1.
Stereographic projection
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In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =
2.
Conformal map
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In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
3.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
4.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
5.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
6.
Ball (mathematics)
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In mathematics, a ball is the space bounded by a sphere. It may be a ball or an open ball. These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, a ball or hyperball in n dimensions is called an n-ball and is bounded by an -sphere. Thus, for example, a ball in the Euclidean plane is the thing as a disk. In Euclidean 3-space, a ball is taken to be the bounded by a 2-dimensional spherical shell. In a one-dimensional space, a ball is a line segment, in other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In Euclidean n-space, an n-ball of radius r and center x is the set of all points of less than r from x. A closed n-ball of radius r is the set of all points of less than or equal to r away from x. In Euclidean n-space, every ball is bounded by a hypersphere, the ball is a bounded interval when n =1, is a disk bounded by a circle when n =2, and is bounded by a sphere when n =3. The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is, V n = π n 2 Γ R n, where Γ is Leonhard Eulers gamma function. These are, V2 k = π k k. R2 k, V2 k +1 =2 k +1 π k. R2 k +1 =2 k. R2 k +1. In the formula for odd-dimensional volumes, the double factorial. is defined for odd integers 2k +1 as, let be a metric space, namely a set M with a metric d. Note in particular that a ball always includes p itself, since the definition requires r >0, the closure of the open ball Br is usually denoted Br. While it is always the case that Br ⊆ Br ⊆ Br, for example, in a metric space X with the discrete metric, one has B1 = and B1 = X, for any p ∈ X. A unit ball is a ball of radius 1, a subset of a metric space is bounded if it is contained in some ball. A set is bounded if, given any positive radius. The open balls of a space are a basis for a topological space. This space is called the topology induced by the metric d, any normed vector space V with norm | · | is also a metric space, with the metric d = | x − y |
7.
Surface (topology)
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In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an abstract surface not embedded in any Euclidean space. For example, the Klein bottle is a surface, which cannot be represented in the three-dimensional Euclidean space without introducing self-intersections, in mathematics, a surface is a geometrical shape that resembles to a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, the exact definition of a surface may depend on the context. Typically, in geometry, a surface may cross itself, while, in topology and differential geometry. A surface is a space, this means that a moving point on a surface may move in two directions. In other words, around almost every point, there is a patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, the concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the properties of an airplane. A surface is a space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a chart and it is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean. In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second countable. It is also assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second countable and these homeomorphisms are also known as charts. The boundary of the upper half-plane is the x-axis, a point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of points is known as the boundary of the surface which is necessarily a one-manifold, that is. On the other hand, a point mapped to above the x-axis is an interior point, the collection of interior points is the interior of the surface which is always non-empty. The closed disk is an example of a surface with boundary
8.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
9.
3-manifold
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In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a shape of the universe. Just as a sphere looks like a plane to a small enough observer and this is made more precise in the definition below. A topological space X is a 3-manifold if it is a second-countable Hausdorff space, 3-manifold theory is considered a part of low-dimensional topology or geometric topology. A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it, Thurstons contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry. The most prevalent geometry is hyperbolic geometry, using a geometry in addition to special surfaces is often fruitful. The fundamental groups of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold, thus, there is an interplay between group theory and topological methods. Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it and this is just the standard 3-dimensional vector space over the real numbers. A 3-sphere is a analogue of a sphere. It consists of the set of points equidistant from a central point in 4-dimensional Euclidean space. Real projective 3-space, or RP3, is the space of lines passing through the origin 0 in R4. It is a compact, smooth manifold of dimension 3, and is a special case Gr of a Grassmannian space. RP3 is SO, hence admits a group structure, the covering map S3 → RP3 is a map of groups Spin → SO, the 3-dimensional torus is the product of 3 circles. That is, T3 = S1 × S1 × S1, the 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer lattice Z3, equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together. A 3-torus in this sense is an example of an 3-dimensional compact manifold and it is also an example of a compact abelian Lie group. This follows from the fact that the circle is a compact abelian Lie group. Group multiplication on the torus is defined by coordinate-wise multiplication
10.
Coordinates
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space, depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems, another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis, for a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
11.
