What are the smooth map and the vector field in the Fig. 8.2, page 182 of John Lee's Smooth manifolds, 2nd 
The purpose of this figure is to show that when $F$ is not surjective, then 
the differential of $F$ can't decide what vector can be assigned to the points outside of Im$F$ and when $F$ is not injective, then for some points of $N$, there may be several different vectors obtained by applying $dF$ to $X$ at different points of M.
But I find it hard to make up a concrete example of such map. I know to many professional geometricians, drawing pictures is enough to give such examples, but a beginner like me is still curious about(or suspect) the existence of such a map. This explains why I want to find an explicit, written-down example. 
Specifically, are there a smooth vector field on $\mathbb S^1$ and a smooth map from $\mathbb S^1$ to $\mathbb R^2$ taking a unit circle to a "figure eight" that make trouble at the cross of "figure eight"?(Please note that the example I want is a smooth map and a vector field defined on the whole unit circle, not just an open arc of it)
 A: Let $f(x,y)=(2xy,y)$, which is clearly smooth on $S^1$ with image figure eight, It is not injective since it sends $(1,0),(-1,0)$ to O. Compute the pushforward(I hope it is correct)
$$df_{(x,y)}(\frac{d}{d\theta})=2(2x^2-1)\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}$$
Then $df_{(1,0)}(\frac{d}{d\theta})=2\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$ and $df_{(-1,0)}(\frac{d}{d\theta})=2\frac{\partial}{\partial x}-\frac{\partial}{\partial y}$, which gives two vectors at the cross.
$f$ is a composition of a stereographic projection, a homeomorphim between an open interval and the real line and a parametrization of figure eight. I thought at the beginning it is not extendable to the whole $S^1$, but after computations I found it is.
A: Yes, take the vector field to be $\partial/\partial \theta$ and the figure eight curve to be an immersion that draws that figure. Then $dF_t(\partial/\partial \theta)$ is just $F'(t)$ (that just being the literal derivative with respect to $t$); if $t_1, t_2$ are the two points in $S^1$ sent to the bad point, then $F'(t_1) \neq F'(t_2)$, since the curve is going different directions! (Of course you could compute this by hand using whatever your parameterization of the figure eight curve is.)
The point of this figure is not to give an explicit counterexample, it's to get an idea of what could go wrong. This is an important skill you should try to develop; having to come  up with explicit counterexamples when the idea suffices can be a crippling problem.
