- testwiki

General

Display information for equation id:math.1355.120 on revision:1355

* Page found: Dynamische Systeme und deterministisches Chaos (eq math.1355.120)

Occurrences on the following pages:

Hash: 38a42d561784fe5d5647402a6ba1c2cd

TeX (original user input):

\begin{align}
  & {{V}_{t}}=\int_{{{U}_{t}}}^{{}}{{{d}^{2f}}x}=\int_{{{U}_{t}}_{0}}^{{}}{{{d}^{2f}}{{x}_{0}}}\det D{{\Phi }_{t}}({{{\bar{x}}}_{0}})=\int_{{{U}_{t}}_{0}}^{{}}{{{d}^{2f}}{{x}_{0}}}\left[ 1+(t-{{t}_{0}})\sum\limits_{i=1}^{2f}{\frac{\partial {{F}_{i}}}{\partial {{x}_{0}}^{i}}+...} \right] \\ 
 & \sum\limits_{i=1}^{2f}{\frac{\partial {{F}_{i}}}{\partial {{x}_{0}}^{i}}={{\left( div\bar{F} \right)}_{{{{\bar{x}}}_{0}}}}} \\ 
 & {{V}_{t}}={{V}_{{{t}_{0}}}}+(t-{{t}_{0}})\int_{{{U}_{t}}_{0}}^{{}}{{{d}^{2f}}{{x}_{0}}}{{\left( div\bar{F} \right)}_{{{{\bar{x}}}_{0}}}}+O{{(t-{{t}_{0}})}^{2}} \\ 
 & \frac{d{{V}_{t}}}{dt}=\begin{matrix}
   \lim   \\
   t->{{t}_{0}}  \\
\end{matrix}\frac{{{V}_{t}}-{{V}_{{{t}_{0}}}}}{(t-{{t}_{0}})}=\int_{{{U}_{t}}_{0}}^{{}}{{{d}^{2f}}{{x}_{0}}}{{\left( div\bar{F} \right)}_{{{{\bar{x}}}_{0}}}}=0 \\ 
 & {{\left( div\bar{F} \right)}_{{{{\bar{x}}}_{0}}}}=0 \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{V}_{t}}=\int _{{U}_{t}}^{}{{{d}^{2f}}x}=\int _{{{U}_{t}}_{0}}^{}{{{d}^{2f}}{{x}_{0}}}\det D{{\Phi }_{t}}({{\bar {x}}_{0}})=\int _{{{U}_{t}}_{0}}^{}{{{d}^{2f}}{{x}_{0}}}\left[1+(t-{{t}_{0}})\sum \limits _{i=1}^{2f}{{\frac {\partial {{F}_{i}}}{\partial {{x}_{0}}^{i}}}+...}\right]\\&\sum \limits _{i=1}^{2f}{{\frac {\partial {{F}_{i}}}{\partial {{x}_{0}}^{i}}}={{\left(div{\bar {F}}\right)}_{{\bar {x}}_{0}}}}\\&{{V}_{t}}={{V}_{{t}_{0}}}+(t-{{t}_{0}})\int _{{{U}_{t}}_{0}}^{}{{{d}^{2f}}{{x}_{0}}}{{\left(div{\bar {F}}\right)}_{{\bar {x}}_{0}}}+O{{(t-{{t}_{0}})}^{2}}\\&{\frac {d{{V}_{t}}}{dt}}={\begin{matrix}\lim \\t->{{t}_{0}}\\\end{matrix}}{\frac {{{V}_{t}}-{{V}_{{t}_{0}}}}{(t-{{t}_{0}})}}=\int _{{{U}_{t}}_{0}}^{}{{{d}^{2f}}{{x}_{0}}}{{\left(div{\bar {F}}\right)}_{{\bar {x}}_{0}}}=0\\&{{\left(div{\bar {F}}\right)}_{{\bar {x}}_{0}}}=0\\\end{aligned}}

LaTeXML (experimental; uses MathML) rendering

MathML (0 B / 8 B) :

SVG image empty. Force Re-Rendering

SVG (0 B / 8 B) :

