shape operator) Properties of the -derivative The -derivative commutes with contraction, satisfies the product rule for any collection of indices and obeys a chain rule for surface restrictions of spatial tensors: Chain rule shows that the -derivatives of spatial "metrics" vanishes where and are covariant and contravariant metric tensors, is the Kronecker delta symbol, and and are the Levi-Civita symbols. The main article on Levi-Civita symbols describes them for Cartesian coordinate systems. The preceding rule is valid in general coordinates, where the definition of the Levi-Civita symbols must include the square root of the determinant of the covariant metric tensor . Differentiation table for the -derivative The derivative of the key surface objects leads to highly concise and attractive formulas. When applied to the covariant surface metric tensor and the contravariant metric tensor , the following identities result where and are the doubly covariant and doubly contravariant curvature tensors. These curvature tensors, as well as for the mixed curvature tensor , satisfy The shift tensor and the normal satisfy Finally, the surface Levi-Civita symbols and satisfy Time differentiation of integrals The CMS provides rules for time differentiation of volume and surface integrals. References Tensors Differential geometry Riemannian geometry Curvature (mathematics) Calculus