifugation steps. For that, each step the centrifugation speed has to be increased until the desired particles are separated. In contrast,

4087-401: The particle is denser than the fluid) is largely a function of the following factors: Larger particles sediment more quickly and at lower centrifugal forces. If a particle is less dense than the fluid (e.g., fats in water), the particle will not sediment, but rather will float, regardless of strength of the g-force experienced by the particle. Centrifugal force separates components not only on

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4154-486: 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

4221-644: 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)}

4288-453: 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

4355-520: 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

4422-512: 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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