Four-dimensional space
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For example, the volume of a rectangular box is found by measuring its length, width, and depth. More than two millennia ago Greek philosophers explored in detail the implications of this uniformity, culminating in Euclids Elements. However, it was not until recent times that a handful of insightful mathematical innovators generalized the concept of dimensions to more than three. The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s, in 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension. Which was notable for explaining the concept of a cube by going through a step-by-step generalization of the properties of lines, squares. The simplest form of Hintons method is to draw two ordinary cubes separated by a distance, and then draw lines between their equivalent vertices. This form can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube, the eight lines connecting the vertices of the two cubes in that case represent a single direction in the unseen fourth dimension. Higher dimensional spaces have become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces, calendar entries for example are usually 4D locations, such as a meeting at time t at the intersection of two streets on some building floor. In list form such a meeting place at the 4D location. Einsteins concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space, when dimensional locations are given as ordered lists of numbers such as they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the richness and geometric complexity of 4D. A hint of that complexity can be seen in the animation of one of simplest possible 4D objects. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time, the possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843 and this associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R, one of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension. Published in the Dublin University magazine and he coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension. Hintons ideas inspired a fantasy about a Church of the Fourth Dimension featured by Martin Gardner in his January 1962 Mathematical Games column in Scientific American, in 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams
12.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
13.
Quaternion
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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843, a feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a space or equivalently as the quotient of two vectors. Quaternions are generally represented in the form, a + bi + cj + dk where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units. In practical applications, they can be used other methods, such as Euler angles and rotation matrices, or as an alternative to them. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, in fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H, or in blackboard bold by H and it can also be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. The unit quaternions can be thought of as a choice of a structure on the 3-sphere S3 that gives the group Spin. Quaternion algebra was introduced by Hamilton in 1843, carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the numbers could be interpreted as points in a plane. Points in space can be represented by their coordinates, which are triples of numbers, however, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i2 = j2 = k2 = ijk = −1, into the stone of Brougham Bridge as he paused on it. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves and this letter was later published in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. xxv, pp 489–95. In the letter, Hamilton states, And here there dawned on me the notion that we must admit, in some sense, an electric circuit seemed to close, and a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, Hamiltons treatment is more geometric than the modern approach, which emphasizes quaternions algebraic properties
14.
Versor
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Versors are an algebraic parametrisation of rotations. In classical quaternion theory a versor is a quaternion of norm one. Each versor has the form q = exp = cos a + r sin a, r 2 = −1, a ∈, in case a = π/2, the versor is termed a right versor. The corresponding 3-dimensional rotation has the angle 2a about the axis r in axis–angle representation, the word is derived from Latin versare = to turn with the suffix -or forming a noun from the verb. It was introduced by William Rowan Hamilton in the context of his quaternion theory, for historical reasons, it sometimes is used synonymously with a unit quaternion without a reference to rotations. In the quaternion algebra a versor q = exp will rotate any quaternion v through the product map v ↦ q v q −1 such that the scalar part of v is preserved. If this scalar part is zero, i. e. v is a Euclidean vector in three dimensions, then the formula above defines the rotation through the angle 2a around the vector r. In other words, qvq−1 rotates the vector part of v around the vector r, see quaternions and spatial rotation for details. A quaternionic versor expressed in the complex 2×2 matrix representation is an element of the unitary group SU. Spin and SU are the same group, angles of rotation in this λ = 1/2 representation are equal to a, there is no 2 factor in angles unlike the λ =1 adjoint representation mentioned above, see representation theory of SU for details. For a fixed r, versors of the form exp where a ∈ (−π, π], in 2003 David W. Lyons wrote the fibers of the Hopf map are circles in S3. Lyons gives an introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions. Hamilton denoted the versor of a quaternion q by the symbol Uq and he was then able to display the general quaternion in polar coordinate form q = Tq Uq, where Tq is the norm of q. The norm of a versor is always equal to one, hence they occupy the unit 3-sphere in H, examples of versors include the eight elements of the quaternion group. Of particular importance are the right versors, which have angle π/2 and these versors have zero scalar part, and so are vectors of length one. The right versors form a sphere of square roots of −1 in the quaternion algebra, the generators i, j, and k are examples of right versors, as well as their additive inverses. Other versors include the twenty-four Hurwitz quaternions that have the norm 1, Hamilton defined a quaternion as the quotient of two vectors. A versor can be defined as the quotient of two unit vectors, for any fixed plane Π the quotient of two unit vectors lying in Π depends only on the angle between them, the same a as in the unit vector–angle representation of a versor explained above
15.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
16.
