Calculus of moving surfaces

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

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

where ${\displaystyle P^{\ast }}$ is the point on ${\displaystyle {\mathrm{\Sigma }}_{t+h}}$ that lies on the straight line perpendicular to ${\displaystyle {\mathrm{\Sigma }}_t}$ 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 ${\displaystyle \overline{P{P}^{\ast }}}$ points in the direction of the chosen normal, and negative otherwise. The relationship between ${\displaystyle {\mathrm{\Sigma }}_t}$ 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 ${\displaystyle \dot{\mathrm{\nabla }}}$ for a scalar field F defined on ${\displaystyle {\mathrm{\Sigma }}_t}$ is the rate of change in ${\displaystyle F}$ in the instantaneously normal direction:

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

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

Analytical definitions

For analytical definitions of ${\displaystyle C}$ and ${\displaystyle \dot{\mathrm{\nabla }}}$, consider the evolution of ${\displaystyle S}$ given by

$${\displaystyle Z^i=Z^i\left(t,S\right)}$$

where ${\displaystyle Z^i}$ are general curvilinear space coordinates and ${\displaystyle S^{\alpha }}$ are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains ${\displaystyle S}$ rather than ${\displaystyle S^{\alpha }}$. The velocity object ${\displaystyle \mathbf{\text{V}}=V^i{\mathbf{\text{Z}}}_i}$ is defined as the partial derivative

$${\displaystyle V^i=\frac{\mathrm{\partial }{Z}^i\left(t,S\right)}{\mathrm{\partial }t}}$$

The velocity ${\displaystyle C}$ can be computed most directly by the formula

$${\displaystyle C=V^i{N}_i}$$

where ${\displaystyle N_i}$ are the covariant components of the normal vector ${\displaystyle \overrightarrow{N}}$.

Also, defining the shift tensor representation of the surface's tangent space ${\displaystyle Z_i^{\alpha }={\mathbf{\text{S}}}^{\alpha }\cdot {\mathbf{\text{Z}}}_i}$ and the tangent velocity as ${\displaystyle V^{\alpha }=Z_i^{\alpha }{V}^i}$ , then the definition of the ${\displaystyle \dot{\mathrm{\nabla }}}$ derivative for an invariant F reads

$${\displaystyle \dot{\mathrm{\nabla }}F=\frac{\mathrm{\partial }F\left(t,S\right)}{\mathrm{\partial }t}-V^{\alpha }{\mathrm{\nabla }}_{\alpha }F}$$

where ${\displaystyle {\mathrm{\nabla }}_{\alpha }}$ is the covariant derivative on S.

For tensors, an appropriate generalization is needed. The proper definition for a representative tensor ${\displaystyle T_{j\beta }^{i\alpha }}$ reads

$${\displaystyle \dot{\mathrm{\nabla }}{T}_{j\beta }^{i\alpha }=\frac{\mathrm{\partial }{T}_{j\beta }^{i\alpha }}{\mathrm{\partial }t}-V^{\eta }{\mathrm{\nabla }}_{\eta }{T}_{j\beta }^{i\alpha }+V^m{\mathrm{\Gamma }}_{mk}^i{T}_{j\beta }^{k\alpha }-V^m{\mathrm{\Gamma }}_{mj}^k{T}_{k\beta }^{i\alpha }+{\dot{\mathrm{\Gamma }}}_{\eta }^{\alpha }{T}_{j\beta }^{i\eta }-{\dot{\mathrm{\Gamma }}}_{\beta }^{\eta }{T}_{j\eta }^{i\alpha }}$$

where ${\displaystyle {\mathrm{\Gamma }}_{mj}^k}$ are Christoffel symbols and ${\displaystyle {\dot{\mathrm{\Gamma }}}_{\beta }^{\alpha }={\mathrm{\nabla }}_{\beta }{V}^{\alpha }-C{B}_{\beta }^{\alpha }}$ is the surface's appropriate temporal symbols (${\displaystyle B_{\beta }^{\alpha }}$ is a matrix representation of the surface's curvature shape operator)