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Theorems of the month

First fundamental theorem of exact differential and integral calculus for line integrals: The function [math]\displaystyle{ F(z)=\int\limits_{\gamma }{f(\zeta )dB\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 [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{ dB(F(z))=\int\limits_{t\in [d,x] \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square }[/math]

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


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


Proof: [math]\displaystyle{ F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[ \, \cap \, C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square }[/math]

Recommended readings

Nonstandard Mathematics