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Let $V,W$ be vector field of $M$, a smooth manifold. Let $\sigma: T_pM\to T_p^*(M)$ be a linear map. Then whether following inequality is true??

$$\sigma(V_p(Wf))= \sigma(V_p)(Wf)\text{ for any } f\in C^\infty(M,\mathbb R)$$

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As $f \in C^\infty(M)$ and $W \in \mathfrak{X}(M)$, it is clear that $g:=Wf \in C^\infty(M)$. I guess the notation $V_p(g)$ means $dg_p(V_p)$. If that's the case then $V_p(g) \in T_{g(p)}M$. Therefore you can't evaluate $\sigma$ at $V_p(g)$ unless $g(p)=p$! –  Mercy Aug 4 '12 at 0:34
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I don't see an inequality here. –  Qiaochu Yuan Aug 4 '12 at 1:01
    
The questioner probably meant "equality." –  dc631 Aug 4 '12 at 5:02

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