Jacques Salomon Hadamard, French Mathematician, 1865–1963 CE
Suppose that is the evolution of the surface indexed by a time-like parameter . The definitions of the surface velocity and the operator are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface in the instantaneous normal direction. The value of at a point is defined as the limit
where is the point on that lies on the straight line perpendicular to at point P. This definition is illustrated in the first geometric figure below. The velocity is a signed quantity: it is positive when points in the direction of the chosen normal, and negative otherwise. The relationship between and 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.
Geometric construction of the surface velocity C
Geometric construction of the -derivative of an invariant field F
The -derivative for a scalar field F defined on is the rate of change in 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 in terms of elementary operations from calculus and differential geometry.
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 .
The definition of the -derivative for an invariantF reads
where is the shift tensor and is the covariant derivative on S.
For tensors, an appropriate generalization is needed. The proper definition for a representative tensor reads
Differentiation table for the δ/δt-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