Calculus of moving surfaces

The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative formula_1 whose original definition was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative formula_2 on differential manifolds in that it produces a tensor when applied to a tensor.
Suppose that formula_3 is the evolution of the surface formula_4 indexed by a time-like parameter formula_5. The definitions of the surface velocity formula_6 and the operator formula_1 are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface formula_4 in the instantaneous normal direction. The value of formula_6 at a point formula_10 is defined as the limit
where formula_12 is the point on formula_13 that lies on the straight line perpendicular to formula_14 at point P. This definition is illustrated in the first geometric figure below. The velocity formula_6 is a signed quantity: it is positive when formula_16 points in the direction of the chosen normal, and negative otherwise. The relationship between formula_14 and formula_6 is analogous to the relationship between location and velocity in elementary calculus: knowing either quantity allows one to construct the other by differentiation or integration.
The Tensorial Time Derivative formula_1 for a scalar field F defined on formula_14 is the rate of change in formula_21 in the instantaneously normal direction:
This definition is also illustrated in second geometric figure.
The above definitions are "geometric". In analytical settings, direct application of these definitions may not be possible. The CMS gives "analytical" definitions of C and formula_1 in terms of elementary operations from calculus and differential geometry.
Analytical definitions.
For analytical definitions of formula_6 and formula_1, consider the evolution of formula_26 given by
where formula_28 are general curvilinear space coordinates and formula_29 are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains formula_26 rather than formula_31. The velocity object formula_32 is defined as the partial derivative
The velocity formula_6 can be computed most directly by the formula
where formula_36 are the covariant components of the normal vector formula_37.
Also, defining the shift tensor representation of the Surface's Tangent Space formula_38 and the Tangent Velocity as formula_39 , then the definition of the formula_1 derivative for an invariant "F" reads
where formula_42 is the covariant derivative on S.
For "tensors", an appropriate generalization is needed. The proper definition for a representative tensor formula_43 reads
where formula_45 are Christoffel symbols and formula_46 is the surface's appropriate temporal symbols (formula_47 is a matrix representation of the surface's curvature shape operator)
Properties of the formula_1-derivative.
The formula_1-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 formula_1-derivatives of spatial "metrics" vanishes
where formula_54 and formula_55 are covariant and contravariant metric tensors, formula_56 is the Kronecker delta symbol, and formula_57 and formula_58 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 formula_54.
Differentiation table for the formula_1-derivative.
The formula_1 derivative of the key surface objects leads to highly concise and attractive formulas. When applied to the covariant surface metric tensor formula_62 and the contravariant metric tensor formula_63, the following identities result
where formula_65 and formula_66 are the doubly covariant and doubly contravariant curvature tensors. These curvature tensors, as well as for the mixed curvature tensor formula_47, satisfy
The shift tensor formula_69 and the normalformula_70 satisfy
Finally, the surface Levi-Civita symbols formula_72 and formula_73 satisfy
Time differentiation of integrals.
The CMS provides rules for time differentiation of volume and surface integrals.