Rotation Groups
Taken from a few problems from Misner, Thorne, Wheeler, "Gravitation":
Exercise 9.13: rotation groups - generators
Exercise 9.13: rotation groups - structure constants: ${c_{\alpha\beta}}^\gamma = -\epsilon_{\alpha\beta\gamma}$
Exercise 10.17: rotation groups - connection coefficients: ${\Gamma^\alpha}_{\beta\gamma} = \frac{1}{2} \epsilon_{\alpha\beta\gamma}$
Exercise 11.12: rotation groups - Riemann curvature: ${R^\alpha}_{\beta\gamma\delta} = \frac{1}{2} \delta^{\alpha\beta}_{\gamma\delta}$
Exercise 13.15: rotation groups - metric (in non-coordinate basis of generators): $g_{\alpha\beta} = \delta_{\alpha\beta}, [e_\alpha, e_\beta] = -\epsilon_{\alpha\beta\gamma} e_\gamma$
$R_x(\theta) = $ $\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & \cos\left( \theta\right) & -{\sin\left( \theta\right)} \\ 0 & \sin\left( \theta\right) & \cos\left( \theta\right)\end{matrix} \right]$
$R_y(\theta) = $ $\left[ \begin{matrix} \cos\left( \theta\right) & 0 & \sin\left( \theta\right) \\ 0 & 1 & 0 \\ -{\sin\left( \theta\right)} & 0 & \cos\left( \theta\right)\end{matrix} \right]$
$R_z(\theta) = $ $\left[ \begin{matrix} \cos\left( \theta\right) & -{\sin\left( \theta\right)} & 0 \\ \sin\left( \theta\right) & \cos\left( \theta\right) & 0 \\ 0 & 0 & 1\end{matrix} \right]$
$R_i(t) = exp(t K_i)$
$\frac{\partial}{\partial t} R_i(t) = K_i R_i(t)$
$K_i = \frac{\partial}{\partial t} R_i(t) R_i(t)^{-1}
= \frac{\partial}{\partial t} R_i(t) R_i(t)^T$
$ = \frac{\partial}{\partial t} R_i(t)|_{t=0}$
$K_x = $ $\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & -{1} \\ 0 & 1 & 0\end{matrix} \right]$
$K_y = $ $\left[ \begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -{1} & 0 & 0\end{matrix} \right]$
$K_z = $ $\left[ \begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]$
$P = R_z(\psi) R_x(\theta) R_z(\phi) = $ $\left[ \begin{matrix} {{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}} - {{{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} & -{\left({{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)} & {{\sin\left( \psi\right)}} {{\sin\left( \theta\right)}} \\ {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}}} + {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}}} & {-{{{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} & -{{{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}} \\ {{\sin\left( \theta\right)}} {{\sin\left( \phi\right)}} & {{\sin\left( \theta\right)}} {{\cos\left( \phi\right)}} & \cos\left( \theta\right)\end{matrix} \right]$
$R_n(\Phi) = $ $\left[ \begin{matrix} {\cos\left( \Phi\right)} + {{{x}^{2}} - {{{{x}^{2}}} {{\cos\left( \Phi\right)}}}} & {-{{{z}} {{\sin\left( \Phi\right)}}}} + {{{{x}} {{y}}} - {{{x}} {{y}} {{\cos\left( \Phi\right)}}}} & {{{y}} {{\sin\left( \Phi\right)}}} + {{{{x}} {{z}}} - {{{x}} {{z}} {{\cos\left( \Phi\right)}}}} \\ {{{z}} {{\sin\left( \Phi\right)}}} + {{{{x}} {{y}}} - {{{x}} {{y}} {{\cos\left( \Phi\right)}}}} & {\cos\left( \Phi\right)} + {{{y}^{2}} - {{{{y}^{2}}} {{\cos\left( \Phi\right)}}}} & {-{{{x}} {{\sin\left( \Phi\right)}}}} + {{{{y}} {{z}}} - {{{y}} {{z}} {{\cos\left( \Phi\right)}}}} \\ {-{{{y}} {{\sin\left( \Phi\right)}}}} + {{{{x}} {{z}}} - {{{x}} {{z}} {{\cos\left( \Phi\right)}}}} & {{{x}} {{\sin\left( \Phi\right)}}} + {{{{y}} {{z}}} - {{{y}} {{z}} {{\cos\left( \Phi\right)}}}} & {\cos\left( \Phi\right)} + {{{z}^{2}} - {{{{z}^{2}}} {{\cos\left( \Phi\right)}}}}\end{matrix} \right]$
$\frac{\partial}{\partial x} R_n(\Phi) = $ $\left[ \begin{matrix} {{2}} {{x}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & {{y}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & {{z}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} \\ {{y}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & 0 & -{\sin\left( \Phi\right)} \\ {{z}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & \sin\left( \Phi\right) & 0\end{matrix} \right]$
$\frac{\partial}{\partial y} R_n(\Phi) = $ $\left[ \begin{matrix} 0 & {{x}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & \sin\left( \Phi\right) \\ {{x}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & {{2}} {{y}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & {{z}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} \\ -{\sin\left( \Phi\right)} & {{z}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & 0\end{matrix} \right]$
