Main Page

From MWiki
Jump to: navigation, search

Welcome to MWiki

Theorems of the month

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

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


Proof: [math]dB(F(z))=\int\limits_{t\in [d,x]C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[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]\gamma: [a, b[C \rightarrow {}^{(\omega)}\mathbb{K}[/math] that


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


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

Recommended readings

Relil - Religion und Lebensweg

Nonstandard Mathematics