Search results
Results from the WOW.Com Content Network
Fundamental lemma of the calculus of variations. Initial result in using test functions to find extremum. In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional ...
Calculus. The calculus of variations (or variational calculus) 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. [a] Functionals are often expressed as definite integrals involving ...
Bounded variation. In mathematical analysis, a function of bounded variation, also known as BV 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. For a continuous function of a single variable, being of bounded variation means that the ...
In other words, every value of y defines a function, denoted f y, which is a function of one variable x. [6] That is, = + +. In this section the subscript notation f y denotes a function contingent on a fixed value of y, and not a partial derivative. Once a value of y is chosen, say a, then f(x,y) determines a function f a which traces a curve ...
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) [1] relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends. In the calculus of variations, functionals are usually expressed ...
e. In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence ...
If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists. In other words, the graph of f has a non-vertical tangent line at the point (x 0, f(x 0)). f is said to be differentiable on U if it is differentiable at every point of U.
Variations. Miscellanea. v. t. e. In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a ...