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{ {}^\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{ \begin{aligned}{\downarrow}F(z) &={\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} &={\uparrow}_{x}{f(\gamma (t))\tfrac{\gamma ({}^\curvearrowright t)-\gamma (t)}{{}^\curvearrowright t-t}{\downarrow}t} \\ &=f(\gamma (x)){{\gamma}^{\prime}}(x){\downarrow}x=\;\;\;\;\;\;\;\;\;\;\;\;\;f(\gamma (x))({}^\curvearrowright\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{ \begin{aligned}F(\gamma (b))-F(\gamma (a)) &={+}_{t\in G}{F({}^\curvearrowright\,\gamma (t))}-F(\gamma (t)) &={+}_{t\in G}{{{F}^{\prime}}(\gamma (t))({}^\curvearrowright\,\gamma (t)-\gamma (t))} \\ &={\uparrow}_{t\in G}{{F^{\prime}}(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t} &={\uparrow}_{\gamma }{{F_{{}^\curvearrowright }^{\prime}}(\zeta ){\downarrow}\zeta }.\square\end{aligned} }[/math] |