$\frac{\partial}{\partial z} R_n(\Phi) = $ $\left[ \begin{matrix} 0 & -{\sin\left( \Phi\right)} & {{x}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} \\ \sin\left( \Phi\right) & 0 & {{y}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} \\ {{x}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & {{y}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}} & {{2}} {{z}} {{\left({{1} - {\cos\left( \Phi\right)}}\right)}}\end{matrix} \right]$
${\frac{\partial P}{\partial \psi}} = {\left[ \begin{matrix} -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}}\right)} & {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}} & {{\sin\left( \theta\right)}} {{\cos\left( \psi\right)}} \\ {{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}} & -{\left({{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)} & {{\sin\left( \theta\right)}} {{\sin\left( \psi\right)}} \\ 0 & 0 & 0\end{matrix} \right]}$
${\frac{\partial P}{\partial \theta}} = {\left[ \begin{matrix} {{\sin\left( \theta\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}} & {{\sin\left( \theta\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} & {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}} \\ -{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}} & -{{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}} & -{{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}} \\ {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}} & {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}} & -{\sin\left( \theta\right)}\end{matrix} \right]}$
${\frac{\partial P}{\partial \phi}} = {\left[ \begin{matrix} -{\left({{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)} & {-{{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} + {{{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} & 0 \\ {-{{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} & -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}}\right)} & 0 \\ {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}} & -{{{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}} & 0\end{matrix} \right]}$
$\Phi = acos((tr P - 1) / 2)$
$\frac{\partial \Phi}{\partial t} = -\frac{1}{2 \sqrt{1 - ((tr P - 1) / 2)^2}} \frac{\partial tr P}{\partial t}$
$n_i = - \frac{1}{2 sin \Phi} \epsilon_{ijk} P_{jk}$
$n_x = (P_{32} - P_{23}) / (2 sin \Phi)$
$n_y = (P_{13} - P_{31}) / (2 sin \Phi)$
$n_z = (P_{21} - P_{12}) / (2 sin \Phi)$
$\frac{\partial n_i}{\partial \psi} = \frac{\partial}{\partial \psi} (-\frac{1}{2 sin\Phi} \epsilon_{ijk} P_{jk})$
$= -\frac{1}{2} \epsilon_{ijk} (-\frac{cos(\Phi)}{(sin\Phi)^2} \frac{\partial \Phi}{\partial \psi} P_{jk} + \frac{1}{sin\Phi} \frac{\partial P_{jk}}{\partial \psi} )$
$= -\frac{1}{2} \epsilon_{ijk} (\frac{cos(\Phi)}{(sin\Phi)^2} \frac{1}{2 \sqrt{1 - ((tr P - 1) / 2)^2}} \frac{\partial tr P}{\partial \psi} P_{jk} + \frac{1}{sin\Phi} \frac{\partial P_{jk}}{\partial \psi} )$
$= -\frac{1}{2} \epsilon_{ijk} (\frac{tr P - 1}{4 (sin\Phi)^3} \frac{\partial tr P}{\partial \psi} P_{jk} + \frac{1}{sin\Phi} \frac{\partial P_{jk}}{\partial \psi} )$
$tr P = $ ${{{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} - {{{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {\cos\left( \theta\right)}$
${\frac{\partial trP}{\partial \psi}} = {-{\left({{{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)}}$
${\frac{\partial trP}{\partial \theta}} = {{{\sin\left( \theta\right)}} {{\left({{{-{1}} - {{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} + {{{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}}}\right)}}}$
${\frac{\partial trP}{\partial \phi}} = {-{\left({{{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)}}$
${\cos\left( \Phi\right)} = {{\frac{1}{2}}{\left({{{{{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} - {{{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {\cos\left( \theta\right)}} - {1}}\right)}}$
${\cos\left( \Phi\right)} = {{\frac{1}{2}}{\left({{-{1}} + {{{{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}} - {{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \phi\right)}}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {\cos\left( \theta\right)}}\right)}}$
