|
Clairaut's Equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form :y(x)=x\frac+f\left(\frac\right) where f is continuously differentiable. It is a particular case of the Lagrange differential equation. It is named after the French mathematician Alexis Clairaut, who introduced it in 1734. Solution To solve Clairaut's equation, one differentiates with respect to x, yielding :\frac=\frac+x\frac+f'\left(\frac\right)\frac, so :\left +f'\left(\frac\right)\rightfrac = 0. Hence, either :\frac = 0 or :x+f'\left(\frac\right) = 0. In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by :y(x)=Cx+f(C),\, the so-called ''general solution'' of Clairaut's equation. The latter case, :x+f'\left(\frac\right) = 0, defines only one solution y(x), the so-called '' singular solution'', whose graph is the envelope of the graphs of the gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexis Clairaut
Alexis Claude Clairaut (; ; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Isaac Newton, Sir Isaac Newton had outlined in the ''Principia Mathematica Philosophiae Naturalis, Principia'' of 1687. Clairaut was one of the key figures in the expedition to Sápmi, Lapland that helped to confirm Newton's theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as "Clairaut's theorem (gravity), Clairaut's theorem". He also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moon's orbit. In mathematics he is also credited with Clairaut's equation and Clairaut's relation. Biography Childhood and early life Clairaut was born in Paris, France, to Jean-Baptiste and Catherine Petit Clairaut. The couple had 20 children, howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Solution
A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In some cases, the term ''singular solution'' is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By ''tangent'' we mean that there is a point ''x'' where ''ys''(''x'') = ''yc''(''x'') and ''y's''(''x'') = ''y'c''(''x'') where ''yc'' is a solution in a family of solutions parameterized by ''c''. This m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envelope (mathematics)
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two " infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions. To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does not apply otherwise, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius. Envelope of a family of curves Let each curve ''C''''t'' in the family be given as the solution of an equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D'Alembert's Equation
In mathematics, d'Alembert's equation, sometimes also known as Lagrange's equation, is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert ( ; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopé .... The equation reads asDavis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962. :y = x f\left( \frac \right) + g\left( \frac\right). After differentiating once, and rearranging with p=dy/dx, we have :\frac + \frac=0 The above equation is linear. When f(p)=p, d'Alembert's equation is reduced to Clairaut's equation. References Eponymous equations of physics Mathematical physics Ordinary differential equations {{Mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chrystal's Equation
In mathematics, Chrystal's equation is a first order nonlinear ordinary differential equation, named after the mathematician George Chrystal, who discussed the singular solution of this equation in 1896. The equation reads asInce, E. L. (1939). Ordinary Differential Equations, London (1927). Google Scholar. :\left(\frac\right)^2 + Ax \frac + By + Cx^2 =0 where A,\ B, \ C are constants, which upon solving for dy/dx, gives :\frac = -\frac x \pm \frac (A^2 x^2 - 4By - 4Cx^2)^. This equation is a generalization particular cases of Clairaut's equation since it reduces to a form of Clairaut's equation under condition as given below. Solution Introducing the transformation 4By=(A^2-4C-z^2)x^2 gives :xz\frac = A^2 + AB - 4C \pm Bz - z^2. Now, the equation is separable, thus :\frac = \frac. The denominator on the left hand side can be factorized if we solve the roots of the equation A^2 + AB - 4C \pm Bz - z^2=0 and the roots are a,\ b = \pm \left[ B +\sqrt \right]/2, therefore :\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Transformation
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function. In physical problems, the Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables. For sufficiently smooth functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eponymous Equations Of Mathematics
An eponym is a noun after which or for which someone or something is, or is believed to be, named. Adjectives derived from the word ''eponym'' include ''eponymous'' and ''eponymic''. Eponyms are commonly used for time periods, places, innovations, biological nomenclature, astronomical objects, works of art and media, and tribal names. Various orthographic conventions are used for eponyms. Usage of the word The term ''eponym'' functions in multiple related ways, all based on an explicit relationship between two named things. ''Eponym'' may refer to a person or, less commonly, a place or thing for which someone or something is, or is believed to be, named. ''Eponym'' may also refer to someone or something named after, or believed to be named after, a person or, less commonly, a place or thing. A person, place, or thing named after a particular person share an eponymous relationship. In this way, Elizabeth I of England is the eponym of the Elizabethan era, but the Elizabethan e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
