Directional Delta Derivative

Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be time scales. Let $f \colon \mathbb{T}_1 \times \mathbb{T}_2 \rightarrow \mathbb{R}$. The directional delta derivative of $f$ at the point $(t^0,s^0)$ in the direction of the vector $\omega$ ("along $\omega$") is defined to be the number $$\dfrac{\partial f(t^0,s^0)}{\Delta \omega}=F^{\Delta}(0),$$ where $F(\xi)=f(t^0+\xi \omega_1, s^0 + \xi \omega_2)$ for $\xi \in \mathbb{T}$.

Properties

Theorem: Suppose the function $f$ is $\sigma_1$ completely delta differentiable at $(t^0,s^0)$. Then the directional derivative of $f$ at $(t^0,s^0)$in the direction of the vector $\omega$ exists and is expressed by $$\dfrac{\partial f(t^0,s^0)}{\Delta \omega} = \dfrac{\partial f(t^0,s^0)}{\Delta_1 t} \omega_1 + \dfrac{\partial f(\omega_1(t^0),s^0)}{\Delta_2s}\omega_2.$$

Proof: █