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Let $E\rightarrow M$ be a vector bundle with connection $\nabla$. Denote by $\Pi_{\gamma(t_{0})}^{\gamma(t_{1})}:E_{\gamma(t_{0})}\rightarrow E_{\gamma(t_{1})}$ the parallel transport map along the curve $\gamma$.

Show that for any $X\in \Gamma(TM)$ and $s\in\Gamma(E) $ $$(\nabla_{X}s)(p)=\lim_{\delta\rightarrow 0}\dfrac{\Pi_{\gamma(\delta)}^{\gamma(0)}(s(\gamma(\delta)))-s(\gamma(0))}{\delta}$$ where $\gamma$ is any curve in $M$ such that $\gamma(0)=p$ amd $\gamma'(0)=X(p)$.

So I have no idea how to do this should any tips for this?

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This shows you, that the connection is completely encoded in it's parallel transport and in this encoding, it's nothing than the regular derivative.

Let $\xi_1,...,\xi_n$ be a basis of $E_p$ and $s_1,...,s_n$ be the unique parallel sections along $\gamma$ with $s_i(0)=\xi_p$. Define $\tilde{S}=s\circ\gamma$ and let $\tilde{S}(t)=\sum\sigma^j(t)s_j(t)$. The following calculation then does the job:

enter image description here

I think you can handle the few changes of notation. Otherwise feel free to ask.

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    $\begingroup$ thanks for this, I forgot to make use of the vector space structure of $E_{p}$ and that turns out to be key. $\endgroup$ – lance wellton Feb 9 '14 at 19:47

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