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z_$bT2ZBI;%s^hfl<*#0SjPG8rbmo(r-`wn<(jWgG4>*0>qIBca{G;FZU#-7=`RvO2 z4S(i_nEGz2=?J#p`e$!Q>F@5R8-uRa&Uz}pMC(`2)5CLrt)EsuWz|P^9r>46_HW<g zv`UPBd(D>M`P*(;u72NL8XKSCxhuGSQ`C~IYVFm}ExxL){<|dW`u~WznxU1PVZXhv z-h01>?^wW<|3xjy+&Qh_B&sLTYP(U-{M*rg76R-G|4F<2e{U!_*+6o664$u{CUe?< q^|!X}kSH#g88Ub8^N)Y*)je4Kr-!dOt-x(;X=K2qs_N?R#svVbvk@c! diff --git a/doc/rcfs.tex b/doc/rcfs.tex index 3a03dc8..626f12e 100644 --- a/doc/rcfs.tex +++ b/doc/rcfs.tex @@ -2848,10 +2848,10 @@ By leveraging this matrix formulation, the curvature in \eqref{eq:curvature0} ca \end{equation} with selection matrices $\bm{S}_A=\bm{s}_{1,1} \bm{s}_{2,2}^\trsp - \bm{s}_{1,2} \bm{s}_{2,1}^\trsp$ and $\bm{S}_B=\bm{s}_{1,1} \bm{s}_{1,1}^\trsp + \bm{s}_{1,2} \bm{s}_{1,2}^\trsp$. -With this formulation, by using the derivative property $(fg)'=f'g+fg'$, the derivatives of \eqref{eq:curvature0} w.r.t $\bm{x}$ form the Jacobian +With this formulation, by using the derivative property $(fg)'=f'g+fg'$, and by observing that $\bm{S}_B$ is a symmetric matrix and that $\bm{S}_A$ is an asymmetric matrix, the derivatives of \eqref{eq:curvature0} w.r.t $\bm{x}$ form the Jacobian \begin{equation} \bm{J}(\bm{x}) = \frac{\partial \kappa(\bm{x})}{\partial\bm{x}} = - 2 \; (\bm{x}^\trsp \bm{S}_B \bm{x})^{-\frac{3}{2}} \; \bm{S}_A \bm{x} - + (\bm{x}^\trsp \bm{S}_B \bm{x})^{-\frac{3}{2}} \; (\bm{S}_A+\bm{S}_A^\trsp) \bm{x} - 3 \; (\bm{x}^\trsp \bm{S}_A \bm{x}) \; (\bm{x}^\trsp \bm{S}_B \bm{x})^{-\frac{5}{2}} \; \bm{S}_B \bm{x}. \label{eq:diff_curvature} \end{equation} -- GitLab