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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 of ...
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [l] is defined as the linear part of the change in the functional, and the second variation [m] is defined as the quadratic part. [22]
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 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.
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It is also called the constant of variation or constant of proportionality. Given such a constant k , the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by { ( a , b ) ∈ A × B : a = k b } . {\displaystyle \{(a,b)\in A\times B:a=kb\}.}
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