- #1
sambo
- 17
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Got the theorem, having trouble with the proof... [SOLVED]
Hi all. OK, so I am trying to prove a theorem that I have for some time been just using as-is. Long story short, it occurred to me that I needed to prove it. So, I have almost done it, but am stuck near the end. The theorem is:
Suppose [itex] \mathcal{X} [/itex] is a smooth vector field on a manifold [itex]\mathcal{M}[/itex]. Assuming that [itex]\mathcal{X}_p\neq0[/itex] at a point [itex]p\in\mathcal{M}[/itex], then there exists a coordinate neighborhood [itex]\left(\mathcal{W};w^i\right)[/itex] about [itex]p[/itex] such that
[tex]
\left.\mathcal{X}\right\vert_\mathcal{W}=\dfrac{\partial}{\partial w^1}
[/tex]
Proof: ?
Now, it's not that I have nothing for the proof, it's just that I'm stuck. As well, since there is more than one way to skin a cat, I figured it would be better to leave the proof empty, rather than potentially confuse anyone with the technique I have employed thus far.
That said, a big thanks in advance for all the help!
Hi all. OK, so I am trying to prove a theorem that I have for some time been just using as-is. Long story short, it occurred to me that I needed to prove it. So, I have almost done it, but am stuck near the end. The theorem is:
Suppose [itex] \mathcal{X} [/itex] is a smooth vector field on a manifold [itex]\mathcal{M}[/itex]. Assuming that [itex]\mathcal{X}_p\neq0[/itex] at a point [itex]p\in\mathcal{M}[/itex], then there exists a coordinate neighborhood [itex]\left(\mathcal{W};w^i\right)[/itex] about [itex]p[/itex] such that
[tex]
\left.\mathcal{X}\right\vert_\mathcal{W}=\dfrac{\partial}{\partial w^1}
[/tex]
Proof: ?
Now, it's not that I have nothing for the proof, it's just that I'm stuck. As well, since there is more than one way to skin a cat, I figured it would be better to leave the proof empty, rather than potentially confuse anyone with the technique I have employed thus far.
That said, a big thanks in advance for all the help!
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