Passage: The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions". Functionals have extrema with respect to the elements y of a given function space defined over a given domain. A functional J [ y ] is said to have an extremum at the function f if ΔJ = J [ y ] − J [ f] has the same sign for all y in an arbitrarily small neighborhood of function. f inaitwa extreme function au extremeal. The J [ f ] f is called a local maximum if ΔJ ≤ 0 in an arbitrarily small neighborhood of f, and a local minimum if fJ ≥ 0. Kwa nafasi ya kuendelea, kazi zinazohusiana na extrema zinaitwa kazi zinazoendelea au zinazoendelea katika uwanja uliopewa. Kwa hivyo, kazi ya J [ y ] inasemekana kuwa na mwisho katika kazi f ikiwa ΔJ = J [ y ] − J [ f ] ina ishara sawa kwa y zote katika ujirani mdogo wa kazi f. f inaitwa kazi ya mwisho au ya mwisho.
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