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List Of Coordinate Charts
This article attempts to conveniently list articles on some of the most useful coordinate charts in some of the most useful examples of Riemannian manifolds. The notion of a coordinate chart is fundamental to various notions of a ''manifold'' which are used in mathematics. In order of increasing ''level of structure:'' *topological manifold *smooth manifold *Riemannian manifold and Pseudo-Riemannian manifold, semi-Riemannian manifold For our purposes, the key feature of the last two examples is that we have defined a metric tensor which we can use to integrate along a curve, such as a geodesic curve. The key difference between Riemannian metrics and semi-Riemannian metrics is that the former arise from fiber bundle, bundling definite bilinear form, positive-definite quadratic forms, whereas the latter arise from bundling Definite quadratic form, indefinite quadratic forms. A four-dimensional semi-Riemannian manifold is often called a Lorentzian manifold, because these provide the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axial Projection Chart
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Axial may refer to: * one of the anatomical directions describing relationships in an animal body * In geometry: :* a geometric term of location :* an axis of rotation * In chemistry, referring to an axial bond * a type of modal frame, in music * axial-flow, a type of fan * the Axial age in China, India, etc. * Axial Seamount and submarine volcano off Oregon, USA * Axial, Colorado, a ghost town See also *Axiality (other) *Axis (other) An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis *Axis (mathematics), a designator for a Cartesian-coordinate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Projection Chart
Central is an adjective usually referring to being in the center of some place or (mathematical) object. Central may also refer to: Directions and generalised locations * Central Africa, a region in the centre of Africa continent, also known as Middle Africa * Central America, a region in the centre of America continent * Central Asia, a region in the centre of Eurasian continent * Central Australia, a region of the Australian continent * Central Belt, an area in the centre of Scotland * Central Europe, a region of the European continent * Central London, the centre of London * Central Region (other) * Central United States, a region of the United States of America Specific locations Countries * Central African Republic, a country in Africa States and provinces * Blue Nile (state) or Central, a state in Sudan * Central Department, Paraguay * Central Province (Kenya) * Central Province (Papua New Guinea) * Central Province (Solomon Islands) * Central Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic Chart
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric (distance preserving) nor equiareal (area preserving). The stereographic projection gives a way to represent a sphere by a plane. The metric induced by the inverse stereographic projection from the plane to the sphere defines a geodesic distance between points in the plane equal to the spherical distance between the spherical points they represent. A two-dimensional coordinate system on the stereograph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxwell Chart
Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (other) * Maxwell baronets, in the Baronetage of Nova Scotia * Maxwell (footballer, born 1979), Brazilian forward * Maxwell (footballer, born 1981), Brazilian left-back * Maxwell (footballer, born 1986), Brazilian striker * Maxwell (footballer, born 1989), Brazilian left-back * Maxwell (footballer, born 1995), Brazilian forward * Maxwell (musician) (born 1973), American R&B and neo-soul singer * Maxwell (rapper) (born 1993), German rapper, member of rap band 187 Strassenbande * Maxwell Jacob Friedman (born 1997) AEW Professional wrestler * Maxwell Silva (born 1953), Sri Lankan Sinhala Catholic cleric, Auxiliary Bishop of Colombo Places United States * Maxwell, California * Maxwell, Indiana * Maxwell, Iowa * Maxwell, Nebraska * Maxwell, New Mexico * Maxwell, Texas * Maxwel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Chart
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Line
In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. Definitions An asymptotic direction is one in which the normal curvature is zero. Which is to say: for a point on an asymptotic curve, take the plane which bears both the curve's tangent and the surface's normal at that point. The curve of intersection of the plane and the surface will have zero curvature at that point. Asymptotic directions can only occur when the Gaussian curvature is negative (or zero). There will be two asymptotic directions through every point with negative Gaussian curvature, bisected by the principal directions. If the surface is minimal, the asymptotic directions are orthogonal to one another. Related notions The direction of the asymptotic direction are the same as the asymptotes of the hyperbola of the Dupin indicatrix. A related ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Chart
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monge Chart
Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During the French Revolution he served as the Minister of the Marine, and was involved in the reform of the French educational system, helping to found the École Polytechnique. Biography Early life Monge was born at Beaune, Côte-d'Or, the son of a merchant. He was educated at the college of the Oratorians at Beaune. In 1762 he went to the Collège de la Trinité at Lyon, where, one year after he had begun studying, he was made a teacher of physics at the age of just seventeen. After finishing his education in 1764 he returned to Beaune, where he made a large-scale plan of the town, inventing the methods of observation and constructing the necessary instruments; the plan was presented to the town, and is still preserved in their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isothermal Chart
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a change in the system occurs slowly enough to allow the system to be continuously adjusted to the temperature of the reservoir through heat exchange (see quasi-equilibrium). In contrast, an '' adiabatic process'' is where a system exchanges no heat with its surroundings (''Q'' = 0). Simply, we can say that in an isothermal process * T = \text * \Delta T = 0 * dT = 0 * For ideal gases only, internal energy \Delta U = 0 while in adiabatic processes: * Q = 0. Etymology The adjective "isothermal" is derived from the Greek words "ἴσος" ("isos") meaning "equal" and "θέρμη" ("therme") meaning "heat". Examples Isothermal processes can occur in any kind of system that has some means of regulating the temperature, includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
