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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 eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 multiple variables ('' multivariate''), rather than just one. Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called ''vector calculus''. Introduction In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces: # There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ... of integration. Works * * * * (posthumous) References 1905 births 1948 deaths 20th-century Japanese mathematicians Mathematical analysts Academic staff of Kyoto University Kyoto University alumni Scientists from Kyoto {{Japan-bio-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. Smale obtained his Bachelor of Science degree in 1952. Despite his grades, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Norton's Dome
Norton's dome is a thought experiment that exhibits a determinism, non-deterministic system within the bounds of Newtonian mechanics. It was devised by John D. Norton in 2003. It is a special limiting case of a more general class of examples from 1997 by Sanjay Bhat and Dennis Bernstein. The Norton's dome problem can be regarded as a problem in physics, mathematics, and philosophy. Description The model consists of an idealized point particle initially sitting motionless at the Apex (geometry), apex of an idealized radially-symmetrical frictionless dome described by the equation h = r^, \quad 0 \leq r T. \end Importantly these two are both solutions to the initial value problem: \ddot = b^2\sqrt, \quad r(0)=0, \quad \dot(0) = 0. Therefore within the framework of Newtonian mechanics this problem has an indeterminate solution, in other words given the initial conditions and there are multiple possible trajectories the particle may take. This is the paradox which implies Newto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 is a static vector field, that is, a vector-valued function with Cartesian coordinates , and that is a parametric curve with Cartesian coordinates . Then is an integral curve of if it is a solution of the autonomous system of ordinary differential equations, \begin \frac &= F_1(x_1,\ldots,x_n) \\ &\;\, \vdots \\ \frac &= F_n(x_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Boundary Value Problem
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called 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, in the linear case, 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 equations is devot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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'(t) = f\left(t, y(t)\right) a continuous, explicit first-order differential equation defined on ''D'', then every initial value problem y\left(t_0\right) = y_0 for ''f'' with (t_0, y_0) \in D has a local solution z\colon I \to \mathbb where I is a neighbourhood of t_0 in \mathbb, such that z'(t) = f\left(t, z(t)\right) for all t \in I . The solution need not be unique: one and the same initial value (t_0, y_0) may give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lyapunov Function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state-space Markov chains usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state, the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems, and conservation laws can often be used to construct Lyapunov fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |