total differential

全微分

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

Translation evidence: Literature usage

Meaning in context

For f from an open subset of R^n to R^m, the total differential at a is the linear map A for which f(a+h)-f(a)-Ah=o(|h|), if such A exists.

欧氏空间开集上的映射若在 a 处的增量可写为某个线性映射作用于 h 加上相对于 |h| 的高阶小量，该线性映射称为全微分。

Example

For $f(x,y)=xy$ at $(2,3)$, $df(h,k)=3h+2k$ and the omitted increment is $hk=o(\sqrt{h^2+k^2})$.

Usage and distinctions

Partial derivatives test coordinate lines; the remainder condition must hold along every direction simultaneously.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in Several Variables (Advanced Calculus)§2.1, equations (2.1.1)–(2.1.4), printed pp.41–42 · 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，全微分 · Checked 2026-10-14

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