Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Properties of Derivations and of Tangent Vectors

**Physics Forums | Science Articles, Homework Help, Discussion**