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In Loring Tu's text An Introduction to Manifolds, Exercise $2.4$ asks us to show that $D_1\circ D_2$ need not be a derivation while $D_1\circ D_2-D_2\circ D_1$ is always a derivation. My question is the following: how should I interpret $D_1\circ D_2$?

For the background, a derivation is a map of real vector spaces $$D:C_p^\infty\to\mathbb{R}$$ that is $\mathbb{R}$-linear and satisfies the Leibniz rule: $$D(fg)=(Df)g(p)+f(p)(Dg),$$ where $p$ is some element in $\mathbb{R}$.

Clearly it doesn't make sense as a composition, since the domain and codomain are different, one not even being a subset of the other, so I'm inclined to think it is a product. However, if it is a product, then the second part of the question becomes trivial by the commutativity of $\mathbb{R}$, hence it is always zero.

Am I misunderstanding something? Please help!

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up vote 3 down vote accepted

Note that the problem is talking about derivations of an algebra, not derivations at a point. The page before the problems start defines what that means. It is composition of maps $A\to A$.

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Thanks, Jonas. I didn't take as much time as I should have; the notation threw me off a bit and the "title" for the exercise made me inclined to think of products, but not for an algebra. Again, thanks! – Clayton Jan 21 '13 at 6:42

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