# 反函数定理

## 定理的表述

$J_{F^{-1}}(F(p)) = [ J_F(p) ]^{-1}$

$J_{G \circ H} (p) = J_G (H(p)) \cdot J_H (p).$

GFHF -1$G \circ H$就是恒等函数，其雅可比矩阵也是单位矩阵。在这个特殊的情况中，上面的公式可以对$J_{F^{-1}}(F(p))$求解。注意链式法则假设了函数H的全导数存在，而反函数定理则证明了F-1在点p具有全导数。

F的反函数存在，等于是说方程组yi = Fj(x1,...,xn)可以对x1，……，xn求解，如果我们把xy分别限制在pF(p)的足够小的邻域内。

## 例子

$\mathbf{F}(x,y)= \begin{bmatrix} {e^x \cos y}\\ {e^x \sin y}\\ \end{bmatrix}.$

$J_F(x,y)= \begin{bmatrix} {e^x \cos y} & {-e^x \sin y}\\ {e^x \sin y} & {e^x \cos y}\\ \end{bmatrix}$

$\det J_F(x,y)= e^{2x} \cos^2 y + e^{2x} \sin^2 y= e^{2x}. \,\!$

## 推广

### 流形

(dF)p : TpM → TF(p)N

M内的某个点p线性同构，那么存在p的一个开邻域U，使得：

F|U : UF(U)

## 注释

1. ^ Michael Spivak, Calculus on Manifolds.
2. ^ John H. Hubbard and Barbara Burke Hubbard, Vector Analysis, Linear Algebra, and Differential Forms: a unified approach, Matrix Editions, 2001.
3. ^ Serge Lang, Differential and Riemannian Manifolds, Springer, 1995, ISBN 0-387-94338-2.
4. ^ Wiilliam M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 2002, ISBN 0-12-116051-3.

## 参考文献

• Nijenhuis, Albert. Strong derivatives and inverse mappings. Amer. Math. Monthly. 1974, 81: 969–980. doi:10.2307/2319298.
• Renardy, Michael and Rogers, Robert C. An introduction to partial differential equations. Texts in Applied Mathematics 13 Second edition. New York: Springer-Verlag. 2004: 337–338. ISBN 0-387-00444-0.
• Rudin, Walter. Principles of mathematical analysis. International Series in Pure and Applied Mathematics Third edition. New York: McGraw-Hill Book Co. 1976: 221–223.