In number theory , the arithmetic derivative , or number derivative , is a function defined for integer s, based on prime factorization , by analogy with the product rule for the derivativederivative of a function that is used in mathematical analysis . Definition For natural numbers defined as follows math p 1 math for any prime math p math . math ab a b , ,ab math for any math a textrm , , b in mathbb N math product rule Leibniz rule . To coincide with the Leibniz rule math 1 math is defined to be math 0 math , as is math 0 math . Explicitly, assume that math x p 1 e 1 cdots p k e k textrm , math where math p 1, , dots, , p k math are distinct primes and math e 1, , dots, , e k math are positive integers. Then math x sum i 1 k e ip 1 e 1 cdots p i 1 e i 1 p i e i 1 p i 1 e i 1 cdots p k e k sum i 1 k e i frac x p i . math The arithmetic derivative also preserves the power rule for primes math p a ap a 1 textrm , math where math p math is prime and math a math is a positive integer. For example, math begin align 81 3 4 & 9 cdot 9 9 cdot 9 9 cdot 9 2 9 3 cdot 3 & 2 9 3 cdot 3 3 cdot 3 2 9 cdot 6 108 4 cdot 3 3. end align math The sequence of number derivatives for k 0, 1, 2, ... begins ... defines the derivative over the integers. Barbeau also further extended it to rational numbers. Victor ... and bounds E. J. Barbeau examined bounds of the arithmetic derivative. He found that the arithmetic derivative of natural numbers is bounded by math n leq frac n log k n k math where k is the least ... to find similar bounds for the arithmetic derivative extended to rational numbers by proving that between ... derivative , Canadian Mathematical Bulletin Vol. 4 1961 , 117 122. Victor Ufnarovski and Bo ... ArithmeticDerivative.html Arithmetic Derivative , Planet Math , accessed 04 15, 9 April 2008 UTC L. Westrick. http web.mit.edu lwest www intmain.pdf Investigations of the Number Derivative . Peterson ... of the arithmetic derivative . Category Number theory Category Generalizations of the derivative numtheory ... more details
In mathematics , the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from derivative calculus . Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit mathematics limit , which is in general impossible to define for a ring mathematics ring . Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not. The primary use of formal differentiation in algebra is to test for multiple roots of a polynomial . Definition The definition of a formal derivative is as follows fix a ring R not necessarily commutative and let A R x be the ring of polynomials over R . Then the formal derivative is an operation on elements of A , where if math f x , ,a n x n cdots a 1 x a 0 math then its formal derivative is math f x , ,Df x n a n x n 1 cdots 2 a 2 x a 1 math just as for polynomials over the real numbers real or complex numbers complex numbers. Properties It can ... also be included as a linearity property. The formal derivative satisfies the Product rule Leibniz ... . Application to finding repeated factors As in calculus, the derivative detects multiple roots ... however, its formal derivative is zero since 3 0 in R and in any extension of R , so when we pass ... to analytic derivative When the ring R of scalars is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus ... with the formal derivative of f as it was defined above. This formulation of the derivative ... continuous at math X math , it will recapture the classical definition of the derivative. If it is carried .... This way differentiation becomes a part of algebra of functions. See also Derivative ... derivative References Lang Algebra edition 3r http arxiv.org abs 0905.3611 Michael Livshits, You could ... more details
lowercase In mathematics , in the area of combinatorics , the q derivative , or Jackson derivative , is a q analog of the ordinary derivative , introduced by Frank Hilton Jackson . It is the inverse of Jackson integral Jackson s q integration Definition The q derivative of a function f x is defined as math left frac d dx right q f x frac f qx f x qx x . math It is also often written as math D qf x math . The q derivative is also known as the Jackson derivative . It is a linear operator math displaystyle D q f x g x D q f x D q g x . math It has product rule analogous to the ordinary derivative product rule which has two equivalent forms math displaystyle D q f x g x g x D q f x f qx D q g x g qx D q f x f x D q g x . math Similarly it satisfies a quotient rule math displaystyle D q f x g x frac g x D q f x f x D q g x g qx g x , quad g x g qx neq 0. math There is also a rule similar to the chain rule for ordinary derivatives. Let math g x c x k math . Then math displaystyle D q f g x D q k f g x D q g x . math Relationship to ordinary derivatives Q differentiation resembles ordinary differentiation, with curious differences. For example, the q derivative of the monomial is math left frac d dz right q z n frac 1 q n 1 q z n 1 n q z n 1 math where math n q math is the q bracket of n . Note that math lim q to 1 n q n math so the ordinary derivative is regained in this limit. The n th q derivative of a function may be given as math D n q f 0 frac f n 0 n frac q q n 1 q n frac f n 0 n n q math provided that the ordinary n th derivative of f exists at x 0. Here, math q q n math is the q Pochhammer symbol , and math n q math is the q factorial . If math f x math is analytic we can apply the Taylor formula to the definition of math D q f x math to get math displaystyle D q f x sum ... See also Derivative generalizations Jackson integral Q exponential Q difference polynomial s Quantum ... 9908140 A note on the q derivative operator , 1999 ArXiv math 9908140 Thomas Ernst, http www.math.uu.se ... more details
