I generally try not to ask HW-like problems (at least in maths), but following problem has been keeping me for hours now and I wanted to ask what is wrong with the following derivation.
Statement to be shown:
Compute the flow of the vector field in plane, $V = y\frac{\partial}{\partial x} + \frac{\partial}{\partial y}.$
This problem is from Lee's Intro to smooth manifolds. I am using "Fundamental Theorem on Flows" from his book. In particular the following:
Given a smooth vector field on $M$, there exists unique maximal flow of that field $\theta : D \rightarrow M$ (field is the infinitesimal generator). Moreover, $\theta^{p}: D^{p} \rightarrow M$ is the unique maximal integral curve of field starting at $p$ for given $p \in M.$ ($\theta^{p}(t) = \theta(t, p)$ wherever defined and domain is all possible time inputs.)
Now given that, letting $\theta^{p} = \theta^{p}_{1} \times \theta^{p}_{2}$, I get $\theta^{p'}_{1}(t) \times \theta^{p'}_{2}(t) = \theta^{p}_2(t) \times 1$
Solving gives me $\theta(t, p) = (x + yt + \frac{t^2}{2}, t + y)$ where $p = (x, y)$ for initial value.
The issue is that, this equation doesn't define 1-parameter group action on the plane always, and I can't figure out what is wrong with derivation. Could you help me please? Thank you.