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Functional Derivative

Functional derivative

In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. The difference is that for the former one differentiates in the direction of a vector, while for the latter in the direction of a function. Both of these can be viewed as extensions of the usual calculus derivative. Two possible, restricted definitions suitable for certain computations are given here. There are more general definitions of functional derivatives. For a functional F mapping (continuous/smooth/with certain boundary conditions/etc.) functions φ from a manifold M to R or C, the functional derivative of F, denoted δF is a distribution such that for all test functions f, :

\delta F[f]=\fracF[\phi+\epsilon f].

Another definition is in terms of a limit and the Dirac delta function, δ: :

\frac=\lim_\frac.

Formal description

The definition of a functional derivative may be made much more mathematically precise and formal by defining the space of functions more carefully. For example, when the space of functions is a Banach space, the functional derivative becomes known as the Gâteaux derivative, and on Fréchet spaces, the Fréchet derivative. Note that the well-known Hilbert space is a special case of a Banach space. The more formal treatment allows many theorems from ordinary calculus and analysis to be generalized to corresponding theorems in functional analysis, as well as numerous new theorems to be stated. Category:Functional analysis

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given unit vector intuitively represents the rate of change of the function in the direction of that vector. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes.

Definition

The directional derivative of a scalar function

f(\vec) = f(x_1, x_2, \ldots, x_n)

along a unit vector

\vec = (v_1, \ldots, v_n)

is the function defined by the limit :

D_ = \lim_.

If the function is differentiable, it can be written in terms of the gradient

\nabla(f)

f

by :

D_ = \nabla(f) \cdot \vec

where

\cdot

denotes the dot product (Euclidean inner product). At any point

p

, the directional derivative of

f

intuitively represents the rate of change in

f

in the direction of

\vec

at the point

p

The directional derivative in differential geometry

A vector field at a point

p

naturally gives rise to linear functionals defined on

p

by evaluating the directional derivative of a differentiable function

f

along the unit vector

\vec/||\vec||

where

\vec

is the vector of the tangent space at

p

assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at

p

in the direction of

\vec

Normal derivative

A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a unit normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition.

Derivative

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.) The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:
- The derivative gives the slope of a tangent to the graph of the function at a point. In this way, derivatives can be used to determine many geometrical properties of the graph, such as concavity or convexity.
- The derivative provides a mathematical formulation of rate of change; it measures the rate at which the function's value changes as the function's argument changes. This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative. The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

In physical terms, differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients :

\frac

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written :

\frac

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dy by dx" or "dy over dx". The form "dy dx" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as: :

\lim_\frac.

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line. tangent tangent To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is :

.

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: :

f'(x)=\lim_.

difference quotient If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x. Since immediately substituting 0 for h results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens easily for polynomials; see calculus with polynomials. For almost all functions however, the result is a mess. Fortunately, many guidelines exist.

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write: : $\frac.$ We can write the derivative of f at the point a in two different ways: : $\frac\left.\right|_ = \left(\frac\right)(a).$ If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as: : $\frac.$ Higher derivatives are expressed as : $\frac$ or $\frac$ for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is: : $\frac$ which we can loosely write as: : $\left(\frac\right)^3 \left(f(x)\right) =
\frac \left(f(x)\right).$ Dropping brackets gives the notation above. Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel: : $\frac = \frac \cdot \frac.$ (In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)
Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name: : $\dot = \frac = x'(t)$ : $\ddot = x$ (t) and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator: This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable: Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.

Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration. For example, if an object's position

p(t) = -16t^2 + 16t + 32

; then, the object's velocity is

\dot p(t) = p'(t) = -32t + 16

; the object's acceleration is

\ddot p(t) = p

(t) = -32; and the object's jerk is $p(t) = 0.$ If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic manipulation
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $cf(x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
- Linearity: $(af + bg)' = af' + bg'$ for all functions f and g and all real numbers a and b.
- Power rule: If $f(x) = x^r$ , for some real number r; $f'(x) = rx^$ .
- Product rule: $(fg)' = f'g + fg'$ for all functions f and g.
- Quotient rule: $(f/g)' = (f'g - fg')/(g^2)$ unless g is zero.
- Chain rule: If $f(x) = h(g(x))$ , then $f'(x) = h'(g(x)) g'(x)$ .
- Inverse function: If $g(x) = f^(x)$ , and $f(x)$ is injective, then $g'(x) = 1/f'(f^(x))$ .
- Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f(t)$ and $y = g(t)$ . Now $d y/d x = (d y/d t)/(d x/d t)$ . This is the chain rule in the Leibniz notation.
- Implicit differentiation: If $f(x,y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y). In addition, the derivatives of some common functions are useful to know. See the table of derivatives. As an example, the derivative of : $f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$ is : $f'(x) = 8x^3 + 2x\cos (x^2) - \frac\;e^x - \ln (x)\;e^x.$
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither. In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3). Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Generalizations
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'. The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles. In order to differentiate all continuous functions and much more, one defines the concept of distribution. For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.
See also