unreferenced date March 2010 In mathematics , a weak derivative is a generalization of the concept of the derivative of a function mathematics function strong derivative for functions not assumed differentiable , but only Integrable function integrable , i.e. to lie in the Lebesgue space math L 1 a,b math . See distribution mathematics distribution s for an even more general definition. Definition Let math u math be a function in the Lebesgue space math L 1 a,b math . We say that math v math in math L 1 a,b math is a weak derivative of math u math if, math int a b u t varphi t dt int a b v t varphi t dt math for all infinitely differentiable function s math varphi math with math varphi a varphi b 0 math . This definition is motivated by the integration technique of Integration by parts . Generalizing ... alpha math is a multi index , we say that math v math is the math alpha th math weak derivative of math ... in math U math . If math u math has a weak derivative, it is often written math D alpha u math since ... is not differentiable at t 0, has a weak derivative v known as the sign function given ... if t 0. end cases math This is not the only weak derivative for u any w that is equal to v almost everywhere is also a weak derivative for u . Usually, this is not a problem, since in the theory of Lp ... yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero, math int 1 mathbb Q t varphi t dt 0 math Thus math v t 0 math is the weak derivative of math 1 mathbb Q ... are equivalent if they are equal almost everywhere, then the weak derivative is unique. Also, if u is differentiable in the conventional sense then its weak derivative is identical in the sense given above to its conventional strong derivative. Thus the weak derivative is a generalization of the strong ... for the weak derivative. Extensions This concept gives rise to the definition of weak solution s in Sobolev ... functions Category Functional analysis Category Generalizations of the derivative ca Derivada ... more details
In continuum mechanics , the material derivative ref name BSLr2 ref name Batchelor cite book first G.K. ... dependent flow velocity velocity field . The material derivative can serve as a link between Continuum .... Then the material derivative describes the temperature evolution of a certain fluid parcel ... other names for this operator, including convective derivative ref name Ockendon cite book first ... 2004 isbn 038740399X page 6 ref advective derivative substantive derivative ref name Granger cite ... 0486683567 page 30 ref substantial derivative ref name BSLr2 cite book last1 Bird first1 R.B. last2 ... isbn 978 0 470 11539 8 page 83 ref Lagrangian derivative ref name Mellor cite book first G.L. last ... 19 ref Stokes derivative ref name Granger particle derivative hydrodynamic derivative ref name BSLr2 derivative following the motion ref name BSLr2 total derivative ref name BSLr2 ref Cite book publisher ... derivative of a vector. In case of the material derivative of a vector field, the term v u can both be interpreted as v u involving the tensor derivative continuum mechanics tensor derivative of u , or as v ... convective derivative is both used for the whole material derivative D&phi Dt or D u Dt , and for only ..., the convective derivative only equals D Dt for time independent flows. These derivatives ... is represented by the vector field v x , t . The total derivative with respect to time of &phi is expanded ... t nabla varphi cdot frac d mathbf x d t math It is apparent that this derivative is dependent ... derivative becomes equal to the partial derivative, which agrees with the definition of a partial derivative a derivative taken with respect to some variable time in this case holding other variables constant space in this case . This makes sense because if math d mathbf x d t 0 math , then the derivative is taken at some constant position. This static position derivative is called the Eulerian derivative. An example of this case is a swimmer standing still and sensing temperature change in a lake ... more details
In mathematics , the metric derivative is a notion of derivative appropriate to Parametric equation parametrized path topology paths in metric space s. It generalizes the notion of speed or absolute velocity to spaces which have a notion of distance i.e. metric spaces but not direction such as vector space s . Definition Let math M, d math be a metric space. Let math E subseteq mathbb R math have a limit point at math t in mathbb R math . Let math gamma E to M math be a path. Then the metric derivative of math gamma math at math t math , denoted math gamma t math , is defined by math gamma t lim s to 0 frac d gamma t s , gamma t s , math if this Limit mathematics limit exists. Properties Recall that absolute continuity AC sup p sup I X is the space of curves I X such that math d left gamma s , gamma t right leq int s t m tau , mathrm d tau mbox for all s, t subseteq I math for some m in the Lp space L sup p sup space L sup p sup I R . For AC sup p sup I X , the metric derivative of exists for Lebesgue measure Lebesgue almost all times in I , and the metric derivative is the smallest m L sup p sup I R such that the above inequality holds. If Euclidean space math mathbb R n math is equipped with its usual Euclidean norm math math , and math dot gamma E to V math is the usual Fr chet derivative with respect to time, then math gamma t dot gamma t , math where math d x, y x y math is the Euclidean metric. References cite book author Ambrosio, L., Gigli, N. & Savar , G. title Gradient Flows in Metric Spaces and in the Space of Probability Measures publisher ETH Z rich, Birkh user Verlag, Basel year 2005 isbn 3 7643 2428 7 Category Differential calculus Category Metric geometry ... more details
