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Display information for equation id:math.1325.226 on revision:1325

* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1325.226)

Occurrences on the following pages:

Poisson- Klammern Eq: math.1979.19

Hash: 3f711fa7c94d8abf3b30a71660d3b2b6

TeX (original user input):

\begin{align}
  & \frac{\partial f}{\partial {{x}_{i}}}=\sum\limits_{k}{\frac{\partial f}{\partial {{y}_{k}}}\frac{\partial {{y}_{k}}}{\partial {{x}_{i}}}=}\sum\limits_{k}{{{M}_{ki}}^{-1}\frac{\partial f}{\partial {{y}_{k}}}\Leftrightarrow {{{\bar{f}}}_{x}}={{\left( {{M}^{-1}} \right)}^{T}}{{{\bar{f}}}_{y}}\Leftrightarrow {{{\bar{f}}}_{x}}^{T}={{{\bar{f}}}_{y}}^{T}\left( {{M}^{-1}} \right)} \\ 
 & {{{\bar{f}}}_{x}}^{T}J{{{\bar{g}}}_{x}}={{{\bar{f}}}_{y}}^{T}\left( {{M}^{-1}} \right)J{{\left( {{M}^{-1}} \right)}^{T}}{{{\bar{g}}}_{y}}={{{\bar{f}}}_{y}}^{T}J{{{\bar{g}}}_{y}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{\frac {\partial f}{\partial {{x}_{i}}}}=\sum \limits _{k}{{\frac {\partial f}{\partial {{y}_{k}}}}{\frac {\partial {{y}_{k}}}{\partial {{x}_{i}}}}=}\sum \limits _{k}{{{M}_{ki}}^{-1}{\frac {\partial f}{\partial {{y}_{k}}}}\Leftrightarrow {{\bar {f}}_{x}}={{\left({{M}^{-1}}\right)}^{T}}{{\bar {f}}_{y}}\Leftrightarrow {{\bar {f}}_{x}}^{T}={{\bar {f}}_{y}}^{T}\left({{M}^{-1}}\right)}\\&{{\bar {f}}_{x}}^{T}J{{\bar {g}}_{x}}={{\bar {f}}_{y}}^{T}\left({{M}^{-1}}\right)J{{\left({{M}^{-1}}\right)}^{T}}{{\bar {g}}_{y}}={{\bar {f}}_{y}}^{T}J{{\bar {g}}_{y}}\\\end{aligned}}

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\begin{aligned} \frac{\partial f}{\partial x_{i}} = \sum_{k} \frac{\partial f}{\partial y_{k}} \frac{\partial y_{k}}{\partial x_{i}} = \sum_{k} {M_{k i}}^{- 1} \frac{\partial f}{\partial y_{k}} \Leftrightarrow {\bar{f}}_{x} = {(M^{- 1})}^{T} {\bar{f}}_{y} \Leftrightarrow {\bar{f}}_{x}^{T} = {\bar{f}}_{y}^{T} (M^{- 1}) \\ {\bar{f}}_{x}^{T} J {\bar{g}}_{x} = {\bar{f}}_{y}^{T} (M^{- 1}) J {(M^{- 1})}^{T} {\bar{g}}_{y} = {\bar{f}}_{y}^{T} J {\bar{g}}_{y} \end{aligned}

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Identifiers

$f$

$x_{i}$

$k$

$f$

$y_{k}$

$y_{k}$

$x_{i}$

$k$

$M_{k i}$

$f$

$y_{k}$

${\bar{f}}_{x}$

$M$

$T$

${\bar{f}}_{y}$

${\bar{f}}_{x}$

$T$

${\bar{f}}_{y}$

$T$

$M$

${\bar{f}}_{x}$

$T$

$J$

${\bar{g}}_{x}$

${\bar{f}}_{y}$

$T$

$M$

$J$

$M$

$T$

${\bar{g}}_{y}$

${\bar{f}}_{y}$

$T$

$J$

${\bar{g}}_{y}$

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