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Unduloid
In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. Formula Let \operatorname(u,k) represent the normal Jacobi sine function and \operatorname(u,k) be the normal Jacobi elliptic function and let \operatorname(z,k) represent the normal elliptic integral of the first kind and \operatorname(z,k) represent the normal elliptic integral of the second kind. Let ''a'' be the length of the ellipse's major axis, and ''e'' be the eccentricity of the ellipse. Let ''k'' be a fixed value between 0 and 1 called the modulus. Given these variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant-mean-curvature Surface
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2003:E1/ref> This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. History In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. In 1853 J. H. Jellet showed that if S is a compact star-shaped surface in \R^3 with constant mean curvature, then it is the standard sphere. Subsequently, A. D. Alexandrov proved that a compact embedded surface in \R^3 with constant mean curvature H \neq 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unduloid
In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. Formula Let \operatorname(u,k) represent the normal Jacobi sine function and \operatorname(u,k) be the normal Jacobi elliptic function and let \operatorname(z,k) represent the normal elliptic integral of the first kind and \operatorname(z,k) represent the normal elliptic integral of the second kind. Let ''a'' be the length of the ellipse's major axis, and ''e'' be the eccentricity of the ellipse. Let ''k'' be a fixed value between 0 and 1 called the modulus. Given these variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics Of Fluids
''Physics of Fluids'' is a monthly peer-reviewed scientific journal covering fluid dynamics, established by the American Institute of Physics in 1958, and is published by AIP Publishing. The journal focus is the dynamics of gases, liquids, and complex or multiphase fluids—and the journal contains original research resulting from theoretical, computational, and experimental studies. History From 1958 through 1988, the journal included plasma physics. From 1989 until 1993, the journal split into ''Physics of Fluids A'' covering fluid dynamics, and ''Physics of Fluids B,'' on plasma physics. In 1994, the latter was renamed ''Physics of Plasmas'', and the former continued under its original name, ''Physics of Fluids''. The journal was originally published by the American Institute of Physics in cooperation with the American Physical Society's Division of Fluid Dynamics. In 2016, the American Institute of Physics became the sole publisher. From 1985–2015, ''Physics of Fluid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferrofluid
Ferrofluid is a liquid that is attracted to the poles of a magnet. It is a colloidal liquid made of nanoscale ferromagnetic or ferrimagnetic particles suspended in a carrier fluid (usually an organic solvent or water). Each magnetic particle is thoroughly coated with a surfactant to inhibit clumping. Large ferromagnetic particles can be ripped out of the homogeneous colloidal mixture, forming a separate clump of magnetic dust when exposed to strong magnetic fields. The magnetic attraction of tiny nanoparticles is weak enough that the surfactant's Van der Waals force is sufficient to prevent magnetic clumping or agglomeration. Ferrofluids usually do not retain magnetization in the absence of an externally applied field and thus are often classified as "superparamagnets" rather than ferromagnets. In contrast to ferrofluids, magnetorheological fluids (MR fluids) are magnetic fluids with larger particles. That is, a ferrofluid contains primarily nanoparticles, while an MR fluid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molybdenum
Molybdenum is a chemical element with the symbol Mo and atomic number 42 which is located in period 5 and group 6. The name is from Neo-Latin ''molybdaenum'', which is based on Ancient Greek ', meaning lead, since its ores were confused with lead ores. Molybdenum minerals have been known throughout history, but the element was discovered (in the sense of differentiating it as a new entity from the mineral salts of other metals) in 1778 by Carl Wilhelm Scheele. The metal was first isolated in 1781 by Peter Jacob Hjelm. Molybdenum does not occur naturally as a free metal on Earth; it is found only in various oxidation states in minerals. The free element, a silvery metal with a grey cast, has the sixth-highest melting point of any element. It readily forms hard, stable carbides in alloys, and for this reason most of the world production of the element (about 80%) is used in steel alloys, including high-strength alloys and superalloys. Most molybdenum compounds have low solubility ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silver
Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical conductivity, thermal conductivity, and reflectivity of any metal. The metal is found in the Earth's crust in the pure, free elemental form ("native silver"), as an alloy with gold and other metals, and in minerals such as argentite and chlorargyrite. Most silver is produced as a byproduct of copper, gold, lead, and zinc refining. Silver has long been valued as a precious metal. Silver metal is used in many bullion coins, sometimes alongside gold: while it is more abundant than gold, it is much less abundant as a native metal. Its purity is typically measured on a per-mille basis; a 94%-pure alloy is described as "0.940 fine". As one of the seven metals of antiquity, silver has had an enduring role in most human cultures. Oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tempering (metallurgy)
Tempering is a process of heat treating, which is used to increase the toughness of iron-based alloys. Tempering is usually performed after hardening, to reduce some of the excess hardness, and is done by heating the metal to some temperature below the critical point for a certain period of time, then allowing it to cool in still air. The exact temperature determines the amount of hardness removed, and depends on both the specific composition of the alloy and on the desired properties in the finished product. For instance, very hard tools are often tempered at low temperatures, while springs are tempered at much higher temperatures. Introduction Tempering is a heat treatment technique applied to ferrous alloys, such as steel or cast iron, to achieve greater toughness by decreasing the hardness of the alloy. The reduction in hardness is usually accompanied by an increase in ductility, thereby decreasing the brittleness of the metal. Tempering is usually performed after quen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clairaut's Relation
In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle then : \rho(\gamma(t)) \sin \psi(\gamma(t)) = \text,\, where ''ρ''(''P'') is the distance from a point ''P'' on the great circle to the ''z''-axis, and ''ψ''(''P'') is the angle between the great circle and the meridian through the point ''P''. The relation remains valid for a geodesic on an arbitrary surface of revolution. A statement of the general version of Clairaut's relation is: Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun ''geodesic'' and the adjective ''geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvatur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eccentricity (mathematics)
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similar if and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is zero. * The eccentricity of an ellipse which is not a circle is greater than zero but less than 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of lines is \infty Definitions Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is orient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