In mathematical analysis , a Pompeiu derivative is a real valued function mathematics function of one real variable that is the derivative mathematics derivative of an everywhere differentiable function and that vanishes in a dense set . Note in particular that a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non identically zero such functions may exist was a problem that arose in the context of early 1900s research on functional differentiability and integrability. The question was affirmatively answered by Dimitrie Pompeiu by constructiong an explicit example these functions are therefore named after him. Pompeiu s construction Pompeiu s construction is described here. Let math sqrt 3 x math denote the real cubic root of the real number math x. math Let math q j j in N math be an enumeration of the rational numbers in the unit interval math 0, ,1 . math Let math a j j in N math be positive real numbers with math textstyle sum j a j infty. math Define, for all math x in 0, ,1 math math g x sum j 0 infty ,a j sqrt 3 x q j . math Since for any math x in 0, ,1 math each term of the series is less than or equal to a sub j sub in absolute value, the series uniform convergence uniformly converges to a continuous, strictly increasing function g x , due to the Weierstrass M test . Moreover, it turns out that the function g is differentiable, with math g prime ... 1 math has a finite derivative at any point, which vanishes at least in the points math g q j j in N ... below . Properties It is known that the zero set of a derivative of any everywhere differentiable ... of Pompeiu functions is a derivative, and vanishes on the set f 0 g ... is a Pompeiou derivative. Indeed, it is a derivative, due to the theorem of limit under the sign of derivative ... s function a theorem of Weil states that generically a Pompeiu derivative assumes both positive ... series, Montreal 1994 . DEFAULTSORT Pompeiu Derivative Category Real analysis ... more details
Unreferenced date August 2009 Parametric derivative is a derivative in calculus that is taken when both the x and y variables traditionally independent variable independent and dependent variable dependent , respectively depend on an independent third variable t , usually thought of as time . Example For example, consider the set of function mathematics function s where math x t 4t 2 , math and math y t 3t. , math The first derivative of the parametric equation s above is given by math frac frac dy dt frac dx dt frac dot y t dot x t , math where the notation math dot x t math denotes the derivative of x with respect to t , for example. To understand why the derivative appears in this way, recall the chain rule for derivatives math frac dy dx frac dy dt cdot frac dt dx , math or in other words math frac dy dx frac frac dy dt frac dx dt . math More formally, by the chain rule math frac dy dt frac dy dx cdot frac dx dt math and dividing both sides by math frac dx dt math gets the equation above. Differentiating both functions with respect to t leads to math frac dx dt 8t math and math frac dy dt 3, math respectively. Substituting these into the formula for the parametric derivative, we obtain math frac dy dx frac dot y dot x frac 3 8t , math where math dot x math and math dot y math are understood to be functions of t . The second derivative of a parametric equation is given by math frac d 2y dx 2 math math frac d dx left frac dy dx right math math frac d dt left frac dy dx right cdot frac dt dx math math frac d dt left frac dot y dot x right frac 1 dot x math math frac dot x ddot y dot y ddot x dot x 3 math by making use of the quotient rule for derivatives. The latter result is useful in the computation of curvature . See also Derivative generalizations Parametric equation DEFAULTSORT Parametric Derivative Category Differential calculus eo Parametra deriva o ... more details
Image 4 fonctions du second degr .svg right thumb 200px The second derivative of a quadratic function is constant function constant . In calculus , the second derivative of a function mathematics function &fnof is the derivative of the derivative of &fnof . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing for example, the second derivative of the position ... at which the velocity of the vehicle is changing. On the graph of a function , the second derivative ... second derivative curves upwards, while the graph of a function with negative second derivative curves downwards. Notation Details Notation for differentiation The second derivative of a function ... for derivatives, the second derivative of a dependent variable y with respect to an independent variable ..., math the derivative of &fnof is the function math f x 3x 2. math The second derivative of &fnof is the derivative ... derivative of a function &fnof measures the concavity of the graph of &fnof . A function whose second derivative is positive will be concave up sometimes referred to as convex , meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative ... will lie above the graph of the function. Inflection points main Inflection point If the second derivative ... derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection. Second derivative test main Second derivative test The relation between the second derivative and the graph can be used ... x math . If math f prime prime x 0 math , the second derivative test says nothing about the point math x math , a possible inflection point. The reason the second derivative produces these results can ... to write a single Limit mathematics limit for the second derivative math f x lim h to 0 frac f ... . Quadratic approximation Just as the first derivative is related to linear approximation s, the second ... more details
