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In mathematics, a conformal map is a function which preserves angles. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
More formally, a map
- w = f(z)
is called conformal (or angle-preserving) at z0, if it preserves oriented angles between curves through z0, as well as their orientation, i.e. direction. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size. An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
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. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
Complex analysis
An important family of examples of conformal maps comes from complex analysis. If U is an open subset of the complex plane, , then a function Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic (that is, the conjugate to a holomorphic function), it still preserves angles, but it reverses their orientation. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
For a non-technical overview of the subject, see Calculus. ...
In mathematics, a function on the complex plane is antiholomorphic at a point if its derivative with respect to z* exists, where here, z* is the complex conjugate. ...
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of admits a bijective conformal map to the open unit disk in . The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
A bijective function. ...
In mathematics, the open unit disk 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...
A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation. Again, for the conjugate, angles are preserved, but orientation is reversed. A rendering of the Riemann Sphere. ...
A surjective function. ...
In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
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 radial coordinate in circular coordinates, keeping the angle the same. See also inversive geometry. This article describes some of the common coordinate systems that appear in elementary mathematics. ...
In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ...
Riemannian geometry In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called conformally equivalent if g=uh for some positive function u on M. The function u is called the conformal factor. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map. Stereographic projection of a circle of radius R onto the x axis. ...
A sphere is a perfectly symmetrical geometrical object. ...
Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...
The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...
Euclidean space Any conformal map from Euclidean space of dimension at least 3 to itself is a composition of a homothetic transformation and an isometry. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
Uses If a function is harmonic (that is, it satisfies Laplace's equation ) over a particular space, and is transformed via a conformal map to another space, the transformation is also harmonic. For this reason, any function which is defined by a potential can be transformed by a conformal map and still remain governed by a potential. Examples in physics of equations defined by a potential include the electromagnetic field, the gravitational field, and, in fluid dynamics, potential flow, which is an approximation to fluid flow assuming constant density, zero viscosity, and irrotational flow. One example of a fluid dynamic application of a conformal map is the Joukowsky transform. 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) which satisfies Laplaces equation, i. ...
In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ...
It has been suggested that this article or section be merged with Scalar potential. ...
The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ...
The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ...
The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ...
Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ...
A potential flow is characterized by an irrotational velocity field. ...
In physics, density is mass m per unit volume V. For the common case of a homogeneous substance, it is expressed as: where, in SI units: Ï (rho) is the density of the substance, measured in kg·m-3 m is the mass of the substance, measured in kg V is...
Viscosity is a measure of the resistance of a fluid to deform under shear stress. ...
In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. ...
Example of a Joukowsky transform. ...
The importance of conformal transformations for electromagnetism was brought to light by Harry Bateman in 1910. Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...
Harry Bateman (May 29, 1882 Manchester, England - January 21, 1946 Pasadena California USA) was a leading English mathematician. ...
Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that exhibit inconvenient geometries. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. For example, one may be desirous of calculating the electric field, E(z), arising from a point charge located near the corner of two conducting planes separated by a certain angle (where z is the complex coordinate of a point in 2-space). This problem per se is quite clumsy to solve in closed form. However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely pi radians, meaning that the corner of two planes is transformed to a straight line. In this new domain, the problem — that of calculating the electric field impressed by a point charge located near a conducting wall — is quite easy to solve. The solution is obtained in this domain, E(w), and then mapped back to the original domain by noting that w was obtained as a function (viz., the composition of E and w) of z, whence E(w) can be viewed as E(w(z)), which is a function of z, the original coordinate basis. Note that this application is not a contradiction to the fact that conformal mappings preserve angles, they do so only for points in the interior of their domain, and not at the boundary. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
See also In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space...
In theoretical physics, a Penrose diagram (named after Roger Penrose who invented them) is usually a two-dimensional diagram that captures the causal relations between different points in spacetime. ...
External links - E.P. Dolzhenko, Conformal mapping, SpringerLink Encyclopaedia of Mathematics (2001)
- Eric W. Weisstein, Conformal Mapping at MathWorld.
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