Difference between revisions of "Delta derivative"
(Created page with "Let $\mathbb{T}$ be a time scale. Define $\mathbb{T}^{\kappa} := \mathbb{T} \setminus \sup \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. We define the del...") |
|||
| Line 4: | Line 4: | ||
\displaystyle\lim_{s \rightarrow t} \dfrac{f(s) - f(t)}{s-t} &\colon \mu(t) = 0. | \displaystyle\lim_{s \rightarrow t} \dfrac{f(s) - f(t)}{s-t} &\colon \mu(t) = 0. | ||
\end{array} \right.$$ | \end{array} \right.$$ | ||
| + | |||
| + | ==Properties of the $\Delta$-derivative== | ||
| + | *$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t)$ | ||
| + | *Sum rule:$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t)$ | ||
| + | *Constant rule:if $\alpha$ is constant with respect to $t$, then | ||
| + | $$\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t)$$ | ||
| + | *Product Rule I | ||
| + | $$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t))$$ | ||
| + | *Product Rule II | ||
| + | $$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t))$$ | ||
| + | *Quotient Rule: | ||
| + | $$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}$$ | ||
Revision as of 20:12, 19 May 2014
Let $\mathbb{T}$ be a time scale. Define $\mathbb{T}^{\kappa} := \mathbb{T} \setminus \sup \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. We define the delta-derivative of $f$ to be the function $f^{\Delta} \colon \mathbb{T}^{\kappa} \rightarrow \mathbb{R}$ by the formula $$f^{\Delta}(t) := \left\{ \begin{array}{ll} \dfrac{f(\sigma(t))-f(t)}{\mu(t)} &\colon \mu(t) > 0 \\ \displaystyle\lim_{s \rightarrow t} \dfrac{f(s) - f(t)}{s-t} &\colon \mu(t) = 0. \end{array} \right.$$
Properties of the $\Delta$-derivative
- $f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t)$
- Sum rule:$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t)$ *Constant rule:if $\alpha$ is constant with respect to $t$, then
$$\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t)$$
- Product Rule I
$$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t))$$
- Product Rule II
$$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t))$$
- Quotient Rule:
$$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}$$