In mathematics , the covariant derivative is a way of specifying a derivative along tangent vector s of a manifold . Alternatively, the covariant derivative is a way of introducing and working with a connection ... Euclidean space , the covariant derivative can be viewed as the orthonormal projection of the Euclidean derivative along a tangent vector onto the manifold s tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component. This article presents an introduction to the covariant derivative of a vector ... system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion ... . Introduction and history Historically, at the turn of the 20th century, the covariant derivative ... a notion of derivative differentiation which generalized the classical directional derivative of vector fields on a manifold. This new derivative the Levi Civita connection was Covariance and contravariance ... , 325 412. ref that a covariant derivative could be defined abstractly without the presence of a metric ... law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory ... the classical notion of covariant derivative in many post 1950 treatments of the subject. Motivation The covariant derivative is a generalization of the directional derivative from vector calculus . As with the directional derivative, the covariant derivative is a rule, math nabla bold u bold ... , defined in a neighborhood of P . ref The covariant derivative is also denoted variously by math ... bold u bold v P math , also at the point P . The primary difference from the usual directional derivative ... transformation . The covariant derivative is required to transform, under a change in coordinates, in the same way as a vector does the covariant derivative must change by a covariant transformation ... more details
Image Marcel Duchamp Mona Lisa LHOOQ.jpg thumb 250px L.H.O.O.Q. 1919 . Derivative work by Marcel Duchamp .... Often used by law professors to illustrate legal concept of derivative work. In United States copyright law , a derivative work is an expressive creation that includes major, copyright protected elements of an original, previously created first work the underlying work . Derivative being something ... law Definition In the United States, the Copyright Act defines derivative work in UnitedStatesCode 17 101 blockquote A derivative work is a work based upon one or more pre existing works, such as a translation ... modifications which, as a whole, represent an original work of authorship, is a derivative work ... or derivative work extends only to the material contributed by the author of such work, as distinguished ... of the following 1 to reproduce the copyrighted work in copies... 2 to prepare derivative works based ... circs circ14.pdf US Copyright Office Circular 14 Derivative Works notes that blockquote A typical example of a derivative work received for registration in the Copyright Office is one that is primarily ... makes the work a derivative work under the copyright law. To be copyrightable, a derivative work must .... blockquote The statutory definition is incomplete and the concept of derivative work must be understood with reference to explanatory case law . Three major copyright law issues arise concerning derivative works 1 what acts are sufficient to cause a copyright protected derivative work to come into existence ... protected derivative work excused from liability by an affirmative defense, such as first sale or fair use ? When does derivative work copyright exist? For copyright protection to attach to a later, allegedly derivative work, it must display some originality of its own. It cannot be a rote, uncreative ... this requirement in relation to derivative works. In Durham Industries, Inc. v. Tomy Corp. ref ... cases fulltext batlinsnydertext.htm ref the Second Circuit held that a derivative work ... more details
In mathematics , the Pincherle derivative of a linear operator math scriptstyle T mathbb K x longrightarrow mathbb K x math on the vector space of polynomial s in the variable math scriptstyle x math over a field mathematics field math scriptstyle mathbb K math is another linear operator math scriptstyle T mathbb K x longrightarrow mathbb K x math defined as math T T,x Tx xT operatorname ad x T, , math so that math T p x T xp x xT p x qquad forall p x in mathbb K x . math In other words, Pincherle derivation is the commutator of math scriptstyle T math with the multiplication by math scriptstyle x math in the endomorphism ring algebra of endomorphisms math scriptstyle operatorname End left mathbb K x right math . This concept is named after the Italian mathematician Salvatore Pincherle 1853&ndash 1936 . Properties The Pincherle derivative, like any commutator , is a derivation abstract algebra derivation , meaning it satisfies the sum and products rules given two linear operator s math scriptstyle S math and math scriptstyle T math belonging to math scriptstyle operatorname End left mathbb K x right math math scriptstyle T S prime T prime S prime math math scriptstyle TS prime T prime ... T,S TS ST math is the usual Lie algebra Lie bracket . The usual derivative, D d dx , is an operator on polynomials. By straightforward computation, its Pincherle derivative is math ... derivative of a differential operator math partial sum a n d n over dx n sum a n D n math is also a differential operator, so that the Pincherle derivative is a derivation of math scriptstyle operatorname ... S h sum n 0 h n over n D n math by the Taylor formula . Its Pincherle derivative is then math S h sum ... derivative, whose spectrum is the whole space of scalars math scriptstyle mathbb K math ... derivative is the shift operator math scriptstyle delta S h math . See also Commutator Delta operator ... Pincherle Derivative . From MathWorld A Wolfram Web Resource. http www history.mcs.st andrews.ac.uk ... more details
