(Solved): Work through the derivation of the coefficient matrices for the linearized force equations, includ ...

Work through the derivation of the coefficient matrices for the linearized force equations, included below and discussed in Section \( 2.6 \) of the textbook, filling in all the steps. (Equation 2.6-11) \[ -\left[\begin{array}{l} \nabla_{\dot{X}} f_{1} \\ \nabla_{\dot{X}} f_{2} \\ \nabla_{\dot{X}} f_{3} \end{array}\right]_{X=X_{e}}=m\left[\begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & V_{T_{e}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & V_{T_{e}}-Z_{\dot{\alpha}} & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \] (Equation 2.6-14) \( =m\left[\begin{array}{ccccccccc}X_{V}+X_{T_{V}} \cos \left(\alpha_{e}+\alpha_{T}\right) & 0 & X_{\alpha} & 0 & -g_{D} \cos \gamma_{e} & 0 & 0 & 0 & 0 \\ 0 & Y_{\beta} & 0 & g_{D} \cos \theta_{e} & 0 & 0 & Y_{p} & 0 & Y_{r}-V_{T_{e}} \\ Z_{V}-X_{T_{V}} \sin \left(\alpha_{e}+\alpha_{T}\right) & 0 & Z_{\alpha} & 0 & -g_{D} \sin \gamma_{e} & 0 & 0 & V_{T_{e}}+Z_{q} & 0\end{array}\right] \) (Equation 2.6-16)