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I am going over J.Lee's Smooth Manifolds, Chapter 3; specifically, Lemmas

3.1, 3.4, in which he states properties of derivations. Lee calls linear maps L with the

Leibniz property (i.e L(fg)(a)=f(a)L(g)+g(a)L(f) ) derivations, when these maps are

defined in a subset of R^n, and he calls these same maps tangent vectors when these

maps are defined on the tangent space of a general (abstract) manifold. So far, so good.

**Now** , what I find confusing is this: that the lemmas are cited separately,

as if the two lemmas need different proofs; I don't see why different proofs

are necessary.

The two properties cited (same properties cited in lemmas 3.1, 3.4 respectively;

a),b) below are in lemma 3.1, and a'), b') are in lemma 3.4) )

are : (f is in C^oo(M) , a is a point in R^n , p is any point on the manifold M)

a) If f is a constant function, then Lf=0

b) If f(a)=g(a) , then L(fg)(a)=0

a') If f is a constant function, then Xf=0

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

The proof for a),b) are straightforward :

a) L(c)=X(1c)=1X(c)+cX(1)=cX(1)+cX(1)

b) L(fg)(p)=f(p)L(g)+g(p)L(f)

Now, why do we need separate proofs of the same facts for a') and b').?

Thanks.