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Ellipse

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 < 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix

A = PDP^T

, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :

\frac + \frac = 1

The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :

x = a\,\cos t

y = b\,\sin t

0 \leq t < 2\pi

which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :

\frac + \frac = 1

where (h,k) is the center. A Gauss-mapped form: :

\left(\frac,\frac\right)

has normal

(\cos\phi,\sin\phi)

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :

e = \sqrt

or where

c

(the linear eccentricity of the ellipse) equals the distance from the center to either focus :

e = \frac

The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted

l\,\!

(lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to

a\,\!

and

b\,\!

(the ellipse's semi-axes) by the formula

al=b^2\,\!

or, if using the eccentricity,

l=a(1-e^2)\,\!

. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation :

r (1 + e \cos \theta) = l \,\!

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

Area

The area enclosed by an ellipse is

\pi ab\,\!

, where

\pi

is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :

c = 2\pi a \left[\right]\!\,

A good approximation is Ramanujan's: :

c \approx \pi \left[3(a+b) - \sqrt\right]\!\,

which can also be written as: :

c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\,

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Stretching and Projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a point). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

Elliptical trainer

An elliptical trainer is a piece of stationary exercise equipment used to simulate walking or running while decreasing the risk of impact injuries. The user stands over the machine on pedals that move in independent elliptical orbits. They can be powered with electricity for assisting the stride and/or tracking progress. To generate resistence aerodynamic, magnetic or motor-assisted braking are commonly used. On some models, the incline can be changed to work different muscle groups in the legs. Electronic models can vary the incline over the course of a workout according to a preset program. Some elliptical trainers have a tall handle attached to each pedal, in order to exercise the arms and upper body in coordination with the legs. The user holds on at about shoulder height, and pulls on the handle while stepping on the opposite pedal.

External links

- [http://nutritionalsupplementguide.com/Body_building_Articles/elliptical_trainers_growing_in_popularity.php Elliptical Trainers Growing in Popularity]
- [http://www.consumersearch.com/www/health_and_fitness/elliptical_trainers/fullstory.html Elliptical Trainers Review]
- [http://www.primusweb.com/fitnesspartner/library/equipment/elliptical.htm Elliptical Trainers: Giving The Treadmill A Run For Its Money?] Category:Exercise equipment

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

Curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).

Definitions

In mathematics, a (topological) curve is defined as follows. Let

I

be an interval of real numbers (i.e. a non-empty connected subset of

\mathbb

). Then a curve

\!\,\gamma

is a continuous mapping

\,\!\gamma : I \rightarrow X

, where

X

is a topological space. The curve

\!\,\gamma

is said to be simple if it is injective, i.e. if for all

x

y

I

, we have

\,\!\gamma(x) = \gamma(y) \rightarrow x = y

. If

I

is a closed bounded interval

\,\![a, b]

, we also allow the possibility

\,\!\gamma(a) = \gamma(b)

(this convention makes it possible to talk about closed simple curve). If

\gamma(x)=\gamma(y)

for some

x\ne y

(other than the extremities of

I

), then

\gamma(x)

is called a double (or: multiple) point of the curve. A curve

\!\,\gamma

is said to be closed or a loop if

\,\!I = [a, b]

and if

\!\,\gamma(a) = \gamma(b)

. A closed curve is thus a continuous mapping of the circle

S^1

; a simple closed curve is also called a Jordan curve. A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is yet another weird example.

Conventions and terminology

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

Length of curves

X

is a metric space with metric

d

, then we can define the length of a curve

\!\,\gamma : [a, b] \rightarrow X

by :

\mbox (\gamma)=\sup \left\

A rectifiable curve is a curve with finite length. A parametrization of

\!\,\gamma

is called natural (or unit speed or parametrised by arc length) if for any

t_1

t_2

[a, b]

, we have :

\mbox (\gamma|_)=|t_2-t_1|

\!\,\gamma

is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of

\!\,\gamma

t_0

as :

\mbox(t_0)=\limsup_

and then :

\mbox(\gamma)=\int_a^b \mbox(t) \, dt

In particular, if

X = \mathbb^n

is Euclidean space and

\gamma : [a, b] \rightarrow \mathbb^n

is differentiable then :

\mbox(\gamma)=\int_a^b \left| \,  \, \right| \, dt

Differential geometry

Main article: differential geometry of curves While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If

X

is a differentiable manifold, then we can define the notion of differentiable curve in

X

. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take

X

to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to

X

by means of this notion of curve. If

X

is a smooth manifold, a smooth curve in

X

is a smooth map :

\!\,\gamma : I \rightarrow X.