$sin(\Phi) = $ $\sqrt{{1} - {{\left({{\frac{1}{2}}{\left({{-{1}} + {{{{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}} - {{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \phi\right)}}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {\cos\left( \theta\right)}}\right)}}\right)}^{2}}}$
$sin(\Phi) = $ ${\frac{1}{2}} {\sqrt{{2} + {{\cos\left( \phi\right)}^{2}} + {{{\cos\left( \psi\right)}^{2}} - {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \phi\right)}^{2}}}}} + {{{4}} {{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{2}} {{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{\cos\left( \theta\right)}}} + {{{{2}} {{{\cos\left( \phi\right)}^{2}}} {{\cos\left( \theta\right)}}} - {{{4}} {{{\cos\left( \phi\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}} {{\cos\left( \theta\right)}}}} + {{{{\cos\left( \phi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{2}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \phi\right)}^{2}}}}} + {{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \phi\right)}}} - {{{2}} {{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}} + {{{{2}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} - {{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}}}}}$
$e_x(P) = \frac{\partial}{\partial t} R_x(t) |_{t=0} \cdot P = K_x \cdot P = $ $\left[ \begin{matrix} 0 & 0 & 0 \\ -{\left({{{{\sin\left( 0\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} + {{{\cos\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}}}\right)} & {{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} - {{{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {{{\sin\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \psi\right)}}} - {{{\cos\left( 0\right)}} {{\cos\left( \theta\right)}}} \\ {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}} + {{{{\cos\left( 0\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & {-{{{\cos\left( 0\right)}} {{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}}} + {{{{\cos\left( 0\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}} - {{{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & -{\left({{{{\cos\left( \psi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}} + {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}}}}\right)}\end{matrix} \right]$
$e_y(P) = \frac{\partial}{\partial t} R_y(t) |_{t=0} \cdot P = K_y \cdot P = $ $\left[ \begin{matrix} {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( 0\right)}}}} + {{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} + {{{\cos\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( 0\right)}}} + {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {-{{{\sin\left( \psi\right)}} {{\sin\left( \theta\right)}} {{\sin\left( 0\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}}} \\ 0 & 0 & 0 \\ {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( 0\right)}}}} + {{{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( 0\right)}}} + {{{{\cos\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} - {{{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}} + {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}}}}\right)}\end{matrix} \right]$
$e_z(P) = \frac{\partial}{\partial t} R_z(t) |_{t=0} \cdot P = K_z \cdot P = $ $\left[ \begin{matrix} {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( 0\right)}}}} + {{{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}}} - {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( 0\right)}}} + {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} & {{\sin\left( \theta\right)}} {{\left({{{{\cos\left( \psi\right)}} {{\cos\left( 0\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}}}\right)}} \\ {{{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( 0\right)}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}}}} - {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\sin\left( 0\right)}}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} & {{-{{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( 0\right)}}}} - {{{\cos\left( \phi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}} + {{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} & {{\sin\left( \theta\right)}} {{\left({{{{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}} + {{{\cos\left( \psi\right)}} {{\sin\left( 0\right)}}}}\right)}} \\ 0 & 0 & 0\end{matrix} \right]$
