> [!definition] > > Let $M$ be an $E$-[[Manifold|manifold]], and $\omega: C^{\text{enough}}(M, \real) \to \real$ be a [[Derivation|derivation]]. $\omega$ s a **derivation at $p$** if it satisfies > $ > \omega(fg) = f(p)\omega(g) + g(p) \omega(f) > $ > > If $M$ is a $n$-manifold, then $\omega$ interpreted with respect to a given [[Atlas|chart]] [[Derivation on Rn|represents]] a [[Directional Derivative|directional derivative]] corresponding a to a vector in its [[Concrete Tangent Space|concrete tangent space]]. The collection of all such interpretations forms an [[Tangent Space|abstract tangent vector]]. > [!theorem] > > Let $M$ be an $E$-manifold, $p \in M$, $f, g \in C^\infty(M)$, and $\omega: C^\infty(M) \to \real$ be a derivation at $p$. > 1. If $f$ is constant, then $\omega f = 0$. > 2. If $f(p) = g(p) = 0$, then $\omega (fg) = 0$. > [!definition] > > Let $X$ be a [[Manifold|manifold]]. A mapping on the [[Function on Manifold|functions]] $D: C^{p}(X) \to C^{p - 1}(X)$ is a **derivation** if it is linear over $\real$ and satisfies the product rule > $ > D(fg) = fDg + gDf > $ > for all $f, g \in C^{p}(X)$.