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Display information for equation id:math.1354.120 on revision:1354
* Page found: Dynamische Systeme und deterministisches Chaos (eq math.1354.120)
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\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}
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data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</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">∑</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>∂</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>∂</mi><msup><msub><mi>x</mi><mrow 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stretchy="false">)</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</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 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data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><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></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>t</mi><mo>−</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">∫</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 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