In mathematics and physics, a scalar field associates a scalar value to every point in a space – physical space. The scalar may either be a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, spin-zero quantum fields, such as the Higgs field; these fields are the subject of scalar field theory. Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U; the region U may be a set in some Euclidean space, Minkowski space, or more a subset of a manifold, it is typical in mathematics to impose further conditions on the field, such that it be continuous or continuously differentiable to some order.
A scalar field is a tensor field of order zero, the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. Physically, a scalar field is additionally distinguished by having units of measurement associated with it. In this context, a scalar field should be independent of the coordinate system used to describe the physical system—that is, any two observers using the same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as vector fields, which associate a vector to every point of a region, as well as tensor fields and spinor fields. More subtly, scalar fields are contrasted with pseudoscalar fields. In physics, scalar fields describe the potential energy associated with a particular force; the force is a vector field, which can be obtained as the gradient of the potential energy scalar field. Examples include: Potential fields, such as the Newtonian gravitational potential, or the electric potential in electrostatics, are scalar fields which describe the more familiar forces.
A temperature, humidity or pressure field, such as those used in meteorology. In quantum field theory, a scalar field is associated with spin-0 particles; the scalar field may be complex valued. Complex scalar fields represent charged particles; these include the charged Higgs field of the Standard Model, as well as the charged pions mediating the strong nuclear interaction. In the Standard Model of elementary particles, a scalar Higgs field is used to give the leptons and massive vector bosons their mass, via a combination of the Yukawa interaction and the spontaneous symmetry breaking; this mechanism is known as the Higgs mechanism. A candidate for the Higgs boson was first detected at CERN in 2012. In scalar theories of gravitation scalar fields are used to describe the gravitational field. Scalar-tensor theories represent the gravitational interaction through a scalar; such attempts are for example the Jordan theory as a generalization of the Kaluza–Klein theory and the Brans–Dicke theory. Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.
This field interacts Yukawa-like with the particles that get mass through it. Scalar fields are found within superstring theories as dilaton fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor. Scalar fields are supposed to cause the accelerated expansion of the universe, helping to solve the horizon problem and giving a hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless scalar fields in this context are known as inflatons. Massive scalar fields are proposed, using for example Higgs-like fields. Vector fields; some examples of vector fields include the electromagnetic field and the Newtonian gravitational field. Tensor fields, which associate a tensor to every point in space. For example, in general relativity gravitation is associated with the tensor field called Einstein tensor. In Kaluza–Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".
The dilaton scalar is found among the massless bosonic fields in string theory. Scalar field theory Vector-valued function
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force exhibits electromagnetic fields such as electric fields, magnetic fields, light, is one of the four fundamental interactions in nature; the other three fundamental interactions are the strong interaction, the weak interaction, gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force. Electromagnetic phenomena are defined in terms of the electromagnetic force, sometimes called the Lorentz force, which includes both electricity and magnetism as different manifestations of the same phenomenon; the electromagnetic force plays a major role in determining the internal properties of most objects encountered in daily life. Ordinary matter takes its form as a result of intermolecular forces between individual atoms and molecules in matter, is a manifestation of the electromagnetic force.
Electrons are bound by the electromagnetic force to atomic nuclei, their orbital shapes and their influence on nearby atoms with their electrons is described by quantum mechanics. The electromagnetic force governs all chemical processes, which arise from interactions between the electrons of neighboring atoms. There are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential and electric current. In Faraday's law, magnetic fields are associated with electromagnetic induction and magnetism, Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents; the theoretical implications of electromagnetism the establishment of the speed of light based on properties of the "medium" of propagation, led to the development of special relativity by Albert Einstein in 1905. Electricity and magnetism were considered to be two separate forces; this view changed, with the publication of James Clerk Maxwell's 1873 A Treatise on Electricity and Magnetism in which the interactions of positive and negative charges were shown to be mediated by one force.
