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Monkey Saddle
In mathematics, the monkey saddle is the surface defined by the equation : z = x^3 - 3xy^2, \, or in cylindrical coordinates :z = \rho^3 \cos(3\varphi). It belongs to the class of saddle surfaces, and its name derives from the observation that a saddle for a monkey would require two depressions for the legs and one for the tail. The point on the monkey saddle corresponds to a degenerate critical point of the function at . The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is strictly negative at all other points. One can relate the rectangular and cylindrical equations using complex numbers x+iy = r e^: : z = x^3 - 3xy^2 = \operatorname x+iy)^3= \operatorname ^3 e^= r^3\cos(3\varphi). By replacing 3 in the cylindrical equation with any integer one can create a saddle with depressions. Another orientation of the monkey saddle is the ''Smelt petal'' defined by x+y+z+xyz=0, so that the ''z-''axis of the monk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Surface
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a ''coordinate patch'' on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles (ideally) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Coordinate System
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference direction ''(axis A)'', and the distance from a chosen reference plane perpendicular to the axis ''(plane containing the purple section)''. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from the ''polar axis'', which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are called ''radial lines''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monkey Saddle Surface (Shaded)
Monkey is a common name that may refer to most mammals of the infraorder Simiiformes, also known as the simians. Traditionally, all animals in the group now known as simians are counted as monkeys except the apes, which constitutes an incomplete paraphyletic grouping; however, in the broader sense based on cladistics, apes (Hominoidea) are also included, making the terms ''monkeys'' and ''simians'' synonyms in regards to their scope. In 1812, Geoffroy grouped the apes and the Cercopithecidae group of monkeys together and established the name Catarrhini, "Old World monkeys", ("''singes de l'Ancien Monde''" in French). The extant sister of the Catarrhini in the monkey ("singes") group is the Platyrrhini (New World monkeys). Some nine million years before the divergence between the Cercopithecidae and the apes, the Platyrrhini emerged within "monkeys" by migration to South America likely by ocean. Apes are thus deep in the tree of extant and extinct monkeys, and any of the ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle
The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not known precisely when riders first began to use some sort of padding or protection, but a blanket attached by some form of surcingle or girth was probably the first "saddle", followed later by more elaborate padded designs. The solid saddle tree was a later invention, and though early stirrup designs predated the invention of the solid tree, the paired stirrup, which attached to the tree, was the last element of the saddle to reach the basic form that is still used today. Today, modern saddles come in a wide variety of styles, each designed for a specific equestrianism discipline, and require careful fit to both the rider and the horse. Proper saddle care can extend the useful life of a saddle, often for decades. The saddle was a crucial step ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monkey
Monkey is a common name that may refer to most mammals of the infraorder Simiiformes, also known as the simians. Traditionally, all animals in the group now known as simians are counted as monkeys except the apes, which constitutes an incomplete paraphyletic grouping; however, in the broader sense based on cladistics, apes (Hominoidea) are also included, making the terms ''monkeys'' and ''simians'' synonyms in regards to their scope. In 1812, Geoffroy grouped the apes and the Cercopithecidae group of monkeys together and established the name Catarrhini, "Old World monkeys", ("''singes de l'Ancien Monde''" in French). The extant sister of the Catarrhini in the monkey ("singes") group is the Platyrrhini (New World monkeys). Some nine million years before the divergence between the Cercopithecidae and the apes, the Platyrrhini emerged within "monkeys" by migration to South America likely by ocean. Apes are thus deep in the tree of extant and extinct monkeys, and any of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (mathematics)
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero. The value of the function at a critical point is a critical value. This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbilical Point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a ''principal direction''. The name "umbilic" comes from the Latin ''umbilicus'' (navel). Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A torus can have no umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. An unproven conjecture of Constantin Carathéodory states that every s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedding, embedded in Euclidean space. This is the content of the ''Theorema egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema egregium'' in 1827. Informal definition At any point on a surface, we can find a Normal (geometry), normal vector that is at right angles to the sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
