|
Jacobi Ellipsoid
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gustav Jacob Jacobi. History Before Jacobi, the Maclaurin spheroid, which was formulated in 1742, was considered to be the only type of ellipsoid which can be in equilibrium. Lagrange in 1811 considered the possibility of a tri-axial ellipsoid being in equilibrium, but concluded that the two equatorial axes of the ellipsoid must be equal, leading back to the solution of Maclaurin spheroid. But Jacobi realized that Lagrange's demonstration is a sufficiency condition, but not necessary. He remarked: "One would make a grave mistake if one supposed that the spheroids of revolution are the only admissible figures of equilibrium even under the restrictive assumption of second degree surfaces" (...) "In fact a simple consideration shows that ellips ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haumea Rotation
, discoverer = , discovered = , earliest_precovery_date = March 22, 1955 , mpc_name = (136108) Haumea , pronounced = , adjectives = Haumean , note = yes , alt_names = , named_after = Haumea , mp_category = , orbit_ref = , epoch = 17 December 2020 ( JD 2459200.5) , uncertainty = 2 , observation_arc = () , aphelion = , perihelion = , time_periastron = ≈ 1 June 2133±2 days , semimajor = , eccentricity = 0.19642 , period = 283.12 yr (103,410 days) , mean_anomaly = 218.205 ° , mean_motion = / day , inclination = 28.2137° , asc_node = 122.167° , arg_peri = 239.041° , avg_speed = , satellites = 2 (Hiʻiaka and Namaka) and ring , dimensions = , mean_radius = , surface_area = ≈ , volume = ≈ , mass = , density = , surface_grav = ≈ m/s2 , escape_velocity = ≈ km/s , sidereal_day = () , right_asc_north_pole = , declination = or , spectral_type = , magnitude = 17.3 (opposition) , abs_magnitude = ( V-ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlson Symmetric Form
In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa. The Carlson elliptic integrals are: R_F(x,y,z) = \tfrac\int_0^\infty \frac R_J(x,y,z,p) = \tfrac\int_0^\infty \frac R_C(x,y) = R_F(x,y,y) = \tfrac \int_0^\infty \frac R_D(x,y,z) = R_J(x,y,z,z) = \tfrac \int_0^\infty \frac Since R_C and R_D are special cases of R_F and R_J, all elliptic integrals can ultimately be evaluated in terms of just R_F and R_J. The term ''symmetric'' refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of R_F(x,y,z) is the same for any permutation of its arguments, and the value of R_J(x,y,z,p) is the same for any permutation of its first three arguments. The Carlson elliptic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrics
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in ''D'' + 1 variables; for example, in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', CRC Press, from The Geometry Center at University of Minnesota : \sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0 which may be compactly written in vector and matrix notation as: : x Q x^\mathrm + P x^\mathrm + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astrophysical Journal
''The Astrophysical Journal'', often abbreviated ''ApJ'' (pronounced "ap jay") in references and speech, is a peer-reviewed scientific journal of astrophysics and astronomy, established in 1895 by American astronomers George Ellery Hale and James Edward Keeler. The journal discontinued its print edition and became an electronic-only journal in 2015. Since 1953 ''The Astrophysical Journal Supplement Series'' (''ApJS'') has been published in conjunction with ''The Astrophysical Journal'', with generally longer articles to supplement the material in the journal. It publishes six volumes per year, with two 280-page issues per volume. ''The Astrophysical Journal Letters'' (''ApJL''), established in 1967 by Subrahmanyan Chandrasekhar as Part 2 of ''The Astrophysical Journal'', is now a separate journal focusing on the rapid publication of high-impact astronomical research. The three journals were published by the University of Chicago Press for the American Astronomical Society unt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid (r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roche Ellipsoid
F. Hoffmann-La Roche AG, commonly known as Roche, is a Swiss multinational healthcare company that operates worldwide under two divisions: Pharmaceuticals and Diagnostics. Its holding company, Roche Holding AG, has shares listed on the SIX Swiss Exchange. The company headquarters are located in Basel. Roche is the fifth largest pharmaceutical company in the world by revenue, and the leading provider of cancer treatments globally. The company controls the American biotechnology company Genentech, which is a wholly owned affiliate, and the Japanese biotechnology company Chugai Pharmaceuticals, as well as the United States-based companies Ventana and Foundation Medicine. Roche's revenues during fiscal year 2020 were 58.32 billion Swiss francs. Descendants of the founding Hoffmann and Oeri families own slightly over half of the bearer shares with voting rights (a pool of family shareholders 45%, and Maja Oeri a further 5% apart), with Swiss pharma firm Novartis owning a fur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Ellipsoid
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Biography Early years Riemann was born on 17 September 1826 in Breselenz, a village near Dan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. If two angles of a triangle h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of vortex rings. Mathematically, the vorticity \vec is the curl of the flow velocity \vec: :\vec \equiv \nabla \times \vec\,, where \nabla is the nabla operator. Conceptually, \vec could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their ''relative'' displacements as they move along the flow. The vorticity \vec would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. In a two-dimensional fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow Velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall). Definition The flow velocity ''u'' of a fluid is a vector field : \mathbf=\mathbf(\mathbf,t), which gives the velocity of an '' element of fluid'' at a position \mathbf\, and time t.\, The flow speed ''q'' is the length of the flow velocity vector :q = \, \mathbf \, and is a scalar field. Uses The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flow The flow of a fluid is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Ellipsoidal Problem
In astrophysics, Dirichlet's ellipsoidal problem, named after Peter Gustav Lejeune Dirichlet, asks under what conditions there can exist an ellipsoidal configuration at all times of a homogeneous rotating fluid mass in which the motion, in an inertial frame, is a linear function of the coordinates. Dirichlet's basic idea was to reduce Euler equations to a system of ordinary differential equations such that the position of a fluid particle in a homogeneous ellipsoid at any time is a linear and homogeneous function of initial position of the fluid particle, using Lagrangian framework instead of the Eulerian framework. History In the winter of 1856–57, Dirichlet found some solutions of Euler equations and he presented those in his lectures on partial differential equations in July 1857 and published the results in the same month. His work was left unfinished at his sudden death in 1859, but his notes were collated and published by Richard Dedekind posthumously in 1860. Bernhar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Haumea Rotation")
