Address review comments: negative sign and AdjointTranspose section

doc/math.lyx

@@ -5086,6 +5086,13 @@ reference "ex:projection"
 
 \begin_layout Subsection
 Derivative of Adjoint
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:pose3_adjoint_deriv"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -5098,7 +5105,7 @@ Consider
 \end_inset
 
 .
- The derivative is notated (see 
+ The derivative is notated (see Section 
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "sec:Derivatives-of-Actions"
@@ -5114,7 +5121,7 @@ noprefix "false"
 \begin_layout Standard
 \begin_inset Formula 
 \[
-Df_{(T,y)}(\xi,\delta y)=D_{1}f_{(T,y)}(\xi)+D_{2}f_{(T,y)}(\delta y)
+Df_{(T,\xi_{b})}(\xi,\delta\xi_{b})=D_{1}f_{(T,\xi_{b})}(\xi)+D_{2}f_{(T,\xi_{b})}(\delta\xi_{b})
 \]
 
 \end_inset
@@ -5149,11 +5156,12 @@ D_{2}f_{(T,\xi_{b})}(\xi_{b})=Ad_{T}
 
 \end_inset
 
-To compute 
+We will derive 
-\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi_{b})$
+\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi)$
 \end_inset
 
-, we'll first define 
+ using two approaches.
+ In the first, we'll define 
 \begin_inset Formula $g(T,\xi)\triangleq T\exp\hat{\xi}$
 \end_inset
 
@@ -5194,18 +5202,30 @@ Now we can use the definition of the Adjoint representation
 \begin_layout Standard
 \begin_inset Formula 
 \begin{align*}
-D_{1}f_{(T,\xi_{b})}(\xi)=D_{1}\left(Ad_{T\exp(\hat{\xi})}\hat{\xi}_{b}\right)(\xi) & =D_{1}\left(g\hat{\xi}g^{-1}\right)(\xi)\\
+D_{1}f_{(T,\xi_{b})}(\xi)=D_{1}\left(Ad_{T\exp(\hat{\xi})}\hat{\xi}_{b}\right)(\xi) & =D_{1}\left(g\hat{\xi}_{b}g^{-1}\right)(\xi)\\
- & =\left(D_{2}g_{(T,\xi)}(\xi)\right)\hat{\xi}g^{-1}(T,0)+g(T,0)\hat{\xi}\left(D_{2}g_{(T,\xi)}^{-1}(\xi)\right)\\
+ & =\left(D_{2}g_{(T,\xi)}(\xi)\right)\hat{\xi}_{b}g^{-1}(T,0)+g(T,0)\hat{\xi}_{b}\left(D_{2}g_{(T,\xi)}^{-1}(\xi)\right)\\
  & =T\hat{\xi}\hat{\xi}_{b}T^{-1}-T\hat{\xi}_{b}\hat{\xi}T^{-1}\\
  & =T\left(\hat{\xi}\hat{\xi}_{b}-\hat{\xi}_{b}\hat{\xi}\right)T^{-1}\\
- & =-Ad_{T}(ad_{\xi_{b}}\hat{\xi})\\
+ & =Ad_{T}(ad_{\hat{\xi}}\hat{\xi}_{b})\\
+ & =-Ad_{T}(ad_{\hat{\xi}_{b}}\hat{\xi})\\
 D_{1}F_{(T,\xi_{b})} & =-(Ad_{T})(ad_{\hat{\xi}_{b}})
 \end{align*}
 
 \end_inset
 
-An alternative, perhaps more intuitive way of deriving this would be to
+Where 
- use the fact that the derivative at the origin 
+\begin_inset Formula $ad_{\hat{\xi}}:\mathfrak{g}\rightarrow\mathfrak{g}$
+\end_inset
+
+ is the adjoint map of the lie algebra.
+\end_layout
+
+\begin_layout Standard
+The second, perhaps more intuitive way of deriving 
+\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi_{b})$
+\end_inset
+
+, would be to use the fact that the derivative at the origin 
 \begin_inset Formula $D_{1}Ad_{I}\hat{\xi}_{b}=ad_{\hat{\xi}_{b}}$
 \end_inset
 
@@ -5224,28 +5244,212 @@ An alternative, perhaps more intuitive way of deriving this would be to
 \begin_layout Standard
 \begin_inset Formula 
 \[
-D_{1}Ad_{T}\hat{\xi}_{b}(\xi)=D_{1}Ad_{T*I}\hat{\xi}_{b}(\xi)=Ad_{T}\left(D_{1}Ad_{I}\hat{\xi}_{b}(\xi)\right)=Ad_{T}\left(ad_{\hat{\xi}_{b}}(\xi)\right)
+D_{1}Ad_{T}\hat{\xi}_{b}(\xi)=D_{1}Ad_{T*I}\hat{\xi}_{b}(\xi)=Ad_{T}\left(D_{1}Ad_{I}\hat{\xi}_{b}(\xi)\right)=Ad_{T}\left(ad_{\hat{\xi}}(\hat{\xi}_{b})\right)=-Ad_{T}\left(ad_{\hat{\xi}_{b}}(\hat{\xi})\right)
 \]
 
 \end_inset
 
-It's difficult to apply a similar procedure to compute the derivative of
+
- 
+\end_layout
-\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$
+
+\begin_layout Subsection
+Derivative of AdjointTranspose
+\end_layout
+
+\begin_layout Standard
+The transpose of the Adjoint, 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $Ad_{T}^{T}:\mathfrak{g^{*}\rightarrow g^{*}}$
 \end_inset
 
- (note the
+, is useful as a way to change the reference frame of vectors in the dual
+ space 
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+(note the 
 \begin_inset Formula $^{*}$
 \end_inset
 
- denoting that we are now in the dual space) because 
+ denoting that we are now in the dual space)
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+.
+ To be more concrete, where
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+as 
+\begin_inset Formula $Ad_{T}\hat{\xi}_{b}$
+\end_inset
+
+ converts the 
+\emph on
+twist
+\emph default
+ 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $\xi_{b}$
+\end_inset
+
+ from the 
+\begin_inset Formula $T$
+\end_inset
+
+ frame,
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+ 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$
+\end_inset
+
+ converts the 
+\family default
+\series default
+\shape default
+\size default
+\emph on
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+wrench
+\emph default
+ 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $\xi_{b}^{*}$
+\end_inset
+
+ from the 
+\begin_inset Formula $T$
+\end_inset
+
+ frame
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+.
+ It's difficult to apply a similar derivation as in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:pose3_adjoint_deriv"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ for the derivative of 
+\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$
+\end_inset
+
+ because 
 \begin_inset Formula $Ad_{T}^{T}$
 \end_inset
 
- cannot be naturally defined as a conjugation so we resort to crunching
+ cannot be naturally defined as a conjugation, so we resort to crunching
  through the algebra.
- The details are omitted but the result is a form vaguely resembling (but
+ The details are omitted but the result is a form that vaguely resembles
- not quite) the 
+ (but does not exactly match) 
 \begin_inset Formula $ad(Ad_{T}^{T}\hat{\xi}_{b}^{*})$
 \end_inset