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Integrating Factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential. Use An integrating factor is any expression that a differential equation is multiplied by to facilitate integration. For example, the nonlinear second order equation : \frac = A y^ admits \frac as an integrating factor: : \frac \frac = A y^ \frac. To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule: : \frac\left(\frac 1 2 \left(\frac\right)^2\right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Equation
In mathematics, 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) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. Weisstein continuously improved the notes and accepted corrections and comments from online readers. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function Q in an orthogonal coordinate system. An exact differential is sometimes also called a ''total differential'', or a ''full differential'', or, in the study of differential geometry, it is termed an exact form. The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics. Overview Definition Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type :A(x,y,z) \,dx + B(x,y,z) \,dy + C(x,y,z) \,dz is called a differential form. This form is called ''exact'' on an open domain D \subset \mathbb^3 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the derivative of is :h'(x) = \frac. It is provable in many ways by using other derivative rules. Examples Example 1: Basic example Given h(x)=\frac, let f(x)=e^x, g(x)=x^2, then using the quotient rule:\begin \frac \left(\frac\right) &= \frac \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac. \end Example 2: Derivatives of tangent and cotangent functions The quotient rule can be used to find the derivative of \tan x = \frac as follows:\begin \frac \tan x &= \frac \left(\frac\right) \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac = \sec^2 x. \endSimilarly, the derivative of \cot x = \frac can be obtained as follows:\begin \frac \cot x &= \frac \left(\frac\right) \\ &= \frac \\ &= \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negligi ... [...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 theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variation Of Parameters
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations. Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the derivative of is :h'(x) = \frac. It is provable in many ways by using other derivative rules. Examples Example 1: Basic example Given h(x)=\frac, let f(x)=e^x, g(x)=x^2, then using the quotient rule:\begin \frac \left(\frac\right) &= \frac \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac. \end Example 2: Derivatives of tangent and cotangent functions The quotient rule can be used to find the derivative of \tan x = \frac as follows:\begin \frac \tan x &= \frac \left(\frac\right) \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac = \sec^2 x. \endSimilarly, the derivative of \cot x = \frac can be obtained as follows:\begin \frac \cot x &= \frac \left(\frac\right) \\ &= \frac \\ &= \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Differential Equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written :f(x,y) \, dy = g(x,y) \, dx, where and are homogeneous functions of the same degree of and . In this case, the change of variable leads to an equation of the form :\frac = h(u) \, du, which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. History The term ''homogeneous'' was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negligi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