MathML (experimental; no images) rendering

MathML (8.936 KB / 806 B) :

\begin{aligned} V_{t} = \int_{U_{t}}^{} d^{2 f} x = \int_{{U_{t}}_{0}}^{} d^{2 f} x_{0} \det D Φ_{t} ({\bar{x}}_{0}) = \int_{{U_{t}}_{0}}^{} d^{2 f} x_{0} [1 + (t - t_{0}) \sum_{i = 1}^{2 f} \frac{\partial F_{i}}{\partial {x_{0}}^{i}} + . . .] \\ \sum_{i = 1}^{2 f} \frac{\partial F_{i}}{\partial {x_{0}}^{i}} = {(d i v \bar{F})}_{{\bar{x}}_{0}} \\ V_{t} = V_{t_{0}} + (t - t_{0}) \int_{{U_{t}}_{0}}^{} d^{2 f} x_{0} {(d i v \bar{F})}_{{\bar{x}}_{0}} + O {(t - t_{0})}^{2} \\ \frac{d V_{t}}{d t} = \begin{matrix} \lim \\ t - > t_{0} \end{matrix} \frac{V_{t} - V_{t_{0}}}{(t - t_{0})} = \int_{{U_{t}}_{0}}^{} d^{2 f} x_{0} {(d i v \bar{F})}_{{\bar{x}}_{0}} = 0 \\ {(d i v \bar{F})}_{{\bar{x}}_{0}} = 0 \end{aligned}

<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><msub><mi>V</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"><msub><mi>U</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></msup><mi>x</mi></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"><msub><msub><mi>U</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mi>det</mi><mo>&#x2061;</mo><mi>D</mi><msub><mi mathvariant="normal">&#x03A6;</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"><msub><msub><mi>U</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mi>t</mi><mo>&#x2212;</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></munderover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>F</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup></mrow></mrow></mfrac></mrow><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><munderover><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></munderover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>F</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup></mrow></mrow></mfrac></mrow><mo>=</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></msub></mrow></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>V</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo>=</mo><msub><mi>V</mi><mrow data-mjx-texclass="ORD"><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mi>t</mi><mo>&#x2212;</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"><msub><msub><mi>U</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></msub><mo>+</mo><mi>O</mi><msup><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>t</mi><mo>&#x2212;</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><msub><mi>V</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>lim</mi></mtd></mtr><mtr><mtd><mi>t</mi><mo>&#x2212;</mo><mo>&gt;</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mi>V</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo>&#x2212;</mo><msub><mi>V</mi><mrow data-mjx-texclass="ORD"><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>t</mi><mo>&#x2212;</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"><msub><msub><mi>U</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>f</mi></mrow></mrow></msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$V_{t}$

$U_{t}$

$f$

$x$

$U_{t} 0$

$f$

$x_{0}$

$D$

$Φ_{t}$

${\bar{x}}_{0}$

$U_{t} 0$

$f$

$x_{0}$

$t$

$t_{0}$

$i$

$f$

$F_{i}$

$x_{0}$

$i$

$i$

$f$

$F_{i}$

$x_{0}$

$i$

$d$

$i$

$v$

$\bar{F}$

${\bar{x}}_{0}$

$V_{t}$

$V_{t_{0}}$

$t$

$t_{0}$

$U_{t} 0$

$f$

$x_{0}$

$d$

$i$

$v$

$\bar{F}$

${\bar{x}}_{0}$

$O$

$t$

$t_{0}$

$d$

$V_{t}$

$d$

$t$

$t$

$t_{0}$

$V_{t}$

$V_{t_{0}}$

$t$

$t_{0}$

$U_{t} 0$

$f$

$x_{0}$

$d$

$i$

$v$

$\bar{F}$

${\bar{x}}_{0}$

$d$

$i$

$v$

$\bar{F}$

${\bar{x}}_{0}$

MathML observations

0results

no statistics present please run the maintenance script ExtractFeatures.php

0 results

General

LaTeXML (experimental; uses MathML) rendering

MathML (experimental; no images) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigation menu

General

LaTeXML (experimental; uses MathML) rendering

MathML (experimental; no images) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigation menu

Search