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While reading the definition of the tangent vector, I found that the tangent vector is a linear form defined on the germs that vanish on the stationary germs. I understand the concept of germs which is the equivalence class of functions that behave identically in a local sense. For the germs to be stationary, $f\in[f]$ must satisfy $(f\circ\phi^{-1})'(\phi(p))=0$. My question is the following:

Since $p$ is a point of a manifold $M$ of dimension $m$, ($f\circ\phi^{-1}):\mathbb{R}^m\rightarrow \mathbb{R}$. If so, $(f\circ\phi^{-1})'$ includes all partial derivatives. Then, what is the meaning of $(f\circ\phi^{-1})'(\phi(p))=0$? $f$ is a constant function (constant germs)?

Any help or suggestions on the references are appreciated.

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$(f \circ \phi^{-1})'$ only has to vanish at the single point $\phi(p)$, not in a neighborhood of that point. So $f$ doesn't have to be constant; it just has to have a critical point (i.e., an extreme value or a saddle point) at $p$.

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  • $\begingroup$ Thank you for your comments. What I still don't see is the meaning of $(f\circ \phi^{-1})'$. Is that prime means the total derivative, since $f$ is a multivariable function? It should be the total derivative, shouldn't it? $\endgroup$ – D.S. Oct 28 '13 at 3:15
  • $\begingroup$ @D.S.: You are correct, it should be the total derivative. $\endgroup$ – Micah Oct 28 '13 at 3:58
  • $\begingroup$ Thank you very much. I need to confirm it because of notations without clear explanations. $\endgroup$ – D.S. Oct 30 '13 at 4:03

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