Angle coordinate vector field on $S^1$ This post has stated why the angle coordinate vector field $\frac{d}{d\theta}$ is global,since we can define a global vector field that admit local defined vector filed $\frac{d}{d\theta}$
I have another question,what $\theta$ mean in the $\frac{d}{d\theta}$?Since $\theta$ only defined local,how can we use it globally?
 A: There is no formal meaning of the $\theta$ in $\frac{d}{d\theta}$. All you can do formally is to think of the symbol $\frac{d}{d\theta}$ as a single unbreakable unit. You cannot assign meaning to its parts in a way that implies its meaning as a whole.
But it does convey an informal meaning: that $\theta$ is part of an informal, and yet useful, abuse of notation, which goes like this.
There is an informal agreement in mathematics to abuse the notation of the $\theta$ symbol, by using that symbol to represent any angle coordinate defined on any proper open subset of $S^1$. We agree to do that even though there are many different proper open subsets of $S^1$, and even though there are many different angle coordinates defined on each of those subsets.
Having agreed to participate in this abuse, the informal meaning of $\frac{d}{d\theta}$ is this: given any angle coordinate defined on any proper open subset $U \subset S^1$, the restriction of the vector field $\frac{d}{d\theta}$ to $U$ is equal to the coordinate vector field of the given angle coordinate.
