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Math Help - function germ and tangential space: show that derivation is zero

  1. #1
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    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.



    Is this the solution already?

    Regards
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  2. #2
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    Re: function germ and tangential space: show that derivation is zero

    Quote Originally Posted by huberscher View Post
    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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  3. #3
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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
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