There are four main effects resulting from these interactions, all of which have been demonstrated by experiments: Electric charges attract or repel one another with a force inversely proportional to the square of the distance between them: unlike charges attract, like ones repel. Magnetic poles attract or repel one another in a manner similar to positive and negative charges and always exist as pairs: every north pole is yoked to a south pole. An electric current inside a wire creates a corresponding circumferential magnetic field outside the wire, its direction depends on the direction of the current in the wire. A current is induced in a loop of wire when it is moved toward or away from a magnetic field, or a magnet is moved towards or away from it. While preparing for an evening lecture on 21 April 1820, Hans Christian Ørsted made a surprising observation; as he was setting up his materials, he noticed a compass needle deflected away from magnetic north when the electric current from the battery he was using was switched on and off.
This deflection convinced him that magnetic fields radiate from all sides of a wire carrying an electric current, just as light and heat do, that it confirmed a direct relationship between electricity and magnetism. At the time of discovery, Ørsted did not suggest any satisfactory explanation of the phenomenon, nor did he try to represent the phenomenon in a mathematical framework. However, three months he began more intensive investigations. Soon thereafter he published his findings, proving that an electric current produces a magnetic field as it flows through a wire; the CGS unit of magnetic induction is named in honor of his contributions to the field of electromagnetism. His findings resulted in intensive research throughout the scientific community in electrodynamics, they influenced French physicist André-Marie Ampère's developments of a single mathematical form to represent the magnetic forces between current-carrying conductors. Ørsted's discovery represented a major step toward a unified concept of energy.
This unification, observed by Michael Faraday, extended by James Clerk Maxwell, reformulated by Oliver Heaviside and Heinrich Hertz, is one of the key accomplishments of 19th century mathematical physics. It has had far-reaching consequences, one of, the understanding of the nature of light. Unlike what was proposed by the electromagnetic theory of that time and other electromagnetic waves are at present seen as taking the form of quantized, self-propagating oscillatory electromagnetic field disturbances called photons. Different frequencies of oscillation give rise to the different forms of electromagnetic radiation, from radio waves at the lowest frequencies, to visible light at intermediate frequencies, to gamma rays at the highest frequencies. Ørsted was not the only person to examine the relationship between magnetism. In 1802, Gian Domenico Romagnosi, an Italian legal scholar, deflected a magnetic needle using a Voltaic pile; the factual setup of the experiment is not clear, so if current flew across the needle or not.
An account of the discovery was published in 1802 in an Italian newspaper, but it was overlooked by the contemporary scientific community, because Romagnosi did not belong to this community. An earlier, neglected, connec
Conservative vector field
In vector calculus, a conservative vector field is a vector field, the gradient of some function. Conservative vector fields have the property that the line integral is path independent, i.e. the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is irrotational. An irrotational vector field is conservative provided that the domain is connected. Conservative vector fields appear in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path, so it is possible to define a potential energy, independent of the actual path taken. In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure.
Therefore, in general, the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements d R that don't have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy; this is. As an example of a non-conservative field, imagine pushing a box from one end of a room to another. Pushing the box in a straight line across the room requires noticeably less work against friction than along a curved path covering a greater distance. M. C. Escher's painting Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground as one moves along the staircase.
It is rotational in that one can keep getting higher or keep getting lower while going around in circles. It is non-conservative in that one can return to one's starting point while ascending more than one descends or vice versa. On a real staircase, the height above the ground is a scalar potential field: If one returns to the same place, one goes upward as much as one goes downward, its gradient is irrotational. The situation depicted in the painting is impossible. A vector field v: U → R n, where U is an open subset of R n, is said to be conservative if and only if there exists a C 1 scalar field φ on U such that v = ∇ φ. Here, ∇ φ denotes the gradient of φ; when the equation above holds, φ is called a scalar potential for v. The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. A key property of a conservative vector field v is that its integral along a path depends only on the endpoints of that path, not the particular route taken.