Polar coordinates
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The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle, the Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth, from the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidges Origin of Polar Coordinates, grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs, in Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the Seventh Manner, For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole, Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoullis work extended to finding the radius of curvature of curves expressed in these coordinates, the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacocks 1816 translation of Lacroixs Differential and Integral Calculus, alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The radial coordinate is often denoted by r or ρ, the angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics, in many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the axis is often drawn horizontal. Adding any number of turns to the angular coordinate does not change the corresponding direction. Also, a radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the point can be expressed with an infinite number of different polar coordinates or
17.
Elliptic geometry
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Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a line must intersect. In fact, the perpendiculars on one side all intersect at the pole of the given line. There are no points in elliptic geometry. Every point corresponds to a polar line of which it is the absolute pole. Any point on this line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant, the distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name elliptic is possibly misleading and it does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes, analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A simple way to picture elliptic geometry is to look at a globe, neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, with this identification of antipodal points, the model satisfies Euclids first postulate, which states that two points uniquely determine a line. Metaphorically, we can imagine geometers who are like living on the surface of a sphere. Even if the ants are unable to move off the surface, they can still construct lines, the existence of a third dimension is irrelevant to the ants ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the axioms is incapable of expressing the distinction between one model and another. In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the figures are similar, i. e. they have the same angles. In elliptic geometry this is not the case, for example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere. A line segment therefore cannot be scaled up indefinitely, a geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space
18.
Compact space
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In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a set of points. This notion is defined for general topological spaces than Euclidean space in various ways. One such generalization is that a space is compact if any infinite sequence of points sampled from the space must frequently get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space, the Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses a number of points in the closed unit interval some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, the same set of points would not accumulate to any point of the open unit interval, so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded, in particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points, the term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Various equivalent notions of compactness, including sequential compactness and limit point compactness, in general topological spaces, however, different notions of compactness are not necessarily equivalent. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, the term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano had been aware that any bounded sequence of points has a subsequence that must eventually get close to some other point. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until it closes down on the limit point. The full significance of Bolzanos theorem, and its method of proof, in the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as points of a generalized space dates back to the investigations of Giulio Ascoli. The uniform limit of this sequence then played precisely the same role as Bolzanos limit point and this ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property, in 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous
19.
Connected space
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In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the topological properties that are used to distinguish topological spaces. A subset of a topological space X is a set if it is a connected space when viewed as a subspace of X. A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets, otherwise, X is said to be connected. A subset of a space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set as a space. For a topological space X the following conditions are equivalent, X is connected, X cannot be divided into two disjoint nonempty closed sets. The only subsets of X which are open and closed are X and the empty set. The only subsets of X with empty boundary are X and the empty set, X cannot be written as the union of two nonempty separated sets. All continuous functions from X to are constant, where is the two-point space endowed with the discrete topology, the maximal connected subsets of a nonempty topological space are called the connected components of the space. The components of any topological space X form a partition of X, they are disjoint, nonempty, Every component is a closed subset of the original space. It follows that, in the case where their number is finite, however, if their number is infinite, this might not be the case, for instance, the connected components of the set of the rational numbers are the one-point sets, which are not open. A space in all components are one-point sets is called totally disconnected. Clearly any totally separated space is totally disconnected, but the converse does not hold, for example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected, however, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being separated is strictly stronger than the condition of being Hausdorff. The closed interval in the subspace topology is connected, although it can, for example, be written as the union of [0, 1) and. The union of is disconnected, both of these intervals are open in the topological space
20.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
21.
