Difference between revisions of "Fundamental theorems of calculus"
Borishaase (talk | contribs) (Fundamental theorems of calculus) |
Borishaase (talk | contribs) (Fundamental theorems of calculus) |
||
| Line 1: | Line 1: | ||
| − | '''First fundamental theorem of exact differential and integral calculus for line integrals:''' The function <math>F(z)={\uparrow}_{\gamma }{f(\zeta ){\downarrow}\zeta }</math> where <math>\gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in G = [a, b[ \, \cap \, C</math>, and choosing <math>{} | + | '''First fundamental theorem of exact differential and integral calculus for line integrals:''' The function <math>F(z)={\uparrow}_{\gamma }{f(\zeta ){\downarrow}\zeta }</math> where <math>\gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in G = [a, b[ \, \cap \, C</math>, and choosing <math>\overset{\rightharpoonup}{\gamma}(x) = \gamma(\overset{\rightharpoonup}{x})</math> is exactly differentiable, and for all <math>x \in G</math> and <math>z = \gamma(x)</math> |
<div style="text-align:center;"><math>F^{\prime}(z) = f(z).</math></div> | <div style="text-align:center;"><math>F^{\prime}(z) = f(z).</math></div> | ||
| − | <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}{\downarrow}F(z) &={\uparrow}_{ | + | <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}{\downarrow}(F(z)) &={\uparrow}_{s\in [d,x] \cap C}{f(\gamma (s)){{\gamma}^{\prime}}(s){\downarrow}s}-{\uparrow}_{s\in [d,x[ \, \cap \, C}{f(\gamma (s)){{\gamma }^{\prime}}(s){\downarrow}s} ={\uparrow}_{x}{f(\gamma (s))\tfrac{\gamma (\overset{\rightharpoonup}{s})-\gamma (s)}{\overset{\rightharpoonup}{s}-s}{\downarrow}s} \\ &=f(\gamma (x)){{\gamma}^{\prime}}(x){\downarrow}x =\,f(\gamma (x))(\overset{\rightharpoonup}{\gamma}(x)-\gamma (x))=f(z){\downarrow}z.\square\end{aligned}</math></td></tr></table> |
'''Second fundamental theorem of exact differential and integral calculus for line integrals:''' Conditions above imply with <math>\gamma: G \rightarrow {}^{(\omega)}\mathbb{K}</math> that | '''Second fundamental theorem of exact differential and integral calculus for line integrals:''' Conditions above imply with <math>\gamma: G \rightarrow {}^{(\omega)}\mathbb{K}</math> that | ||
| Line 12: | Line 12: | ||
| − | <table style="width:100%"><tr><td style="vertical-align: top; padding-top: 0.4em;">'''Proof:'''</td><td style="text-align: center; font-size: 84%;"><math> | + | <table style="width:100%"><tr><td style="vertical-align: top; padding-top: 0.4em;">'''Proof:'''</td><td style="text-align: center; font-size: 84%;"><math>F(\gamma (b))-F(\gamma (a))</math> <math>={\Large{+}}_{s\in G}{F(\overset{\rightharpoonup}{\gamma}(s))}-F(\gamma (s))</math> <math>={\Large{+}}_{s\in G}{{{F}^{\prime}}(\gamma (s))(\overset{\rightharpoonup}{\gamma}(s)-\gamma(s))}</math> <math>={\uparrow}_{s\in G}{{{F}^{\prime}}(\gamma (s)){{\gamma }^{\prime}}(s){\downarrow}s}</math> <math>={\uparrow}_{\gamma }{{{F}^{\prime}}(\zeta ){\downarrow}\zeta }.\square</math></td></tr></table> |
== See also == | == See also == | ||
Latest revision as of 18:04, 1 May 2024
First fundamental theorem of exact differential and integral calculus for line integrals: The function [math]\displaystyle{ F(z)={\uparrow}_{\gamma }{f(\zeta ){\downarrow}\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 G = [a, b[ \, \cap \, C }[/math], and choosing [math]\displaystyle{ \overset{\rightharpoonup}{\gamma}(x) = \gamma(\overset{\rightharpoonup}{x}) }[/math] is exactly differentiable, and for all [math]\displaystyle{ x \in G }[/math] and [math]\displaystyle{ z = \gamma(x) }[/math]
[math]\displaystyle{ F^{\prime}(z) = f(z). }[/math]
| Proof: | [math]\displaystyle{ \begin{aligned}{\downarrow}(F(z)) &={\uparrow}_{s\in [d,x] \cap C}{f(\gamma (s)){{\gamma}^{\prime}}(s){\downarrow}s}-{\uparrow}_{s\in [d,x[ \, \cap \, C}{f(\gamma (s)){{\gamma }^{\prime}}(s){\downarrow}s} ={\uparrow}_{x}{f(\gamma (s))\tfrac{\gamma (\overset{\rightharpoonup}{s})-\gamma (s)}{\overset{\rightharpoonup}{s}-s}{\downarrow}s} \\ &=f(\gamma (x)){{\gamma}^{\prime}}(x){\downarrow}x =\,f(\gamma (x))(\overset{\rightharpoonup}{\gamma}(x)-\gamma (x))=f(z){\downarrow}z.\square\end{aligned} }[/math] |
Second fundamental theorem of exact differential and integral calculus for line integrals: Conditions above imply with [math]\displaystyle{ \gamma: G \rightarrow {}^{(\omega)}\mathbb{K} }[/math] that
[math]\displaystyle{ F(\gamma (b))-F(\gamma (a))={\uparrow}_{\gamma }{{F^{\prime}}(\zeta ){\downarrow}\zeta }. }[/math]
| Proof: | [math]\displaystyle{ F(\gamma (b))-F(\gamma (a)) }[/math] [math]\displaystyle{ ={\Large{+}}_{s\in G}{F(\overset{\rightharpoonup}{\gamma}(s))}-F(\gamma (s)) }[/math] [math]\displaystyle{ ={\Large{+}}_{s\in G}{{{F}^{\prime}}(\gamma (s))(\overset{\rightharpoonup}{\gamma}(s)-\gamma(s))} }[/math] [math]\displaystyle{ ={\uparrow}_{s\in G}{{{F}^{\prime}}(\gamma (s)){{\gamma }^{\prime}}(s){\downarrow}s} }[/math] [math]\displaystyle{ ={\uparrow}_{\gamma }{{{F}^{\prime}}(\zeta ){\downarrow}\zeta }.\square }[/math] |