Let *n* be fixed and for all \(\mathbb{T}_{i}\) be time scales where \(i \in I=\{1,2,\ldots,n \}\). An *n*-dimensional time scale can be defined by the Cartesian product as follows:

$$\Lambda^{n}=\mathbb{T}_{1} \times\cdots\times \mathbb{T}_{n}=\bigl\{ (t_{1},\ldots, t_{n}) \mid t_{i} \in\mathbb{T}_{i}, \forall i \in I \bigr\} . $$

For \(u \in\mathbb{T}_{i}\), the forward and backward jump operators can be defined as \(\sigma_{i}(u)=\inf\{v \in\mathbb{T}_{i} \mid v>u\}\) and \(\rho_{i}(u)=\sup\{v \in\mathbb{T} \mid v< u\}\), respectively. If \(\mathbb{T}_{i}\) has a left scattered maximum *M* and right scattered minimum *m*, then \((\mathbb{T}_{i} )^{\kappa}_{\kappa}=\mathbb{T}_{i} \backslash\{M,m\}\), \((\mathbb{T}_{i} )^{\kappa}_{\kappa}=\mathbb {T}_{i}\); otherwise, see [11].

Let \(f: \Lambda^{n} \to\mathbb{R}\) be a real-valued function. The symmetric partial derivative of *f* can be defined as

$$\mathop{\mathop{\lim_{s_{i} \to t_{i}}}_{\rho_{i}(t_{i}) \neq s_{i}}}_{\sigma_{i}(t_{i}) \neq s_{i}} \frac{f^{\sigma_{i}}(s_{i})-f(t_{1},\ldots,t_{i},\ldots ,t_{n})+f(t_{1},\ldots,2s_{i}-t_{i},\ldots,t_{n})-f^{\rho_{i}}(s_{i})}{\sigma (t_{i})+2s_{i}-2t_{i}-\rho(t_{i})} $$

existing as a finite number, and is denoted by \(f^{\Diamond_{i}}(t)\) or \(\frac{\partial f(t)}{\Diamond_{i} t_{i}}\), where \(f^{\sigma_{i}}(s_{i})=f(t_{1},\ldots,\sigma_{i}(s_{i}), \ldots,t_{n})\), \(f^{\rho _{i}}(s_{i})=f(t_{1},\ldots,\rho_{i}(s_{i}),\ldots,t_{n})\), and \(t = (t_{1},\ldots,t_{n}) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa }\times\cdots\times (\mathbb{T}_{n} )^{\kappa}_{\kappa}\).

### Definition 1

A function \(f : \Lambda^{n} \to\mathbb{R}\) is symmetric differentiable at a point \(t^{0}=(t^{0}_{1},\ldots,t^{0}_{n})\in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times\cdots\times (\mathbb{T}_{n} )^{\kappa }_{\kappa}\) if there exist numbers \(A_{1},\ldots,A_{n}\) independent of \(t=(t_{1},\ldots,t_{n})\in\Lambda^{n}\) such that for all \(t \in U_{\delta}(t^{0})\) and \(i \in\{1,\ldots,n\}\),

$$\begin{aligned} &f\bigl(t^{0}_{1},\ldots,\sigma_{i} \bigl(t^{0}_{i}\bigr),\ldots,t^{0}_{n} \bigr)-f(t_{1},\ldots ,t_{n})+f\bigl(2t^{0}_{1}-t_{1}, \ldots,2t^{0}_{i}-t_{i},\ldots,2t^{0}_{n}-t_{n} \bigr) \\ &\qquad{}-f\bigl(t^{0}_{1},\ldots,\rho_{i} \bigl(t^{0}_{i}\bigr),\ldots,t^{0}_{n} \bigr) \\ &\quad= \sum_{i=1}^{n} A_{i} \bigl[\sigma_{i}\bigl(t^{0}_{i} \bigr)+2t^{0}_{i}-2t_{i}-\rho _{i} \bigl(t^{0}_{i}\bigr)\bigr]+\sum _{i=1}^{n} \alpha_{i} \bigl[ \sigma_{i}\bigl(t^{0}_{i}\bigr)+2t^{0}_{i}-2t_{i}- \rho_{i}\bigl(t^{0}_{i}\bigr)\bigr], \end{aligned}$$

