Difference between revisions of "Fundamental theorems of calculus"
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− | '''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> ist mit <math>\gamma: [d, x[ \; \cap \; C \rightarrow A \subseteq | + | '''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> ist mit <math>\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>\curvearrowright B \gamma(x) = \gamma(\curvearrowright D x)</math> is exactly <math>B</math>-differentiable, and for all <math>x \in [a, b[ \; \cap \; C</math> and <math>z = \gamma(x)</math> |
<div style="text-align:center;"><math>F' \curvearrowright B(z) = f(z).</math></div> | <div style="text-align:center;"><math>F' \curvearrowright B(z) = f(z).</math></div> | ||
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<table style="width:100%"><tr><td style="vertical-align: top; padding-top: 1em;">'''Proof:'''</td><td style="text-align: center; font-size: 84%;"><math>\begin{aligned}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\end{aligned}</math></td></tr></table> | <table style="width:100%"><tr><td style="vertical-align: top; padding-top: 1em;">'''Proof:'''</td><td style="text-align: center; font-size: 84%;"><math>\begin{aligned}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\end{aligned}</math></td></tr></table> | ||
− | '''Second fundamental theorem of exact differential and integral calculus for line integrals:''' According to the conditions from above, it holds with <math>\gamma: [a, b[ \; \cap \; C \rightarrow | + | '''Second fundamental theorem of exact differential and integral calculus for line integrals:''' According to the conditions from above, it holds with <math>\gamma: [a, b[ \; \cap \; C \rightarrow \mathbb{K}</math> that |
Revision as of 06:57, 28 April 2020
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] 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]
Proof: | [math]\displaystyle{ \begin{aligned}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\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
Proof: | [math]\displaystyle{ \begin{aligned}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\end{aligned} }[/math] |