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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 Fundamental increment lemma ： ウィキペディア英語版 Fundamental increment lemma In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative ''f''(''a'') of a function ''f'' at a point ''a'': :$f\text{'}(a)\; =\; \backslash lim\_\; \backslash frac.$ The lemma asserts that the existence of this derivative implies the existence of a function $\backslash varphi$ such that :$\backslash lim\_\; \backslash varphi(h)\; =\; 0\; \backslash qquad\; \backslash text\; \backslash qquad\; f(a+h)\; =\; f(a)\; +\; f\text{'}(a)h\; +\; \backslash varphi(h)h$ for sufficiently small but non-zero ''h''. For a proof, it suffices to define :$\backslash varphi(h)\; =\; \backslash frac\; -\; f\text{'}(a)$ and verify this $\backslash varphi$ meets the requirements. == Differentiability in higher dimensions == In that the existence of $\backslash varphi$ uniquely characterises the number $f\text{'}(a)$, the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose ''f'' maps some subset of $\backslash mathbb^n$ to $\backslash mathbb$. Then ''f'' is said to be differentiable at a if there is a linear function :$M:\; \backslash mathbb^n\; \backslash to\; \backslash mathbb$ and a function :$\backslash Phi:\; D\; \backslash to\; \backslash mathbb,\; \backslash qquad\; D\; \backslash subseteq\; \backslash mathbb^n\; \backslash smallsetminus\; \backslash ,$ such that :$\backslash lim\_\; \backslash Phi(\backslash bold)\; =\; 0\; \backslash qquad\; \backslash text\; \backslash qquad\; f(\backslash bold+\backslash bold)\; =\; f(\backslash bold)\; +\; M(\backslash bold)\; +\; \backslash Phi(\backslash bold)\; \backslash cdot\; \backslash Vert\backslash bold\backslash Vert$ for non-zero h sufficiently close to 0. In this case, ''M'' is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of ''f'' at a. Notably, ''M'' is given by the Jacobian matrix of ''f'' evaluated at a. 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「Fundamental increment lemma」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.