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Smooth Map

Smooth map

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.

See also


- quasi-analytic function Category:Mathematical analysis ko:매끈한 함수

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.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

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

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

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.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

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

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

In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the output is not defined), the function is said to be discontinuous (or to have a discontinuity). The context in this entry is real-valued functions on the real domain or on topological or metric spaces other than the complex numbers; for complex-valued functions see complex analysis. The notable difference in approach is that in the present context, the points in the domain that would be regarded as singularities (points of discontinuity) in the complex domain are usually assumed to be absent, or they are explicitly excluded, so as to leave a function that is continuous on a disconnected real domain. As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous (unless the flower is cut). As another example, if T(x) denotes the air temperature at height x, then this function is also continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous. There are also some more special usages of continuity in some mathematical disciplines. Probably the most common one, found in topology, is described in the article on continuity (topology). In order theory, especially in domain theory, one considers a notion derived from this basic definition, which is known as Scott continuity.

Real-valued continuous functions

Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the three functions h, T and M from above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper. To be more precise, we say that the function f is continuous at some point c when the following two requirements are satisfied:
- f(c) must be defined (i.e. c must be an element of the domain of f).
- The limit of f(x) as x approaches c must exist and be equal to f(c). (If the point c in the domain of f is not an accumulation point of the domain, then this condition is vacuously true, since x cannot approach c. Thus, for example, every function whose domain is the set of all integers is continuous, merely for lack of opportunity to be otherwise.) We call the function everywhere continuous, or simply continuous, if it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset.

Epsilon-delta definition

Without resorting to limits, one can define continuity of real functions as follows. Again consider a function f that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of f. The function f is said to be continuous at the point c if (and only if) the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c − δ < x < c + δ, the value of f(x) will satisfy f(c) − ε < f(x) < f(c) + ε. Alternatively written: Given I,D\subset\mathbb (that is, I and D are subsets of the real numbers), continuity of f:I \to D (read f maps I into D) at c\in\mathbb means that for all \varepsilon>0 there exists a \delta>0 such that |x-c|<\delta and x\in I imply that |f(x)-f(c)|<\varepsilon. This "epsilon-delta definition" of continuity was first given by Cauchy. More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(c), we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter how small the f(x) neighborhood is; f(x) is then continuous at c.

Heine definition of continuity

The following definition of continuity is due to Heine. :A real function f is continuous if for any sequence (x_n) such that ::\lim\limits_ x_n=x_0, :it holds that ::\lim\limits_ f(x_n)=f(x_0). :(We assume that all points x_n, x_0 belong to the domain of f.) One can say briefly, that a function is continuous if and only if it preserves limits. Cauchy's and Heine's definition of continuity are equivalent. The usual (easier) proof makes use of the axiom of choice, but in the case of real functions it was proved by Wacław Sierpiński that the axiom of choice is not actually needed. [http://www.apronus.com/math/cauchyheine.htm] In more general setting of topological spaces, the concept analogous to Heine definition of continuity is called sequential continuity. In general, sequential continuity is not equivalent to the analogue of Cauchy continuity, which is just called continuity (see continuity (topology) for details).

Examples


- All polynomials are continuous.
- If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions f(x)=1/x and g(x)=(sin x)/x. Neither function is defined at x=0, so each has domain R\, and each function is continuous. The question of continuity at x=0 does not arise, since it is not in the domain. The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of g at 0 is 1, g can be extended continuously to R by defining its value at 0 to be 1. A point in the domain that can filled in so that the resulting function is continuous is called a removable singularity. Whether this can be done is not the same as continuity.
- The rational functions, exponential functions, logarithms, square root function, trigonometric functions and absolute value function are continuous.
- An example of a discontinuous function is the function f defined by f(x) = 1 if x > 0, f(x) = 0 if x ≤ 0. Pick for instance ε = 1/2. There is no δ-neighborhood around x=0 that will force all the f(x) values to be within ε of f(0). Intuitively we can think of this type of discontinuity as a sudden jump in function values.
- Another example of a discontinuous function is the sign function.
- A more complicated example of a discontinuous function is the popcorn function.

