implicit function theorem

隐函数定理

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

Translation evidence: Literature usage

Meaning in context

If F is a C^1 map near (a,b) in R^n times R^m, F(a,b)=0, and its derivative with respect to the second variable is invertible there, then nearby solutions F(x,y)=0 uniquely have y=g(x) for a C^1 map g with g(a)=b.

C^1 映射 F 在 (a,b) 为零，且关于第二组变量的导数可逆时，附近的零点唯一写为 y=g(x)，其中 g 是 C^1 映射并满足 g(a)=b。

Example

For F(x,y)=y3+x−8F(x,y)=y^3+x-8 at (0,2)(0,2), Fy=12F_y=12, and the local branch is y=(8−x)1/3y=(8-x)^{1/3} with g′(0)=−1/12g'(0)=-1/12.

Usage and distinctions

The C^1 conclusion is local and depends on an invertible partial derivative; at (8,0) in this example, F_y=0 and the branch y=(8-x)^{1/3} is not differentiable.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in Several Variables (Advanced Calculus)§2.2, Theorem 2.2.5 and equations (2.2.38)–(2.2.45) · 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：三、隐函数定理 · Checked 2026-10-14
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