|
Stationary Point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface (mathematics), surface of the graph where all its partial derivatives are zero (equivalently, the gradient has zero vector norm, norm). The notion of stationary points of a real-valued function is generalized as ''Critical point (mathematics), critical points'' for complex-valued functions. Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., Parallel (geometry), parallel to the Abscissa, -axis). For a function of two variables, they correspond to the points on the gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation (mathematics), transformation. Specifically, for function (mathematics), functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain of a function, domain and the codomain of , and . In particular, cannot have any fixed point if its domain is disjoint from its codomain. If is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point corresponds to an intersection of the curve with the line , cf. picture. For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abscissa
In mathematics, the abscissa (; plural ''abscissae'' or ''abscissas'') and the ordinate are respectively the first and second coordinate of a point in a Cartesian coordinate system: : abscissa \equiv x-axis (horizontal) coordinate : ordinate \equiv y-axis (vertical) coordinate Together they form an ordered pair which defines the location of a point in two-dimensional rectangular space. More technically, the abscissa of a point is the signed measure of its projection on the primary axis. Its absolute value is the distance between the projection and the origin of the axis, and its sign is given by the location on the projection relative to the origin (before: negative; after: positive). Similarly, the ordinate of a point is the signed measure of its projection on the secondary axis. In three dimensions, the third direction is sometimes referred to as the '' applicate''. Etymology Though the word "abscissa" () has been used at least since ''De Practica Geometrie'' (1220) by F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxima And Minima
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concave Function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be ''concave'' if, for any x and y in the interval and for any \alpha \in ,1/math>, :f((1-\alpha )x+\alpha y)\geq (1-\alpha ) f(x)+\alpha f(y) A function is called ''strictly concave'' if :f((1-\alpha )x+\alpha y) > (1-\alpha ) f(x)+\alpha f(y) for any \alpha \in (0,1) and x \neq y. For a function f: \mathbb \to \mathbb, this second definition merely states that for ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary And Inflection Pts
Stationary may refer to: * "Stationary", a song from ''Copacetic'' (Knuckle Puck album) * a common misspelling of "stationery", meaning office supplies * stationary process In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ..., in mathematics and statistics, a process whose probability distribution does not change over time. See also * * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Derivative Test
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. First-derivative test The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch" and remains increasing or remains decreasing, then no highest or least value is achieved. One can examine a fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extrema Example Original
Extrema may refer to: * Extrema (mathematics), maxima and minima values * Extremities (other) * Extrema, Minas Gerais, town in Brazil * Extrema, Rondônia, town in Brazil * Extrema (band), Italian thrash/groove metal band {{disambig, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflection Point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class (its first derivative , and its second derivative , exist and are continuous), the condition can also be used to find an inflection point since a point of must be passed to change from a positive value (concave upward) to a negative value (concave downward) or vice versa as is continuous; an inflection point of the curve is where and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isolated Point
In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton is an open set in the topological space (considered as a subspace of ). Another equivalent formulation is: an element of is an isolated point of if and only if it is not a limit point of . If the space is a metric space, for example a Euclidean space, then an element of is an isolated point of if there exists an open ball around that contains only finitely many elements of . A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space). Related notions Any discrete subset of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of may be mapped injective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ecliptic Circle
The ecliptic or ecliptic plane is the orbital plane of Earth around the Sun. It was a central concept in a number of ancient sciences, providing the framework for key measurements in astronomy, astrology and calendar-making. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars – specifically the Zodiac constellations. The planets of the Solar System can also be seen along the ecliptic, because their orbital planes are very close to Earth's. The Moon's orbital plane is also similar to Earth's; the ecliptic is so named because the ancients noted that eclipses only occur when the Moon is crossing it. The ecliptic is an important reference plane and is the basis of the ecliptic coordinate system. Ancient scientists were able to calculate Earth's axial tilt by comparing the ecliptic plane to that of the equator. Sun's apparent motion The eclipt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apparent Retrograde Motion
Apparent retrograde motion is the apparent motion of a planet in a direction opposite to that of other bodies within its system, as observed from a particular vantage point. Direct motion or prograde motion is motion in the same direction as other bodies. While the terms ''direct'' and ''prograde'' are equivalent in this context, the former is the traditional term in astronomy. The earliest recorded use of ''prograde'' was in the early 18th century, although the term is now less common. Etymology and history The term ''retrograde'' is from the Latin word – "backward-step", the affix meaning "backwards" and "step". ''Retrograde'' is most commonly an adjective used to describe the path of a planet as it travels through the night sky, with respect to the zodiac, stars, and other bodies of the celestial canopy. In this context, the term refers to planets, as they appear from Earth, stopping briefly and reversing direction at certain times, though in reality, of course, we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
