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Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): :Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. The theorem is "remarkable" because the starting ''definition'' of Gaussian curvature mak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler. Soap film attached to twin circular rings will take the shape of a catenoid. Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa. Geometry The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste Meusnier. There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid. The cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat *Surface (differential geometry), a differentiable two-dimensional manifold *Surface (topology), a two-dimensional manifold * Algebraic surface, an algebraic variety of dimension two *Coordinate surfaces *Fractal surface, generated using a stochastic algorithm *Polyhedral surface * Surface area *Surface integral Arts and entertainment * Surface (band), an American R&B and pop trio ** ''Surface'' (Surface album), 1986 *Surfaces (band), American musical duo * ''Surface'' (Circle album), 1998 * "Surface" (Aero Chord song), 2014 * ''Surface'' (2005 TV series), an American science fiction show, 2005–2006 * ''Surface'' (2022 TV series), an American psychological thriller miniseries that began streaming in 2022 *'' The Surface'', an American film, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Fundamental Form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold. Surface in R3 Motivation The second fundamental form of a parametric surface in was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, , and that the plane is tangent to the surface at the origin. Then and its partial derivatives with respect to and vanish at (0,0). Therefore, the Taylor expansion of ''f'' at (0,0) starts with quadratic terms: : z=L\frac + Mxy + N\frac + \text\,, and the second fundamental form at the origin in the coordinates is the qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potato Chip
A potato chip (North American English; often just chip) or crisp (British and Irish English) is a thin slice of potato that has been either deep fried, baked, or air fried until crunchy. They are commonly served as a snack, side dish, or appetizer. The basic chips are cooked and salted; additional varieties are manufactured using various flavorings and ingredients including herbs, spices, cheeses, other natural flavors, artificial flavors, and additives. Potato chips form a large part of the snack food and convenience food market in Western countries. The global potato chip market generated total revenue of US$16.49 billion in 2005. This accounted for 35.5% of the total savory snacks market in that year ($46.1 billion). History The earliest known recipe for something similar to today's potato chips is in William Kitchiner's book '' The Cook's Oracle'' published in 1817, which was a bestseller in the United Kingdom and the United States. The 1822 edition's recipe for "Pota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corrugated Galvanised Iron
Corrugated galvanised iron or steel, colloquially corrugated iron (near universal), wriggly tin (taken from UK military slang), pailing (in Caribbean English), corrugated sheet metal (in North America) and occasionally abbreviated CGI is a building material composed of sheets of hot-dip galvanised mild steel, cold-rolled to produce a linear ridged pattern in them. Although it is still popularly called "iron" in the UK, the material used is actually steel (which is iron alloyed with carbon for strength, commonly 0.3% carbon), and only the surviving vintage sheets may actually be made up of 100% iron. The corrugations increase the bending strength of the sheet in the direction perpendicular to the corrugations, but not parallel to them, because the steel must be stretched to bend perpendicular to the corrugations. Normally each sheet is manufactured longer in its strong direction. CGI is lightweight and easily transported. It was and still is widely used especially in rural a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corrugated Fiberboard
Corrugated fiberboard or corrugated cardboard is a type of packaging material consisting of a fluted corrugated sheet and one or two flat linerboards. It is made on "flute lamination machines" or "corrugators" and is used for making corrugated boxes. The corrugated medium sheet and the linerboard(s) are made of kraft containerboard, a paperboard material usually over thick. History Corrugated (also called pleated) paper was patented in England in 1856, and used as a liner for tall hats, but corrugated boxboard was not patented and used as a shipping material until 20 December 1871. The patent was issued to Albert Jones of New York City for single-sided (single-face) corrugated board. Jones used the corrugated board for wrapping bottles and glass lantern chimneys. The first machine for producing large quantities of corrugated board was built in 1874 by G. Smyth, and in the same year Oliver Long improved upon Jones' design by inventing corrugated board with liner sheets on both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corrugated
The term corrugated, describing a series of parallel ridges and furrows, may refer to the following: Materials *Corrugated fiberboard, also called corrugated cardboard *Corrugated galvanised iron, a building material composed of sheets of cold-rolled hot-dip galvanised mild steel *Corrugated plastic, a wide range of extruded twinwall plastic-sheet products produced from high-impact polypropylene resin *Corrugated stainless steel tubing, tubing made of stainless steel with corrugation on the inside or outside Animals *Corrugated darter, a species of fish endemic to the eastern United States *Corrugated pipefish, a marine fish of the family Syngnathidae *Corrugated frog, a species of frog in the family Dicroglossidae *Corrugated water frog, a species of frog in the family Nyctibatrachidae *Corrugated nutmeg, a species of sea snail Other uses * Corrugaphone or whirly tube, an experimental musical instrument or toy * Corrugated road, a form of damage prone to develop in the surface ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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