|
Dirichlet Principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. Formal statement Dirichlet's principle states that, if the function u ( x ) is the solution to Poisson's equation :\Delta u + f = 0 on a domain \Omega of \mathbb^n with boundary condition :u=g on the boundary \partial\Omega, then ''u'' can be obtained as the minimizer of the Dirichlet energy :E (x)= \int_\Omega \left(\frac, \nabla v, ^2 - vf\right)\,\mathrmx amongst all twice differentiable functions v such that v=g on \partial\Omega (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet. History The name "Dirichlet's principle" is due to Bernhard Riemann, who applied it in the study of complex analytic functions. Riemann (and others such as Carl Friedrich Gauss and P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics. He later received an honorary doctorate and became professor of mathematics in Berlin. Among many other contributions, Weierstrass formalized the definition of the continuity of a function and complex analysis, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals. Biography Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh, in the Province of Westphalia. Karl Weierstrass was the son of Wilhelm Weierstrass and Theodora Vonderforst, the former of whom was a government official and both of whom were Cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Identities
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem. Green's first identity This identity is derived from the divergence theorem applied to the vector field while using an extension of the product rule that : Let and be scalar functions defined on some region , and suppose that is twice continuously differentiable, and is once continuously differentiable. Using the product rule above, but letting , integrate over . Then \int_U \left( \psi \, \Delta \varphi + \nabla \psi \cdot \nabla \varphi \right)\, dV = \oint_ \psi \left( \nabla \varphi \cdot \mathbf \right)\, dS=\oint_\psi\,\nabla\varphi\cdot d\mathbf where is the Laplace operator, is the boundary of region , is the outward pointing unit normal to the surface element and is the oriented surface element. This the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plateau's Problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory. History Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó. Their methods were quite different; Radó's work built on the previous work of René Garnier and held only for rectifiable simple closed curves, whereas Douglas used completely new ideas with his result holding for an arbitrary simple closed curve. Both relied on setting up minimization problems; Douglas minimized the now-named Douglas integral while Radó minimized the "energy". Douglas went on to be awa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Twentieth Problem
Hilbert's twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether all boundary value problems can be solved (that is, do variational problems with certain boundary conditions have solutions). Introduction Hilbert noted that there existed methods for solving partial differential equations where the function's values were given at the boundary, but the problem asked for methods for solving partial differential equations with more complicated conditions on the boundary (e.g., involving derivatives of the function), or for solving calculus of variation problems in more than 1 dimension (for example, minimal surface problems or minimal curvature problems) Problem statement The original problem statement in its entirety is as follows: An important problem closely connected with the foregoing Hilbert's_nineteenth_problem.html" ;"title="eferring to Hilbert's nineteenth problem">eferring to Hilbert's nineteenth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Problem
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: :Given a function ''f'' that has values everywhere on the boundary of a region in \mathbb^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u=f on the boundary? This requirement is called the Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proven using the maximum principle. History The Dirichlet problem goes back to George Green, who studied the problem on general domains with general boundary conditions in his ''Essay on the Application o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Method In The Calculus Of Variations
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy. The method The calculus of variations deals with functionals J:V \to \bar, where V is some function space and \bar = \mathbb \cup \ . The main interest of the subject is to find ''minimizers'' for such functionals, that is, functions v \in V such that J(v) \leq J(u) for all u \in V . The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforeha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