In mathematics , the Lie derivative IPAc en icon l i , named after Sophus Lie by W adys aw lebodzi ski ... is coordinate invariant and therefore the Lie derivative is defined on any manifold differentiable manifold . The Lie derivative along a vector field is the evaluation of the vector field on functions ... field s over a manifold M . It also commutes with contraction and the exterior derivative on differential forms . This uniquely determines the Lie derivative and it follows that for vector fields the Lie derivative is the commutator math mathcal L X Y X, Y math It also shows that the Lie derivatives ... mathematics connection and vector valued differential forms . Definition The Lie derivative ... the Lie derivative acting on scalar functions and vector fields. The Lie derivative can also be defined to act on general tensors, as developed later in the article. The Lie derivative of a function Einstein summation convention There are several equivalent definitions of a Lie derivative of a function. The Lie derivative can be defined in terms of the definition of vector field s as first ... derivative math mathcal L X f math of a function along a vector field math X math is simply the application of the vector field. It can be interpreted as the directional derivative of f along ... a f p math recovering the original definition. Alternatively, the Lie derivative can be defined ... Frobenius theorem . The Lie derivative of a vector field The Lie derivative can be defined for vector .... Regardless of the chosen definition, one then defines the Lie derivative of the vector ... the tangent map derivative operator math mathcal L X Y x lim t to 0 frac mathrm d Phi X t Y Phi X t x ... derivative of differential forms The Lie derivative can also be defined on differential forms . In this context, it is closely related to the exterior derivative . Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences ... more details
A shareholder derivative suit is a lawsuit brought by a shareholder on behalf of a corporation against a third party. Often, the third party is an insider of the corporation, such as an executive officer or director. Shareholder derivative suits are unique because under traditional corporation corporate law , management is responsible for bringing and defending the corporation against suit. Shareholder derivative suits permit a shareholder to initiate a suit when management has failed to do so. Because Clarify date February 2010 derivative suits vary the traditional roles Citation needed date February 2010 of management and shareholders, many jurisdictions have implemented various procedural requirements to derivative suits. Purpose and difficulties Under traditional corporate business law, shareholders are the owners of a corporation. However, they are not empowered to control the day to day operations of the corporation. Instead, shareholders appoint directors, and the directors in turn appoint officers or executives to manage day to day operations. Derivative suits permit a shareholder to bring an lawsuit action in the name of the corporation against parties allegedly causing harm to the corporation. If the directors, officers, or employees of the corporation are not willing to file an action, a shareholder may first petition them to proceed. If such petition fails, the shareholder ... fees in the event that he does not prevail. Derivative suits in the United States In the United States ..., institute a number of barriers to derivative suits. Under the Model Business Corporation Act MBCA , the procedure of a derivative suit is as follows. There has been harm to the corporation but the board ... states adhere to the MBCA procedures to varying degrees. In New York , for example, derivative ... Court , originated with a shareholder derivate suit against Greyhound Bus Lines . Derivative suits .... Roberts v Gill & Co Solicitors 2010 http www.bailii.org uk cases UKSC 2010 22.html UKSC 22 Derivative ... more details
In mathematics , the reduced derivative is a generalization of the notion of derivative that is well suited to the study of functions of bounded variation . Although functions of bounded variation have derivatives in the sense of Radon measure s, it is desirable to have a derivative that takes values in the same space as the functions themselves. Although the precise definition of the reduced derivative is quite involved, its key properties are quite easy to remember it is a multiple of the usual derivative wherever it exists at jump points, it is a multiple of the jump vector. The notion of reduced derivative appears to have been introduced by Alexander Mielke and Florian Theil in 2004. Definition Let X be a separable space separable , reflexive space reflexive Banach space with norm mathematics norm and fix T > 0. Let BV sub &minus sub 0, T X denote the space of all continuous function left continuous functions z 0, T &rarr X with bounded variation on 0, T . For any function of time f , use subscripts &minus to denote the right left continuous versions of f , i.e. math f t lim s downarrow t f s math math f t lim s uparrow t f s . math For any sub interval a , b of 0, T , let Var z , a , b denote the variation of z over a , b , i.e., the supremum math mathrm Var z, a, b sup left left. sum i 1 k z t i z t i 1 right a t 0 t 1 cdots t k b, k in mathbb N right . math The first step in the construction of the reduced derivative is the &ldquo stretch&rdquo time so that z can be linearly interpolated ... T X math math left frac mathrm d hat z mathrm d tau right L infty 0, hat T X leq 1. math The derivative ... . The reduced derivative of z is the pull back dn date August 2011 of this derivative by the stretching ... d tau left frac hat tau t hat tau t 2 right . math Associated with this pull back of the derivative ... t 1 , t 2 mathrm d z . math Properties The reduced derivative rd z is defined only &mu sub z sub almost ... more details
