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Display information for equation id:math.3053.0 on revision:3053

* Page found: Nakajima-Zwanzig-Gleichung (eq math.3053.0)

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Hash: 8150c8758587803be200344e5387ce2d

TeX (original user input):

\begin{align}
  & {{d}_{t}}\chi =L\chi  \\ 
 & \chi =\mathcal{P}\chi +\mathcal{Q}\chi  \\ 
 & {{d}_{t}}\left( \begin{matrix}
   \mathcal{P}  \\
   \mathcal{Q}  \\
\end{matrix} \right)\chi =\left( \begin{matrix}
   \mathcal{P}  \\
   \mathcal{Q}  \\
\end{matrix} \right)L\left( \begin{matrix}
   \mathcal{P}  \\
   \mathcal{Q}  \\
\end{matrix} \right)\chi +\left( \begin{matrix}
   \mathcal{P}  \\
   \mathcal{Q}  \\
\end{matrix} \right)L\left( \begin{matrix}
   \mathcal{Q}  \\
   \mathcal{P}  \\
\end{matrix} \right)\chi  \\ 
 & \Rightarrow \mathcal{Q}\chi ={{e}^{\mathcal{Q}Lt}}Q{{\chi }_{0}}+\int '{{e}^{\mathcal{Q}Lt}}\mathcal{Q}L\mathcal{P}\chi (t-{t}') \\ 
 & \Rightarrow {{\text{d}}_{t}}\mathcal{P}\chi =\mathcal{P}L\mathcal{P}\chi +\underbrace{\mathcal{P}{{e}^{\mathcal{Q}Lt}}Q{{\chi }_{0}}}_{=0}+\mathcal{P}L\int '{{e}^{\mathcal{Q}Lt}}\mathcal{Q}L\mathcal{P}\chi (t-{t}') \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{d}_{t}}\chi =L\chi \\&\chi ={\mathcal {P}}\chi +{\mathcal {Q}}\chi \\&{{d}_{t}}\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)\chi =\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)L\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)\chi +\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)L\left({\begin{matrix}{\mathcal {Q}}\\{\mathcal {P}}\\\end{matrix}}\right)\chi \\&\Rightarrow {\mathcal {Q}}\chi ={{e}^{{\mathcal {Q}}Lt}}Q{{\chi }_{0}}+\int '{{e}^{{\mathcal {Q}}Lt}}{\mathcal {Q}}L{\mathcal {P}}\chi (t-{t}')\\&\Rightarrow {{\text{d}}_{t}}{\mathcal {P}}\chi ={\mathcal {P}}L{\mathcal {P}}\chi +\underbrace {{\mathcal {P}}{{e}^{{\mathcal {Q}}Lt}}Q{{\chi }_{0}}} _{=0}+{\mathcal {P}}L\int '{{e}^{{\mathcal {Q}}Lt}}{\mathcal {Q}}L{\mathcal {P}}\chi (t-{t}')\\\end{aligned}}

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\begin{aligned} d_{t} χ = L χ \\ χ = 𝒫 χ + 𝒬 χ \\ d_{t} (\begin{matrix} 𝒫 \\ 𝒬 \end{matrix}) χ = (\begin{matrix} 𝒫 \\ 𝒬 \end{matrix}) L (\begin{matrix} 𝒫 \\ 𝒬 \end{matrix}) χ + (\begin{matrix} 𝒫 \\ 𝒬 \end{matrix}) L (\begin{matrix} 𝒬 \\ 𝒫 \end{matrix}) χ \\ \Rightarrow 𝒬 χ = e^{𝒬 L t} Q χ_{0} + \int^{'} e^{𝒬 L t} 𝒬 L 𝒫 χ (t - {t^{'}}^{'}) \\ \Rightarrow d_{t} 𝒫 χ = 𝒫 L 𝒫 χ + \underset{= 0}{\underset{⏟}{𝒫 e^{𝒬 L t} Q χ_{0}}} + 𝒫 L \int^{'} e^{𝒬 L t} 𝒬 L 𝒫 χ (t - {t^{'}}^{'}) \end{aligned}

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Identifiers

$d_{t}$

$χ$

$L$

$χ$

$χ$

$𝒫$

$χ$

$𝒬$

$χ$

$d_{t}$

$𝒫$

$𝒬$

$χ$

$𝒫$

$𝒬$

$L$

$𝒫$

$𝒬$

$χ$

$𝒫$

$𝒬$

$L$

$𝒬$

$𝒫$

$χ$

$𝒬$

$χ$

$e$

$𝒬$

$L$

$t$

$Q$

$χ_{0}$

$e$

$𝒬$

$L$

$t$

$𝒬$

$L$

$𝒫$

$χ$

$t$

$t$

$t$

$𝒫$

$χ$

$𝒫$

$L$

$𝒫$

$χ$

$𝒫$

$e$

$𝒬$

$L$

$t$

$Q$

$χ_{0}$

$𝒫$

$L$

$e$

$𝒬$

$L$

$t$

$𝒬$

$L$

$𝒫$

$χ$

$t$

$t$

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