Calculus of moving surfaces

The calculus of moving surfaces (CMS) [1] is an extension of the classical tensor calculus to deforming manifolds.

Central to the CMS is the tensorial time derivative

whose original definition [2] was put forth by Jacques Hadamard.

It plays the role analogous to that of the covariant derivative

indexed by a time-like parameter

The velocity C is the rate of deformation of the surface

that lies on the straight line perpendicular to

at point P. This definition is illustrated in the first geometric figure below.

is a signed quantity: it is positive when

points in the direction of the chosen normal, and negative otherwise.

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.

in the instantaneously normal direction: This definition is also illustrated in second geometric figure.

In analytical settings, direct application of these definitions may not be possible.

in terms of elementary operations from calculus and differential geometry.

are general curvilinear space coordinates and

By convention, tensor indices of function arguments are dropped.

is defined as the partial derivative The velocity

are the covariant components of the normal vector

Also, defining the shift tensor representation of the surface's tangent space

is the covariant derivative on S. For tensors, an appropriate generalization is needed.

The proper definition for a representative tensor

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

are covariant and contravariant metric tensors,

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

derivative of the key surface objects leads to highly concise and attractive formulas.

When applied to the covariant surface metric tensor

satisfy Finally, the surface Levi-Civita symbols

satisfy The CMS provides rules for time differentiation of volume and surface integrals.