Re: Derivations on manifolds. J. Lee, p.65

Tangent Vectors/ Derivations on mflds. Lee's Smooth
Mflds, p.65

Hi:

I am trying toshow that for M a smooth mfld., p in M
and

X in T_pM (tangent space at p) , we have that:

(a) If f is constant, then Xf=0

(b) If f(p)=g(p)=0, then X(fg)=0 .

There is a proof of a similar result for X in IR^n,
but this is for

any mfld. M .

Problem I have is this:

A derivation in M is defined as a linear map
X:C^oo(M)->IR^k ,

No, it is a linear map C^oo(M) --> R, satisfying the product rule given below

such that : for f,g in C^oo(M):

X(fg)(p)=f(p)Xg+ g(p)Xf.

BUT: At this point (Lee's book, p.65)

We have not defined Xg for g in a manifold; we have
only

defined Xg for g :U<IR^n-->IR^n .

g is an element of C^oo(M), hence a smooth map M --> R
X is a derivation at p in M, hence a map C^oo(M) --> R,
so Xg is a real number.

--Martin Wanvik
.

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