Difference between revisions of "Delta derivative"
(→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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==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>== | ==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>== | ||
| − | + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | |
| − | + | <strong>Theorem:</strong> If $f$ is $\Delta$-differentiable at $t$, then $f$ is [[continuity | continuous]] at $t$. | |
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> If $f$ is continuous at $t$ and $t$ is right-scattered, then | ||
$$f^{\Delta}(t) = \dfrac{f(\sigma(t))-f(t)}{\mu(t)}$$ | $$f^{\Delta}(t) = \dfrac{f(\sigma(t))-f(t)}{\mu(t)}$$ | ||
| − | + | <div class="mw-collapsible-content"> | |
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> If $t$ is right-dense, then (if it exists), | ||
$$f^{\Delta}(t) = \displaystyle\lim_{s \rightarrow t}\dfrac{f(t)-f(s)}{t-s}.$$ | $$f^{\Delta}(t) = \displaystyle\lim_{s \rightarrow t}\dfrac{f(t)-f(s)}{t-s}.$$ | ||
| − | + | <div class="mw-collapsible-content"> | |
| − | $$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t)$$ | + | <strong>Proof:</strong> █ |
| − | + | </div> | |
| − | $$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t)$$ | + | </div> |
| − | + | ||
| − | $$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t)$$ | + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> |
| − | + | <strong>Theorem:</strong> If $f$ is differentiable at $t$, then | |
| − | $$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t))$$ | + | $$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t).$$ |
| − | + | <div class="mw-collapsible-content"> | |
| − | $$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t))$$ | + | <strong>Proof:</strong> █ |
| − | + | </div> | |
| − | $$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}$$ | + | </div> |
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> Sum rule: | ||
| + | $$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Constant rule):</strong> If $\alpha$ is constant with respect to $t$, then | ||
| + | $$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Product rule,I):</strong> The following formula holds: | ||
| + | $$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t)).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t)).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
==Interesting Examples== | ==Interesting Examples== | ||
Revision as of 20:58, 20 October 2014
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<ref>Bohner, Martin ; Peterson, Allan. Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston, Inc., Boston, MA, 2001,p.8.</ref>
Theorem: If $f$ is $\Delta$-differentiable at $t$, then $f$ is continuous at $t$.
Proof: █
Theorem: If $f$ is continuous at $t$ and $t$ is right-scattered, then $$f^{\Delta}(t) = \dfrac{f(\sigma(t))-f(t)}{\mu(t)}$$
Proof: █
Theorem: If $t$ is right-dense, then (if it exists), $$f^{\Delta}(t) = \displaystyle\lim_{s \rightarrow t}\dfrac{f(t)-f(s)}{t-s}.$$
Proof: █
Theorem: If $f$ is differentiable at $t$, then $$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t).$$
Proof: █
Theorem: Sum rule: $$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t).$$
Proof: █
Theorem (Constant rule): If $\alpha$ is constant with respect to $t$, then $$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t).$$
Proof: █
Theorem (Product rule,I): The following formula holds: $$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t)).$$
Proof: █
Theorem: The following formula holds: $$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t)).$$
Proof: █
Theorem: The following formula holds: $$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}.$$
Proof: █
Interesting Examples
- 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
$$\sigma(t) = \left\{ \begin{array}{ll} 0 &; t \in [0,1) \\ 1 &; t \in \{1,2,3,\ldots\}. \end{array}\right.$$ This function is clearly not continuous at $t=1$ and hence it is not $\Delta$-differentiable at $t=1$.
References
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