Simply connected space
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If a space is not simply-connected, it is convenient to measure the extent to which it fails to be simply-connected, this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space, if there are no holes, the group is trivial — equivalently. Informally, an object in our space is simply-connected if it consists of one piece. For example, neither a doughnut nor a cup is simply connected. In two dimensions, a circle is not simply-connected, but a disk and a line are, spaces that are connected but not simply connected are called non–simply connected or, in a somewhat old-fashioned term, multiply connected. To illustrate the notion of connectedness, suppose we are considering an object in three dimensions, for example, an object in the shape of a box, a doughnut. Think of the object as a strangely shaped aquarium full of water, with rigid sides. Now think of a diver who takes a piece of string and trails it through the water inside the aquarium, in whatever way he pleases. Now the loop begins to contract on itself, getting smaller and smaller, if the loop can always shrink all the way to a point, then the aquariums interior is simply connected. If sometimes the loop gets caught — for example, around the hole in the doughnut — then the object is not simply-connected. Notice that the only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point. The stronger condition, that the object has no holes of any dimension, is called contractibility, intuitively, this means that p can be continuously deformed to get q while keeping the endpoints fixed. Hence the term simply connected, for any two points in X, there is one and essentially only one path connecting them. A third way to express the same, X is simply-connected if and only if X is path-connected and the fundamental group of X at each of its points is trivial, i. e. consists only of the identity element. Yet another formulation is used in complex analysis, an open subset X of C is simply-connected if. It might also be worth pointing out that a relaxation of the requirement that X be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a set has connected extended complement exactly when each of its connected components are simply-connected
22.
Grigori Perelman
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Grigori Yakovlevich Perelman is a Russian mathematician. He was the winner of the all-Russian mathematical olympiad and he made a landmark contribution to Riemannian geometry and geometric topology. In 1994, Perelman proved the soul conjecture, in 2003, he proved Thurstons geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, on 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. Hamilton, the mathematician who pioneered the Ricci flow with the aim of attacking the conjecture and he also turned down the prestigious prize of the European Mathematical Society. Grigori Perelman was born in Leningrad, Soviet Union on 13 June 1966, grigoris mother Lyubov gave up graduate work in mathematics to raise him. Grigoris mathematical talent became apparent at the age of ten, and his mathematical education continued at the Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education and he continued as a student of School of Mathematics and Mechanics at the Leningrad State University, without admission examinations and enrolled to the university. In the late 1980s and early 1990s, with a recommendation from the celebrated geometer Mikhail Gromov. In 1991 Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrovs spaces of curvature bounded from below. In 1992, he was invited to spend a semester each at the Courant Institute in New York University, from there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley in 1993. Until late 2002, Perelman was best known for his work in comparison theorems in Riemannian geometry, among his notable achievements was a short and elegant proof of the soul conjecture. The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was one of key problems in topology, any loop on a 3-sphere—as exemplified by the set of points at a distance of 1 from the origin in four-dimensional Euclidean space—can be contracted into a point. The Poincaré conjecture asserts that any closed three-dimensional manifold such that any loop can be contracted into a point is topologically a 3-sphere, the analogous result has been known to be true in dimensions greater than or equal to five since 1960 as in the work of Stephen Smale. The four-dimensional case resisted longer, finally being solved in 1982 by Michael Freedman, but the case of three-manifolds turned out to be the hardest of them all. Roughly speaking, this is because in topologically manipulating a three-manifold there are too few dimensions to move problematic regions out of the way without interfering with something else, the most fundamental contribution to the three-dimensional case had been produced by Richard S. Hamilton. The role of Perelman was to complete the Hamilton program, Perelman modified Richard S. Hamiltons program for a proof of the conjecture. The central idea is the notion of the Ricci flow, the heat equation describes the behavior of scalar quantities such as temperature
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Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
24.
