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Conformal map

Index Conformal map

In mathematics, a conformal map is a function that preserves angles locally. [1]

94 relations: Angle, Antiholomorphic function, Big Bang, Bijection, Boundary value problem, Carathéodory's theorem (conformal mapping), Cartography, Cinderella (software), Complex analysis, Complex conjugate, Complex number, Complex plane, Conformal geometry, Conformal map projection, Conservative vector field, Coordinate system, Curvature, Curve, Density, Derivative, Determinant, Diffeomorphism, Differential geometry of surfaces, Domain of a function, Dual number, Ebenezer Cunningham, Electromagnetic field, Euclidean space, Fluid dynamics, Function (mathematics), Function composition, General relativity, Geodesic, Gravitational field, Gravitational singularity, Harmonic function, Harry Bateman, Holomorphic function, Homothetic transformation, Hyperbolic angle, If and only if, Image (mathematics), Inversive geometry, Isometry, Jacobian matrix and determinant, James Jeans, Joseph Liouville, Joukowsky transform, Laplace's equation, Linear fractional transformation, ..., Liouville's theorem (conformal mappings), Map projection, Mathematics, Maxwell's equations, Möbius transformation, Mercator projection, Method of image charges, Metric tensor (general relativity), Open set, Orientation (vector space), Partial derivative, Penrose diagram, Physics, Plane (geometry), Point at infinity, Potential, Potential flow, Pseudo-Riemannian manifold, Riemann mapping theorem, Riemann sphere, Riemannian geometry, Riemannian manifold, Rotation, Rotation matrix, Schwarz–Christoffel mapping, Shear mapping, Similarity (geometry), Simply connected space, Slope, Slosh dynamics, Special conformal transformation, Sphere, Spherical wave transformation, Split-complex number, Squeeze mapping, Stereographic projection, Subring, Surjective function, Trinity College Dublin, Unit disk, University of Chicago Press, Viscosity, Wolfram Demonstrations Project, 2 × 2 real matrices. Expand index (44 more) »

Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

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Antiholomorphic function

In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions.

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Big Bang

The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution.

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Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

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Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.

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Carathéodory's theorem (conformal mapping)

In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem.

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Cartography

Cartography (from Greek χάρτης chartēs, "papyrus, sheet of paper, map"; and γράφειν graphein, "write") is the study and practice of making maps.

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Cinderella (software)

Cinderella is a proprietary interactive geometry software, written in Java programming language.

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Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

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Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

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Conformal map projection

In cartography, a conformal map projection is one in which any angle on Earth (a sphere or an ellipsoid) is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense.

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Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

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Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

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Density

The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

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Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

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Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

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Dual number

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2.

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Ebenezer Cunningham

Ebenezer Cunningham (7 May 1881, Hackney, London – 12 February 1977) was a British mathematician who is remembered for his research and exposition at the dawn of special relativity.

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Electromagnetic field

An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

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General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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Gravitational field

In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body.

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Gravitational singularity

A gravitational singularity or spacetime singularity is a location in spacetime where the gravitational field of a celestial body becomes infinite in a way that does not depend on the coordinate system.

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Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R where U is an open subset of Rn that satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.

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Harry Bateman

Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician.

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Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

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Homothetic transformation

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if) or reverse (if) the direction of all vectors.

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Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola. A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1. The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is ln x. Note that unlike circular angle, hyperbolic angle is unbounded, as is the function ln x, a fact related to the unbounded nature of the harmonic series. The hyperbolic angle in standard position is considered to be negative when 0 a > 1 so that (a, b) and (c, d) determine an interval on the hyperbola xy.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

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Inversive geometry

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

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Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

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James Jeans

Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician.

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Joseph Liouville

Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French mathematician.

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Joukowsky transform

In applied mathematics, the Joukowsky transform, named after Nikolai Zhukovsky (who published it in 1910), is a conformal map historically used to understand some principles of airfoil design.

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Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

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Linear fractional transformation

In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers.

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Liouville's theorem (conformal mappings)

In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space.

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Map projection

A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

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Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

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Mercator projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569.

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Method of image charges

The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics.

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Metric tensor (general relativity)

In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study.

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

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Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

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Penrose diagram

In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime.

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Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

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Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

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Potential

Potential generally refers to a currently unrealized ability.

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Potential flow

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential.

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Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.

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Riemann mapping theorem

In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk This mapping is known as a Riemann mapping.

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Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

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Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.

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Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

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Rotation

A rotation is a circular movement of an object around a center (or point) of rotation.

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Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

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Schwarz–Christoffel mapping

In complex analysis, a Schwarz–Christoffel mapping is a conformal transformation of the upper half-plane onto the interior of a simple polygon.

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Shear mapping

In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.

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Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

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Slosh dynamics

In fluid dynamics, slosh refers to the movement of liquid inside another object (which is, typically, also undergoing motion).

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Special conformal transformation

In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation.

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Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

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Spherical wave transformation

Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames.

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Split-complex number

In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.

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Squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Subring

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).

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Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

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Trinity College Dublin

Trinity College (Coláiste na Tríonóide), officially the College of the Holy and Undivided Trinity of Queen Elizabeth near Dublin, is the sole constituent college of the University of Dublin, a research university located in Dublin, Ireland.

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Unit disk

In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal to one: Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself.

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University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

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Viscosity

The viscosity of a fluid is the measure of its resistance to gradual deformation by shear stress or tensile stress.

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Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.

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2 × 2 real matrices

In mathematics, the associative algebra of real matrices is denoted by M(2, R).

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Angle-preserving, Angle-preserving transformation, Conformal mapping, Conformal mapping theorem, Conformal projection, Conformal transform, Conformal transformation, Conformality.

References

[1] https://en.wikipedia.org/wiki/Conformal_map

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