This is a basic notion. There are less and more restricted ideas, too. If

X

is a

C^k

manifold (i.e., a manifold whose charts are

k

times continuously differentiable), then a

C^k

curve in

X

is such a curve which is only assumed to be

C^k

(i.e.

k

times continuously differentiable). If

X

is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and

\!\,\gamma

is an analytic map, then

\!\,\gamma

is said to be an analytic curve. A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two

C^k

differentiable curves :

\!\,\gamma_1 :I \rightarrow X

and :

\!\,\gamma_2 : J \rightarrow X

are said to be equivalent if there is a bijective

C^k

map :

\!\,p : J \rightarrow I

such that the inverse map :

\!\,p^ : I \rightarrow J

is also

C^k

, and :

\!\,\gamma_(t) = \gamma_(p(t))

for all

t

. The map

\!\,\gamma_2

is called a reparametrisation of

\!\,\gamma_1

; and this makes an equivalence relation on the set of all

C^k

differentiable curves in

X

. A

C^k

arc is an equivalence class of

C^k

curves under the relation of reparametrisation.

Algebraic curve

Main article: Algebraic curve In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces.

History

A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond ruler-and-compass constructions. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.

External links

- [http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html List of famous curves] Category:Curves Category:Metric geometry Category:Topology Category:General topology ko:곡선 ja:曲線

Distance

:This article is about distance in the mathematical / physical sense. For other uses, see distance (disambiguation). The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious. Distance as mentioned above is sometimes not symmetric, hence not a metric (see below): this applies to distance by car in the case of one-way streets, and also in the case the distance is expressed in terms of the time to cover it (a road may be more crowded in one direction than in the other, for a ship upstream and downstream makes a difference). As opposed to a position coordinate, a distance can not be negative. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction. In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.

Distance covered

Pythagorean theorem The distance covered by a vehicle (often recorded by a odometer), person, animal, object, etc. should be distinguished from the distance from starting point to end point, even if latter is taken to mean e.g. the shortest distance along the road, because a detour could be made, and the end point can even coincide with the starting point.

Formal definition

A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space. We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B. Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).

The distance formula

The (Euclidean) distance, d, between two points expressed in Cartesian coordinates equals the square root of the sum of the squares of the changes of each coordinate. Thus, in a two-dimensional space :

d = \sqrt,

and in a three-dimensional space: :

d = \sqrt.

Here, "Δ" (delta) refers to the change in a variable. Thus, Δx is the change in x, pronounced as such, or as "delta-x". In mathematical terms,

\Delta x = \left|x_1 - x_0\right|

, and so

(\Delta x)^2 = (x_1 - x_2)^2

. This distance formula can be seen as a specialized form of the Pythagorean theorem; it can also be expanded into the arc-length formula.

Generalized distance in arbitrary dimensions: Norms

In the Euclidean space Rⁿ, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are sometimes used instead. For a point (x₁, x₂, ...,x_n) and a point (y₁, y₂, ...,y_n), the Minkowski distance of order p (p-norm distance) is defined as: p need not be an integer, but it cannot be less than 1, because then the triangle inequality does not hold. There is no such thing as a negative distance. The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance. The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets). The infinity norm distance is also called Chebyshev distance. In 2D it represents the distance kings, queens, and bishops must travel between two squares on a chessboard. Also, if you measure the strength of each of the n links in a chain (where larger numbers mean weaker links), then because a chain is only as strong as its weakest link, the strength of the chain will be the infinity-norm distance from the list of measurements to the origin. The p-norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse. In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation.

Distances in other spaces

- Mahalanobis distance is used in statistics.
- Hamming distance is used in coding theory.
- Levenshtein distance

Focus (geometry)

In geometry, the focus (pl. foci) is a special point used in describing conic sections. A conic section can be defined as the set of points whose distance to its focus is equal to the eccentricity times the distance to the corresponding directrix. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section. Note that (non-circular) ellipses and hyperbolas each have a pair of foci. An ellipse can be described as the set of points for which the sum of the distances to the foci is constant, while a hyperbola is the set of points for which the absolute value of the difference of the distances to the foci is constant. In the gravitational two-body problem, the orbits of the two bodies are described by conic sections with foci at the center of mass. Category:Conic sections ja:焦点

Cone (geometry)

Suppose

V

is a real (or complex) vector space with a subset

C

. If

\lambda C \subset C

for any real

\lambda >0

, then

C

is a cone. If the origin belongs to a cone, then the cone is called pointed. Otherwise, the cone is called blunt. A pointed cone is salient, if it contains no 1-dimensional vector subspace of