Topological space
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Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion, the branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The utility of the notion of a topology is shown by the fact there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application, the most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so this is given first. Note, A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy and this axiomatization is due to Felix Hausdorff. Let X be a set, the elements of X are usually called points, let N be a function assigning to each x in X a non-empty collection N of subsets of X. The elements of N will be called neighbourhoods of x with respect to N, the function N is called a neighbourhood topology if the axioms below are satisfied, and then X with N is called a topological space. If N is a neighbourhood of x, then x ∈ N, in other words, each point belongs to every one of its neighbourhoods. If N is a subset of X and includes a neighbourhood of x, I. e. every superset of a neighbourhood of a point x in X is again a neighbourhood of x. The intersection of two neighbourhoods of x is a neighbourhood of x, any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M. The first three axioms for neighbourhoods have a clear meaning, the fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. A topological space is a pair, where X is a set and τ is a collection of subsets of X, satisfying the following axioms, The empty set. Any union of members of τ still belongs to τ, the intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X, given X =, the collection τ = of only the two subsets of X required by the axioms forms a topology of X, the trivial topology. Given X =, the collection τ = of six subsets of X forms another topology of X, given X = and the collection τ = P, is a topological space. τ is called the discrete topology, using de Morgans laws, the above axioms defining open sets become axioms defining closed sets, The empty set and X are closed. The intersection of any collection of closed sets is also closed, the union of any finite number of closed sets is also closed
25.
Homology (mathematics)
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In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. Homology groups were defined in algebraic topology. Similar constructions are available in a variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes, however, because a hole is not there, it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold, loosely speaking, a cycle is a closed submanifold, a boundary is the boundary of a submanifold with boundary, and a homology class is an equivalence class of cycles modulo boundaries. There are many different homology theories, a particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has an interpretation like topological spaces do. In general, most homology groups or modules arise as derived functors on appropriate abelian categories and they provide concrete descriptions of the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category, Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. This was followed by Riemanns definition of genus and n-fold connectedness numerical invariants in 1857 and these cycles are also sometimes thought of as cuts which can be glued back together, or as zippers which can be fastened and unfastened. For example, a line drawn on a surface represents a 1-cycle, on the ordinary sphere S2, the cycle b in the diagram can be shrunk to the pole and even the equatorial great circle a can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as c can be shrunk to a point. All cycles on the sphere can therefore be continuously transformed into each other and they are said to be homologous to zero. Cutting a manifold along a cycle homologous to zero separates the manifold into two or more components, for example, cutting the sphere along a produces two hemispheres. This is not generally true of cycles on other surfaces, the torus T2 has cycles which cannot be continuously deformed into each other, for example in the diagram none of the cycles a, b or c can be. Cycles a and b cannot be shrunk significantly, cycle c can be shrunk to a point, so it is homologous to zero. If the torus surface is cut along both a and b, it can be opened out and flattened into a rectangle or, more conveniently, one opposite pair of sides represents the cut along a, and the other opposite pair represents the cut along b. The edges of the square may then be glued together in different ways
26.
Cyclic group
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In algebra, a cyclic group or monogenous group is a group that is generated by a single element. Each element can be written as a power of g in multiplicative notation and this element g is called a generator of the group. Every infinite cyclic group is isomorphic to the group of Z. Every finite cyclic group of n is isomorphic to the additive group of Z/nZ. Every cyclic group is a group, and every finitely generated abelian group is a direct product of cyclic groups. A group G is called if there exists an element g in G such that G = ⟨g⟩ =. Since any group generated by an element in a group is a subgroup of that group, for example, if G = is a group of order 6, then g6 = g0, and G is cyclic. In fact, G is essentially the same as the set with addition modulo 6, for example,1 +2 ≡3 corresponds to g1 · g2 = g3, and 2 +5 ≡1 corresponds to g2 · g5 = g7 = g1, and so on. One can use the isomorphism χ defined by χ = i, the name cyclic may be misleading, it is possible to generate infinitely many elements and not form any literal cycles, that is, every gn is distinct. A group generated in this way is called a cyclic group. The French mathematicians known as Nicolas Bourbaki referred to a group as a monogenous group. The set of integers, with the operation of addition, forms a group and it is an infinite cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group,1 and −1 are the only generators, every infinite cyclic group is isomorphic to this group. For every positive n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group. An element g is a generator of this group if g is relatively prime to n, thus, the number of different generators is φ, where φ is the Euler totient function, the function that counts the number of numbers modulo n that are relatively prime to n. Every finite cyclic group is isomorphic to a group Z/n, where n is the order of the group, the integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/n. If p is a prime, then Z/p is a finite field, every field with p elements is isomorphic to this one. For every positive n, the subset of the integers modulo n that are relatively prime to n, with the operation of multiplication
27.
Knot theory
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In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants
28.
