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I'm just trying to understand better this way of seeing the tangent space. Given a manifold $M$, it's possible to define the tangent space as $(\mathfrak{I}/\mathfrak{I^2})^*$ , being $\mathfrak{I} = \{f \in C^\infty (M, p); f(p) = 0 \}$ and $C^\infty (M, p)$ the germ of functions on $p$. Or, for a given local ring $R$, the cotangent space is defined analogously as $\mathfrak{m}/\mathfrak{m^2}$, being $\mathfrak{m}$ the unique maximal ideal in this ring .

I've already know that $(\mathfrak{I}/\mathfrak{I^2})^*$ is isomorphic to the set of derivations in $C^\infty (M, p)$ by picking the linear functional $l_v(f) = v(f - f(m) + \mathfrak{I^2})$. Furthermore, $\mathfrak{I^2}$ is the zero because always $v(f^2) = 0$. Despite these stuffs, this definition seems very unnatural to me. Is there any intuitive way to see the tangent space via this definition?

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A tangent vector can be identified with its directional derivative, which can be thought of as a linear functional of $\mathcal{J}$. Now by product rule, you can imagine that the derivative at $p$ of two functions vanishing at $p$ should be 0, thus the functional should factor through $\mathcal{J}^2$, i.e. the functional is in $(\mathcal{J}/\mathcal{J}^2)^*$. It turns out that every such functional is a directional derivative.

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