let "v" be some submanifold of "x" other than "z". if "v" is disjoint from "z", then it is essentially unaffected by blowing up along "z". however, if it intersects "z", then there are two distinct analogues of "v" in the blow-up formula_40. one is the proper (or strict) transform, which is the closure of formula_42; its normal bundle in formula_40 is typically different from that of "v" in "x". the other is the total transform, which incorporates some or all of "e"; it is essentially the pullback of "v" in cohomology.