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Sextic
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form: :ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\, where and the ''coefficients'' may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function. Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sextic Graph
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form: :ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\, where and the ''coefficients'' may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function. Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kampé De Fériet Function
In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet Marie-Joseph Kampé de Fériet (Paris, 14 May 1893 – Villeneuve d'Ascq, 6 April 1982) was professor at Université Lille Nord de France from 1919 to 1969. Besides his works on mathematics and fluid mechanics, he directed the ''Institut de mé .... The Kampé de Fériet function is given by : ^F_\left( \begin a_1,\cdots,a_p\colon b_1,b_1';\cdots;b_q,b_q'; \\ c_1,\cdots,c_r\colon d_1,d_1';\cdots;d_s,d_s'; \end x,y\right)= \sum_^\infty\sum_^\infty\frac\frac\cdot\frac. Applications The general sextic equation can be solved in terms of Kampé de Fériet functions. References * * * External links * Hypergeometric functions {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Watt's Curve
In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)). A line segment of length 2''c'' attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate partially back and forth or completely around. It arose in connection with James Watt's pioneering work on the steam engine. The equation of the curve can be given in polar coordinates as :r^2=b^2-\left \sin\theta\pm\sqrt\right2. Derivation Polar coordinates The polar equation for the curve can be derived as follows: Working in the complex plane, let the centers of the circles be at ''a'' and ''−a'', and the connecting segment have endpoints at ''−a''+''be''''i'' λ and ''a''+''be''''i'' ρ. Let the angle of inclination of the segment be ψ with its midpoint at ''re''''i'' θ. Then the endpoints are also given by '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F'' "). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stabilizer (group Theory)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steam Engine
A steam engine is a heat engine that performs mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a cylinder. This pushing force can be transformed, by a connecting rod and crank, into rotational force for work. The term "steam engine" is generally applied only to reciprocating engines as just described, not to the steam turbine. Steam engines are external combustion engines, where the working fluid is separated from the combustion products. The ideal thermodynamic cycle used to analyze this process is called the Rankine cycle. In general usage, the term ''steam engine'' can refer to either complete steam plants (including boilers etc.), such as railway steam locomotives and portable engines, or may refer to the piston or turbine machinery alone, as in the beam engine and stationary steam engine. Although steam-driven devices were known as early as the aeolipile in the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quintic Equation
In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function. Setting and assuming produces a quintic equation of the form: :ax^5+bx^4+cx^3+dx^2+ex+f=0.\, Solving quintic equations in terms of radicals (''n''th roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem. Finding roots of a quintic equa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
