In John Lee's discussion of integral curves, he managed to confuse me when it comes to the uniqueness of integral curves.
In Example 9.1:
With standard coordinates $(x,y)$ on the manifold $\mathbb{R}^2 $, and vector field $V = \frac{\partial}{\partial x}$ he computes the integral curves to be $\gamma(t)=(a+t,b)$ for any constants $a$ and $b$.
He then makes the following statement:
"Thus there is a unique integral curve starting at each point of the plane, and thus the images of different integral curves are either identical or disjoint."
I am confused about what he means by this statement. In particular, the curves $\gamma: (-1,1) \rightarrow \mathbb{R}^2$ and $\tilde{\gamma}:(-2,2) \rightarrow \mathbb{R}^2$ both defined by the formula $t \mapsto (a+t,b)$ are two different curves (because their domains are different) and their images are not disjoint, nor are they the same.
Later on in his proof of proposition 9.13 he makes a statement about "unique" integral curves. Based on my confusion of the previous statement, I am confused about the meaning of this.