Let $M$ be a smooth manifold, $p \in M$. Let $I_p$ be the subspace of $C^{\infty}\left ( M \right )$ consisting of smooth functions that vanish at $p$, and let $I_p^2$ be the product of $I_p$ with itself. In order to understand $T_p^*M$ as the quotient $I_p/I_p^2$ I must show first that, for $f \in I_p$, these statements are equivalent:
- $f \in I_p^2$
- In any smooth local coordinates, its first order Taylor polynomial at $p$ is zero.
Now, one implication is clear, but i have some problems showing that if (2) is true, then (1) is true. I understand that i can use Hadamard's theorem (because (2) is equivalent to $df_p=0$), but that only shows me that $f$ can be expressed as a finite sum of products of elements in $I_p$ locally (in certain neighbourhood of $p$), but i need to show that such expression is true in all of $M$. I tried using the extension lemma, but i can't quite reach the answer.
Any help is appreciated.