Folding of the cerebral cortex is a critical phase of brain development in higher mammals but the biomechanics of folding remain incompletely understood. During folding, the growth of the cortical surface is heterogeneous and anisotropic. We developed and applied a new technique to measure spatial and directional variations in surface growth from longitudinal magnetic resonance imaging (MRI) studies of a single animal or human subject. MRI provides high resolution 3D image volumes of the brain at different stages of development. Surface representations of the cerebral cortex are obtained by segmentation of these volumes. Estimation of local surface growth between two times requires establishment of a point-to-point correspondence ("registration") between surfaces measured at those times. Here we present a novel approach for the registration of two surfaces in which an energy function is minimized by solving a partial differential equation on a spherical surface. The energy function includes a strain-energy term due to distortion and an "error energy" term due to mismatch between surface features. This algorithm, implemented with the finite element method, brings surface features into approximate alignment while minimizing deformation in regions without explicit matching criteria. The method was validated by application to three simulated test cases and applied to characterize growth of the ferret cortex during folding. Cortical surfaces were created from MRI data acquired in vivo at 14 days, 21 days, and 28 days of life. Deformation gradient and Lagrangian strain tensors describe the kinematics of growth over this interval. These quantitative results illuminate the spatial, temporal, and directional patterns of growth during cortical folding.