Normal bundle of a regular zero locus ($N_V\cong E{|}_V$)

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 $K_V=(K_M\otimes \det E){|}_V$.