In the book Griffiths and Harris, they states that: $N^*_V=[-V]|_V$, here $V$ a smooth hypersurface, $N^*_V$ means the conormal bundle w.r.t. $V$.
They define the transition function of $V$ as $g_{\alpha\beta}=f_{\alpha}/f_{\beta}$. The local section of $N^*_V$ on $U_{\alpha}$given by $df_{\alpha}$.And they calculate that:
$df_{\alpha}=g_{\alpha\beta}df_{\beta}$.
Doesn't this formula means that the transition function of $N^*_V$ is $g_{\alpha\beta}$, hence it's isomorphic to $[V]$? I'm quite confused about this part, I would be appreciated if any one can help me.