In the mathematics mathematical field of differential calculus , the term total derivative has a number of closely related meanings. ul li The total derivative full derivative of a function math f math ... input variables, e.g., math t math , is different from its partial derivative partial derivative math partial math . Calculation of the total derivative of math f math with respect to math t math ... the other arguments to depend on math t math . The total derivative adds in these indirect dependencies to find the overall dependency of math f math on math t math . For example, the total derivative ... on math t math , some of that change will be due to the partial derivative of math f math with respect ... dx frac partial partial y j , math which computes the total derivative of a function with respect ... infinitesimally, or by using differential forms and the exterior derivative . li ul It is another name for the derivative as a linear map, i.e., if f is a differentiable function Differentiability ... derivative or differential of f at x &isin R sup n sup is the linear map from R sup n sup to R sup ... the derivative of a function from R sup n sup to R . It is sometimes used as a synonym for the material derivative , math frac D mathbf u Dt math , in fluid mechanics. Differentiation with indirect ... the partial derivative of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y . The total derivative takes such dependencies ... to x is usually the partial derivative of f with respect to x in this case, math frac partial f partial x y math . However, if y depends on x , the partial derivative does not give the true rate ... y x math then math f x,y f x,x x 2 math . In that case, the total derivative of f with respect to x ... to the partial derivative math frac mathrm d f mathrm d x 2 x neq frac partial f ... time derivative of M is math operatorname d M over operatorname d t frac operatorname ... more details
In chemistry , a derivative is a Chemical compound compound that is wikt derive Verb derived from a similar compound by some chemical or physical process. In the past it was also used to mean a compound that can be imagined to arise from another compound, if one atom is replaced with another atom or group of atoms, ref name chemicool cite web title Definition of Derivative publisher Chemicool date 2007 09 18 url http www.chemicool.com definition derivative.html accessdate 2007 09 18 ref but modern chemical language now uses the term structural analogue for this meaning thus eliminating ambiguity of both terms. The term structural analogue is common in organic chemistry . In biochemistry , the word is used for compounds that at least theoretically can be formed from the wiktionary precursor precursor compound. ref name Oxford cite book title Oxford Dictionary of Biochemistry and Molecular Biology publisher Oxford University Press isbn 0 19 850673 2 ref Chemical derivatives may be used to facilitate analysis. For example, melting point MP analysis can assist in identification of many organic compounds. A crystalline derivative may be prepared, such as a semicarbazone or 2,4 dinitrophenylhydrazone derived from aldehyde s ketone s , as a simple way of verifying the identity of the original compound, assuming that a table of derivative MP values is available. ref name Williamson cite book last Williamson first Kenneth L. title Macroscale and Microscale Organic Experiments, 3rd ed. publisher Houghton Mifflin year 1999 location Boston pages 426 7 isbn 0 395 90220 7 ref Prior to the advent of spectroscopic analysis , such methods were widely used. See also Derivatization References references Category Chemical compounds cs Deriv t chemie da Derivat kemi de Derivat Chemie et Derivaat ko hr Derivat he nl Derivaat scheikunde ja pl Pochodna chemia pt Derivado qu mica sk Deriv t ch mia sl Derivat sr Derivat hemija sv Derivat zh ... more details
calculus cTopic Multivariable calculus In mathematics , a partial derivative of a function mathematics function of several variables is its derivative with respect to one of those variables, Ceteris paribus with the others held constant as opposed to the total derivative , in which all variables are allowed ... derivative of a function f with respect to the variable x is variously denoted by math f prime x, f x, f ,x , partial x f, text or frac partial f partial x math The partial derivative symbol is . The notation ... sup 2 sup . For the partial derivative at nowrap 1, 1, 3 that leaves y constant, the corresponding ... it, we see the way the function looks on the plane nowrap y 1 . By finding the derivative of the equation .... Therefore math frac part z part x 3 math at the point. nowrap 1, 1, 3 . That is, the partial derivative ... y . Consequently, the definition of the derivative for a function of one variable applies math ... math frac part f part y x,y x 2y. , math This is the partial derivative of f with respect to y . Here is a rounded d called the partial derivative symbol . To distinguish it from the letter d , is sometimes pronounced del or partial instead of dee . In general, the partial derivative of a function ... space R sup n sup e.g., on R sup 2 sup or R sup 3 sup . In this case f has a partial derivative ... x n bigg mathbf hat e n math Formal definition Like ordinary derivatives, the partial derivative ... U R a function. The partial derivative of f at the point a a sub 1 sub , ..., a sub n sub U with respect ... neighborhood of a and are continuous there, then f is total derivative totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C ... using a componentwise argument. The partial derivative math frac partial f partial x math can ... derivative of V with respect to r is math frac partial V partial r frac 2 pi r h 3 , math which represents .... The partial derivative with respect to h is math frac partial V partial h frac pi r 2 3 , math ... more details