Homotopy group
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In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the group, which records information about loops in a space. Intuitively, homotopy groups record information about the shape, or holes. To define the n-th homotopy group, the maps from an n-dimensional sphere into a given space are collected into equivalence classes. Two mappings are homotopic if one can be deformed into the other. These homotopy classes form a group, called the homotopy group, πn. Topological spaces with differing homotopy groups are never equivalent, but the converse is not true, the notion of homotopy of paths was introduced by Camille Jordan. In modern mathematics it is common to study a category by associating to every object of this category an object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associating groups to topological spaces and that link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two objects have different homotopy groups, they cant have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere, the torus has a hole, however, since continuity only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure, as for the example, the first homotopy group of the torus T is π1=Z2, because the universal cover of the torus is the complex plane C, mapping to the torus T ≅ C / Z2. Here the quotient is in the category of spaces, rather than groups or rings. On the other hand, the sphere S2 satisfies π1=0, because every loop can be contracted to a constant map, hence the torus is not homeomorphic to the sphere. In the n-sphere Sn we choose a point a. For a space X with base point b, we define πn to be the set of classes of maps f. In particular, the classes are given by homotopies that are constant on the basepoint of the sphere. Equivalently, we can define πn to be the group of classes of maps g, n → X from the n-cube to X that take the boundary of the n-cube to b
29.
Finite abelian group
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That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers and they are named after Niels Henrik Abel. The concept of a group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules. The theory of groups is generally simpler than that of their non-abelian counterparts. On the other hand, the theory of abelian groups is an area of current research. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b, the symbol • is a general placeholder for a concretely given operation. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, commutativity For all a, b in A, a • b = b • a. A group in which the operation is not commutative is called a non-abelian group or non-commutative group. There are two main conventions for abelian groups – additive and multiplicative. Generally, the notation is the usual notation for groups, while the additive notation is the usual notation for modules. To verify that a group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is G = under the operation ⋅, the th entry of this contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal and this is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the th entry of the table equals the th entry, every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form a group under addition, as do the integers modulo n. Every ring is a group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group, in particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication
30.
Homotopy groups of spheres
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In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as Sn — generalizes the familiar circle, the n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n +1 located at a unit distance from the origin. The i-th homotopy group πi summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be deformed to the other, thus, only equivalence classes of mappings are summarized. An addition operation defined on these equivalence classes makes the set of classes into an abelian group. The problem of determining πi falls into three regimes, depending on whether i is less than, equal to, or greater than n. For 0 < i < n, any mapping from Si to Sn is homotopic to a constant mapping, therefore the homotopy group is the trivial group. When i = n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself and this degree identifies the homotopy group πn with the group of integers under addition. The most interesting and surprising results occur when i > n, one of the main discoveries is that the homotopy groups πn+k are independent of n for n ≥ k +2. These are called the homotopy groups of spheres and have been computed for values of k up to 64. The stable homotopy groups form the coefficient ring of a cohomology theory. The unstable homotopy groups are more erratic, nevertheless, they have been tabulated for k <20, most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet remains unknown. The study of groups of spheres builds on a great deal of background material. Algebraic topology provides the context, itself built on topology and abstract algebra. An ordinary sphere in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology, geometry defines a sphere rigidly, as a shape. Implicit surface, x02 + x12 + x22 =1 This is the set of points in 3-dimensional Euclidean space found exactly one unit away from the origin and it is called the 2-sphere, S2, for reasons given below
31.
Differentiable manifold
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In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas, one may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart, in formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. In other words, where the domains of overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the charts to one another are called transition maps. Differentiability means different things in different contexts including, continuously differentiable, k times differentiable, smooth, furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for theories such as classical mechanics, general relativity. It is possible to develop a calculus for differentiable manifolds and this leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry, the emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen and these ideas found a key application in Einsteins theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces, the widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space. This formalizes the notion of patching together pieces of a space to make a manifold – the manifold produced also contains the data of how it has been patched together, However, different atlases may produce the same manifold, a manifold does not come with a preferred atlas. And, thus, one defines a manifold to be a space as above with an equivalence class of atlases. There are a number of different types of manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following, a differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable
32.