- Derivative (examples)
- Derivative (generalizations)
- Partial derivative
- Total derivative
- Table of derivatives
- Smooth function
- Differintegral
- Automatic differentiation
External links

- [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives; this software enables also interactive exercises.
References

- Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896. Explains why all this works.
- Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner. "What one fool can do, another can."
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.
- Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.
- ko:미분 ja:微分 simple:Derivative th:อนุพันธ์

Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. Intuitively, a function is continuous if it maps nearby points to nearby points. For metric spaces, nearness is measured in terms of distance, leading to the ε-δ definition used in real analysis. For more general topological spaces, nearness is measured less directly in terms of open sets, leading to the definition below. If a topological space has the metric topology, the two definitions coincide. Given a set X a partial ordering can be defined on the possible topologies on X. A continuous functions between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space. Thus we can consider the continuity of a given function a topological property, depending only on the topologies of its domain and codomain spaces. A continuous function can be visualized as weakening the topology of the domain space. In real analysis continuity of functions is commonly defined using the ε-δ definition which builds on the property of the real line being a metric space. As topological spaces generally do not have this property a more general definition is needed which reduces to the ε-δ definition in case of the real line.

Definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Open and closed set definition

The most common one defines continuous functions as those functions where the preimages of open sets are open. Similar to the open set formulation is the closed set formulation, which says that preimages of closed sets are closed.

Neighborhood definition

Definition based on preimages are often difficult to use directly. Instead, suppose we have a function f from X to Y, where X,Y are topological spaces. We say f is continuous at x for some

x \in X

if for any neighborhood V of f(x), there is a neighborhood U of x such that

f(U) \subseteq V

. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can find a small U containing x that will map inside it. If f is continuous at every

x \in X

, then we simply say f is continuous. Continuity of a function at a point In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. This leads to the standard ε-δ definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance.

Closure and interior operator definition

Given two topological spaces (X,cl) and (X ' ,cl ') where cl and cl ' are two closure operators then a function :

f:(X,\mathrm) \to (X' ,\mathrm')

is continuous if for all subsets A of X :

f(\mathrm(A)) \subseteq \mathrm'(f(A)).

Similarly given two topological spaces (X,int) and (X ' ,int ') where int and int ' are two interior operators then a function :

f:(X,\mathrm) \to (X' ,\mathrm')

is continuous if for all subsets A of X :

f(\mathrm(A)) \subseteq \mathrm'(f(A)).

Closeness relation definition

Given two topological spaces (X,δ) and (X ' ,δ ') where δ and δ ' are two closeness relations then a function :

f:(X,\delta) \to (X' ,\delta')

is continuous if for all points x and y of X :

x \delta y \Leftrightarrow f(x)\delta'f(y).

Useful properties of continuous maps

Some facts about continuous maps between topological spaces:
- If f : X → Y and g : Y → Z are continuous, then so is the composition g o f : X → Z.
- If f : X → Y is continuous and
- X is compact, then f(X) is compact.
- X is connected, then f(X) is connected.
- X is path-connected, then f(X) is path-connected.
- If f : X → Y is continuous and a sequence (x_n) in X converges to a limit x, then the sequence (f(x_n)) obtained by applying f to each element converges to f(x). We say continuous functions take limits to limits. This also holds if sequences are replaced by general nets.
- If X is a first-countable space, then the converse also holds: any function taking limits of sequences to limits of sequences is continuous. In particular, the converse holds if X is a metric space. When using nets instead of sequences, this converse holds for a general topological space X.

