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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Newton's theorem about ovals ： ウィキペディア英語版 Newton's theorem about ovals In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI of book 1 of Newton's ''Principia'', and used it to show that the position of a planet moving in an orbit is not an algebraic function of time. There has been some controversy about whether or not this theorem is correct because Newton did not state exactly what he meant by an oval, and for some interpretations of the word oval the theorem is correct, while for others it is false. If "oval" means "continuous convex curve", then there are counterexamples, such as triangles or one of the lobes of Huygens lemniscate ''y''2 = ''x''2 − ''x''4, while pointed that if "oval" means "infinitely differentiable convex curve" then Newton's claim is correct and his argument has the essential steps of a rigorous proof. generalized Newton's theorem to higher dimensions. ==Statement== An English translation Newton's original statement is: : "There is no oval figure whose area, cut off by right lines at pleasure, can be universally found by means of equations of any number of finite terms and dimensions." In modern mathematical language, Newton essentially proved the following theorem: : There is no convex smooth (meaning infinitely differentiable) curve such that the area cut off by a line ''ax'' + ''by'' = ''c'' is an algebraic function of ''a'', ''b'', and ''c''. In other words, "oval" in Newton's statement should mean "convex smooth curve". The infinite differentiability at all points is necessary: For any positive integer ''n'' there are algebraic curves that are smooth at all but one point and differentiable ''n'' times at the remaining point for which the area cut off by a secant is algebraic. Newton observed that a similar argument shows that the arclength of a (smooth convex) oval between two points is not given by an algebraic function of the points. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Newton's theorem about ovals」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.