where *δ* is a sufficiently small positive number, \(U_{\delta}(t^{0})\) is the *δ*-neighborhood of \(t^{0}\), and \(\alpha_{i}=\alpha_{i}(t^{0},t)\) are defined on \(U_{\delta}(t^{0})\) such that it is equal to zero for \(t=t^{0}\) and \(\lim_{t \to t^{0}}\alpha_{i} = 0\) for all \(i \in \{1,\ldots,n\}\).

If \(\mathbb{T}_{1}=\cdots=\mathbb{T}_{n}=\mathbb{R}\), then Definition 1 coincides with the classical symmetric differentiability, see [13, 14]. To point out why we do not need a restriction such as ‘complete’ in the definition of symmetric differentiation, let us consider a one-dimensional case. If \(t_{0}\) is left dense and right scattered, *i.e.*, \(\sigma(t_{0})>t_{0}\) and \(\rho(t_{0})=t_{0}\), then

$$f\bigl(\sigma(t_{0})\bigr)-f(t)+f(2t_{0}-t)-f(t_{0})=A \bigl[\sigma(t_{0})+t_{0}-2t\bigr]+\alpha\bigl[\sigma (t_{0})+t_{0}-2t\bigr]. $$

This equation leads us to

$$A=f^{\Diamond}=\gamma(t)f^{\Delta}(t)+\bigl(1-\gamma(t) \bigr)f'_{-}(t), $$

where \(\lim_{t \to t^{0}}\alpha= 0\) and \(\gamma (t)=\lim_{t \to t_{0}} \frac{\sigma(t_{0})-t}{\sigma(t_{0})+t_{0}-2t}\) as in [10, Proposition 4.2]. For right dense and left scattered and isolated points, the *A* values become

$$A=f^{\Diamond}=\gamma(t)f'_{+}+\bigl(1-\gamma(t) \bigr)f^{\nabla}(t) $$

and

$$A=f^{\Diamond}=\gamma(t)f^{\Delta}(t)+\bigl(1-\gamma(t) \bigr)f^{\nabla}(t), $$

respectively.

Since we restrict our interest to curves and surfaces on time scales, we will consider a two-dimensional case, *i.e.*, \(n=2\), throughout the study. Interested readers can simply extend this idea to a higher-dimensional case.

### Definition 2

Let \(f : \mathbb{T}_{1} \times\mathbb{T}_{2} \to\mathbb{R}\) be a real-valued function and \((t_{0},s_{0}) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times (\mathbb{T}_{2} )^{\kappa}_{\kappa}\). For all \(\varepsilon_{1} > 0\), there is an open (relative to the topology of \(\mathbb{T}_{1} \times\mathbb{T}_{2}\)) neighborhood \(U_{1}\) of \((t_{0},s)\) such that for all \((t,s) \in U_{1}\),

$$\begin{aligned} &\bigl\vert \bigl[f\bigl(\sigma_{1}(t_{0}),s \bigr)-f(t,s)+f(2t_{0}-t,s)-f\bigl(\rho _{1}(t_{0}),s \bigr) \bigr]-f^{\Diamond_{1}} \bigl[\sigma_{1}(t_{0})+2t_{0}-2t- \rho_{1}(t_{0}) \bigr] \bigr\vert \\ &\quad\leq\varepsilon_{1} \bigl\vert \sigma_{1}(t_{0})+2t_{0}-2t- \rho _{1}(t_{0}) \bigr\vert . \end{aligned}$$

### Definition 3

Let \(f : \mathbb{T}_{1} \times\mathbb{T}_{2} \to\mathbb{R}\) be a real-valued function and \((t_{0},s_{0}) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times (\mathbb{T}_{2} )^{\kappa}_{\kappa}\). For all \(\varepsilon_{2} > 0\), there is an open (relative to the topology of \(\mathbb{T}_{1} \times\mathbb{T}_{2}\)) neighborhood \(U_{2}\) of \((t,s_{0})\) such that for all \((t,s) \in U_{2}\),

$$\begin{aligned} &\bigl\vert \bigl[f\bigl(t,\sigma_{2}(s_{0}) \bigr)-f(t,s)+f(t,2s_{0}-s)-f\bigl(t,\rho_{2}(s_{0}) \bigr) \bigr]-f^{\Diamond_{2}} \bigl[\sigma_{2}(s_{0})+2s_{0}-2s- \rho_{2}(s_{0}) \bigr] \bigr\vert \\ &\quad\leq\varepsilon_{2} \bigl\vert \sigma_{2}(s_{0})+2s_{0}-2s- \rho_{2}(s_{0}) \bigr\vert . \end{aligned}$$

Note that higher order or mixed symmetric partial derivatives of a function defined on \(\mathbb{T}_{1} \times\mathbb{T}_{2}\) can be defined in the same sense as

$$\frac{\partial^{2} f(t_{0},s_{0})}{\Diamond_{1} t^{2}}=\frac{\partial}{\Diamond _{1} t} \biggl( \frac{\partial f(t_{0},s_{0})}{\Diamond_{1} t} \biggr),\qquad \frac{\partial^{2} f(t_{0},s_{0})}{\Diamond_{2} s^{2}}=\frac{\partial}{\Diamond _{2} s} \biggl( \frac{\partial f(t_{0},s_{0})}{\Diamond_{2} s} \biggr) $$

and

$$\frac{\partial^{2} f(t_{0},s_{0})}{\Diamond_{1} t \Diamond_{2} s}=\frac{\partial }{\Diamond_{2} s} \biggl( \frac{\partial f(t_{0},s_{0})}{\Diamond_{1} t} \biggr),\qquad \frac{\partial^{2} f(t_{0},s_{0})}{\Diamond_{2} s\Diamond_{1} t}=\frac{\partial }{\Diamond_{1} t} \biggl( \frac{\partial f(t_{0},s_{0})}{\Diamond_{2} s} \biggr) . $$

### Proposition 4

*If*
\(f: \mathbb{T}_{1}\times\mathbb{T}_{2} \to\mathbb{R}\)
*is delta and nabla differentiable*, *then*
*f*
*is symmetric differentiable for each*
\((t,s) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times (\mathbb{T}_{2} )^{\kappa}_{\kappa}\)
*with*

$$\frac{\partial f(t,s)}{\Diamond_{1} t}=\gamma_{1}(t_{0}) \frac{\partial f(t_{0},s_{0})}{\Delta_{1} t}+ \bigl(1-\gamma_{1}(t_{0})\bigr)\frac{\partial f(t_{0},s_{0})}{\nabla_{1} t} $$

*and*

$$\frac{\partial f(t,s)}{\Diamond_{2} s}=\gamma_{2}(s_{0}) \frac{\partial f(t_{0},s_{0})}{\Delta_{2} s}+ \bigl(1-\gamma_{2}(s_{0})\bigr)\frac{\partial f(t_{0},s_{0})}{\nabla_{2} s}, $$

*where*

$$\gamma_{1}(t_{0})=\lim_{t \to t_{0}} \frac{\sigma _{1}(t_{0})-t}{\sigma_{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})} \quad\textit{and}\quad \gamma_{2}(s_{0})=\lim _{s \to s_{0}}\frac{\sigma _{2}(s_{0})-s}{\sigma_{1}(s_{0})+2s_{0}-2s-\rho_{1}(s_{0})}. $$

