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Initial Value Problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. Definition An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb \times \mathbb^n \to \mathbb^n where \Omega is an open set of \mathbb \times \mathbb^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In higher dimensions, the differential equation is replaced with a family of equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariable Calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus. Typical operations Limits and continuity A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function. :f(x,y) = \frac approaches zero whenever the point (0,0) is approached along lines through the origin (y=kx). However, when the origin is appr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hiroshi Okamura
was a Japanese mathematician who made contributions to analysis and the theory of differential equations. He was a professor at Kyoto University.''Funkcialaj Ekvacioj'', 2 (1959), Profesoro Hirosi OKAMURA, nekrologo (''E-e'') He discovered the necessary and sufficient conditions on initial value problems of ordinary differential equations for the solution to be unique. He also refined the second mean value theorem of integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, .... Works * * * * (posthumous) References 1905 births 1948 deaths 20th-century Japanese mathematicians Mathematical analysts Academic staff of Kyoto University Kyoto University alumni {{Japan-bio-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis. Education and career Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morris W
Morris may refer to: Places Australia *St Morris, South Australia, place in South Australia Canada * Morris Township, Ontario, now part of the municipality of Morris-Turnberry * Rural Municipality of Morris, Manitoba ** Morris, Manitoba, a town mostly surrounded by the municipality * Morris (electoral district), Manitoba (defunct) * Rural Municipality of Morris No. 312, Saskatchewan United States ;Communities * Morris, Alabama, a town * Morris, Connecticut, a town * Morris, Georgia, an unincorporated community * Morris, Illinois, a city * Morris, Indiana, an unincorporated community * Morris, Minnesota, a city * Morristown, New Jersey, a town * Morris (town), New York ** Morris (village), New York * Morris, Oklahoma, a city * Morris, Pennsylvania, an unincorporated community * Morris, West Virginia, an unincorporated community * Morris, Kanawha County, West Virginia, a ghost town * Morris, Wisconsin, a town * Morris Township (other) ;Counties and other * Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines. In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits. Definition Suppose that F is a static vector field, that is, a vector-valued function with Cartesian coordinates (''F''1,''F''2,...,''F''''n''), and that x(''t'') is a parametric curve with Cartesian coordinates (''x''1(''t''),''x''2(''t''),...,''x''''n''(''t'')). Then x(''t'') is an integral curve of F if it is a solution of the autonomous system of ordinary differential equations, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Of Integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives. More specifically, if a function f(x) is defined on an interval, and F(x) is an antiderivative of f(x), then the set of ''all'' antiderivatives of f(x) is given by the functions F(x) + C, where C is an arbitrary constant (meaning that ''any'' value of C would make F(x) + C a valid antiderivative). For that reason, the indefinite integral is often written as \int f(x) \, dx = F(x) + C, although the constant of integration might be sometimes omitted in lists of integrals for simplicity. Origin The derivative of any constant function is zero. Once one has found one antiderivative F(x) for a function f(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Value Problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Existence Theorem
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. History Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations. Theorem Let D be an open subset of \mathbb\times\mathbb with f\colon D \to \mathbb a continuous function and y'(x) = f\left(x,y(x)\right) a continuous, explicit first-order differential equation defined on ''D'', then every initial value problem y\left(x_0\right) = y_0 for ''f'' with (x_0, y_0) \in D has a local solution z\colon I \to \mathbb where I is a neighbourhood of x_0 in \mathbb, such that z'(x) = f\left(x,z(x)\right) for all x \in I . The solution need not be unique: one and the same initial value (x_0,y_0) may give ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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