Difference between revisions of "Delta derivative"

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(→‎Properties of the $\Delta$-derivativeBohner, Martin ; Peterson, Allan. Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston, Inc., Boston, MA, 2001,p.8.)
 
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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<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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$,
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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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$,
 
$$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
 
$$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
 
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 $f^{\Delta}(t)$.
  
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==Properties of the $\Delta$-derivative<ref>Bohner, Martin ; Peterson, Allan. Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston, Inc., Boston, MA, 2001,p.8.</ref>==
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==Properties of the $\Delta$-derivative==
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[[Delta derivative of constant]]<br />
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[[Delta differentiable implies continuous]]<br />
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[[Delta derivative at right-scattered]]<br />
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[[Delta derivative at right-dense]]<br />
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[[Delta simple useful formula]]<br />
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[[Delta derivative of sum]]<br />
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[[Delta derivative of constant multiple]]<br />
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[[Delta derivative of product (1)]]<br />
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[[Delta derivative of product (2)]]<br />
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[[Delta derivative of reciprocal]]<br />
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[[Delta derivative of quotient]]<br />
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[[Delta derivative of squaring function]]<br />
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[[Delta derivative of classical polynomial]]<br />
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[[Delta derivative of reciprocal of classical polynomial]]<br />
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[[Relationship between nabla derivative and delta derivative]]<br />
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[[Relationship between delta derivative and nabla derivative]]<br />
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[[Delta mean value theorem]]<br />
  
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{{:Delta derivative implies continuous}}
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=See also=
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[[Nabla derivative]]<br />
  
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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== References ==
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<strong>Theorem:</strong> If $f$ is continuous at $t$ and $t$ is right-scattered, then
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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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$$f^{\Delta}(t) = \dfrac{f(\sigma(t))-f(t)}{\mu(t)}.$$
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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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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem:</strong> If $t$ is right-dense, then (if it exists),
 
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$$f^{\Delta}(t) = \displaystyle\lim_{s \rightarrow t}\dfrac{f(t)-f(s)}{t-s}.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem:</strong> If $f$ is differentiable at $t$, then
 
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$$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem (Sum rule):</strong>
 
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$$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Category:Definition]]
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<strong>Theorem (Constant rule):</strong> If $\alpha$ is constant with respect to $t$, then
 
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$$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem (Product rule,I):</strong> The following formula holds:
 
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$$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t)).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem (Product rule,II):</strong> The following formula holds:
 
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$$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t)).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem (Quotient rule):</strong> The following formula holds:
 
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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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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>  █
 
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</div>
 
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</div>
 
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==Interesting Examples==
 
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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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$$\sigma(t) = \left\{ \begin{array}{ll}
 
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0 &; t \in [0,1) \\
 
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1 &; t \in \{1,2,3,\ldots\}.
 
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\end{array}\right.$$
 
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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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== References ==
 
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<references/>
 

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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