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.

I am facing the following problem:

Let $\alpha:\mathbb{R}\rightarrow\mathbb{R}^2$ and $\beta:\mathbb{R}\rightarrow\mathbb{R}^2$ be $C^1$ curves with $\alpha(0)=(0,0)=\beta (0)$, such that $\alpha '(0)$ and $\beta '(0)$ are linearly independent. Show that there are open sets $U$ and $V$ in $\mathbb{R}^2$ and a $C^1$ diffeomorphism $\phi :U\rightarrow V$ such that $\phi(0,0)=(0,0)$, $\phi (\alpha(x))=(x,0)$ and $\phi (\beta (y))=(0,y)$ whenever $\alpha (x)$ and $\beta (y)$ are in $U$.

Using the Theorem for Local Form of Immersions (I don't know the english name for this theorem, I'm using a portuguese Analysis book and haven't found it elsewhere in the internet) I can find $C^1$ homeomorphisms $h_\alpha:\mathbb{R}^2\rightarrow\mathbb{R}^2$ and $h_\beta:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $h_\alpha\circ\alpha (t) = (t,0)$ and $h_\beta \circ \beta (t)=(0,t)$.

Now, I'm trying to define a function $h:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that it solves the problem; it should involve $h_\alpha$ and $h_\beta$ though I don't know how to assemble these parts together. Any help would be appreciated.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Let $f:{\Bbb R}^2\to {\Bbb R}^2$ be $(x,y)\mapsto \alpha(x)+\beta(y)$. Then, use the inverse function theorem on $f$ at $(0,0)$.

share|improve this answer
    
Well, this is just perfect! :) Thanks! –  Marra Feb 17 '13 at 11:22

Your Answer

 
discard

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.