nowhere $0$ vector field on $\mathbb{R}$ such that integral curves can be defined only on neighborhood of $0$

Asked Oct 21 '19 at 21:53

Active Oct 21 '19 at 22:19

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This is question 2 from Spivak Vol 1. Chapter 5:

Find a nowhere $0$ vector field on $\mathbb{R}$ such that all integral curves can be defined only on some interval around $0$.

I don't quite understand what the question is asking...

Is $x \mapsto (x^2 + 1)\partial /\partial x$, with integral curve $t \mapsto t^3/3 + x$ a solution?

edited Oct 21 '19 at 22:19

asked Oct 21 '19 at 21:53

du.Du.DU

You'd better review how to solve basic ODE. What is the ODE that this vector field gives you? – Ted Shifrin Oct 21 '19 at 21:58
@TedShifrin You are right.... Is it $c'(t) = f(c(t)) = c(t)^2 + 1$? – du.Du.DU Oct 21 '19 at 22:09
Yes. And now the solutions ... ? – Ted Shifrin Oct 21 '19 at 22:09
c(t) = tan (t)? – du.Du.DU Oct 21 '19 at 22:13
That's the integral curve with initial condition $c(0)=0$. Generally? – Ted Shifrin Oct 21 '19 at 22:14
c(t) = tan(t + c). And it still is a solution to this question? – du.Du.DU Oct 21 '19 at 22:18
Well, what do you think? – Ted Shifrin Oct 21 '19 at 22:33
@TedShifrin Yes, because it clearly went through all points. – du.Du.DU Oct 21 '19 at 22:43
I do not understand your comment. – Ted Shifrin Oct 21 '19 at 22:49
Because the image of the map $c(t) = \tan(t)$ is the whole $\mathbb{R}$? – du.Du.DU Oct 21 '19 at 22:50
That is not relevant. What did the question ask? – Ted Shifrin Oct 21 '19 at 22:51
Well, that is exactly why I ask this question here: I don't know what the author means by: can be defined only on some interval around 0. I interpret it as all integral curve must contain a neighborhood of $0$. – du.Du.DU Oct 21 '19 at 22:53
The function $c(t)$ (for some particular initial condition) is only defined for $t$ in some interval $(-\delta,\delta)$, not for all $t$. – Ted Shifrin Oct 21 '19 at 22:54
Sorry, I am even more confused, can you explain what is the question asking? and maybe a hint on how to approach it? – du.Du.DU Oct 21 '19 at 22:59

nowhere $0$ vector field on $\mathbb{R}$ such that integral curves can be defined only on neighborhood of $0$

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