Other notes

If a set is given the discrete topology, all functions with that space as a domain are continuous. If the domain set is given the indiscrete topology and the range set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous. Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection need not be continuous, but if it is, this special function is called a homeomorphism. A continuous bijection is a homeomorphism if its domain is compact and its codomain is Hausdorff. Category:General topology

Smooth function

In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i.e., has derivatives of all finite orders. A function is called C or more commonly C⁰ if it is a continuous function. A function is C¹ if it has a derivative that is continuous; such functions are also called continuously differentiable. A function is called Cⁿ for n ≥ 1 if it can be differentiated n times, with a continuous n-th derivative. The smooth functions are therefore those that lie in the class Cⁿ for all n. They are also called C^∞ functions. For example, the exponential function is evidently smooth because the derivative of the exponential function is the exponential function itself.

Constructing smooth functions to specifications

It is often useful to construct smooth functions that are zero outside a given interval, but not inside it. This is possible; on the other hand it is impossible that a power series can have that property. This shows that there is a large gap between smooth and analytic functions; so that Taylor's theorem cannot in general be applied to expand smooth functions. To give an explicit construct of such functions, we can start with a function such as :f(x) = exp(−1/x²), defined initially for x > 0. Not only do we have :f(x) → 0 as x → 0 from above, we have :P(x)f(x) → 0 for any polynomial P — because exponential growth with a negative exponent dominates. That means that setting f(x) = 0 for x < 0 gives a smooth function. Combinations such as f(x)f(1-x) can then be made with any required interval as support; in this case the interval [0,1]. Such functions have an extremely slow 'lift-off' from 0. See also an infinitely differentiable function that is not analytic.

Relation to analytic function theory

Thinking in terms of complex analysis, a function like :g(z) = exp(−1/z²) is smooth for z taking real values, but has an essential singularity at z = 0. That is, the behaviour near z = 0 is bad; but it happens that one cannot see that, by looking at real arguments alone.

Smooth partitions of unity

Smooth functions with given closed support are used in the construction of smooth partitions of unity (see topology glossary for partition of unity); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [a,b] and such that :f(x) > 0 for a < x < b. Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals (-∞, c] and [d,+∞) to cover the whole line, such that the sum of the functions is always 1. From what has just been said, partitions of unity don't apply to holomorphic functions; their different behaviour relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

Smooth maps of manifolds

Smooth maps between smooth manifolds may be defined by means of charts, since the idea of smoothness of function is independent of the particular chart used. Such a map has a first derivative defined on tangent vectors; it gives a fibre-wise linear mapping on the level of tangent bundles.

Advanced definitions

When one needs to talk about the set of all infinitely differentiable functions, and how elements of that space behave when differentiated and integrated, summed and taken limits of, then it turns out that the space of all smooth functions is an inappropriate choice, as it fails to be complete and closed under these operations. For a proper treatment in this case, the concept of a Sobolev space must be used.

Boundary condition

In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. In the problems most frequently considered, only one of the infinitely many solutions of the differential equation satisfies the boundary conditions. In a physical model simulation, the boundary conditions describe the behavior of the simulation at the edges of the simulation region. There are many kinds of possible conditions, depending on the formulation of the problem, number of variables involved, and (crucially) the mathematical nature of the equation. Conditions imposed at a time t = 0 are called initial conditions. One may also impose limiting conditions, for example under the limit of time t → +∞. Conditions in problems with a physical science origin usually match what is expected to determine a unique, well-defined physical situation. For example when a vibrating string is modelled, we assume that the two ends are held fixed: this accords with physical intuition. With the function to be found representing the displacement as function of position on the string, this implies the solution should take the value 0 at two points through all time. Another example from this kind is a whip—a string held fixed at one end, with the other end free to vibrate. Physical analysis of the loose end implies that the appropriate boundary condition is that the solution's derivative at this end is equal to 0 all time. The general picture is of a boundary (in one or several parts) where solutions are specified. For partial differential equations boundary conditions are usually defined on a continuous perimeter or surface, rather than at discrete points. Famous in potential theory (typical of elliptic PDEs) are the Dirichlet and Neumann boundary conditions, on a boundary enclosing a compact region. For a wave (hyperbolic) PDE one assumes waves propagate from an initial disturbance along some surface.