$\frac{\partial}{\partial x} P =e_{x} P =\frac{\partial \psi}{\partial x} \frac{\partial}{\partial \psi} P+\frac{\partial \theta}{\partial x} \frac{\partial}{\partial \theta} P+\frac{\partial \phi}{\partial x} \frac{\partial}{\partial \phi} P={e_x}^{\psi} \frac{\partial}{\partial \psi} P+{e_x}^{\theta} \frac{\partial}{\partial \theta} P+{e_x}^{\phi} \frac{\partial}{\partial \phi} P= K_x \cdot P$
$\frac{\partial}{\partial y} P =e_{y} P =\frac{\partial \psi}{\partial y} \frac{\partial}{\partial \psi} P+\frac{\partial \theta}{\partial y} \frac{\partial}{\partial \theta} P+\frac{\partial \phi}{\partial y} \frac{\partial}{\partial \phi} P={e_y}^{\psi} \frac{\partial}{\partial \psi} P+{e_y}^{\theta} \frac{\partial}{\partial \theta} P+{e_y}^{\phi} \frac{\partial}{\partial \phi} P= K_y \cdot P$
$\frac{\partial}{\partial z} P =e_{z} P =\frac{\partial \psi}{\partial z} \frac{\partial}{\partial \psi} P+\frac{\partial \theta}{\partial z} \frac{\partial}{\partial \theta} P+\frac{\partial \phi}{\partial z} \frac{\partial}{\partial \phi} P={e_z}^{\psi} \frac{\partial}{\partial \psi} P+{e_z}^{\theta} \frac{\partial}{\partial \theta} P+{e_z}^{\phi} \frac{\partial}{\partial \phi} P= K_z \cdot P$
$\left[\matrix{
\frac{\partial P}{\partial \psi} |
\frac{\partial P}{\partial \theta} |
\frac{\partial P}{\partial \phi}
}\right] \left[\matrix{
e_x | e_y | e_z
}\right] = \left[\matrix{
K_x P | K_y P | K_z P
}\right]$
determining ${e_a}^I$ via linear combination of $K_i P$:
${ {\left[ \begin{matrix} -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}}\right)} & {{\sin\left( \theta\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}} & -{\left({{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)} \\ {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}} & {{\sin\left( \theta\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} & {-{{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} + {{{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} \\ {{\sin\left( \theta\right)}} {{\cos\left( \psi\right)}} & {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}} & 0 \\ {{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}} & -{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}} & {-{{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} \\ -{\left({{{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} + {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}}\right)} & -{{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}} & -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}}\right)} \\ {{\sin\left( \theta\right)}} {{\sin\left( \psi\right)}} & -{{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}} & 0 \\ 0 & {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}} & {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}} \\ 0 & {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}} & -{{{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}} \\ 0 & -{\sin\left( \theta\right)} & 0\end{matrix} \right]} {\left[ \begin{matrix} {{e_x}^{\psi}} & {{e_y}^{\psi}} & {{e_z}^{\psi}} \\ {{e_x}^{\theta}} & {{e_y}^{\theta}} & {{e_z}^{\theta}} \\ {{e_x}^{\phi}} & {{e_y}^{\phi}} & {{e_z}^{\phi}}\end{matrix} \right]}} = {\left[ \begin{matrix} 0 & {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( 0\right)}}}} + {{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} + {{{\cos\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( 0\right)}}}} + {{{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}}} - {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}}} \\ 0 & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( 0\right)}}} + {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( 0\right)}}} + {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} \\ 0 & {-{{{\sin\left( \psi\right)}} {{\sin\left( \theta\right)}} {{\sin\left( 0\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}}} & {{\sin\left( \theta\right)}} {{\left({{{{\cos\left( \psi\right)}} {{\cos\left( 