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- Mar 10, 2012

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*Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$.*

**QUESTION:**Let $M$ be the set of all the points $\mathbf x$ such that $f(\mathbf x)=\mathbf 0$ and $N$ be the set of all the points $\mathbf x$ such that $$f_1(\mathbf x)=\cdots=f_{n-1}(\mathbf x)=0\text{ and } f_n(\mathbf x)\geq 0$$

Assume $M$ is non-empty.

1) Assume $\text{rank} Df(\mathbf x)=n$ for all $\mathbf x\in M$ and show that $M$ is a $k$-manifold without boundary in $\mathbb R^{n+k}$.

2) Assume that the matrix $\displaystyle\frac{\partial(f_1,\ldots,f_{n-1})}{\partial \mathbf x}$ has rank $n-1$ for all $\mathbf x\in N$ and show that $N$ is a $(k+1)$-manifold with boundary in $\mathbb R^{n+k}$.

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*I am trying to show $(2)$ and I am not sure if the hypothesis of $(1)$ is required to do that.*

I have approached this question using the constant rank theorem which dictates:

**Let $U$ be open in $\mathbb R^n$ and $\mathbf a$ be any point in $U$. Let $f:U\to \mathbb R^m$ be a function of class $\mathscr C^p$ such that $\text{rank } Df(\mathbf z) =r$ for all $\mathbf z\in U$. Then there exist open sets $U_1,U_2\subseteq U$ and $V\subseteq \mathbb R^m$ such that $\mathbf a\in U_1$ and $f(\mathbf a)\in V$, and $\mathscr C^p$-diffeomorphisms $\phi:U_1\to U_2$ and $\psi:V\to V$ such that $$(\psi\circ f\circ \phi^{-1})(\mathbf z)=(z_1,\ldots,z_r,0,\ldots,0)$$**

Constant Rank Theorem:Constant Rank Theorem:

for all $\mathbf z\in U_2$.

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What I did was define a function $g:\mathbb R^{n+k}\to \mathbb R^{n-1}$ as $$g(\mathbf x)=(f_1(\mathbf x),\ldots,f_{n-1}(\mathbf x))$$

Then $\text{rank }Dg(\mathbf x)=n-1$ for all $\mathbf x\in N$.

Let $\mathbf z_0\in N$.

I can show that there exists an open set $U\subseteq \mathbb R^{n+k}$ such that $\mathbf z_0\in U$ and $\text{rank }Dg(\mathbf z)=n-1$ for all $\mathbf z\in U$.

Thereby, using the conastant rank theorem I get $U_1, U_2,\psi$ and $\phi$ such that $(\psi\circ g\circ\phi^{-1})(\mathbf x)=(x_1,\ldots,x_{n-1})$

Can somebody guide me what to do from here?