A mathematically closed form expression for the electric field between arbitrarily shaped surfaces is derived utilizing the techniques of differential geometry. This solution is then applied to the problem of Fowler-Nordheim emission from non-planar surfaces. Knowledge of the field everywhere on the emitting surface allows calculation of the current emitted. A comparison of these calculated values, with measurements made on textured polysilicon emission test structures, is made. The agreement is within 10% across eight orders of magnitude in current. The topological features of the emitting surface predicted by the computations are in agreement with those observed in scanning electron micrographs (SEM's).