# function germ and tangential space: show that derivation is zero

• Feb 22nd 2014, 09:42 AM
huberscher
function germ and tangential space: show that derivation is zero
Dear mathematicians

I don't know if derivation is the right expression but we have the following to solve:

Let $f,g \in F_p$ where p is a point of a smooth manifold M. Furthermore assume that f(p)=g(p)=0. Now show that for all $v \in T_p M$ v(fg)=0.

According to the lecture we defined $F_p$ to be the function germ in point p. $T_p M$ was defined as the tangential vector in p that is a function $v: F_p \rightarrow \mathbb{R}$. A tangential vector v fulfills:

(1.) $v(f)=0$ if f stationary
(2.) For scalars $\alpha, \beta$ and f,g function germs we have $v(\alpha *f + \beta*g)=\alpha*v(f)+\beta*v(g)$

I try to go by contradiction. f,g are $C^{\infty}$.

v(fg)=v(f)*g(p)+f(p)*v(g)=v(f)*0+0*v(g)=0 by the derivative rule.

Regards
• Feb 22nd 2014, 10:16 AM
romsek
Re: function germ and tangential space: show that derivation is zero
Quote:

Originally Posted by huberscher
Dear mathematicians

I don't know if derivation is the right expression but we have the following to solve:

Let $$f,g \in F_p$$ where p is a point of a smooth manifold M. Furthermore assume that f(p)=g(p)=0. Now show that for all $$v \in T_p M$$ v(fg)=0.

According to the lecture we defined $$F_p$$ to be the function germ in point p. $$T_p M$$ was defined as the tangential vector in p that is a function $$v: F_p \rightarrow \mathbb{R}$$. A tangential vector v fulfills:

(1.) $$v(f)=0$$ if f stationary
(2.) For scalars $$\alpha, \beta$$ and f,g function germs we have $$v(\alpha *f + \beta*g)=\alpha*v(f)+\beta*v(g)$$

I try to go by contradiction. f,g are $$C^{\infty}$$.

v(fg)=v(f)*g(p)+f(p)*v(g)=v(f)*0+0*v(g)=0 by the derivative rule.

Regards

Alright you mean I should use the Dollar-sign to express a $\sum$ for example, right?
$v(fg)=v(f)*g(p)+f(p)*v(g)=v(f)*0+0*v(g)=0$