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The question how to describe effectively the motion of a deformable surface in ordinary Euclidean space is discussed. Two alternative formulations are supplied, both based on the assumption that the Riemannian metric of the moving surface must appear explicitly in the system describing the motion. The motivation for this assumption is to divide the variables responsible for the evolution of the intrinsic geometry (strains) of the surface from those responsible for the evolution of its extrinsic geometry (bending). Exemplary application of these results to the large deflection/small strain class of deformations of thin shells is considered.