Differential geometric approximation of the gradient and Hessian on a triangulated manifold

In a number of medical imaging modalities, including measurements or estimates of electrical activity on cortical or cardiac surfaces, it is often useful to estimate spatial derivatives of data on curved anatomical surfaces represented by triangulated meshes. Assuming the triangle vertices are points on a smooth manifold, we derive a method for estimating gradients and Hessians on locally 2D surfaces embedded in 3D directly in the global coordinate system. Accuracy of the method is validated through simulations on both smooth and corrugated surfaces.