Bounded variation and Calculus of variations

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Difference between Bounded variation and Calculus of variations

Bounded variation vs. Calculus of variations

In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.

Arc length

Determining the length of an irregular arc segment is also called rectification of a curve.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Direct method in the calculus of variations

In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900.

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Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

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First-order partial differential equation

In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Function space

In mathematics, a function space is a set of functions between two fixed sets.

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Functional (mathematics)

In mathematics, the term functional (as a noun) has at least two meanings.

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Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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Leonida Tonelli

Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calculus of variations.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Maxima and minima

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

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Minimal surface

In mathematics, a minimal surface is a surface that locally minimizes its area.

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Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Young measure

In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions.

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