Bibliography

- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition). ISBN 1-58488-297-2
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9 Category:Differential equations Category:Partial differential equations ja:境界条件

Distribution'

Distribution can mean:
- In mathematics, there are several distinct concepts given the name of distribution:
- For generalized functions, see distribution (mathematics).
- In probability, see probability distribution.
- In Carnot-Carathéodory manifolds, see sub-Riemannian manifold.
- In physics, a distribution function, for example the Maxwell-Boltzmann distribution, describes the number of particles per unit volume in phase space.
- For business operations and logistics, see distribution (business).
- For the computer science concept, see distributed computing.
- For the meaning of distribution in the terminology of the Linux operating system, see Linux distribution.
- For electric power, see electricity distribution.
- For linguistics, see complementary distribution.
- In the film industry, see distribution (film).

Dirac delta function

The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. The integral from minus infinity to plus infinity is 1. The discrete analog of the delta function is the degenerate distribution which is sometimes known as a delta function.

Overview

Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph of the delta function can be usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function.) Despite its name, the delta function is not a function as defined in the strictest mathematical sense. One reason for this is because the functions f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are ostensibly different. According to Lebesgue integration theory, if f, g are functions such that f = g almost everywhere, then f is integrable iff g is integrable and the integrals of f and g are the same. Precise treatment of the Dirac delta requires measure theory or the theory of distributions. The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse). It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball. The Dirac delta function was named after the Kronecker delta, since it can be used as a continuous analogue of the discrete Kronecker delta.

Formal introduction

The Dirac delta is often introduced with the property: :

\int_^\infty f(x) \, \delta(x) \, dx
= f(0)

valid for any continuous function f. However, there is no actual function δ(x) with this property. The Dirac delta is not a function; but it can be usefully treated as a distribution, as well as a measure. As a distribution, the Dirac delta is defined by :

\delta[\phi] = \phi(0)\,

for every test function

\phi \

. It is a distribution with compact support (the support being ). Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience, and not a true integral. As a measure,

\delta (A)=1

0\in A

, and

\delta (A)=0

otherwise. Then, :

\int_^\infty f(x) \, d\delta(x) 
= f(0)

for all continuous f. As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution.

Fourier transform

The continuous Fourier transform of the Dirac delta is the constant function

\frac

. The inverse transform of this constant function will be the Dirac delta again, yielding the orthogonality property for the Fourier kernel: :

\frac\int_^\infty e^\,dx=\delta(k)

From the convolution theorem for the Fourier transform, the convolution of δ with any distribution S yields S.

The Dirac delta function as a probability density function

The Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function is the Heaviside step function.

Derivatives of the delta function

The derivative of the Dirac delta is the distribution δ' defined by :

\delta'[\phi] = -\phi'(0)\,

for every test function

\phi \

. From this it follows that :

\delta'(x)=-\frac

The n-th derivative δ⁽ⁿ⁾ is given by :

\delta^[\phi] = (-1)^n \phi^(0)\,

The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials. A helpful identity is :

\delta(g(x)) = \sum_\frac

where x_i are the roots of g(x). In the integral form it is equivalent to :

\int_^\infty f(x) \, \delta(g(x)) \, dx
= \sum_\frac

The derivative of a Dirac delta is called a doublet.

Equivalent definition

The Dirac delta function

\delta : \mathbb \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb)

is a distribution

\delta ( \xi )

whose indefinite integral is the function :

h : \mathbb \ni \xi \longrightarrow \frac \in \mathbb,

usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation :

\int^_ \delta (t) dt = h(x) \equiv \frac

for all real numbers x.

Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions :

\delta (x) = \lim_ \delta_a(x),

where

\delta_a(x)

is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions. Some nascent delta functions are: :
Note: If δ(a, x) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always non-negative between -∞ and +∞) then another nascent delta function δ_φ(a, x) can be built from its characteristic function as follows: :

\delta_\varphi(a,x)=\frac~\frac

where :

\varphi(a,k)=\int_^\infty \delta(a,x)e^\,dx

is the characteristic function of the nascent delta function δ(a, x). This result is related to the localization property of the continuous Fourier transform.

External links

- [http://mathworld.wolfram.com/DeltaFunction.html Delta Function] on MathWorld
- [http://planetmath.org/encyclopedia/DiracDeltaFunction.html Dirac Delta Function] on PlanetMath Category:Mathematical analysis ja:ディラックのデルタ関数

Banach space

In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions.

Definition

Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm ||.|| such that every Cauchy sequence (with respect to the metric d(x, y) = ||x - y||) in V has a limit in V. Since the norm induces a topology on the vector space, a Banach space provides an example of a topological vector space.

