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Oblate Spheroidal Coordinates
Oblate spheroidal coordinates are a three-dimensional orthogonal coordinates, orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinates, elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius a in the ''x''-''y'' plane. (Rotation about the other axis produces prolate spheroidal coordinates.) Oblate spheroidal coordinates can also be considered as a limiting case (mathematics), limiting case of ellipsoidal coordinates in which the two largest semi-axis, semi-axes are equal in length. Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid, hyperboloid of revolution. For example, they played an important role in the calculation of the Perrin friction factors, which contributed to the awarding of the 1926 Nobel Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroidal Coordinates Half Hyperboloid
In Christianity (especially in the Roman Catholic Church, Roman Catholic, Eastern Orthodox Church, Orthodox, Anglicanism, Anglican and Methodism, Methodist traditions), an oblate is a person who is specifically dedicated to God or to God's service. Oblates are individuals, either laity, laypersons or clergy, normally living in general society, who, while not profession (religious), professed monks or nuns, have individually affiliated themselves with a monastic community of their choice. They make a formal, private promise (annually renewable or for life, depending on the monastery with which they are affiliated) to follow the monasticism, Rule of the Order in their private lives as closely as their individual circumstances and prior commitments permit. Such oblates are considered an extended part of the monastic community; for example, Benedictine oblates also often include the post-nominal letters 'OblSB' or 'ObSB' after their names on documents. They are comparable to the Third ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry. If the ellipse is rotated about its major axis, the result is a ''prolate spheroid'', elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an ''oblate spheroid'', flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid, instead of a spher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosh (mathematical Function)
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 from which are derived: * hyperbolic tangent "" (), * hyperbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroidal Coordinates Full Hyperboloid
In Christianity (especially in the Roman Catholic, Orthodox, Anglican and Methodist traditions), an oblate is a person who is specifically dedicated to God or to God's service. Oblates are individuals, either laypersons or clergy, normally living in general society, who, while not professed monks or nuns, have individually affiliated themselves with a monastic community of their choice. They make a formal, private promise (annually renewable or for life, depending on the monastery with which they are affiliated) to follow the Rule of the Order in their private lives as closely as their individual circumstances and prior commitments permit. Such oblates are considered an extended part of the monastic community; for example, Benedictine oblates also often include the post-nominal letters 'OblSB' or 'ObSB' after their names on documents. They are comparable to the tertiaries associated with the various mendicant orders. The term "oblate" is also used in the official name of some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Legendre Polynomial
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently \frac \left \left(1 - x^2\right) \frac P_\ell^m(x) \right+ \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, where the indices ''ℓ'' and ''m'' (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if ''ℓ'' and ''m'' are integers with 0 ≤ ''m'' ≤ ''ℓ'', or with trivially equivalent negative values. When in addition ''m'' is even, the function is a polynomial. When ''m'' is zero and ''ℓ'' integer, these functions are identical to the Legendre polynomials. In general, when ''ℓ'' and ''m'' are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Of Variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) Suppose a differential equation can be written in the form :\frac f(x) = g(x)h(f(x)) which we can write more simply by letting y = f(x): :\frac=g(x)h(y). As long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, so that the two variables ''x'' and ''y'' have been separated. ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who dislike Leibniz's notation may prefer to write this as :\frac \frac = g(x), but that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measured from a fixed zenith direction, and the ''azimuthal angle'' of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called '' colatitude'', ''zenith angle'', '' normal angle'', or ''inclination angle''. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates. The use of symbols and the order of the coordinates differs among sources and disciplines. This article will us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
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] |
Asymptote
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 the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-minor Axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both focus (geometry), foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major wikt:semiaxis, semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus (geometry), focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the eccentricity (mathematics), eccentricity and the semi-latus rectum \ell, as follows: The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
