變分法
Calculus of variations

context: zh_tw: 變分法的關鍵定理是尤拉－拉格朗日方程。它對應於泛函的臨界點。在尋找函式的極大和極小值時，在一個解附近的微小變化的分析給出一階的一個近似。它不能分辨是找到了最大值或者最小值（或者都不是）。
en: 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  f .  The function f is called an extremal function or extremal. The extremum J [ f ] is called a local maximum if ΔJ ≤ 0 everywhere in an arbitrarily small neighborhood of f , and a local minimum if ΔJ ≥ 0 there. For a function space of continuous functions, extrema of corresponding functionals are called weak extrema or strong extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.Both strong and weak extrema of functionals are for a space of continuous functions but weak extrema have the additional requirement that the first derivatives of the functions in the space be continuous.  Thus a strong extremum is also a weak extremum, but the converse may not hold.  Finding strong extrema is more difficult than finding weak extrema. An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.
qas: question: zh_tw: 最大值又稱為什麼？
en: What is another term for maxima?
answers: zh_tw: text: 極大
en: text: extrema

context: zh_tw: 勒讓德（1786）確定了一種方法，但在對極大和極小的區別不完全令人滿意。牛頓和萊布尼茨也是在早期關注這一學科，對於這兩者的區別 Vincenzo Brunacci（1810）、高斯（1829）、泊松（1831）、Mikhail Ostrogradsky（1834）、和雅可比（1837）都曾做出過貢獻。Sarrus（1842）的由柯西（1844）濃縮和修改的是一個重要的具有一般性的成就。Strauch（1849）、Jellett（1850）、Otto Hesse（1857）、Alfred Clebsch（1858）、和 Carll（1885）寫了一些其他有價值的論文和研究報告，但可能那個世紀最重要的成果是 Weierstrass 所取得的。他關於這個理論的著名教材是劃時代的，並且他可能是第一個將變分法置於一個穩固而不容置疑的基礎上的。1900 年希爾伯特發表的 23 個問題中的第 20 和 23 個問題促進了其更深遠的發展。
en: The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but  Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.In the 20th century David Hilbert, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming of Richard Bellman is an alternative to the calculus of variations.
qas: question: zh_tw: 用於阿爾弗雷德・克萊佈施公司的理論叫什麼？
en: What is the name of the theory that Alfred Clebsch placed on his firm?
answers: zh_tw: text: 劃時代的
en: text: epoch-making