# Potential Typo: Derivation On The Algebra Of Germs

This is from Lee's Introduction To Smooth Manifolds Second Edition, page 76. Should $$\Phi: T_pM \rightarrow \mathcal{D}_pM$$, not $$\Phi: \mathcal{D}_pM \rightarrow T_pM$$? Just looking at how $$\Phi$$ is defined, it looks like it is producing a derivation on a germ.

The notation can be confusing, but actually, I do not think this is a typo. $$v$$ is derivation on a germ ($$v \in \mathcal{D}_pM$$) and $$\Phi$$ transforms $$v$$ into a derivation of a smooth function ($$\Phi v \in T_pM$$) as follows: The value of the smooth function derivation $$\Phi v$$ at a smooth function $$f$$ is defined to be equal to the value of the germ derivation $$v$$ on the germ $$[f]_p$$.

• That's exactly right. – Jack Lee Oct 15 '18 at 19:19