Let $C$ be a $1$-dimensional compact differentiable manifold with boundary in $\mathbb{R}^{3}$.
In easy examples, it looks like we can always parametrize such a manifold with only two charts: usually one that covers almost everything and a last one to cover a missing point.
Is this always true? Is there any theorem that bounds the quantity of minimum charts for a manifold?
I ask this because I am doing a homework about work (physics) and I'm not sure how I would integrate over a manifold that has more than one chart. (I think that when there are two as above, we can just integrate using one chart because the other one is only a point so has measure zero).
Thanks.