In mathematics , the directional derivative of a multivariate differentiable function along a given vector ... derivative , in which the direction is always taken parallel to one of the coordinate axes . The directional derivative is a special case of the G teaux derivative . Definition The directional derivative of a scalar function math f bold x f x 1, x 2, ldots, x n math along a unit vector math ... at x , then the directional derivative exists along any unit vector u , and one has math ... derivative of f intuitively represents the derivative rate of change in math f math along ... derivative to be taken in the direction of v , where v is any nonzero vector. In this case, one must ... derivative hold for the directional derivative. These include, for any functions f and g defined in a neighborhood mathematics neighborhood of, and total derivative differentiable at, p ol li The Sum ... in a neighborhood of p , and total derivative differentiable at p . If v is a tangent vector to M at p , then the directional derivative of f along v , denoted variously as math nabla v f p math see covariant derivative , math L v f p math see Lie derivative , or math v p f math see Tangent space ... 0 p and &prime 0 v . Then the directional derivative is defined by math nabla v f p left. frac ..., provided is selected in the prescribed manner so that &prime 0 v . Normal derivative A normal derivative is a directional derivative taken in the direction normal that is, orthogonal to some surface ... derivative of a function is sometimes denoted as math frac partial f partial n ... v math be a real valued function of the vector math mathbf v math . Then the derivative of math f mathbf ... math mathbf v math . Then the derivative of math mathbf f mathbf v math with respect to math mathbf ... math boldsymbol S math . Then the derivative of math f boldsymbol S math with respect to math boldsymbol ... tensor valued function of the second order tensor math boldsymbol S math . Then the derivative of math ... more details
In mathematics , the G teaux differential or G teaux derivative is a generalization of the concept of directional derivative in differential calculus . Named after Ren G teaux , a French mathematician ... space s such as Banach space s. Like the Fr chet derivative on a Banach space, the G teaux differential is often used to formalize the functional derivative commonly used in the calculus of variations ... between the G teaux differential which may be nonlinear and the G teaux derivative which they take ... is taken instead of a strong limit, which leads to the notion of a weak G teaux derivative. Linearity ... differential may fail to be linear, unlike the Fr chet derivative . Even if linear, it may fail ... is linear but not continuous. Relation with the Fr chet derivative If F is Fr chet differentiable, then it is also ... is clearly not true, since the G teaux derivative may fail to be linear or continuous. In fact, it is even possible for the G teaux derivative to be linear and continuous but for the Fr chet derivative ... space Y , the G teaux derivative where the limit is taken over complex tending to zero as in the definition ... of harvtxt Zorn 1945 . Furthermore, if F is complex G teaux differentiable at each u U with derivative math DF u psi mapsto dF u psi math then F is Fr chet differentiable on U with Fr chet derivative ... L X , L sup n 1 sup X , Y , higher order G teaux derivative cannot be defined in this way. Instead the n th order G teaux derivative of a function F U X Y in the direction ... candidate for the definition of the higher order derivative, the function NumBlk math ... math is continuous in the product topology, and moreover that the second derivative defined by EquationRef ... order derivative D sup 2 sup F u with the differential d sup 2 sup F u . Similar conclusions ... for the G teaux derivative of F , provided F is assumed to be sufficiently continuously differentiable. Specifically Suppose that F X Y is C sup 1 sup in the sense that the G teaux derivative is a continuous ... more details
In mathematics , the Fox derivative is an algebra ic construction in the theory of free group s which bears many similarities to the conventional derivative of calculus . The Fox derivative and related concepts are often referred to as the Fox calculus , or Fox s original term the free differential calculus . The Fox derivative was developed in a series of five papers by mathematician Ralph Fox , published in Annals of Mathematics beginning in 1953. Definition If G is a free group with identity element e and generating set of a group generators g sub i sub , then the Fox derivative with respect to g sub i sub is a function from G into the integral group ring Z G which is denoted math frac partial partial g i math , and obeys the following axiom s math frac partial partial g i g j delta ij math , where math delta ij math is the Kronecker delta math frac partial partial g i e 0 math math frac partial partial g i uv frac partial partial g i u u frac partial partial g i v math for any elements u and v of G . The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule . As a consequence of the axioms, we have the following formula for inverses math frac partial partial g i u 1 u 1 frac partial partial g i u math for any element u of G . Applications The Fox derivative has applications in group cohomology , knot theory and covering space theory, among other areas of mathematics. See Also Alexander polynomial Alexander module Free group Ring mathematics Integral domain References Cite book first Kenneth S. last1 Brown title Cohomology of Groups publisher Springer Verlag year 1972 isbn 0 387 90688 6 series Graduate Texts in Mathematics id MathSciNet id 0672956 volume 87 postscript None cite journal last Fox first Ralph authorlink Ralph Fox year 1953 month May title Free Differential Calculus, I Derivation in the Free Group Ring journal Annals of Mathematics volume 57 issue 3 pages 547 ... more details
