Fundamental theorems of calculus
First fundamental theorem of exact differential and integral calculus for line integrals: 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{ \overset{\rightharpoonup}{\gamma}(x) = \gamma(\overset{\rightharpoonup}{x}) }[/math] is exactly differentiable, and for all [math]\displaystyle{ x \in G }[/math] and [math]\displaystyle{ z = \gamma(x) }[/math]
| Proof: | [math]\displaystyle{ \begin{aligned}{\downarrow}(F(z)) &={\uparrow}_{s\in [d,x] \cap C}{f(\gamma (s)){{\gamma}^{\prime}}(s){\downarrow}s}-{\uparrow}_{s\in [d,x[ \, \cap \, C}{f(\gamma (s)){{\gamma }^{\prime}}(s){\downarrow}s} ={\uparrow}_{x}{f(\gamma (s))\tfrac{\gamma (\overset{\rightharpoonup}{s})-\gamma (s)}{\overset{\rightharpoonup}{s}-s}{\downarrow}s} \\ &=f(\gamma (x)){{\gamma}^{\prime}}(x){\downarrow}x =\,f(\gamma (x))(\overset{\rightharpoonup}{\gamma}(x)-\gamma (x))=f(z){\downarrow}z.\square\end{aligned} }[/math] |
Second fundamental theorem of exact differential and integral calculus for line integrals: 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{ ={\Large{+}}_{s\in G}{F(\overset{\rightharpoonup}{\gamma}(s))}-F(\gamma (s)) }[/math] [math]\displaystyle{ ={\Large{+}}_{s\in G}{{{F}^{\prime}}(\gamma (s))(\overset{\rightharpoonup}{\gamma}(s)-\gamma(s))} }[/math] [math]\displaystyle{ ={\uparrow}_{s\in G}{{{F}^{\prime}}(\gamma (s)){{\gamma }^{\prime}}(s){\downarrow}s} }[/math] [math]\displaystyle{ ={\uparrow}_{\gamma }{{{F}^{\prime}}(\zeta ){\downarrow}\zeta }.\square }[/math] |