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Theorems of the month
First fundamental theorem of exact differential and integral calculus for LIs
The function [math]\displaystyle{ F(z)={\uparrow}_{\gamma }{f(\zeta ){\downarrow}\zeta } }[/math] where [math]\displaystyle{ \gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in G = [a, b[ \, \cap \, C }[/math], and choosing [math]\displaystyle{ {}^\curvearrowright \gamma(x) = \gamma({}^\curvearrowright x) }[/math] is exactly differentiable, and for all [math]\displaystyle{ x \in G }[/math] and [math]\displaystyle{ z = \gamma(x) }[/math]
Proof
[math]\displaystyle{ {\downarrow}F(z) }[/math] [math]\displaystyle{ ={\uparrow}_{t\in [d,x] \cap C}{f(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t}-{\uparrow}_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t} }[/math] [math]\displaystyle{ ={\uparrow}_{x}{f(\gamma (t))\tfrac{\gamma ({}^\curvearrowright t)-\gamma (t)}{{}^\curvearrowright t-t}{\downarrow}t} }[/math] [math]\displaystyle{ =f(\gamma (x)){{\gamma}^{\prime}}(x){\downarrow}x= }[/math] [math]\displaystyle{ \,f(\gamma (x))({}^\curvearrowright\gamma (x)-\gamma (x)) }[/math] [math]\displaystyle{ =f(z){\downarrow}z.\square }[/math]
Second fundamental theorem of exact differential and integral calculus for LIs
Conditions above imply with [math]\displaystyle{ \gamma: G \rightarrow {}^{(\omega)}\mathbb{K} }[/math] that
Proof
[math]\displaystyle{ F(\gamma (b))-F(\gamma (a)) }[/math] [math]\displaystyle{ ={+}_{t\in G}{F({}^\curvearrowright\,\gamma (t))}-F(\gamma (t)) }[/math] [math]\displaystyle{ ={+}_{t\in G}{{{F}^{\prime}}(\gamma (t))({}^\curvearrowright\,\gamma (t)-\gamma (t))} }[/math] [math]\displaystyle{ ={\uparrow}_{t\in G}{{F^{\prime}}(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t} }[/math] [math]\displaystyle{ ={\uparrow}_{\gamma }{{F_{{}^\curvearrowright }^{\prime}}(\zeta ){\downarrow}\zeta }.\square }[/math]
Approximation theorem
The derivatives [math]\displaystyle{ f^{(s)}(x) \in {}^{\omega}\mathbb{R} }[/math] for [math]\displaystyle{ x \in {}^{\omega}\mathbb{R} }[/math] allow computing the interpolating function [math]\displaystyle{ g(x) := {+}_{r=0}^{\acute{m}}{\chi_{]x_r, x_{\grave{r}}[}(x)((x_{\grave{r}}-x)p_r(x)+(x-x_r)p_{\grave{r}}(x))/(x_{\grave{r}}-x_r)}+{+}_{r=0}^m{\chi_{\{x_r\}}(x)p_r(x)} }[/math] for [math]\displaystyle{ m, n \in {}^{\nu}\mathbb{N} }[/math] and [math]\displaystyle{ p_r(x) := {+}_{s=0}^n{f^{(s)}(x_r){(x-x_r)}^s/s!} }[/math] in [math]\displaystyle{ \mathcal{O}(\sigma mn) }[/math] where [math]\displaystyle{ f^{(s)}(x_r) = g^{(s)}(x_r) }[/math] holds for every [math]\displaystyle{ x_r \in {}^{\omega}\mathbb{R} }[/math]. Replace in the complex case [math]\displaystyle{ {}^{\omega}\mathbb{R} }[/math] by [math]\displaystyle{ {}^{\omega}\mathbb{C} }[/math] and put [math]\displaystyle{ x = \gamma(t) \in {}^{\omega}\mathbb{C} }[/math] for the path [math]\displaystyle{ \gamma(t) }[/math] where [math]\displaystyle{ t \in {}^{\omega}\mathbb{R}.\square }[/math]