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 )dB\zeta } }[/math] ist mit [math]\displaystyle{ \gamma: [d, x[ \; \cap \; C \rightarrow A \subseteq \mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow \mathbb{K}, d \in [a, b[ \; \cap \; C }[/math], and choosing [math]\displaystyle{ \curvearrowright B \gamma(x) = \gamma(\curvearrowright D x) }[/math] is exactly [math]\displaystyle{ B }[/math]-differentiable, and for all [math]\displaystyle{ x \in [a, b[ \; \cap \; C }[/math] and [math]\displaystyle{ z = \gamma(x) }[/math]

[math]\displaystyle{ F' \curvearrowright B(z) = f(z). }[/math]

Proof:

[math]\displaystyle{ \begin{aligned}{\downarrow}B(F(z))&={\uparrow}_{t\in [d,x] \; \cap \; C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t){\downarrow}Dt}\;\,\;\;-{\uparrow}_{t\in [d,x[ \; \cap \; C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t){\downarrow}Dt} \\ &={\uparrow}_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}{\downarrow}Dt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x){\downarrow}Dx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z){\downarrow}Bz.\square\end{aligned} }[/math]

Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, it holds with [math]\displaystyle{ \gamma: [a, b[ \; \cap \; C \rightarrow \mathbb{K} }[/math] that

[math]\displaystyle{ F(\gamma (b))-F(\gamma (a))={\uparrow}_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta ){\downarrow}B\zeta }. }[/math]

Proof:

[math]\displaystyle{ \begin{aligned}F(\gamma (b))-F(\gamma (a))&={+}_{t\in [a,b[ \; \cap \; C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))\;\,={+}_{t\in [a,b[ \; \cap \; C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))} \\ &={\uparrow}_{t\in [a,b[ \; \cap \; C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t){\downarrow}Dt}={\uparrow}_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta ){\downarrow}B\zeta }.\square\end{aligned} }[/math]

See also

• List of mathematical symbols
• Fundamental theorem of calculus