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In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case is ordinary differential equations (ODEs), which deal with functi

Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] A visualisation of a solution to the two-dimensional heat equation with temperature represented by the third dimension <span>In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe

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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,

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What Kolmogorov did was to say that the new real analysis that had started with the PhD thesis of Henri Lebesgue (1902) and had been rapidly generalized to integrals of real-valued functions on arbitrary spaces by Radon, Fr´echet, and others (called Lebesgue integration or abstract integration) should also be used in probability theory

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Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces.

n abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. <span>Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tool

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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's <span>direct effect as well as indirect effects via the other arguments of the function. <span><body><html>

integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous ar

integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals ge

ion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals. <span>In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Differential calculus[edit source] Main article: Differential calculus [imagelink] Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise ov

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The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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derivative defined by limit considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero

e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di

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∫ 2 x d x = x 2 + C . {\displaystyle \int 2x\,dx=x^{2}+C.} <span>The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration. Fundamental theorem[edit source] Main article: Fundamental theorem of calculus The fundamental theorem of calculus states that differentiation and integration are inverse operatio

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n as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) <span>The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangl

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∫ f ( x ) d x . {\displaystyle \int f(x)\,dx.} Functions differing by only a constant have the same derivative, and it can be shown that <span>the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x 2 + C, where C is any constant, is y′ = 2x, the antiderivative of the latter given by: ∫ 2 x

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tyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).} This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the proliferation of analytic results after their work became known. <span>The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences. Applications[edit

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The definite integral is an operator that inputs a function and outputs a number

n as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) <span>The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangl

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the area, as an output. The terminating differential, dx , is not a number, and is not being multiplied by f(x) , although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. <span>Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. <span><body><html>

t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di

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Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.

t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di

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Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.

t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di

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the antiderivative of a given function is actually a family of functions differing only by a constant.

∫ f ( x ) d x . {\displaystyle \int f(x)\,dx.} Functions differing by only a constant have the same derivative, and it can be shown that <span>the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x 2 + C, where C is any constant, is y′ = 2x, the antiderivative of the latter given by: ∫ 2 x

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The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration.

∫ 2 x d x = x 2 + C . {\displaystyle \int 2x\,dx=x^{2}+C.} <span>The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration. Fundamental theorem[edit source] Main article: Fundamental theorem of calculus The fundamental theorem of calculus states that differentiation and integration are inverse operatio

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The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives.

tyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).} This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the proliferation of analytic results after their work became known. <span>The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences. Applications[edit