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Isocline
300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. The solution curves are y = C e^. Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some member of the family attains a given slope. The word comes from the Greek words ἴσος (isos), meaning "same", and the κλίνειν (klenein), meaning "make to slope". Generally, an isocline will itself have the shape of a curve or the union of a small number of curves. Isoclines are often used as a graphical method of solving ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives .... In an equation of the form ''y' = f''(''x'', ''y''), the isoclines are lines in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isocline 3
300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. The solution curves are y = C e^. Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some member of the family attains a given slope. The word comes from the Greek words ἴσος (isos), meaning "same", and the κλίνειν (klenein), meaning "make to slope". Generally, an isocline will itself have the shape of a curve or the union of a small number of curves. Isoclines are often used as a graphical method of solving ordinary differential equations. In an equation of the form ''y' = f''(''x'', ''y''), the isoclines are lines in the (''x'', ''y'') plane obtained by setting ''f''(''x'', ''y'') equal to a constant. This gives a series of lines (for different constants) along which the solution curves have the same gradient. By calculating this gradient for each isocl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nullcline
In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations :x_1'=f_1(x_1, \ldots, x_n) :x_2'=f_2(x_1, \ldots, x_n) ::\vdots :x_n'=f_n(x_1, \ldots, x_n) where x' here represents a derivative of x with respect to another parameter, such as time t. The j'th nullcline is the geometric shape for which x_j'=0. The equilibrium points of the system are located where all of the nullclines intersect. In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves. History The definition, though with the name 'directivity curve', was used in a 1967 article by Endre Simonyi.E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967 This article also defined 'directivity vector' as \mathbf = \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Curves
In geometry, a family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more complicated than a simple linear transformation. Sets of curves given by an implicit relation may also represent families of curves. Families of curves appear frequently in solutions of differential equations; when an additive constant of integration is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation. Families of curves may also arise in other areas. For example, all non-degenerate conic sections can be represented using a single polar equation with one parameter, the eccentricity of the curve: :r(\theta) = as the value of changes, the appearance of the curve varies in a relatively complicated way. Applications Families of curves may arise in various top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. The line may be physical – as set by a Surveying, road surveyor, pictorial as in a diagram of a road or roof, or Pure mathematics, abstract. An application of the mathematical concept is found in the grade (slope), grade or gradient in geography and civil engineering. The ''steepness'', incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows: *An "increasing" or "ascending" line goes from left to right and has positive slope: m>0. *A "decreasing" or "descending" line goes from left to right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Language
Greek (, ; , ) is an Indo-European languages, Indo-European language, constituting an independent Hellenic languages, Hellenic branch within the Indo-European language family. It is native to Greece, Cyprus, Italy (in Calabria and Salento), southern Albania, and other regions of the Balkans, Caucasus, the Black Sea coast, Asia Minor, and the Eastern Mediterranean. It has the list of languages by first written accounts, longest documented history of any Indo-European language, spanning at least 3,400 years of written records. Its writing system is the Greek alphabet, which has been used for approximately 2,800 years; previously, Greek was recorded in writing systems such as Linear B and the Cypriot syllabary. The Greek language holds a very important place in the history of the Western world. Beginning with the epics of Homer, ancient Greek literature includes many works of lasting importance in the European canon. Greek is also the language in which many of the foundational texts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of Set (mathematics), sets is the set of all element (set theory), elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of Zero, zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the List of mathematical symbols, table of mathematical symbols. Binary union The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, : A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y',\ldots, y^ are the successive derivatives of the unknown function y of the variable x. Among ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
