# Welcome to MWiki

## Theorems of the month

First fundamental theorem of exact differential and integral calculus for line integrals: The function $\displaystyle{ F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta } }$ where $\displaystyle{ \gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in [a, b[ \, \cap \, C }$, and choosing $\displaystyle{ \curvearrowright B \gamma(x) = \gamma(\curvearrowright D x) }$ is exactly $\displaystyle{ B }$-differentiable, and for all $\displaystyle{ x \in [a, b[ \, \cap \, C }$ and $\displaystyle{ z = \gamma(x) }$

$\displaystyle{ F' \curvearrowright B(z) = f(z). }$

Proof: $\displaystyle{ dB(F(z))=\int\limits_{t\in [d,x] \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square }$

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

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

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