V

. If

C-x_0

is a cone for some

x_0 \in V

, then

C

is a cone with vertex at

x_0

. A proper cone is a cone

C \subset \R^n

that satisfies the following:
-

C

is convex;
-

C

is closed;
-

C

is solid, meaning it has nonempty interior;
-

C

is pointed, meaning

x, -x\in C\Rightarrow x=0

. A proper cone

C

induces a partial ordering "<=" on

\R^n

: :

a <= b\Leftrightarrow b-a\in C

Examples

#In

\R^1

, the set

x>0

is a salient blunt cone. #Suppose

x\in \R^n

. Then for any

\varepsilon>0

, the set

C=\bigcup \

is an open cone. If

|x| < \varepsilon

, then

C=\R^n

. Here,

B_x(\varepsilon)

is the open ball at

x

with radius

\varepsilon

Properties

#The union and intersection of a collection of cones is a cone. #A set

C

in a real (or complex) vector space is a convex cone if and only if #:

\lambda C \subset C,

for all

\lambda>0,

C+C\subset  C.

#For a convex pointed cone

C

, the set

C\cap (-C)

is the largest vector subspace contained in

C

. #A pointed convex cone

C

is salient if and only if

C\cap (-C)=\.

References

-
- Category:geometry

Right angle

Angle#Types of angles

Semimajor axis

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. It is related to the semi-minor axis

b\,\!

through the eccentricity

e\,\!

and the semi-latus rectum

\ell\,\!

, as follows: :

b = a \sqrt\,\!

\ell=a(1-e^2)\,\!

. :

a\ell=b^2\,\!

. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping

\ell\,\!

fixed. Thus

a\,\!

and

b\,\!

tend to infinity,

a\,\!

faster than

b\,\!

. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, :

r(1-e\cos\theta)=l\,\!

The mean value of

r=\,\!

and

r=\,\!

, is

a=\,\!

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is in the x-direction the equation is:

\frac - \frac = 1

In terms of the semi-latus rectum and the eccentricity we have

a=

Astronomy

Orbital period

In astrodynamics the orbital period

T\,

of a small body orbiting a central body in a circular or elliptical orbit is: :

T = 2\pi\sqrt

where: :

a\,

is the length of the orbit's semi-major axis :

\mu

is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), :

P^2=a^3\,

where P is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton: :

P^2= \fraca^3\,

where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.
- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis

b = a \sqrt\,\!

.
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman):

a (1 + \frac)\,\!

. The time-average of the inverse of the radius,

r^\,\!

, is

a^\,\!

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis

a \,

can be calculated from orbital state vectors:

a = \,

for an elliptical orbit and

a = \,

for a hyperbolic trajectory and

\epsilon =  -

(specific orbital energy) and

\mu = GM \,

(standard gravitational parameter), where:
-

v\,

is orbital velocity from velocity vector of an orbiting object,
-

\mathbf\,

is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
-

G \,

is the gravitational constant,
-

M \,

the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

References

- [http://orca.phys.uvic.ca/~tatum/celmechs/celm9.pdf Jeremy B. Tatum, Celestial Mechanics, Chapter 9 - The Two Body Problem in Two Dimensions (2004)]
- [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000071000011001198000001&idtype=cvips&gifs=yes Darren M. Williams, Average distance between a star and planet in an eccentric orbit, American Journal of Physics, November 2003, Volume 71, Issue 11, pp. 1198-1200] Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Circle

:This article is about the shape and mathematical concept of circle; for other meanings, see Circle (disambiguation). Circle (disambiguation) In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). The points can only be those that are part of a conic section; within the set of a plane bisecting a cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually however, the circumference means the length of the circle, and the interior of the circle is called a disk or disc. An arc is any continuous portion of a circle.

Mathematical definitions

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that :

\left( x - a \right)^2 + \left( y - b \right)^2=r^2.

If the circle is centered at the origin (0, 0), then this formula can be simplified to :

x^2 + y^2 = r^2.

The circle centered at the origin with radius 1 is called the unit circle. Expressed in parametric equations, (x, y) can be written as :x = a + r cos(t) :y = b + r sin(t). The slope a circle at a point (x, y) can be expressed with the following formula, assuming the center is at the origin and (x, y) is on the circle: :

y' = - \frac.