Facts about continuous functions

If two functions f and g are continuous, then f + g and fg are continuous. If g(x) ≠ 0 for all x in the domain, then f/g is also continuous. The composition f o g of two continuous functions is continuous. The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states: "If the real-valued function f is continuous on the closed interval [a, b] and k is some number between f(a) and f(b), then there is some number c in [a, b] such that f(c) = k. For example, if a child undergoes continuous growth from 1m to 1.5m between the ages of 2 years and 6 years, then, at some time between 2 years and 6 years of age, the child's height must have been 1.25m. As a consequence, if f is continuous on [a, b] and f(a) and f(b) differ in sign, then, at some point c, f(c) must equal zero. If a function f is defined on a closed interval [a,b] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c ∈ [a,b] with f(c) ≥ f(x) for all x ∈ [a,b]. The same is true of the minimum of f. (Note that these statements are false if our function is defined on an open interval (a,b) (or any set that is not both closed and bounded). Consider for instance the continuous function f(x) = 1/x defined on the open interval (0,1).) If a function is differentiable at some point c of its domain, then it is also continuous at c. The converse is not true: a function that's continuous at c need not be differentiable there. Consider for instance the absolute value function at c = 0.

Continuous functions between metric spaces

Now consider a function f from one metric space (X, dX) to another metric space (Y, dY). Then f is continuous at the point c in X if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying dX(x, c) < δ will also satisfy dY(f(x), f(c)) < ε. This can also be formulated in terms of sequences and limits: the function f is continuous at the point c if for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = f(c). Continuous functions transform limits into limits. This latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (xn) in X with limit c, the sequence (f(xn)) is a Cauchy sequence. Continuous functions transform convergent sequences into Cauchy sequences.

Continuous functions between topological spaces

The above definitions of continuous functions can be generalized to functions from one topological spaces to another in a natural way; a function f : X → Y, where X and Y are topological spaces, is continuous iff for every open set V ⊆ Y, f −1(V) is open in X.

See also


- semicontinuity
- classification of discontinuities
- uniform continuity
- absolute continuity
- equicontinuity
- Lipschitz continuity
- normal function
- bounded linear operator
- continuous functor

References


- [http://archives.math.utk.edu/visual.calculus/ Visual Calculus] by Lawrence S. Husch, University of Tennessee (2001) Category:CalculusCategory:General topology ko:연속함수 ja:連続 (数学) th:ฟังก์ชันต่อเนื่อง

Interval (mathematics)

In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers.

Algebra

In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is the notation in which permitted values for a variable are expressed as ranging over a a certain interval; "5 < x < 9" is an example of the application of interval notation. In conventional interval notation, parentheses ( () ) indicate exclusion while square brackets ( [] ) indicate inclusion. For example, the interval "(10,20)" indicates the set of all real numbers between 10 and 20 but does not include 10 or 20, the first and last numbers of the interval, respectively. On the other hand, the interval "[10,20]" includes both every number between 10 and 20 as well as 10 and 20. Other possibilities are listed below.

Higher mathematics

In higher mathematics, a formal definition is the following: An interval is a subset S of a totally ordered set T with the property that whenever x and y are in S and x < z < y then z is in S. As mentioned above, a particularly important case is when T = R, the set of real numbers. Intervals of R are of the following eleven different types (where a and b are real numbers, with a < b): # (a,b) = # [a,b] = # [a,b) = # (a,b] = # (a,∞) = # [a,∞) = # (-∞,b) = # (-∞,b] = # (-∞,∞) = R itself, the set of all real numbers # # the empty set In each case where they appear above, a and b are known as endpoints of the interval. Note that a square bracket [ or ] indicates that the endpoint is included in the interval, while a round bracket ( or ) indicates that it is not. For more information about the notation used above, see Naive set theory. Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets). Intervals (3) and (4) are sometimes called half-closed (or, not surprisingly, half-open) intervals. Notice that intervals (9) and (11) are both open and closed, which is not the same thing as being half-open and half-closed. Intervals (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded intervals. Interval (10) is also known as a singleton. The length of the bounded intervals (1), (2), (3), (4) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5. Intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define (see above). The concept of measure can then be extended to more complicated sets, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are precisely the connected subsets of R. They are also precisely the convex subsets of R. Since a continuous image of a connected set is connected, it follows that if f: R→R is a continuous function and I is an interval, then its image f(I) is also an interval. This is one formulation of the intermediate value theorem.

Intervals in partial orders

In order theory, one usually considers partially ordered sets. However, the above notations and definitions can immediately be applied to this general case as well. Of special interest in this general setting are intervals of the form [a,b]. For a partially ordered set (P, ≤) and two elements a and b of P, one defines the set : [a, b] = One may choose to restrict this definition to pairs of elements with the property that a ≤ b. Alternatively, the intervals without this condition will just coincide with the empty set, which in the former case would not be considered as an interval.

