Normal bundle of a smooth analytic hypersurface ($N_{Y/X}\cong \mathcal{O}(Y){|}_Y$)

For a smooth complex analytic hypersurface $Y\subset X$, its holomorphic normal bundle is naturally the restriction of its holomorphic line bundle associated to a divisor:If reduced local defining functions satisfy $f_i=g_{ij}{f}_j$, then $d{f}_i{|}_Y=g_{ij}d{f}_j{|}_Y$. Pairing a normal vector with $d{f}_i$ and using the reciprocal transition functions of $\mathcal{O}(Y)$ gives the isomorphism.