In the complex plane, a circle with a center at c and radius r has the equation

|z-c|^2 = r^2

. Since

|z-c|^2 = z\overline-\overlinez-c\overline+c\overline

, the slightly generalized equation

pz\overline + gz + \overline = q

for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles. All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively. In other words:
- Length of a circle's circumference =

2\pi \times r.

- Area of a circle =

\pi \times r^2.

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the center of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

Properties

limit limit

Chord properties

- Chords equidistant from the centre of a circle are equal.
- Equal chords are equidistant from the centre.
- A line from the centre, perpendicular to a chord, bisects the chord.
- The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord.
- The perpendicular bisector of a chord passes through the centre of a circle.

Tangent properties

- The line drawn perpendicular to the end point of a radius is a tangent to the circle.
- A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
- Tangents drawn from a point outside the circle are equal in length.
- Two tangents can always be drawn from a point outside of the circle.

Inscribed angle theorem

- If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
- If two angles are inscribed on the same chord and on the same side of the chord , then they are equal.
- An inscribed angle subtended by a semicircle is a right angle.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

Secant, tangent, and chord properties

- The chord theorem states that if two chords, CD and EF, intersect at G, then

CG \times DG = EG \times FG

. (Chord Theorem)
- If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then

DC^2 = DG \times DE

. (Tangent Secant Theorem)
- If two secants, DG and DE, also cut the circle at H and F respectively, then

DH \times DG = DF \times DE

. (Corollary of the Tangent Secant Theorem)
- The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent Chord Property)
- If the angle subtended by the chord at the centre is 90 degrees then l = sqrt(2)
- r, where l is the length of the chord and r is the radius of the circle.

External links

- [http://agutie.homestead.com/files/clifford1.htm Clifford's Circle Chain Theorems.] This is a step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression. by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- [http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml Munching on Circles] at cut-the-knot Category:Conic sections ja:円 (数学) simple:Circle

Origin

The origin of something (from the Latin origo, "beginning") is where it came from, in the sense of a physical location or a metaphysical source.
- In mathematics:
- the origin of a coordinate system is the point where the axes of the system intersect.
- The following branches of science and philosophy deal with origins.
- Cosmology -- origin and evolution of the universe
- Epistemology -- origin of knowledge
- Etymology -- origin of words
- Paleoanthropology -- the study of human origin
- Toponymy -- origin of place names
- In games:
- Origin can mean Origin Systems, a computer game developer
- Origins can mean Origins International Game Expo, a gaming convention
- In music:
- Origin is an album by Evanescence
- Origin is a death metal band from Kansas, United States.
- In literature:
- The Origin may refer to Charles Darwin's 1859 book The Origin of Species
- Origin is the name of a biographical novel by writer Irving Stone based on the life of Charles Darwin.
- In comic books, an origin story tells how a character gained his special abilities and/or how he became a superhero or supervillain. Such stories are considered very significant by fans; characters whose origins have never been told are subject to wide speculation.
- Origin is a comic book mini-series published by Marvel Comics in 2002, telling the origin of the superhero Wolverine, Marvel's ultimate "man without a past".
- Secret Origins was a comic book series published by DC Comics that told the origins of different characters in each issue.
- Origin or Manifold: Origin is a science fiction book by Stephen Baxter.
- Origins can also refer to a brand by Estée Lauder Inc.
- You may also be looking for Origen, the Christian scholar and theologian.

Symmetric matrix

In linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if :

A^\textrm = A,

which implies that A is a square matrix. The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (a_ij), then :

a_ = a_

for all indices i and j. The following 3-by-3 matrix is symmetric: :

\begin
1 & 2 & 3\\
2 & -4 & 5\\
3 & 5 & 6\end

Any diagonal matrix is symmetric, since all its off-diagonal entries are zero. A matrix is called skew-symmetric if its transpose is the same as its negative.

Properties

One of the basic theorems concerning such matrices is the finite-dimensional spectral theorem, which says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: to every symmetric real matrix A there exists a real orthogonal matrix Q such that D = Q^TAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. Another way of stating the spectral theorem is that the eigenvectors of a symmetric matrix are orthogonal. Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the above diagonal matrix D, and therefore D is uniquely determined by A, up to the order of its entries.) Essentially, the property of symmetry of real matrices corresponds to the property of being Hermitian for complex matrices. Every square real matrix X can be written in a unique way as the sum of a symmetric and a skew-symmetric matrix. This is done in the following way: :

X=\frac\left(X+X^\textrm\right)+\frac\left(X-X^\textrm\right).

