Uisng sequences of Voyager 1 high-resolution images of Jupiter's Great Red Spot (GRS) and White Oval BC we map the flow fields within the GRS and Oval BC. We compute relative vorticity within these features as a function of semi-major axis length and position angle in a coordinate system consisting of concentric ellipses of equal eccentricity. Both the velocity and the relative vorticity profiles are nearly identical for Oval BC and the outer portion of the GRS. Wind speeds of 110--120 m/s are observed near the outer edges of both features. Along their minor axes relative vorticity profiles reach a maximum of ~6¿10-5 s-1. This is several times greater than the ambient 1.5¿10-5 s-1 meridional shear of zonal winds at the latitudes of the GRS and Oval BC. Maximum Rossby numbers of 0.36 are computed for flows within both the GRS and the Oval BC. Generally, the Rossby numbers within these features are much lower, indicating strongly geostrophic constraints on the flow. The difference in streamline curvature within the GRS and the Oval BC is found to compensate for the difference in planetary vorticity at the respective latitudes of the features. Motions within the central region of the GRS are much slower and more random than around the spot's outer portion.