Suppose that P is a rectifiable path in U with initial point A and terminal point B. If v = ∇ φ for some C 1 scalar field φ so that v is a conservative vector field the gradient theorem states that ∫ P v ⋅ d r = φ − φ; this holds as the fundamental theorem of calculus. An equivalent formulation of this is that ∮ C v ⋅ d r = 0 for every rectifiable simple closed path C in U; the converse of this statement is true: If the circulation of v around every rectifiable simple closed path in U is 0 v is a conservative vector field. Let n =
Kurt Otto Friedrichs
Kurt Otto Friedrichs was a noted German American mathematician. He was the co-founder of the Courant Institute at New York University, a recipient of the National Medal of Science. Friedrichs was born in Kiel, Schleswig-Holstein on September 28, 1901, his family soon moved to Düsseldorf. He attended several different universities in Germany studying the philosophical works of Heidegger and Husserl, but decided that mathematics was his real calling. During the 1920s, Friedrichs pursued this field in Göttingen, which had a renowned Mathematical Institute under the direction of Richard Courant. Courant became a close colleague and lifelong friend of Friedrichs. In 1931, Friedrichs became a full professor of mathematics at the Technische Hochschule in Braunschweig. In early February 1933, a few days after Hitler became the Chancellor of Germany, Friedrichs met and fell in love with a young Jewish student, Nellie Bruell, their relationship became challenging and difficult because of the anti-Semitic Nuremberg Laws of Hitler's government.
In 1937, both Friedrichs and Nellie Bruell managed to emigrate separately to New York City where they married. Their long and happy marriage produced five children. Courant had left Germany in 1933 and had founded an institute for graduate studies in mathematics at New York University. Friedrichs joined him when he remained there for forty years, he was instrumental in the development of the Courant Institute of Mathematical Sciences, which became one of the most distinguished research institutes for applied mathematics in the world. Friedrichs died in New Rochelle, New York on December 31, 1982. Friedrichs's greatest contribution to applied mathematics was his work on partial differential equations, he did major research and wrote many books and papers on existence theory, numerical methods, differential operators in Hilbert space, non-linear buckling of plates, flows past wings, solitary waves, shock waves, magneto-fluid dynamical shock waves, relativistic flows, quantum field theory, perturbation of the continuous spectrum, scattering theory, symmetric hyperbolic equations.
With Cartan, Friedrichs gave a "geometrized" formulation of Newtonian gravitation theory—also known as “Newton–Cartan theory”— and developed by Dautcourt, Dixon and Horneffer, Havas, Künzle, Lottermoser and others. A member of the National Academy of Sciences since 1959, Friedrichs received many honorary degrees and awards for his work. There is a student prize named after Friedrichs at NYU; the American Mathematical Society selected him as the Josiah Willards Gibbs lecturer for 1954. In November 1977, Friedrichs received the National Medal of Science from President Jimmy Carter "for bringing the powers of modern mathematics to bear on problems in physics, fluid dynamics, elasticity." R. von Mises and K. O. Friedrichs, Fluid Dynamics, Springer-Verlag. K. O. Friedrichs, Perturbation of Spectra in Hilbert Space, American Mathematical Society. K. O. Friedrichs, Mathematical aspects of the quantum theory of fields, Interscience. K. O. Friedrichs, Spectral Theory of Operators in Hilbert Space, Springer-Verlag.
Friedrichs, Kurt Otto, Cathleen S. ed. Selecta, Contemporary Mathematicians, Boston–Basel–Stuttgart: Birkhäuser Verlag, ISBN 0-8176-3270-0. A selection from Friedrichs works with a biography and commentaries of David Isaacson, Fritz John, Tosio Kato, Peter Lax, Louis Nirenberg, Wolfgang Wasow, Harold Weitzner. Mollifier Kurt Otto Friedrichs at the Mathematics Genealogy Project O'Connor, John J.. "Kurt Otto Friedrichs", MacTutor History of Mathematics archive, University of St Andrews. National Academy of Sciences Biographical Memoir Literature by and about Kurt Otto Friedrichs in the German National Library catalogue
The velocity of an object is the rate of change of its position with respect to a frame of reference, is a function of time. Velocity is equivalent to a specification of an object's direction of motion. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; the scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI as metres per second or as the SI base unit of. For example, "5 metres per second" is a scalar. If there is a change in speed, direction or both the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes.