Interval arithmetic

Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, was introduced by mathematicians in the 1950s and 1960s as an approach to putting bounds on rounding errors in mathematical computation. Where classical arithmetic defines operations on individual numbers, interval arithmetic defines a set of operations on intervals: :T · S = . The basic operations of interval arithmetic are, for two intervals [a,b] and [c,d] that are subsets of the real line (-∞, ∞),
- [a,b] + [c,d] = [a+c, b+d]
- [a,b] - [c,d] = [a-d, b-c]
- [a,b]
- [c,d] = [min (ac, ad, bc, bd), max (ac, ad, bc, bd)]
- [a,b] / [c,d] = [min (a/c, a/d, b/c, b/d), max (a/c, a/d, b/c, b/d)] Division by an interval containing zero is not defined under the basic interval arithmetic. The addition and multiplication operations are commutative, associative and sub-distributive: the set X ( Y + Z ) is a subset of XY + XZ.

Relational operations

Relational operations on intervals can be defined in tri-state logic :
- T · S is true if for any x in T, and any y in S, x · y is true
- T · S is false if for any x in T, and any y in S, x · y is false
- otherwise T · S is uncertain Often intervals are considered as estimations of some individual numbers. In that case for both arithmetic and relational interval operations the following is true: if x in T and y in S, then the result of T · S contains x · y.

Alternative notation

An alternative way of writing intervals, commonly seen in France and some other European countries, is shown below:
- ]a,b[ =
- [a,b] =
- [a,b[ =
- ]a,b] =

External links


- [http://www.cs.utep.edu/interval-comp/main.html Interval computations website]
- [http://www.cs.utep.edu/interval-comp/icompwww.html Interval computations research centers] Category:Order theory Category:Topology ko:구간 ja:区間 (数学)

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions. There exist real analytic functions and complex analytic functions, which have similarities as well as differences.

Definitions

Formally, function f is real analytic on an open set D in the real line if for any x0 in D one can write : in which the coefficients a0, a1, ... are real numbers and the series is convergent for x in a neighborhood of x0. Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x0 in its domain : T(x) = \sum_^ \frac (x-x_0)^ is convergent for x close enough to x0 and its value equals f(x). The definition of a complex analytic function is obtained by replacing everywhere above "real" with "complex".

Examples


- Any polynomial (real or complex) is an analytic function. This because if a polynomial has degree n, any terms of degree larger than n in its Taylor series will vanish, so this series will be trivially convergent.
- The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 but rather for all values of x.
- Any combination of elementary functions (trigonometric functions, logarithm, power functions) is an analytic function in any open set contained within its domain of definition.
- The absolute value is not analytic, as it is not differentiable. Piecewise defined functions (functions given by different formulas in different regions) are in general not analytic.

Properties of analytic functions


- The sum, product, and composition of analytic functions are analytic.
- The reciprocal of an analytic function that is nowhere zero, is analytic, as is the inverse of an invertible analytic function whose derivative is nowhere zero.
- Any analytic function is smooth. A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function f has an accumulation point inside its domain, then f is zero everywhere on the connected component containing the accumulation point. More formally this can be stated as follows. If (rn) is a sequence of distinct numbers such that f(rn) = 0 for all n and this sequence converges to a point r in the domain of D, then f is identically zero on the connected component of D containing r. Also, if all the derivatives of an analytic function at a point are zero, the same conclusion holds. These statements imply that while analytic functions do have more degrees of freedom than polynomials, they are still quite inflexible.

Analyticity and differentiability

Any analytic function (real or complex) is differentiable, actually infinitely differentiable (that is, smooth). There exist smooth real functions which are not analytic, see the following example. The real analytic functions are much "fewer" than the real (infinitely) differentiable functions. The situation is quite different for complex analytic functions. It can be proved that any complex function differentiable in an open set is analytic. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.

Real versus complex analytic functions

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Complex analytic functions are more rigid in many ways. According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. This statement is clearly false for real analytic functions, as illustrated by :f(x)=\frac. Also, if a complex analytic function is defined in an open ball around a point x0, its power series expansion at x0 is convergent in the whole ball. This is not true in general for real analytic functions. (Note that an open ball in the complex plane would be a disk, while on the real line it would be an interval.) Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not any real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function f (x) defined in the paragraph above is a counterexample.

Analytic functions of several variables

One can define analytic functions in several variables by means of power series in those variables (see power series). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in many dimensions. Category:Mathematical analysis Category:Complex analysis Category:Real analysis

Taylor's theorem

In calculus, Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. This result was first discovered 41 years earlier in 1671 by James Gregory.

