gradient of a real function

梯度

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

Translation evidence: Literature usage

Meaning in context

For a differentiable real scalar function f on an open subset of Euclidean R^n, its gradient at a is the vector of its coordinate partial derivatives, satisfying Df(a)h=gradient f(a) dot h.

可微实值函数在欧氏空间中的梯度，是由各坐标偏导数组成的向量；与增量向量作内积可得全微分。

Example

For $f(x,y)=xy+x^2$, $\nabla f(2,3)=(7,2)$.

Usage and distinctions

The gradient uses an inner product to represent the derivative; a derivative as a linear functional does not need an inner product.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in Several Variables (Advanced Calculus)§3.2, equation (3.2.85) and following gradient definition · Checked 2026-10-14
2. OpenStax — Calculus Volume 3§4.6, Gradient, Definition (4.39) and Theorem 4.12 · 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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