First of all, let me just say that the graphs that Arnold puts in his ODE book… they are highly non-trivial. And it just so happens that this problem that I was stuck on was completely solved by looking carefully at figures.
Question: Can every smooth direction field in a domain of the plane be extended to a smooth vector field?
Ans: No, if the domain is not simply connected.
We’re going to start by assigning a non-zero vector to each line and see what will go wrong.
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