Taylor's theorem in one variable

The most basic example of Taylor's theorem is the approximation of the exponential function \textrm^x near x = 0. Namely, : \textrm^x \approx 1 + x + \frac + \frac + \cdots + \frac. The precise statement of the theorem is as follows: If n ≥ 0 is an integer and f is a function which is n times continuously differentiable on the closed interval [a, x] and n + 1 times differentiable on the open interval (a, x), then we have : f(x) = f(a) + \frac(x - a) + \frac(x - a)^2 + \cdots + \frac(x - a)^n + R_n Here, n! denotes the factorial of n, and Rn is a remainder term which depends on x and is small if x is close enough to a. Several expressions for Rn are available. factorial The Lagrange form of the remainder term states that there exists a number ξ between a and x such that : R_n = \frac (x-a)^. This exposes Taylor's theorem as a generalization of the mean value theorem. In fact, the mean value theorem is used to prove Taylor's theorem with the Lagrange remainder term. The Cauchy form of the remainder term is : R_n(x) = \int_a^x \frac (x - t)^n \, dt. This shows the theorem to be a generalization of the fundamental theorem of calculus. For some functions f(x), one can show that the remainder term Rn approaches zero as n approaches ∞; those functions can be expressed as a Taylor series in a neighbourhood of the point a and are called analytic. Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function f has complex values or vector values. Furthermore, there is a version of Taylor's theorem for functions in several variables. For complex functions analytic in a region containing a circle C surrounding a and its interior, we have a contour integral expression for the remainder : R_n(x) = \frac\int_C \fracdz valid inside of C.

Taylor's theorem for several variables

Using multi-index notation (see also Taylor series in several variables), Taylor's theorem can be generalized to several variables as follows. Let B be a ball in RN centered at a point a, and f be a real-valued function defined on the closure \bar having n+1 continuous partial derivatives at every point. Taylor's theorem asserts that for any x\in B, :f(x)=\sum_^n\frac(x-a)^\alpha+\sum_R_(x)(x-a)^\alpha where the summation extends over multi-indices α. The remainder terms satisfy the inequality :|R_(x)|\le\sup_\left|\frac\right| for all α with |α|=n+1. As was the case with one variable, the remainder terms can be described explicitly. See the proof for details.

Proof: Taylor's theorem in one variable

We first prove Taylor's theorem with the integral remainder term. The fundamental theorem of calculus states that :f(x) = f(a) + \int_a^x (x-t)^0 \, f'(t) \, dt. This proves the theorem for n = 0. Integration by parts yields the case n = 1: :f(x) = f(a) +f'(a)\,(x-a)+\int_a^x (x-t)^1 \, f(t) \, dt. By repeating this process, we may derive Taylor's theorem for higher values of n. This can be formalized by applying the technique of induction. So, suppose that Taylor's theorem holds for a particular n, that is, suppose that : f(x) = f(a) + \frac(x - a) + \cdots + \frac(x - a)^n + \int_a^x \frac (x - t)^n \, dt. \qquad(
- ) We can again rewrite the integral using integration by parts. An antiderivative of (x − t)n as a function of t is given by −(x−t)n+1 / (n + 1), so : \int_a^x \frac (x - t)^n \, dt :: = - \left[ \frac (x - t)^ \right]_a^x + \int_a^x \frac (x - t)^ \, dt :: = \frac (x - a)^ + \int_a^x \frac (x - t)^ \, dt. Substituting this in (
- ) proves Taylor's theorem for n + 1, and hence for all nonnegative integers n. The remainder term in the Lagrange form can be derived by the mean value theorem in the following way: : R_n = \int_a^x \frac (x - t)^n \, dt =f^(\xi) \int_a^x \frac \, dt. The last integral can be solved immediately, which leads to : R_n = \frac (x-a)^.

Proof: several variables

Let x=(x1,...,xN) lie in the ball B with center a. Parametrize the line segment between a and x by u(t)=a+t(x-a). We apply the one-variable version of Taylor's theorem to the function f(u(t)): :f(x)=f(u(1))=f(a)+\sum_^n\left.\frac\frac\right|_f(u(t))\ +\ \int_0^1 \left. \frac\frac\right|_ f(u(s))ds. By the chain rule for several variables, :\fracf(a+t(x-a))=\sum_\left(\begini\\ \alpha\end\right)(D^\alpha f)(a+t(x-a)) where \left(\begini\\ \alpha\end\right) is the multinomial coefficient for the multi-index α. Since \frac\left(\begini\\ \alpha\end\right)=\frac, we get :f(x)= f(a)+\sum_^n\fracD^\alpha f(a)+\sum_\frac\int_0^1 D^\alpha f(a+s(x-a))ds. The remainder term is given by :\sum_\frac\int_0^1 D^\alpha f(a+s(x-a))ds, The terms of this summation are explicit forms for the Rα in the statement of the theorem. These are easily seen to satisfy the required estimate.

