Difference between revisions of "Partial delta derivative"

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Let $\mathbb{T}_1,\ldots,\mathbb{T}_n$ be [[time scale|time scales]] and define
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$$\Lambda^n = \mathbb{T}_1 \times \mathbb{T}_2 \times \ldots \mathbb{T}_n$$
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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
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$$\displaystyle\lim_{\stackrel{s_i\rightarrow t_i}{s_i \neq \sigma_i(t_i)}} \dfrac{f(t_1,\ldots,t_{i-1},\sigma_i(t_i),t_{i+1},\ldots,t_n)-f(t_1,\ldots,t_n)}{\sigma_i(t_i)-s_i}$$
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and is denoted by multiple different notations:
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$$\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \dfrac{\partial f(t)}{\partial_i t_i}, \dfrac{\partial f}{\Delta_i t_i}, \dfrac{\partial f}{\Delta_i t_i}(t), f^{\Delta_i}_{t_i}(t).$$
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=See also=
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[[Delta derivative]]<br />
  
 
=References=
 
=References=
 
[http://web.mst.edu/~bohner/papers/pdots.pdf Partial differentiation on time scales]
 
[http://web.mst.edu/~bohner/papers/pdots.pdf Partial differentiation on time scales]
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* {{PaperReference|Partial dynamic equations on time scales|2006|Billy Jackson||prev=|next=}}: Definition 1
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[[Category:Definition]]

Latest revision as of 14:12, 28 January 2023

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: $$\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \dfrac{\partial f(t)}{\partial_i t_i}, \dfrac{\partial f}{\Delta_i t_i}, \dfrac{\partial f}{\Delta_i t_i}(t), f^{\Delta_i}_{t_i}(t).$$

See also

Delta derivative

References

Partial differentiation on time scales