For analytical definitions of and , consider the evolution of given by
where are general curvilinear space coordinates and are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains rather than . The velocity object is defined as the partial derivative
The velocity can be computed most directly by the formula
where are the covariant components of the normal vector .
Also, defining the shift tensor representation of the Surface's Tangent Space and the Tangent Velocity as , then the definition of the derivative for an invariant F reads
where is the covariant derivative on S.
For tensors, an appropriate generalization is needed. The proper definition for a representative tensor reads
where are Christoffel symbols and is the surface's appropriate temporal symbols ( is a matrix representation of the surface's curvature shape operator)
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 .
J. Hadamard, Leçons Sur La Propagation Des Ondes Et Les Équations de l'Hydrodynamique. Paris: Hermann, 1903.