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 | ||
| + | $$\Lambda^n = \mathbb{T}_1 \times \mathbb{T}_2 \times \ldots \mathbb{T}_n$$ | ||
| + | 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: | ||
| + | $$\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \dfrac{\partial f(t)}{\partial_i t_i}, \dfrac{\partial f}{\Delta_i t_i}, \dfrac{\partial f}{\Delta_i t_i}(t), f^{\Delta_i}_{t_i}(t).$$ | ||
=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] | ||
Revision as of 07:48, 29 March 2015
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: $$\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \dfrac{\partial f(t)}{\partial_i t_i}, \dfrac{\partial f}{\Delta_i t_i}, \dfrac{\partial f}{\Delta_i t_i}(t), f^{\Delta_i}_{t_i}(t).$$
