In some books, when discussing the relation between partial/directional derivatives and tangent vectors, one makes a generalization called a "derivation". A derivation at ##\vec{a} \in \mathbb{R}^n## is defined as a linear map ##D: C^{\infty}(\mathbb{R}^n) \to \mathbb{R}## which for ##f,g \in C^\infty(\mathbb{R}^n)## that satisfies(adsbygoogle = window.adsbygoogle || []).push({});

##D(fg) = f(a) Dg + g(a) Df.##

On the other hand, some books just stick to directional derivatives, so I wondered: what is the virtue of introducing derivations?

Can someone give an example of something that is a derivation, but not a directional derivative?

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# Derivations vs Directional derivatives

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