Open string mirror maps from Picard- Fuchs equations on relative cohomology

A method for computing the open string mirror map and superpotential for noncompact Calabi-Yaus, following the physical computations of Lerche and Mayr, is presented. It is also shown that the obvious extension of these techniques to the compact case is not consistent. As an example, the local CP^2 case is worked out in 2 ways.