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Display information for equation id:math.2045.51 on revision:2045

* Page found: Stabilität und Langzeitverhalten (eq math.2045.51)

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Stabilität und Langzeitverhalten Eq: math.2045.51

Hash: 615f848fe8fb2dd52e79172d4ae54cbc

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}}

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\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}

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Identifiers

$V_{t}$

$U_{t}$

$f$

$x$

$U_{t} 0$

$f$

$x_{0}$

$D$

$Φ_{t}$

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

$U_{t} 0$

$f$

$x_{0}$

$t$

$t_{0}$

$i$

$f$

$F_{i}$

$x_{0}$

$i$

$i$

$f$

$F_{i}$

$x_{0}$

$i$

$d$

$i$

$v$

$\bar{F}$

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

$V_{t}$

$V_{t_{0}}$

$t$

$t_{0}$

$U_{t} 0$

$f$

$x_{0}$

$d$

$i$

$v$

$\bar{F}$

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

$O$

$t$

$t_{0}$

$d$

$V_{t}$

$d$

$t$

$t$

$t_{0}$

$V_{t}$

$V_{t_{0}}$

$t$

$t_{0}$

$U_{t} 0$

$f$

$x_{0}$

$d$

$i$

$v$

$\bar{F}$

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

$d$

$i$

$v$

$\bar{F}$

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

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