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 I have a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ in the $xy$-plane given by $t \mapsto \gamma(t)=(\gamma_1(t),\gamma_2(t))$ which intersects the $x$-axis transversely. Is it then possible to locally express $\gamma$ in terms of $\gamma_2$?

I have not been able to construct a counter example yet but I have not been able to come up with a proof either. Any help is welcome.

share|improve this question
1  
The implicit function theorem should help you out here. –  Kris Aug 8 '12 at 21:57
add comment

2 Answers

Let $t_0 \in \mathbb{R}$ be such that $\gamma_2(t_0)=0$. Since $\gamma$ intersects transversally the $x$-axis, we have $\dot{\gamma}_2(t_0) \ne 0$. Thanks to the inverse function theorem, there exist $\epsilon,\delta>0$ such that $\gamma_2: (t_0-\epsilon,t_0+\epsilon)\to (-\delta,\delta)$ is a diffeomorphism.

Hence $\gamma(\gamma_2^{-1}(\tau))=(\gamma_1\circ\gamma_2^{-1}(\tau),\tau) \quad \forall\ \tau \in (-\delta,\delta)$.

share|improve this answer
    
Suppose I have the curve $\gamma:t \mapsto (t^3,t^3)$ then this curve intersects the $x$-axis transversely but $\dot{\gamma}_2(0)=0$! –  Novo Aug 9 '12 at 17:59
    
It's just a matter of parametrization. The curve $\gamma: t \mapsto (t^3,t^3)$ can be reparametrize as $s \mapsto (s,s)$. –  Mercy Aug 9 '12 at 19:15
add comment

To flesh out my comment into an answer: the condition that $\gamma$ intersects the $x$-axis transversely is equivalent to stating that $\frac{\partial \gamma}{\partial \gamma_1}$ is non-zero, and so by the implicit function theorem we can write $\gamma_1$ as a function of $\gamma_2$ and hence $\gamma$ as a function of $\gamma_2$.

share|improve this answer
    
Notice that formally $\partial\gamma/\partial\gamma_1=(1,0) \ne 0$ for any curve $\gamma=(\gamma_1,\gamma_2)$, therefore you are saying that every curve $\gamma=(\gamma_1,\gamma_2)$ intersects transversally the $x$-axis, which of course is not true. –  Mercy Aug 8 '12 at 22:33
    
Guess you mean $\frac{\partial \gamma_2}{\partial \gamma_1}$ is non-zero. However, my problem is that the latter can be undefined. For example consider $\gamma=(t,t^2)$ at $t=0$. Does the implicit function theorem in these cases still hold? –  Novo Aug 9 '12 at 18:14
add comment

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.