V2EX level curve

Level Curve

Definition / 释义

Level curve(等值曲线/等高线;数学中也称“水平曲线”)指在平面上由满足 f(x, y) = c(常数)的一组点组成的曲线。它表示二维函数在某个固定函数值下的“轮廓线”,常用于微积分、地形图、物理场与经济学等。

Pronunciation / 发音

/lvl kv/

Examples / 例句

A level curve shows where a function has the same value.
等值曲线表示函数在何处取到相同的值。

On the map, each level curve corresponds to a constant height, and closer level curves indicate a steeper slope.
在地图上,每一条等值曲线对应一个固定的海拔高度,而等值曲线越密集表示坡度越陡。

Etymology / 词源

level 原义与“水平、同一高度/程度”相关,curve 表示“曲线”。合起来,level curve 字面意思就是“保持同一水平(同一数值)的曲线”,用于描述函数值不变时形成的曲线(类似地形图上的等高线概念)。

Related Words / 相关词

Literary Works / 文学作品

  • Calculus: Early Transcendentals(James Stewart)常用“level curves”讲解多元函数图像与等值线。
  • Vector Calculus(Jerrold E. Marsden & Anthony Tromba)在讨论梯度与等值集合时频繁出现该术语。
  • Calculus on Manifolds(Michael Spivak)涉及以等值集合/等值曲线理解隐函数与几何结构的表述。
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