Mathematics
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz. For example, the following laws and theories respect Lorentz symmetry: The kinematical laws of special relativity Maxwell's field equations in the theory of electromagnetism The Dirac equation in the theory of the electron The Standard model of particle physicsThe Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In general relativity physics, in cases involving small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as that of special relativity physics; the Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are isometries that leave the origin fixed.
Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations. Mathematically, the Lorentz group may be described as the generalized orthogonal group O, the matrix Lie group that preserves the quadratic form ↦ t 2 − x 2 − y 2 − z 2 on R4; this quadratic form is, when put on matrix form, interpreted in physics as the metric tensor of Minkowski spacetime. The Lorentz group is a six-dimensional noncompact non-abelian real Lie group, not connected; the four connected components are not connected, but rather doubly connected. The identity component of the Lorentz group is itself a group, is called the restricted Lorentz group, is denoted SO+; the restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time.
The restricted Lorentz group has been presented through a facility of biquaternion algebra. The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the point symmetry group of a certain ordinary differential equation; this fact has physical significance. Because it is a Lie group, the Lorentz group O is both a group and admits a topological description as a smooth manifold; as a manifold, it has four connected components. Intuitively, this means; the four connected components can be categorized by two transformation properties its elements have: some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein Lorentz transformations that preserve the direction of time are called orthochronous. The subgroup of orthochronous transformations is denoted O+.
Those that preserve orientation are called proper, as linear transformations they have determinant +1. The subgroup of proper Lorentz transformations is denoted SO; the subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, is denoted by SO+. The set of the four connected components can be given a group structure as the quotient group O/SO+, isomorphic to the Klein four-group; every element in O can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group where P and T are the space inversion and time reversal operators: P = diag T = diag. Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components; this pattern is typical of finite-dimensional Lie groups. The restricted Lorentz group is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a continuous curve lying in the group.
The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension, in this case with dimension six. The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts. Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost, it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation; this is one way. The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO; the set of all boosts, does not form a sub
Homogeneous space
In mathematics in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism; some authors insist. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, making X into a single G-orbit. Let X be a non-empty set and G a group. X is called a G-space if it is equipped with an action of G on X. Note that automatically G acts by automorphisms on the set. If X in addition belongs to some category the elements of G are assumed to act as automorphisms in the same category.
Thus, the maps on X effected by G are structure preserving. A homogeneous space is a G-space. Succinctly, if X is an object of the category C the structure of a G-space is a homomorphism: ρ: G → A u t C into the group of automorphisms of the object X in the category C; the pair defines a homogeneous space provided ρ is a transitive group of symmetries of the underlying set of X. For example, if X is a topological space group elements are assumed to act as homeomorphisms on X; the structure of a G-space is a group homomorphism ρ: G → Homeo into the homeomorphism group of X. Similarly, if X is a differentiable manifold the group elements are diffeomorphisms; the structure of a G-space is a group homomorphism ρ: G → Diffeo into the diffeomorphism group of X. Riemannian symmetric spaces are an important class of homogeneous spaces, include many of the examples listed below. Concrete examples include: Isometry groupsPositive curvature:Sphere: S n − 1 ≅ O / O; this is true because of the following observations: First, S n − 1 is the set of vectors in R n with norm 1.
If we consider one of these vectors as a base vector any other vector can be constructed using an orthogonal transformation. If we consider the span of this vector as a one dimensional subspace of R n the complement is an -dimensional vector space, invariant under an orthogonal transformation from O; this shows us. Oriented sphere: S n − 1 ≅ S O / S O Projective space: P n − 1 ≅ P O / P O Flat:Euclidean space: An ≅ E/ONegative curvature:Hyperbolic space: Hn ≅ O+/O Oriented hyperbolic space: SO+/SO Anti-de Sitter space: AdSn+1 = O/OOthersAffine space: An = Aff/GL. Grassmannian: G r = O / Topological vector spaces From the point of view of the Erlangen program, one may understand that "all points are the same", in the geometry of X; this was true of all geometries proposed before Riemannian geometry, in the middle of the nineteenth century. Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups; the same is true of the models found of non-Euclidean geometry of constant curvature, such as hyperbolic space.
A further classical example is the space of lines in projective space of three dimensions. It is simple linear algebra to show that GL
Sphere
A sphere is a round geometrical object in three-dimensional space, the surface of a round ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in a three-dimensional space; this distance r is the radius of the ball, made up from all points with a distance less than r from the given point, the center of the mathematical ball. These are referred to as the radius and center of the sphere, respectively; the longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, a two-dimensional closed surface, embedded in a three-dimensional Euclidean space, a ball, a three-dimensional shape that includes the sphere and everything inside the sphere, or, more just the points inside, but not on the sphere.
The distinction between ball and sphere has not always been maintained and older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can be confounded. In analytic geometry, a sphere with center and radius r is the locus of all points such that 2 + 2 + 2 = r 2. Let a, b, c, d, e be real numbers with a ≠ 0 and put x 0 = − b a, y 0 = − c a, z 0 = − d a, ρ = b 2 + c 2 + d 2 − a e a 2; the equation f = a + 2 + e = 0 has no real points as solutions if ρ < 0 and is called the equation of an imaginary sphere. If ρ = 0 the only solution of f = 0 is the point P 0 = and the equation is said to be the equation of a point sphere. In the case ρ > 0, f = 0 is an equation of a sphere whose center is P 0 and whose radius is ρ. If a in the above equation is zero f = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius; the points on the sphere with radius r > 0 and center can be parameterized via x = x 0 + r sin θ cos φ y = y 0 + r sin θ sin φ z = z 0 + r cos θ The parameter θ {
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions.
Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.
The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though
Differentiable manifold
In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts known as an atlas. One may 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 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. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts 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 various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable and holomorphic. 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 tensor and vector fields. Differentiable manifolds are important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, Yang–Mills theory, it is possible to develop a calculus for differentiable manifolds. 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 credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen, he motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, presciently described the role of coordinate systems and charts in subsequent formal developments: Having constructed the notion of a manifoldness of n dimensions, found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude... – B. RiemannThe works of physicists such as James Clerk Maxwell, mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one, invariant with respect to coordinate transformations; these ideas found a key application in Einstein's 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 accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a topological manifold is a second countable Hausdorff space, locally homeomorphic to a vector space, by a collection of homeomorphisms called charts; the composition of one chart with the inverse of another chart is a function called a transition map, defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced contains the data of how it has been patched together. However, different atlases may produce "the same" manifold. Thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below. There are a number of different types of differentiable 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. More a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable. A smooth manifold or C∞-manifold is a differentiable manifold for which all the transition maps are smooth; that is, derivatives of all orders exist. An equivalence class of such atlases is said to be a smooth structure. An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is convergent and equals the function on some open ball. A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic. While there is a meaningful notion of a Ck atlas, there is no distinct notion of a Ck manifold other than C0 and C∞, because for every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure – a result of Whitney.
In fact, eve
Minkowski space
In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity. Minkowski space is associated with Einstein's theory of special relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events; because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.
In 3-dimensional Euclidean space, the isometry group is the Euclidean group. It is generated by rotations and translations; when time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, the group of all these transformations is called the Galilean group. All Galilean transformations preserve the 3-dimensional Euclidean distance; this distance is purely spatial. Time differences are separately preserved as well; this changes in the spacetime of special relativity, where time are interwoven. Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context; the Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space; the analogue of the Galilean group for Minkowski space, preserving the spacetime interval is the Poincaré group.
In summary, Galilean spacetime and Minkowski spacetime are, when viewed as manifolds the same. They differ in; the former has the Euclidean distance function and time together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations. In 1905–06 Henri Poincaré showed that by taking time to be an imaginary fourth spacetime coordinate ict, where c is the speed of light and i is the imaginary unit, a Lorentz transformation can formally be regarded as a rotation of coordinates in a four-dimensional space with three real coordinates representing space, one imaginary coordinate representing time, as the fourth dimension. In physical spacetime special relativity stipulates that the quantity − t 2 + x 2 + y 2 + z 2 is invariant under coordinate changes from one inertial frame to another, i. e. under Lorentz transformations. Here the speed of light c is, following Poincaré, set to unity.
In the space suggested by him where physical spacetime is coordinatized by ↦, call it coordinate space, Lorentz transformations appear as ordinary rotations preserving the quadratic form x 2 + y 2 + z 2 + t 2 on coordinate space. The naming and ordering of coordinates, with the same labels for space coordinates, but with the imaginary time coordinate as the fourth coordinate, is conventional; the above expression, while making the former expression more familiar, may be confusing because it is not the same t that appears in the latter as in the former. Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime appear as Euclidean rotations and are interpreted in the ordinary sense; the "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with real inertial coordinates. The analogy with Euclidean rotations is thus only partial.
This idea was elaborated by Hermann Minkowski, who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of special relativity of Einstein. From this he concluded that time and space should be treated and so arose his concept of events taking place in a unified four-dimensional spacetime continuum. In a further development in his 1908 "Space and Time" lecture, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables of space and time in coordinate form in a four dimensional real vector space. Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, events not on the light-cone are classified by their relation to the apex as spacelike or timel