Hence, the car is considered to be undergoing an acceleration. Speed describes only how fast an object is moving, whereas velocity gives both how fast it is and in which direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified; the big difference can be noticed. When something moves in a circular path and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle; this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, which may be referred to as the instantaneous velocity to emphasize the distinction from the average velocity.
In some applications the "average velocity" of an object might be needed, to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v, over some time period Δt. Average velocity can be calculated as: v ¯ = Δ x Δ t; the average velocity is always equal to the average speed of an object. This can be seen by realizing that while distance is always increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time graph, the instantaneous velocity can be thought of as the slope of the tangent line to the curve at any point, the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity; the average velocity is the same as the velocity averaged over time –, to say, its time-weighted average, which may be calculated as the time integral of the velocity: v ¯ = 1 t 1 − t 0 ∫ t 0 t 1 v d t, where we may identify Δ x = ∫ t 0 t 1 v d t and Δ t = t 1 − t 0.
If we consider v as velocity and x as the displacement vector we can express the velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time: v = lim Δ t → 0 Δ x Δ t = d x d t. From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity function v is the displacement function x. In the figure, this corresponds to the yellow area under the curve labeled s. X = ∫ v d t. Since the derivative of the position with respect to time gives the change in position divided by the change in time, velocity is measured in metres per second. Although the concept of an instantaneous velocity might at first seem counter-intuitive, it
OCLC Online Computer Library Center, Incorporated d/b/a OCLC is an American nonprofit cooperative organization "dedicated to the public purposes of furthering access to the world's information and reducing information costs". It was founded in 1967 as the Ohio College Library Center. OCLC and its member libraries cooperatively produce and maintain WorldCat, the largest online public access catalog in the world. OCLC is funded by the fees that libraries have to pay for its services. OCLC maintains the Dewey Decimal Classification system. OCLC began in 1967, as the Ohio College Library Center, through a collaboration of university presidents, vice presidents, library directors who wanted to create a cooperative computerized network for libraries in the state of Ohio; the group first met on July 5, 1967 on the campus of the Ohio State University to sign the articles of incorporation for the nonprofit organization, hired Frederick G. Kilgour, a former Yale University medical school librarian, to design the shared cataloging system.
Kilgour wished to merge the latest information storage and retrieval system of the time, the computer, with the oldest, the library. The plan was to merge the catalogs of Ohio libraries electronically through a computer network and database to streamline operations, control costs, increase efficiency in library management, bringing libraries together to cooperatively keep track of the world's information in order to best serve researchers and scholars; the first library to do online cataloging through OCLC was the Alden Library at Ohio University on August 26, 1971. This was the first online cataloging by any library worldwide. Membership in OCLC is based on use of services and contribution of data. Between 1967 and 1977, OCLC membership was limited to institutions in Ohio, but in 1978, a new governance structure was established that allowed institutions from other states to join. In 2002, the governance structure was again modified to accommodate participation from outside the United States.
As OCLC expanded services in the United States outside Ohio, it relied on establishing strategic partnerships with "networks", organizations that provided training and marketing services. By 2008, there were 15 independent United States regional service providers. OCLC networks played a key role in OCLC governance, with networks electing delegates to serve on the OCLC Members Council. During 2008, OCLC commissioned two studies to look at distribution channels. In early 2009, OCLC negotiated new contracts with the former networks and opened a centralized support center. OCLC provides bibliographic and full-text information to anyone. OCLC and its member libraries cooperatively produce and maintain WorldCat—the OCLC Online Union Catalog, the largest online public access catalog in the world. WorldCat has holding records from private libraries worldwide; the Open WorldCat program, launched in late 2003, exposed a subset of WorldCat records to Web users via popular Internet search and bookselling sites.
In October 2005, the OCLC technical staff began a wiki project, WikiD, allowing readers to add commentary and structured-field information associated with any WorldCat record. WikiD was phased out; the Online Computer Library Center acquired the trademark and copyrights associated with the Dewey Decimal Classification System when it bought Forest Press in 1988. A browser for books with their Dewey Decimal Classifications was available until July 2013; until August 2009, when it was sold to Backstage Library Works, OCLC owned a preservation microfilm and digitization operation called the OCLC Preservation Service Center, with its principal office in Bethlehem, Pennsylvania. The reference management service QuestionPoint provides libraries with tools to communicate with users; this around-the-clock reference service is provided by a cooperative of participating global libraries. Starting in 1971, OCLC produced catalog cards for members alongside its shared online catalog. OCLC commercially sells software, such as CONTENTdm for managing digital collections.
