# Properties of Differentials, Smooth Manifolds.

1. Feb 24, 2013

### BrainHurts

I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better

Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p$\in$M

Proposition: TpF : TpM → TF(p) is linear

ok I know that v$\in$TpM means that

v:C(M)→ℝ is a derivation and that TpM is a vector space.

Does this mean that the image of (av+bw) under TpF where v,w $\in$ TpM and a,b $\in$ ℝ

is aTpF(v) + bTpF(w) which means TpF is linear?

2. Feb 24, 2013

### micromass

Staff Emeritus
Yes, that's what it means.