Difference between revisions of "Delta derivative"
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We sometimes use the notation $\dfrac{\Delta}{\Delta t} f(t)$ or $\dfrac{\Delta f}{\Delta t}$ for $f^{\Delta}(t)$. | 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 | + | ==Properties of the $\Delta$-derivative== |
| − | + | [[Delta derivative implies continuous]]<br /> | |
| − | + | [[Relationship between nabla derivative and delta derivative]]<br /> | |
| − | + | [[Relationship between delta derivative and nabla derivative]]<br /> | |
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== References == | == References == | ||
| − | + | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Induction on time scales|next=Delta differentiable implies continuous}}: Definition 1.10 | |
Revision as of 05:12, 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<ref>Bohner, Martin ; Peterson, Allan. Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston, Inc., Boston, MA, 2001,p.5.</ref> 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 derivative 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