In differential geometry , the exterior derivative extends the concept of the pushforward differential ... form was invented by lie Cartan . The exterior derivative d has the property that nowrap 1 d sup ..., the exterior derivative is the dual of the Chain complex Formal definition boundary map on singular simplices. Definition The exterior derivative of a differential form of degree k is a differential form of degree nowrap 1 k 1. There are a variety of equivalent definitions of the exterior derivative. Exterior derivative of a function If is a smooth function, then the exterior derivative of is the Pushforward ... smooth vector field X , nowrap 1 d &fnof X X&fnof , where X is the directional derivative of in the direction of X . Thus the exterior derivative of a function or 0 form is a one form. Exterior derivative of a k form The exterior derivative is defined to be the unique R linear mapping from ... are forms of degree 0. Exterior derivative in local coordinates Alternatively, one can work entirely ... 1 1 &le p &le k , the exterior derivative of a k form math omega f I text d x I f i 1,i 2 cdots ... index I run over all the values in 1, ..., n , the definition of the exterior derivative ... 1 d x sub i sub &and d x sub I sub 0 see wedge product . The definition of the exterior derivative ... I sub as a zero form, and then applied the properties of the exterior derivative. Invariant formula Alternatively, an explicit formula can be given for the exterior derivative of a k form , when ... 2 sup over a 1 form basis nowrap 1 d x sup 1 sup ,...,d x sup n sup . The exterior derivative is math ... in contractible regions, by the Poincar lemma . Naturality The exterior derivative is natural ... from sup k sup to sup k 1 sup . The exterior derivative in calculus Most vector calculus .... Gradient A smooth function f R sup n sup R is a 0 form. The exterior derivative of this 0 form ... is the flux of V over that hypersurface. The exterior derivative of this n &minus 1 form ... more details
The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. In a certain sense, it is arguably a more natural generalization of the single variable derivative. It allows a generalization of the single variable fundamental theorem of calculus to higher dimensions, in a different vein than the generalization that is Stokes theorem . Formal definition Let math G math be a Lie group , and let math mathfrak g math be its Lie algebra . The Maurer Cartan form , math omega G math , is the smooth math mathfrak g math valued math 1 math form on math G math cf. Lie algebra valued form defined by math omega G X g T g L g 1 X g math for all math g in G math and math X g in T g G math . Here math L g math denotes left multiplication by the element math g in G math and math T g L g math is its derivative at math g math . Let math f M to G math be a smooth function between a smooth manifold math M math and math G math . Then the Darboux derivative of math f math is the smooth math mathfrak g math valued math 1 math form math omega f f omega G, math the pullback differential geometry pullback of math omega G math by math f math . The map math f math is called an integral or primitive of math omega f math . More natural? The reason that one might call the Darboux derivative a more natural generalization of the derivative of single variable calculus is this. In single variable calculus, the derivative math f math of a function math f mathbb R to mathbb R math assigns to each point in the domain a single number. According to the more general manifold ideas of derivatives, the derivative assigns to each point in the domain a linear map from the tangent space at the domain point to the tangent space at the image point. This derivative encapsulates ... i.e. a number . One way to justify this convention of retaining only the linear map aspect of the derivative ... are simply translation. By post composing the manifold type derivative with the tangent space ... more details
In mathematics , the Schwarzian derivative , named after the German mathematician Hermann Schwarz , is a certain operator that is invariant under all linear fractional transformation s. Thus, it occurs in the theory of the complex projective line , and in particular, in the theory of modular forms and hypergeometric functions . It plays an important role in the theory of univalent function s, conformal mapping and Teichm ller space s. Definition The Schwarzian derivative of a function of one complex ... notation math f,z Sf z , math is frequently used. Properties The Schwarzian derivative of any fractional ... transformations are the only functions with this property. Thus, the Schwarzian derivative precisely ... has the same Schwarzian derivative as math VAR f VAR . On the other hand, the Schwarzian derivative ... the Schwarzian derivative an important tool in one dimensional Dynamical system dynamics ref http mathworld.wolfram.com SchwarzianDerivative.html Weisstein, Eric W. Schwarzian Derivative. From MathWorld ... Schiffer 1966 ref math F z,w log f z f w over z w , math its second mixed partial derivative is given ... w 2 , math and the Schwarzian derivative is given by the formula math Sf z left. 6 cdot partial 2 F z,w over partial z partial w right vert z w . math The Schwarzian derivative has a simple inversion ... equation The Schwarzian derivative has a fundamental relation with a second order linear ... math S f le 2. math Conformal mapping of circular arc polygons The Schwarzian derivative and associated ... Christoffel mapping , which can be derived directly without using the Schwarzian derivative. The accessory ... hemisphere with D , Lipman Bers used the Schwarzian derivative to define a Bers embedding mapping ... 1 sup invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill s operator ... The Schwarzian derivative and schlicht functions doi 10.1090 S0002 9904 1949 09241 8 id MathSciNet ... title What Is . . . the Schwarzian Derivative? journal AMS Notices volume 56 issue 01 pages 34 36 year ... more details
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