Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose we know Definition 1 and Theorem 1, and want to prove Theorem 2 given below.

Definition 1: A subset $M$ of $R^n$ is called an k-dimensional manifold (in $R^n$) if for each point $x\in M$ the following condition is satisfied:

(A) There is an open set $U$ containing $x$, an open set V$\subset$$R^n$, and a diffeomorphism $h:U\rightarrow V$ such that $$h(U\cap M)=V\cap (R^k \times \lbrace0\rbrace)=\lbrace y\in V:y^{k+1}=\cdot\cdot\cdot\cdot\cdot\cdot=y^n=0\rbrace$$ Theorem 1: Let $f:R^n \rightarrow R^p$ be continuously differentiable in an open set containing $a$, where $p\le n$. If $f(a)=0$ and the $p\times n$ matrix $(D_jf^i(a))$ has rank $p$, then there is an open set $A\subset R^n$ containing $a$ and a differentiable function $h:A\rightarrow R^n$ with differentiable inverse such that $$f\circ h(x^1,....,x^n)=(x^{n-p+1},.....x^n)$$ Theorem 2: Let $A\subset R^n$ be open and let $g:A\rightarrow R^p$ be a differentiable function such that $g\prime(x)$ has rank $p$ whenever $g(x)=0$. Then $g^{-1}(0)$ is an $n-p$ dimensional manifold in $R^n.$

How to prove theorem 2 using Theorem 1 and Definition 1?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.