External link


- [http://www.cut-the-knot.org/Curriculum/Calculus/TaylorSeries.shtml Taylor Series Approximation to Cosine] at cut-the-knot Category:Calculus Category:Theorems Category:Mathematical series ja:テイラーの定理

Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders. Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions. In linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. With the advent of computers, polynomials have been replaced by splines in many areas in numerical analysis. Splines are piecewise defined polynomials and provide more flexibility than ordinary polynomials when defining simple and smooth functions. They are used in spline interpolation and computer graphics.

History

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has a root: this is the statement of the fundamental theorem of algebra. There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynomials of degree up to 4 have been known since the 16th century (see quadratic equation, Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers for a long time. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relations among roots of polynomials. The difference engine of Charles Babbage was designed to create large tables of values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's difference method.

Definition

For given constants (i.e., numbers) a0, …, an in some field (possibly but not limited to R or C) with an non-zero, for n > 0, then a polynomial (function) of degree n is a function of the form :f(x) = a_0 + a_1 x + \cdots + a_ x^ + a_n x^n. More concisely, the polynomial can be written in sigma notation as : f(x) = \sum_^ a_ x^. The constants a0, …, an are called the coefficients of the polynomial. a0 is called the constant coefficient and an is called the leading coefficient. When the leading coefficient is 1, the polynomial is called monic or normed. Each summand ai xi of the polynomial is called a term. A polynomial with one, two or three terms is called monomial, binomial or trinomial respectively. Polynomial functions of
- degree 0 are called constant functions (excluding the zero polynomial, which has indeterminate degree),
- degree 1 are called linear functions,
- degree 2 are called quadratic functions,
- degree 3 are called cubic functions,
- degree 4 are called quartic functions and
- degree 5 are called quintic functions.

Graphs


- The graph of a constant function :f(x) = a_0 is a horizontal line with y-intercept a_0.
- The graph of a degree 1 polynomial function (or linear function) :f(x) = a_0 + a_1 x, where a_1 \neq 0 is an oblique line with y-intercept a_0 and slope a_1.
- The graph of a degree 2 or higher polynomial function :f(x) = a_0 + a_1 x + \cdots + a_ x^ + a_n x^n, where a_n \neq 0 and n \geq 2 is a continuous non-linear curve. The best way to analyze the graph of a degree 2 or higher polynomial function is by its end behavior, the number of x-intercepts and the number of turning points. End behavior There are four end behaviors which are direct results of whether a_n, the leading coefficient, is positive or negative and whether n, the degree of the polynomial, is even or odd.
- If a_n is positive and n is even, the right end of the polynomial is in quadrant I while the left end is in quadrant II.
- If a_n is negative and n is even, the right end is in quadrant IV while the left end is in quadrant III.
- If a_n is positive and n is odd, the right end is in quadrant I while the left end is in quadrant III.
- If a_n is negative and n is odd, the right end is in quadrant IV while the left end is in quadrant II. Number of x-intercepts From the Fundamental theorem of algebra, a polynomial of degree n has exactly n complex roots, which may or may not be real. Therefore, the number of x-intercepts can't exceed n. It also follows from the Fundamental Theorem of Algebra that the complex roots of a polynomial must exist in conjugate pairs. This implies that an even-degree polynomial may have no x-intercepts (because all its roots may be complex); an odd-degree polynomial, on the other hand, must have at least one x-intercept, since any pairing of roots into conjugate pairs will necessarily leave at least one unpaired for odd n. These "unpaired" roots must therefore be real. For example, a degree 4 polynomial function can have 0, 2 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 3 or 5 x-intercepts. Number of turning points The number of turning points of an even-degree polynomial is any odd number less than the degree, while the number of turning points of an odd-degree polynomial is any even number less than the degree. For example, a degree 4 polynomial function can have 1 or 3 turning points whereas a degree 5 polynomial function can have 0, 2, or 4 turning points. The following are some examples of polynomials of low degree.

Examples

Fundamental theorem of algebra