Difference between revisions of "Delta derivative"
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==Properties of the $\Delta$-derivative== | ==Properties of the $\Delta$-derivative== | ||
| − | [[Delta | + | [[Delta differentiable implies continuous]]<br /> |
[[Relationship between nabla derivative and delta derivative]]<br /> | [[Relationship between nabla derivative and delta derivative]]<br /> | ||
[[Relationship between delta derivative and nabla derivative]]<br /> | [[Relationship between delta derivative and nabla derivative]]<br /> | ||
Revision as of 05:18, 10 June 2016
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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$, $$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$ We sometimes use the notation $\dfrac{\Delta}{\Delta t} f(t)$ or $\dfrac{\Delta f}{\Delta t}$ for $f^{\Delta}(t)$.
Properties of the $\Delta$-derivative
Delta differentiable implies continuous
Relationship between nabla derivative and delta derivative
Relationship between delta derivative and nabla derivative
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Definition 1.10