0\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}}}\right)}} \\ -{\left({{{{\sin\left( 0\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} + {{{\cos\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}}}\right)} & 0 & {{{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( 0\right)}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}}}} - {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\sin\left( 0\right)}}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} \\ {{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} - {{{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & 0 & {{-{{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( 0\right)}}}} - {{{\cos\left( \phi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}} + {{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} \\ {{{\sin\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \psi\right)}}} - {{{\cos\left( 0\right)}} {{\cos\left( \theta\right)}}} & 0 & {{\sin\left( \theta\right)}} {{\left({{{{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}} + {{{\cos\left( \psi\right)}} {{\sin\left( 0\right)}}}}\right)}} \\ {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}} + {{{{\cos\left( 0\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( 0\right)}}}} + {{{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & 0 \\ {-{{{\cos\left( 0\right)}} {{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}}} + {{{{\cos\left( 0\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}} - {{{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( 0\right)}}} + {{{{\cos\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} - {{{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & 0 \\ -{\left({{{{\cos\left( \psi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}} + {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}}}}\right)} & -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}} + {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}}}}\right)} & 0\end{matrix} \right]}$
${\left[ \begin{matrix} {{e_x}^{\psi}} & {{e_y}^{\psi}} & {{e_z}^{\psi}} \\ {{e_x}^{\theta}} & {{e_y}^{\theta}} & {{e_z}^{\theta}} \\ {{e_x}^{\phi}} & {{e_y}^{\phi}} & {{e_z}^{\phi}}\end{matrix} \right]} = { {\left[ \begin{matrix} -{{\frac{1}{2}} {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}}} & {\frac{1}{2}} {{{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}} & \frac{\cos\left( \psi\right)}{{{2}} {{\sin\left( \theta\right)}}} & {\frac{1}{2}} {{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}}} & -{{\frac{1}{2}} {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}}} & \frac{\sin\left( \psi\right)}{{{2}} {{\sin\left( \theta\right)}}} & -{\frac{{{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\sin\left( \theta\right)}}}} & \frac{{{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\sin\left( \theta\right)}}} & 0 \\ {\frac{1}{2}} {{{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {\frac{1}{2}} {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {\frac{1}{2}} {{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}} & -{{\frac{1}{2}} {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}}} & -{{\frac{1}{2}} {{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \theta\right)}}}} & -{{\frac{1}{2}} {{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}}} & {\frac{1}{2}} {{{\cos\left( \theta\right)}} {{\sin\left( \phi\right)}}} & {\frac{1}{2}} {{{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} & -{{\frac{1}{2}} {\sin\left( \theta\right)}} \\ -{{\frac{1}{2}} {{{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}}} & -{{\frac{1}{2}} {{{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} & -{\frac{{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}}{{{2}} {{\sin\left( \theta\right)}}}} & -{{\frac{1}{2}} {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}}}} & -{{\frac{1}{2}} {{{\sin\left( \psi\right)}} {{\cos\left( \phi\right)}}}} & -{\frac{{{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}{{{2}} {{\sin\left( \theta\right)}}}} & \frac{\cos\left( \phi\right)}{{{2}} {{\sin\left( \theta\right)}}} & -{\frac{\sin\left( \phi\right)}{{{2}} {{\sin\left( \theta\right)}}}} & 