It offers the bibliographic discovery system WorldCat Discovery, which allows for library patrons to use a single search interface to access an institution's catalog, database subscriptions and more. OCLC has been conducting research for the library community for more than 30 years. In accordance with its mission, OCLC makes its research outcomes known through various publications; these publications, including journal articles, reports and presentations, are available through the organization's website. OCLC Publications – Research articles from various journals including Code4Lib Journal, OCLC Research, Reference & User Services Quarterly, College & Research Libraries News, Art Libraries Journal, National Education Association Newsletter; the most recent publications are displayed first, all archived resources, starting in 1970, are available. Membership Reports – A number of significant reports on topics ranging from virtual reference in libraries to perceptions about library funding. Newsletters – Current and archived newsletters for the library and archive community.
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Simply connected space
In topology, a topological space is called connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be connected: a path-connected topological space is connected if and only if its fundamental group is trivial. A topological space X is called connected if it is path-connected and any loop in X defined by f: S1 → X can be contracted to a point: there exists a continuous map F: D2 → X such that F restricted to S1 is f. Here, S1 and D2 closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is connected if and only if it is path-connected, whenever p: → X and q: → X are two paths with the same start and endpoint p can be continuously deformed into q while keeping both endpoints fixed. Explicitly, there exists a continuous homotopy F: × → X such that F = F = q.
A topological space X is connected if and only if X is path-connected and the fundamental group of X at each point is trivial, i.e. consists only of the identity element. X is connected if and only if for all points x, y ∈ X, the set of morphisms Hom Π in the fundamental groupoid of X has only one element. In complex analysis: an open subset X ⊆ C is connected if and only if both X and its complement in the Riemann sphere are connected; the set of complex numbers with imaginary part greater than zero and less than one, furnishes a nice example of an unbounded, open subset of the plane whose complement is not connected. It is simply connected, it might be worth pointing out that a relaxation of the requirement that X be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a open set has connected extended complement when each of its connected components are connected. Informally, an object in our space is connected if it consists of one piece and does not have any "holes" that pass all the way through it.
For example, neither a doughnut nor a coffee cup is connected, but a hollow rubber ball is connected. In two dimensions, a circle is not connected, but a disk and a line are. Spaces that are connected but not connected are called non-simply connected or multiply connected; the definition only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point though it has a "hole" in the hollow center; the stronger condition, that the object has no holes of any dimension, is called contractibility. The Euclidean plane R2 is connected, but R2 minus the origin is not. If n > 2 both Rn and Rn minus the origin are connected. Analogously: the n-dimensional sphere Sn is connected if and only if n > 2. Every convex subset of Rn is connected. A torus, the cylinder, the Möbius strip, the projective plane and the Klein bottle are not connected; every topological vector space is connected. For n ≥ 2, the special orthogonal group SO is not connected and the special unitary group SU is connected.
The one-point compactification of R is not connected. The long line L is connected, but its compactification, the extended long line L* is not. A surface is connected if and only if it is connected and its genus is 0. A universal cover of any space X is a connected space which maps to X via a covering map. If X and Y are homotopy equivalent and X is connected so is Y; the image of a connected set under a continuous function need not be connected. Take for example the complex plane under the exponential map: the image is C -, not connected; the notion of simple connectedness is important in complex analysis because of the following facts: The Cauchy's integral theorem states that if U is a connected open subset of the complex plane C, f: U → C is a holomorphic function f has an antiderivative F on U, the value of every line integral in U with integrand f depends only on the end points u and v of the path, can be computed as F - F. The integral thus does not depend on the particular path connecting u and v.
The Riemann mapping theorem states that any non-empty open connected subset of C is conformally eq