Perturbing the Mean Value Theorem: Implicit Functions, the Morse Lemma, and Beyond
Perturbing the Mean Value Theorem: Implicit Functions, the Morse Lemma, and Beyond
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扰动中值定理：隐式函数、莫尔斯引理及其他
DOI：
10.1080/00029890.2021.1840879
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Miles H. Wheeler
中科院分区：
文献类型：
--
作者：
David Lowry;Miles H. Wheeler
微积分的中值定理指出，给定一个区间上的可微函数f，至少存在一个均值横坐标c，使得切线的斜率等于通过和的割线的斜率。在这篇文章中，我们研究了c的选择如何与右端点B的变化相关。特别是，我们问：当我们可以写c作为一个连续的函数的B在某些区间？当我们探讨这个问题时，我们触及隐函数定理，一个简化版本的莫尔斯引理，和理论的解析函数。
Abstract The mean value theorem of calculus states that, given a differentiable function f on an interval , there exists at least one mean value abscissa c such that the slope of the tangent line at is equal to the slope of the secant line through and . In this article, we study how the choices of c relate to varying the right endpoint b. In particular, we ask: When we can write c as a continuous function of b in some interval? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions.