# Partial Delta Derivative

Let $\mathbb{T}_1,\ldots,\mathbb{T}_n$ be time scales and define $$\Lambda^n = \mathbb{T}_1 \times \mathbb{T}_2 \times \ldots \mathbb{T}_n$$ to be an $n$-dimensional time scale. Let $f \colon \Lambda^n \rightarrow \mathbb{R}$ be a function. The partial derivative of $f$ with respect to $t_i \in \mathbb{T}^{\kappa}_i$ is defined by the limit $$\displaystyle\lim_{\stackrel{s_i\rightarrow t_i}{s_i \neq \sigma_i(t_i)}} \dfrac{f(t_1,\ldots,t_{i-1},\sigma_i(t_i),t_{i+1},\ldots,t_n)-f(t_1,\ldots,t_n)}{\sigma_i(t_i)-s_i}$$ and is denoted by multiple different notations: $$\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \dfrac{\partial f(t)}{\partial_i t_i}, \dfrac{\partial f}{\Delta_i t_i}, \dfrac{\partial f}{\Delta_i t_i}(t), f^{\Delta_i}_{t_i}(t).$$