For analytical definitions of ${\displaystyle C}$  and ${\displaystyle {\dot {\nabla }}}$ , 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 ${\displaystyle {\textbf {V}}=V^{i}{\textbf {Z}}_{i}}$  is defined as the partial derivative

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

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 ${\displaystyle {\vec {N}}}$ .

Also, defining the shift tensor representation of the Surface's Tangent Space ${\displaystyle Z_{i}^{\alpha }={\textbf {S}}^{\alpha }\cdot {\textbf {Z}}_{i}}$  and the Tangent Velocity as ${\displaystyle V^{\alpha }=Z_{i}^{\alpha }V^{i}}$  , then the definition of the ${\displaystyle {\dot {\nabla }}}$  derivative for an invariant F reads

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

where ${\displaystyle \nabla _{\alpha }}$  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

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

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

The ${\displaystyle {\dot {\nabla }}}$ -derivative commutes with contraction, satisfies the product rule for any collection of indices

${\displaystyle {\dot {\nabla }}(S_{\alpha }^{i}T_{j}^{\beta })=T_{j}^{\beta }{\dot {\nabla }}S_{\alpha }^{i}+S_{\alpha }^{i}{\dot {\nabla }}T_{j}^{\beta }}$ 

and obeys a chain rule for surface restrictions of spatial tensors:

${\displaystyle {\dot {\nabla }}F_{k}^{j}(Z,t)={\frac {\partial F_{k}^{j}}{\partial t}}+CN^{i}\nabla _{i}F_{k}^{j}}$ 

Chain rule shows that the ${\displaystyle {\dot {\nabla }}}$ -derivatives of spatial "metrics" vanishes

${\displaystyle {\dot {\nabla }}\delta _{j}^{i}=0,{\dot {\nabla }}Z_{ij}=0,{\dot {\nabla }}Z^{ij}=0,{\dot {\nabla }}\varepsilon _{ijk}=0,{\dot {\nabla }}\varepsilon ^{ijk}=0}$ 

where ${\displaystyle Z_{ij}}$  and ${\displaystyle Z^{ij}}$  are covariant and contravariant metric tensors, ${\displaystyle \delta _{j}^{i}}$  is the Kronecker delta symbol, and ${\displaystyle \varepsilon _{ijk}}$  and ${\displaystyle \varepsilon ^{ijk}}$  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 ${\displaystyle Z_{ij}}$ .

The ${\displaystyle {\dot {\nabla }}}$  derivative of the key surface objects leads to highly concise and attractive formulas. When applied to the covariant surface metric tensor ${\displaystyle S_{\alpha \beta }}$  and the contravariant metric tensor ${\displaystyle S^{\alpha \beta }}$ , the following identities result

${\displaystyle {\begin{aligned}{\dot {\nabla }}S_{\alpha \beta }&=0\\[8pt]{\dot {\nabla }}S^{\alpha \beta }&=0\end{aligned}}}$ 

where ${\displaystyle B_{\alpha \beta }}$  and ${\displaystyle B^{\alpha \beta }}$  are the doubly covariant and doubly contravariant curvature tensors. These curvature tensors, as well as for the mixed curvature tensor ${\displaystyle B_{\beta }^{\alpha }}$ , satisfy

${\displaystyle {\begin{aligned}{\dot {\nabla }}B_{\alpha \beta }&=\nabla _{\alpha }\nabla _{\beta }C+CB_{\alpha \gamma }B_{\beta }^{\gamma }\\[8pt]{\dot {\nabla }}B_{\beta }^{\alpha }&=\nabla _{\beta }\nabla ^{\alpha }C+CB_{\gamma }^{\alpha }B_{\beta }^{\gamma }\\[8pt]{\dot {\nabla }}B^{\alpha \beta }&=\nabla ^{\alpha }\nabla ^{\beta }C+CB^{\gamma \alpha }B_{\gamma }^{\beta }\end{aligned}}}$ 

The shift tensor ${\displaystyle Z_{\alpha }^{i}}$  and the normal${\displaystyle N^{i}}$  satisfy

${\displaystyle {\begin{aligned}{\dot {\nabla }}Z_{\alpha }^{i}&=N^{i}\nabla _{\alpha }C\\[8pt]{\dot {\nabla }}N^{i}&=-Z_{\alpha }^{i}\nabla ^{\alpha }C\end{aligned}}}$ 

Finally, the surface Levi-Civita symbols ${\displaystyle \varepsilon _{\alpha \beta }}$  and ${\displaystyle \varepsilon ^{\alpha \beta }}$  satisfy

${\displaystyle {\begin{aligned}{\dot {\nabla }}\varepsilon _{\alpha \beta }&=0\\[8pt]{\dot {\nabla }}\varepsilon ^{\alpha \beta }&=0\end{aligned}}}$ 

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