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.

Is this the solution already?

Regards