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Slope Field
Slope fields (also called direction fields) are a graphical representation of the solutions to a first-order differential equation of a scalar function. Solutions to a slope field are functions drawn as solid curves. A slope field shows the slope of a differential equation at certain vertical and horizontal intervals on the x-y plane, and can be used to determine the approximate tangent slope at a point on a curve, where the curve is some solution to the differential equation. Definition Standard case The slope field can be defined for the following type of differential equations :y' = f(x, y), which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution (''integral curve'') at each point (''x'', ''y'') as a function of the point coordinates. It can be viewed as a creative way to plot a real-valued function of two real variables f(x,y) as a planar picture. Specifically, for a given pair x,y, a vector with the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope Field
Slope fields (also called direction fields) are a graphical representation of the solutions to a first-order differential equation of a scalar function. Solutions to a slope field are functions drawn as solid curves. A slope field shows the slope of a differential equation at certain vertical and horizontal intervals on the x-y plane, and can be used to determine the approximate tangent slope at a point on a curve, where the curve is some solution to the differential equation. Definition Standard case The slope field can be defined for the following type of differential equations :y' = f(x, y), which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution (''integral curve'') at each point (''x'', ''y'') as a function of the point coordinates. It can be viewed as a creative way to plot a real-valued function of two real variables f(x,y) as a planar picture. Specifically, for a given pair x,y, a vector with the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxima (software)
Maxima () is a computer algebra system (CAS) based on a 1982 version of Macsyma. It is written in Common Lisp and runs on all POSIX platforms such as macOS, Unix, BSD, and Linux, as well as under Microsoft Windows and Android. It is free software released under the terms of the GNU General Public License (GPL). History Maxima is based on a 1982 version of Macsyma, which was developed at Massachusetts Institute of Technology, MIT with funding from the United States Department of Energy and other government agencies. A version of Macsyma was maintained by Bill Schelter from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release his version under the GPL. That version, now called Maxima, is maintained by an independent group of users and developers. Maxima does not include any of the many modifications and enhancements made to the commercial version of Macsyma during 1982–1999. Though the core functionality remains similar, cod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including (ε, δ)-definition of limit, codify ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert L
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Qualitative Theory Of Differential Equations
In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively few differential equations that can be solved explicitly, but using tools from analysis and topology, one can "solve" them in the qualitative sense, obtaining information about their properties. References Further reading *Viktor Vladimirovich Nemytskii, Vyacheslav Stepanov, ''Qualitative theory of differential equations'', Princeton University Press, Princeton, 1960. Original references *Henri Poincaré, "Mémoire sur les courbes définies par une équation différentielle", ''Journal de Mathématiques Pures et Appliquées The ''Journal de Mathématiques Pures et Appliquées'' () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Dynamical Systems And Differential Equations Topics
This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. Dynamical systems, in general *Deterministic system (mathematics) *Linear system * Partial differential equation *Dynamical systems and chaos theory * Chaos theory ** Chaos argument ** Butterfly effect ** 0-1 test for chaos * Bifurcation diagram *Feigenbaum constant *Sharkovskii's theorem *Attractor ** Strange nonchaotic attractor * Stability theory **Mechanical equilibrium ** Astable ** Monostable ** Bistability **Metastability * Feedback ** Negative feedback **Positive feedback **Homeostasis * Damping ratio *Dissipative system *Spontaneous symmetry breaking *Turbulence *Perturbation theory *Control theory ** Non-linear control **Adaptive control ** Hierarchical control **Intelligent control ** Optimal control **Dynamic programming ** Robust control ** Stochastic control * System dynamics, system analysis *Takens' theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Transform Applied To Differential Equations
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform: :\mathcal\=s\mathcal\-f(0) :\mathcal\=s^2\mathcal\-sf(0)-f'(0) One can prove by induction that :\mathcal\=s^n\mathcal\-\sum_^s^f^(0) Now we consider the following differential equation: :\sum_^a_if^(t)=\phi(t) with given initial conditions :f^(0)=c_i Using the linearity of the Laplace transform it is equivalent to rewrite the equation as :\sum_^a_i\mathcal\=\mathcal\ obtaining :\mathcal\\sum_^a_is^i-\sum_^\sum_^a_is^f^(0)=\mathcal\ Solving the equation for \mathcal\ and substituting f^(0) with c_i one obtains :\mathcal\=\frac The solution for ''f''(''t'') is obtained by applying the inverse Laplace transform to \mathcal\. Note that if the initial condition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Examples Of Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics. The first version of SageMath was released on 24 February 2005 as free and open-source software under the terms of the GNU General Public License version 2, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB". The originator and leader of the SageMath project, William Stein, was a mathematician at the University of Washington. SageMath uses a syntax resembling Python's, supporting procedural, functional and object-oriented constructs. Development Stein realized when designing Sage that there were many open-source mathematics software packages already written in different languages, namely C, C++, Common Lisp, Fortran and Python. Rather tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. __TOC__ Notebook interface Wolfram Mathematica (called ''Mathematica'' by some of its users) is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
