Normal bundle of a regular zero locus
= Normal bundle of a regular zero locus
{title2=$N_V\cong E|_V$}
The derivative of a section defining a <regular zero locus> patches intrinsically along that locus because changes of trivialization introduce derivative terms multiplied by the vanishing section. Its tangent kernel and surjectivity give $N_V\cong E|_V$. Combining this with <adjunction for a smooth submanifold> gives $K_V=(K_M\otimes\det E)|_V$.