Difference between revisions of "Delta derivative"

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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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$,
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Let $\mathbb{T}$ be a [[time_scale | 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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$,
 
$$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
 
$$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
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We sometimes use the notation $\dfrac{\Delta}{\Delta t} f(t)$ or $\dfrac{\Delta f}{\Delta t}$ for $f^{\Delta}(t)$.
  
 
==Properties of the $\Delta$-derivative==
 
==Properties of the $\Delta$-derivative==
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*If $f$ is $\Delta$-differentiable at $t$, then $f$is [[continuity | continuous]] at $t$.
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[[Delta derivative of constant]]<br />
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*If $f$ is continuous at $t$ and $t$ is right-scattered, then
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[[Delta differentiable implies continuous]]<br />
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$$f^{\Delta}(t) = \dfrac{f(\sigma(t))-f(t)}{\mu(t)}$$
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[[Delta derivative at right-scattered]]<br />
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*If $t$ is right-dense, then (if it exists),
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[[Delta derivative at right-dense]]<br />
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$$f^{\Delta}(t) = \displaystyle\lim_{s \rightarrow t}\dfrac{f(t)-f(s)}{t-s}.$$
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[[Delta simple useful formula]]<br />
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*If $f$ is differentiable at $t$, then
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[[Delta derivative of sum]]<br />
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$$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t)$$
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[[Delta derivative of constant multiple]]<br />
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*Sum rule:
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[[Delta derivative of product (1)]]<br />
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$$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t)$$
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[[Delta derivative of product (2)]]<br />
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*Constant rule:if $\alpha$ is constant with respect to $t$, then
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[[Delta derivative of reciprocal]]<br />
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$$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t)$$
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[[Delta derivative of quotient]]<br />
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*Product Rule I
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[[Delta derivative of squaring function]]<br />
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$$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t))$$
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[[Delta derivative of classical polynomial]]<br />
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*Product Rule II
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[[Delta derivative of reciprocal of classical polynomial]]<br />
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$$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t))$$
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[[Relationship between nabla derivative and delta derivative]]<br />
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*Quotient Rule:
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[[Relationship between delta derivative and nabla derivative]]<br />
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$$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}$$
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[[Delta mean value theorem]]<br />
  
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==Interesting Examples==
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=See also=
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The jump operator $\sigma$ is not $\Delta$-differentiable on all time scales. Consider $\mathbb{T}=[0,1] \bigcup \{2,3,4,\ldots\}$, then we see
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[[Nabla derivative]]<br />
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$$\sigma(t) = \left\{ \begin{array}{ll}
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0 &; t \in [0,1) \\
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== References ==
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1 &; t \in \{1,2,3,\ldots\}.
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* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Induction on time scales|next=Delta differentiable implies continuous}}: Definition 1.10
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\end{array}\right.$$
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* {{PaperReference|Functional series on time scales|2008|Dorota Mozyrska|author2=Ewa Pawluszewicz|prev=Left dense|next=Regulated}}: Definition 2.1
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This function is clearly not continuous at $t=1$ and hence it is not $\Delta$-differentiable at $t=1$.
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[[Category:Definition]]

Latest revision as of 15:19, 21 January 2023

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 derivative of constant
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
Delta derivative of squaring function
Delta derivative of classical polynomial
Delta derivative of reciprocal of classical polynomial
Relationship between nabla derivative and delta derivative
Relationship between delta derivative and nabla derivative
Delta mean value theorem

See also

Nabla derivative

References

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