(This is true more generally for every square matrix X with entries from any field whose characteristic is different from 2.) The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices A and B, then AB is symmetric if and only if A and B commute, i.e. if AB = BA. Two real symmetric matrices commute if and only if they have the same eigenspaces. Denote with <,> the standard inner product on Rⁿ. The real n-by-n matrix A is symmetric if and only if :

\langle Ax,y \rangle = \langle x, Ay\rangle \quad \mboxx,y\in\Bbb^n

. Using the Jordan normal form, one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices. (Bosch, 1986)

Occurrence

Symmetric real n-by-n matrices appear as the Hessian of twice continuously differentiable functions of n real variables. Every quadratic form q on Rⁿ can be uniquely written in the form q(x) = x^TAx with a symmetric n-by-n matrix A. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of Rⁿ, "looks like" :

q(x_1,\ldots,x_n)=\sum_^n \lambda_i x_i^2

with real numbers λ_i. This considerably simplifies the study of quadratic forms, as well as the study of the level sets which are generalizations of conic sections. This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of Taylor's theorem.

References

- Category:Matrices ja:対称行列

Eigenvalues

, the picture was deformed in such a way that its central vertical axis was not modified. (Note: The corners have been cropped on the right hand picture.) The blue vector, from her chest to her shoulder, has changed direction, but the red one, from her chest to her chin, is unchanged. The red vector is thus an eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors along the same vertical line are also eigenvectors, with the same eigenvalue. They form the eigenspace for this eigenvalue.]] In mathematics, an eigenvector of a transformation is a vector whose direction is unchanged by that transformation. The factor by which the magnitude is scaled is called the eigenvalue of that vector. A pictorial example is provided in Fig. 1. Often, a transformation is completely described by its eigenvalues and eigenvectors. An eigenspace is a set of eigenvectors with the same eigenvalue. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and even a variety of nonlinear situations. The German word eigen was first used in this context by Hilbert in 1904 (there was an earlier related usage by Helmholtz). "Eigen" can be translated as "own", "peculiar to", "characteristic" or "individual"—emphasizing how important eigenvalues are to defining the unique nature of a specific transformation. In English mathematical jargon, the closest translation would be "characteristic"; and some older references do use expressions like "characteristic value" and "characteristic vector", or even "eigenwert", German for eigenvalue. In the past, the standard translation used to be "proper". Today the more distinctive term "eigenvalue" is standard.

Definitions

Transformations of space—such as translation (or shifting the origin), rotation, reflection, stretching, compression, or any combination of these; other transformations could also be listed—may be visualized by the effect they produce on vectors. Vectors can be visualised as arrows pointing from one point to another.
- Eigenvectors of transformations are vectors which are either left unaffected or simply multiplied by a scale factor after the transformation.
- An eigenvector's eigenvalue is the scale factor that it has been multiplied by.
- An eigenspace is a space consisting of all eigenvectors which have the same eigenvalue.
- The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.
- The spectrum of a transformation is the set of all its eigenvalues. For instance, an eigenvector of a rotation in three dimensions is a vector located within the axis around which the rotation is performed. The corresponding eigenvalue is 1 and the corresponding eigenspace contains all the vectors parallel to the axis. As this is a one-dimensional space, its geometric multiplicity is one. This is the only eigenvalue of the spectrum (of this rotation) that is a real number.

Examples

As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows that lie on the axis of rotation. Consider the transformation of the Earth after one hour of rotation: An arrow from the center of the Earth to the Geographic South Pole would be an eigenvector of this transformation, but an arrow from the center of the Earth to anywhere on the equator would not be an eigenvector. Since the arrow pointing at the pole is not stretched by the rotation of the Earth, its eigenvalue is 1. Another example is provided by a thin metal sheet expanding uniformly about a fixed point in such a way that the distances from any point of the sheet to the fixed point are doubled. This expansion is a transformation with eigenvalue 2. Every vector from the fixed point to a point on the sheet is an eigenvector, and the eigenspace is the set of all these vectors. equator is scaled but its shape is not modified. In this case the eigenvalue is time dependent.]] However, three-dimensional geometric space is not the only vector space. For example, consider a stressed rope fixed at both ends, like the vibrating strings of a string instrument (Fig. 2). The distances of atoms of the vibrating rope from their positions when the rope is at rest can be seen as the components of a vector in a space with as many dimensions as there are atoms in the rope. If one considers the transformation of the rope as time passes, its eigenvectors, or eigenfunctions, if one assumes the rope is a continuous medium, are its standing waves—the things that, mediated by the surrounding air, humans can experience as the twang of a bow string or the plink of a guitar. The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as time passes. Each com

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