Difference between revisions of "Delta derivative"
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==Properties of the $\Delta$-derivative== | ==Properties of the $\Delta$-derivative== | ||
[[Delta differentiable implies continuous]]<br /> | [[Delta differentiable implies continuous]]<br /> | ||
| + | [[Delta derivative at right-scattered]]<br /> | ||
| + | [[Delta derivative at right-dense]]<br /> | ||
| + | [[Delta simple useful formula]]<br /> | ||
| + | [[Delta derivative of sum]]<br /> | ||
| + | [[Delta derivative of constant multiple]]<br /> | ||
| + | [[Delta derivative of product (1)]]<br /> | ||
| + | [[Delta derivative of product (2)]]<br /> | ||
| + | [[Delta derivative of reciprocal]]<br /> | ||
| + | [[Delta derivative of quotient]]<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 05:47, 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
Delta derivative at right-scattered
Delta derivative at right-dense
Delta simple useful formula
Delta derivative of sum
Delta derivative of constant multiple
Delta derivative of product (1)
Delta derivative of product (2)
Delta derivative of reciprocal
Delta derivative of quotient
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
