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

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

(1.) $\displaystyle v(f)=0$ if f stationary

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

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

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

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.

Is this the solution already?

Regards

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

Alright you mean I should use the Dollar-sign to express a $\sum$ for example, right?

But what do you think about my solution? Is the line

$v(fg)=v(f)*g(p)+f(p)*v(g)=v(f)*0+0*v(g)=0$

already sufficiently solving this exercise?

Regards