directional derivative

方向导数

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For f on an open subset of R^n and a specified unit vector v, the directional derivative at a is the finite two-sided limit [f(a+tv)-f(a)]/t as t tends to zero, when it exists.

沿指定单位向量 v，函数的增量商在参数双侧趋于零时若有有限极限，则该极限为此方向的方向导数。

Example

For $f(x,y)=x^2+y^2$ at $(1,2)$ and $v=(3/5,4/5)$, $D_vf=2(3/5)+4(4/5)=22/5$.

Usage and distinctions

Unit directions measure change per unit distance; allowing arbitrary vectors scales the parameter derivative by their lengths.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. OpenStax — Calculus Volume 3§4.6, Directional Derivatives, Definition (4.36); three-dimensional Definition (4.41) · Checked 2026-10-14

Translation sources

1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第二部分，多元微积分：二、多元函数微分学，方向导数与梯度 · Checked 2026-10-14

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