Examples

Throughout, let K stand for one of the fields R or C. The familiar Euclidean spaces Kⁿ, where the Euclidean norm of x = (x₁, ..., x_n) is given by ||x|| = (∑ |x_i|²)^1/2, are Banach spaces. The space of all continuous functions f : [a, b] → K defined on a closed interval [a, b] becomes a Banach space if we define the norm of such a function as ||f|| = sup . This is indeed a norm since continuous functions defined on a closed interval are bounded. The space is complete under this norm, and the resulting Banach space is denoted by C[a, b]. This example can be generalized to the space C(X) of all continuous functions X → K, where X is a compact space, or to the space of all bounded continuous functions X → K, where X is any topological space, or indeed to the space B(X) of all bounded functions X → K, where X is any set. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact unital Banach algebras. If p ≥ 1 is a real number, we can consider the space of all infinite sequences (x₁, x₂, x₃, ...) of elements in K such that the infinite series ∑_i |x_i|^p is finite. The p-th root of this series' value is then defined to be the p-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by l^p. The Banach space l^∞ consists of all bounded sequences of elements in K; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members. Again, if p ≥ 1 is a real number, we can consider all functions f : [a, b] → K such that |f|^p is Lebesgue integrable. The p-th root of this integral is then defined to be the norm of f. By itself, this space is not a Banach space because there are non-zero functions whose norm is zero. We define an equivalence relation as follows: f and g are equivalent if and only if the norm of f - g is zero. The set of equivalence classes then forms a Banach space; it is denoted by L^p[a, b]. It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space. These examples can be generalized; see L^p spaces for details. If X and Y are two Banach spaces, then we can form their direct sum X ⊕ Y, which is again a Banach space. This construction can be generalized to the direct sum of arbitrarily many Banach spaces. If M is a closed subspace of the Banach space X, then the quotient space X/M is again a Banach space. Finally, every Hilbert space is a Banach space. The converse is not true.

Linear operators

If V and W are Banach spaces over the same ground field K, the set of all continuous K-linear maps A : V → W is denoted by L(V, W). Note that in infinite-dimensional spaces, not all linear maps are automatically continuous. L(V, W) is a vector space, and by defining the norm ||A|| = sup it can be turned into a Banach space. The space L(V) = L(V, V) even forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

Dual space

If V is a Banach space and K is the underlying field (either the real or the complex numbers), then K is itself a Banach space (using the absolute value as norm) and we can define the dual space V as V = L(V, K). This is again a Banach space. It can be used to define a new topology on V: the weak topology. There is a natural map F from V to V' (the dual of the dual) defined by :F(x)(f) = f(x) for all x in V and f in V. Because F(x) is a map from V to K, it is an element of V'. The map F: x → F(x) is thus a map V → V'. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space V is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology. For example, l^p is reflexive for 1 but l¹ and l^∞ are not reflexive. The dual of l^p is l^q where p and q are related by the formula (1/p) + (1/q) = 1. See Lp spaces for details.
Relationship to Hilbert spaces
As mentioned above, every Hilbert space is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product. Not every Banach space is a Hilbert space. A necessary and sufficient condition for a Banach space V to also be a Hilbert space is the parallelogram identity: : $\|u+v\|^2 + \|u-v\|^2 = 2(\|u\|^2 + \|v\|^2)$ for all u and v in V, and where ||
- || is the norm on V. If the norm of a Banach space satisfies this identity, the inner product which makes it into a Hilbert space is given by the polarization identity. If V is a real Banach space, then the polarization identity is : $(u,v) = \frac (\|u+v\|^2 - \|u-v\|^2)$ whereas if V is a complex Banach space, then the polarization identity is given by : $(u,v) = \frac \left(\|u+v\|^2 - \|u-v\|^2 - i(\|u+iv\|^2 - \|u-iv\|^2)\right).$ To see why the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product, one verifies algebraically that this form is additive, whence it follows by induction that the form is linear over the integers and rationals. Then since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line. In the complex case, one can check also that the bilinear form is linear over i in one argument, and conjugate linear in the other.
Derivatives
Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gâteaux derivative.
Generalizations
Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.
Literature
Historical monographs in English, French and Polish:
- Stefan Banach: [http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10 Théorie des opérations linéaires]. -- Warszawa 1932. (Monografie Matematyczne; 1) [http://www-irma.u-strasbg.fr/math-cgi-bin/zmen/ZMATH/en/quick.html?format=complete&type=html&an=0005.20901 Zbl 0005.20901]
External links
For historical references see the Banach space entry in
- [http://members.aol.com/jeff570/b.html Earliest known uses of some of the words of mathematics: B]
- ko:바나흐 공간 ja:バナッハ空間

Gâteaux derivative

In mathematics, the Gâteaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. It is named for R. Gâteaux, a French mathematician who died young in World War I. Defined for locally convex topological vector spaces, it should be contrasted to the derivative on Banach spaces, the Fréchet derivative. Either derivative is often used to formalize the concept of the functional derivative that is frequently used in physics and in particular, in quantum field theory.

Definition

Suppose

X

and

Y

are locally convex topological vector spaces (for example, Banach spaces),

U\subset X

is open, and :

F:X\rightarrow Y.

The Gâteaux derivative

dF(u,\psi)

F

u\in U

in the direction

\psi\in X

is defined as :

dF(u,\psi)=\lim_\frac=\left.\fracF(u+\tau \psi)\right|_

if the limit exists. If the limit exists for all

\psi \in X

, then one says that

F

has Gâteaux derivative at

u\in U

. If the Gâteaux derivative exists, it is unique. One says that

F

is continuously differentiable in

U

if :

dF:U\times X \rightarrow Y

is continuous.

Example

Let

X

and

Y

be Hilbert spaces. Let

x\in\Omega\subset\mathbb^N

. The functional :

E:X\rightarrow \mathbb,

given by

E(u)=\int_\Omega F\left( u(x) \right)dx

where

F^\prime=f

, has Gâteaux derivative :

dE(u,\psi)=(f(u),\psi)\,.

Notice that :

\frac = \frac \left( \int_\Omega F(u+\tau\psi)dx - \int_\Omega F(u)dx \right)

\,
 \quad\quad =\frac \left( \int_\Omega\int_0^1 \frac F(u+s\tau\psi) \,ds\,dx \right)

\,
\quad \quad =\int_\Omega\int_0^1 f(u+s\tau\psi)\psi \,ds\,dx.

After suitable justification, letting

\tau\rightarrow 0

yields

(f(u),\psi)

. Category:Functional analysis

Fréchet space

:This article deals with Fréchet spaces in functional analysis. For Fréchet spaces in general topology, see T1 space. In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces, normed vector spaces which are complete with respect to the metric induced by the norm. Fréchet spaces, in contrast, are locally convex spaces which are complete with respect to a translation invariant metric. Fréchet spaces are studied because even though their topological structure is more complicated due to the lack of a norm, many important results in functional analysis, like the open mapping theorem and the Banach-Steinhaus theorem, still hold. Spaces of infinitely often differentiable functions defined on compact sets are typical examples of Fréchet spaces.

Definitions

Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of semi-norms. A topological vector space X is a Fréchet space iff it satisfies the following three properties:
- it is complete as a uniform space
- it is locally convex
- its topology can be induced by a translation invariant metric, i.e. a metric d : X × X → R such that d(x,y) = d(x+a, y+a) for all a,x,y in X. This means that a subset U of X is open if and only if for every u in U there exists an ε > 0 such that is a subset of U. Note that there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology. The alternative and somewhat more practical definition is the following: a topological vector space X is a Fréchet space iff it satisfies the following two properties:
- it is complete as a uniform space
- its topology may be ind

Functional derivative

Formal description

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

Directional derivative

Definition

The directional derivative in differential geometry

Normal derivative

See also

Derivative

Differentiation and differentiability

Newton's difference quotient

Notations for differentiation

Lagrange's notation

Leibniz's notation

Newton's notation

Euler's notation

Critical points

Physics

Algebraic manipulation

Using derivatives to graph functions

Generalizations

See also

External links

References

Continuous function (topology)

Definitions

Open and closed set definition

Neighborhood definition

Closure and interior operator definition

Closeness relation definition

Useful properties of continuous maps

Other notes

Smooth function

Constructing smooth functions to specifications

Relation to analytic function theory

Smooth partitions of unity

Smooth maps of manifolds

Advanced definitions

See also

Boundary condition

See also

Bibliography

Distribution'

Dirac delta function

Overview

Formal introduction

Fourier transform

The Dirac delta function as a probability density function

Derivatives of the delta function

Equivalent definition

Representations of the delta function

See also

External links

Banach space

Definition

Examples

Linear operators

Dual space

Relationship to Hilbert spaces