### Proof

$$\begin{aligned} \frac{\partial f}{\Diamond_{1} t} =&\lim_{(t,s)\to(t_{0},s)} \frac{f(\sigma _{1}(t_{0}),s)-f(t,s)+f(2t_{0}-t,s)-f(\rho_{1}(t_{0}),s)}{\sigma _{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})} \\ =& \lim_{(t,s)\to(t_{0},s)} \frac{\sigma_{1}(t_{0})-t}{\sigma _{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})}\frac{f(\sigma_{1}(t_{0}),s)-f(t,s)}{\sigma _{1}(t_{0})-t} \\ &{} + \lim_{(t,s)\to(t_{0},s)} \frac{2t_{0}-t-\rho_{1}(t_{0})}{\sigma _{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})}\frac{f(2t_{0}-t,s)-f(\rho _{1}(t_{0}),s)}{2t_{0}-t-\rho_{1}(t_{0})} \\ =& \lim_{(t,s)\to(t_{0},s)} \frac{\sigma_{1}(t_{0})-t}{\sigma _{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})} \frac{\partial f(t_{0},s_{0})}{\Delta_{1} t} \\ &{} + \lim_{(t,s)\to(t_{0},s)}\frac{2t_{0}-t-\rho_{1}(t_{0})}{\sigma _{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})} + \frac{\partial f(t_{0},s_{0})}{\nabla_{1} t} \\ =& \gamma_{1}(t_{0}) \frac{\partial f(t_{0},s_{0})}{\Delta_{1} t} + \bar{\gamma }_{1}(t_{0}) + \frac{\partial f(t_{0},s_{0})}{\nabla_{1} t}, \end{aligned}$$

where \(\gamma_{1}(t_{0})=\lim_{t\to t_{0}} \frac{\sigma_{1}(t_{0})-t}{\sigma _{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})}\) and \(\bar{\gamma}_{1}(t_{0})=\lim_{t\to t_{0}}\frac{2t_{0}-t-\rho_{1}(t_{0})}{\sigma_{1}(t_{0})+2t_{0}-2t-\rho_{1}(t_{0})}\) for all \(t_{0} \in\mathbb{T}_{1}\). It is straightforward that \(\gamma_{1} + \bar {\gamma}_{1} := 1\).

If \(t_{0}\) is left dense and right scattered, then \(\gamma_{1}(t_{0}) = \frac {\sigma_{1}(t_{0})-t_{0}}{\sigma_{1}(t_{0})-t_{0}}=1 \) and \(f^{\Diamond _{1}}(t_{0},s_{0})=f^{\Delta_{1}}(t_{0},s_{0})\).

If \(t_{0}\) is right dense and left scattered, then \(\sigma_{1}(t_{0})=t_{0}\) implies \(\gamma_{1}(t_{0})=0\) and therefore \(f^{\Diamond _{1}}(t_{0},s_{0})=f^{\nabla_{1}}(t_{0},s_{0})\). Also if \(t_{0}\) is dense \(\gamma_{1}(t_{0})= \lim_{t \to t_{0}}\frac {t_{0}-t}{2t_{0}-t}=\frac{1}{2}\) and \(f^{\Diamond_{1}}(t_{0},s_{0})=\frac {1}{2}f^{\Delta_{1}}(t_{0},s_{0})+\frac{1}{2}f^{\nabla_{1}}(t_{0},s_{0})\).

Moreover, if \(t_{0}\) is an isolated point, then \(\gamma_{1}(t_{0})=\frac {\sigma_{1}(t_{0})-t_{0}}{\sigma_{1}(t_{0})-\rho_{1}(t_{0})}\) and \(f^{\Diamond _{1}}(t_{0},s_{0})=\frac{\sigma_{1}(t_{0})-t_{0}}{\sigma_{1}(t_{0})-\rho_{1}(t_{0})} f^{\Delta _{1}}(t_{0},s_{0})+\frac{t_{0}-\rho_{1}(t_{0})}{\sigma_{1}(t_{0})-\rho_{1}(t_{0})}f^{\nabla _{1}}(t_{0},s_{0})\).

The same procedure can be followed to obtain a similar result \(\frac {\partial f(t,s)}{\Diamond_{2} s}=\gamma_{2}(s_{0}) \frac{\partial f(t_{0},s_{0})}{\Delta_{2} s}+(1-\gamma_{2}(s_{0}))\frac{\partial f(t_{0},s_{0})}{\nabla_{2} s}\). □