Difference between revisions of "Delta derivative"
| Line 1: | Line 1: | ||
| − | Let $\mathbb{T}$ be a [[ | + | Let $\mathbb{T}$ be a [[time_scale]] and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ and let $t \in \mathbb{T}^{\kappa}$. We define the $\Delta$-derivative of $f$ at $t$ to be the number $f^{\Delta}(t)$ (if it exists) so that there exists a $\delta >0$ so that for all $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$, |
| − | $ | + | $$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$ |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | $$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|$$ | ||
==Properties of the $\Delta$-derivative== | ==Properties of the $\Delta$-derivative== | ||
Revision as of 20:57, 19 May 2014
Let $\mathbb{T}$ be a time_scale and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ and let $t \in \mathbb{T}^{\kappa}$. We define the $\Delta$-derivative of $f$ at $t$ to be the number $f^{\Delta}(t)$ (if it exists) so that there exists a $\delta >0$ so that for all $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$, $$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
Properties of the $\Delta$-derivative
- $f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t)$
- Sum rule:
$$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t)$$
- Constant rule:if $\alpha$ is constant with respect to $t$, then
$$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t)$$
- Product Rule I
$$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t))$$
- Product Rule II
$$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t))$$
- Quotient Rule:
$$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}$$
Interesting Examples
The jump operator $\sigma$ is not $\Delta$-differentiable on all time scales. Consider $\mathbb{T}=
