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Updates to the notes.

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This commit is contained in:
Dave Marchant
2014-02-12 15:56:19 -08:00
parent 434a4232d5
commit b42f523db0
1 changed files with 7 additions and 6 deletions
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notes/tem/tem.tex
+7 -6
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@@ -203,11 +203,12 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three steps
\begin{align}
\vec{p}^{(n)} = \left[
\begin{array}{c}
0 \\
\vec{p}_b^{(n)} \\
\vec{p}_e^{(n)}
\end{array}
\right] \\
\vec{p}_e^{(n)} = - \diag{\e^{(n)}} \Ace \diag{V} m
\vec{p}_b^{(n)} = 0 \\
\vec{p}_e^{(n)} = - \diag{\e^{(n)}} \Ace \diag{V} m
\end{align}
\end{subequations}
@@ -215,14 +216,14 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three steps
\begin{subequations}
\begin{align}
\dcurl \vec{y}_{e}^{(1)} + \frac{1}{\delta t} \vec{y}_{b}^{(1)} = 0 \\
\dcurl \vec{y}_{e}^{(1)} + \frac{1}{\delta t} \vec{y}_{b}^{(1)} = \vec{p}_b^{(1)} \\
\dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig \vec{y}_e^{(1)} = \vec{p}_e^{(1)}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(1)} = \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(1)} \\
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(1)} = \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(1)} + \MfMui \vec{p}_b^{(1)} \\
\vec{y}_e^{(1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig^{-1} \vec{p}_e^{(1)}
\end{align}
\end{subequations}
@@ -233,7 +234,7 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three steps
\begin{align}
\dcurl \vec{y}_{e}^{(t+1)} + \frac{1}{\delta t} \vec{y}_{b}^{(t+1)}
{\color{red}- \frac{1}{\delta t} \vec{y}_{b}^{(t)} }
= 0 \\
= \vec{p}_b^{(t+1)} \\
\dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+1)} = \vec{p}_e^{(t+1)}
\end{align}
\end{subequations}
@@ -242,7 +243,7 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three steps
\begin{align}
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} =
{\color{red} \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} }
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} \\
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \MfMui \vec{p}_b^{(t+1)} \\
\vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)}
\end{align}
\end{subequations}