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 ${\displaystyle {\dot {\nabla }}}$ whose original definition ^[2] was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative ${\displaystyle \nabla _{\alpha }}$ on differential manifolds in that it produces a tensor when applied to a tensor.

Suppose that ${\displaystyle \Sigma _{t}}$ is the evolution of the surface ${\displaystyle \Sigma }$ indexed by a time-like parameter ${\displaystyle t}$. The definitions of the surface velocity ${\displaystyle C}$ and the operator ${\displaystyle {\dot {\nabla }}}$ are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface ${\displaystyle \Sigma }$ in the instantaneous normal direction. The value of ${\displaystyle C}$ at a point ${\displaystyle P}$ is defined as the limit

${\displaystyle C=\lim _{h\to 0}{\frac {{\text{Distance}}(P,P^{*})}{h}}}$

where ${\displaystyle P^{*}}$ is the point on ${\displaystyle \Sigma _{t+h}}$ that lies on the straight line perpendicular to ${\displaystyle \Sigma _{t}}$ at point P. This definition is illustrated in the first geometric figure below. The velocity ${\displaystyle C}$ is a signed quantity: it is positive when ${\displaystyle {\overline {PP^{*}}}}$ points in the direction of the chosen normal, and negative otherwise. The relationship between ${\displaystyle \Sigma _{t}}$ and ${\displaystyle C}$ 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 ${\displaystyle {\dot {\nabla }}}$ for a scalar field F defined on ${\displaystyle \Sigma _{t}}$ is the rate of change in ${\displaystyle F}$ in the instantaneously normal direction:

${\displaystyle {\frac {\delta F}{\delta t}}=\lim _{h\to 0}{\frac {F(P^{*})-F(P)}{h}}}$

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 ${\displaystyle {\dot {\nabla }}}$ in terms of elementary operations from calculus and differential geometry.