Submanifold
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In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required, different authors often have different definitions. In the following we assume all manifolds are manifolds of class Cr for a fixed r ≥1. Given any injective immersion f, N → M the image of N in M can be given the structure of an immersed submanifold so that f, N → f is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions, the submanifold topology on an immersed submanifold need not be the relative topology inherited from M. In general, it will be finer than the subspace topology, immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds. An embedded submanifold, is an immersed submanifold for which the map is a topological embedding. That is, the topology on S is the same as the subspace topology. Given any embedding f, N → M of a manifold N in M the image f naturally has the structure of an embedded submanifold and that is, embedded submanifolds are precisely the images of embeddings. There is a definition of an embedded submanifold which is often useful. Let M be a manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart containing p such that φ is the intersection of a plane with φ. The pairs form an atlas for the structure on S. Alexanders theorem. There are some variations of submanifolds used in the literature. A neat submanifold is a manifold whose boundary agrees with the boundary of the entire manifold, sharpe defines a type of submanifold which lies somewhere between an embedded submanifold and an immersed submanifold. Many authors define topological submanifolds also and these are the same as Cr submanifolds with r =0. An embedded topological submanifold is not necessarily regular in the sense of the existence of a chart at each point extending the embedding. Counterexamples include wild arcs and wild knots, given any immersed submanifold S of M, the tangent space to a point p in S can naturally be thought of as a linear subspace of the tangent space to p in M
33.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
34.
Circle group
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The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers. Since C× is abelian, it follows that T is as well, the circle group is also the group U of 1×1 unitary matrices, these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle θ of rotation by θ ↦ z = e i θ = cos θ + i sin θ and this is the exponential map for the circle group. The circle group plays a role in Pontryagin duality. The notation T for the circle group stems from the fact that, with the standard topology, more generally Tn is geometrically an n-torus. One way to think about the group is that it describes how to add angles. For example, the diagram illustrates how to add 150° to 270°, the answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to forget the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360° which gives 420° = 60°, another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed. To achieve this, we might need to throw away digits occurring before the decimal point. For example, when we work out 0.784 +0.925 +0.446, the answer should be 2.155, the circle group is more than just an abstract algebraic object. It has a natural topology when regarded as a subspace of the complex plane, since multiplication and inversion are continuous functions on C×, the circle group has the structure of a topological group. Moreover, since the circle is a closed subset of the complex plane. The circle is a 1-dimensional real manifold and multiplication and inversion are real-analytic maps on the circle and this gives the circle group the structure of a one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, moreover, every n-dimensional compact, connected, abelian Lie group is isomorphic to Tn. The circle group shows up in a variety of forms in mathematics and we list some of the more common forms here. Specifically, we show that T ≅ U ≅ R / Z ≅ SO, note that the slash denotes here quotient group. The set of all 1×1 unitary matrices clearly coincides with the circle group, therefore, the circle group is canonically isomorphic to U, the first unitary group. The exponential function gives rise to a group homomorphism exp, R → T from the real numbers R to the circle group T via the map θ ↦ e i θ = cos θ + i sin θ
35.
Vector field
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In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as, the elements of differential and integral calculus extend naturally to vector fields. Vector fields can usefully be thought of as representing the velocity of a flow in space. In coordinates, a field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point. More generally, vector fields are defined on manifolds, which are spaces that look like Euclidean space on small scales. In this setting, a field gives a tangent vector at each point of the manifold. Vector fields are one kind of tensor field, given a subset S in Rn, a vector field is represented by a vector-valued function V, S → Rn in standard Cartesian coordinates. If each component of V is continuous, then V is a vector field. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space, in physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a distinct entity from a simple list of scalars. Thus, suppose that is a choice of Cartesian coordinates, in terms of which the components of the vector V are V x =, then the components of the vector V in the new coordinates are required to satisfy the transformation law Such a transformation law is called contravariant. Given a differentiable manifold M, a field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that p ∘ F is the identity mapping where p denotes the projection from TM to M, in other words, a vector field is a section of the tangent bundle. If the manifold M is smooth or analytic—that is, the change of coordinates is smooth —then one can make sense of the notion of vector fields. The collection of all vector fields on a smooth manifold M is often denoted by Γ or C∞. A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind, the length of the arrow will be an indication of the wind speed