0\end{matrix} \right]} {\left[ \begin{matrix} 0 & {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( 0\right)}}}} + {{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} + {{{\cos\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\sin\left( 0\right)}}}} + {{{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}}} - {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}}} \\ 0 & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( 0\right)}}} + {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\sin\left( 0\right)}}} + {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} + {{{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} \\ 0 & {-{{{\sin\left( \psi\right)}} {{\sin\left( \theta\right)}} {{\sin\left( 0\right)}}}} + {{{\cos\left( \theta\right)}} {{\cos\left( 0\right)}}} & {{\sin\left( \theta\right)}} {{\left({{{{\cos\left( \psi\right)}} {{\cos\left( 0\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}}}\right)}} \\ -{\left({{{{\sin\left( 0\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} + {{{\cos\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} + {{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}}}\right)} & 0 & {{{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( 0\right)}}} - {{{\sin\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}}}} - {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\sin\left( 0\right)}}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \psi\right)}}} \\ {{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} - {{{\cos\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}} & 0 & {{-{{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( 0\right)}}}} - {{{\cos\left( \phi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}} + {{{{\sin\left( \phi\right)}} {{\sin\left( 0\right)}} {{\sin\left( \psi\right)}}} - {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \psi\right)}}}} \\ {{{\sin\left( \theta\right)}} {{\sin\left( 0\right)}} {{\cos\left( \psi\right)}}} - {{{\cos\left( 0\right)}} {{\cos\left( \theta\right)}}} & 0 & {{\sin\left( \theta\right)}} {{\left({{{{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}} + {{{\cos\left( \psi\right)}} {{\sin\left( 0\right)}}}}\right)}} \\ {{{\cos\left( \phi\right)}} {{\sin\left( \psi\right)}} {{\cos\left( 0\right)}}} + {{{{\cos\left( 0\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & {-{{{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( 0\right)}}}} + {{{{\sin\left( \phi\right)}} {{\cos\left( 0\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}} - {{{\sin\left( 0\right)}} {{\sin\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & 0 \\ {-{{{\cos\left( 0\right)}} {{\sin\left( \psi\right)}} {{\sin\left( \phi\right)}}}} + {{{{\cos\left( 0\right)}} {{\cos\left( \psi\right)}} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}}} - {{{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & {{{\cos\left( \psi\right)}} {{\sin\left( \phi\right)}} {{\cos\left( 0\right)}}} + {{{{\cos\left( 0\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} - {{{\sin\left( 0\right)}} {{\cos\left( \phi\right)}} {{\sin\left( \theta\right)}}}} & 0 \\ -{\left({{{{\cos\left( \psi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}} + {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}}}}\right)} & -{\left({{{{\sin\left( \psi\right)}} {{\cos\left( 0\right)}} {{\sin\left( \theta\right)}}} + {{{\cos\left( \theta\right)}} {{\sin\left( 0\right)}}}}\right)} & 0\end{matrix} \right]}}$
${\left[ \begin{matrix} {{e_x}^{\psi}} & {{e_y}^{\psi}} & {{e_z}^{\psi}} \\ {{e_x}^{\theta}} & {{e_y}^{\theta}} & {{e_z}^{\theta}} \\ {{e_x}^{\phi}} & {{e_y}^{\phi}} & {{e_z}^{\phi}}\end{matrix} \right]} = {\left[ \begin{matrix} -{\frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} & \frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)} & 1 \\ \cos\left( \psi\right) & \sin\left( \psi\right) & 0 \\ \frac{\sin\left( \psi\right)}{\sin\left( \theta\right)} & -{\frac{\cos\left( \psi\right)}{\sin\left( \theta\right)}} & 0\end{matrix} \right]}$
coordinates: ${e_x}$ ${e_y}$ ${e_z}$
base coords: $\psi$ $\theta$ $\phi$
embedding: $x$ $y$ $z$
determining ${e_a}^I$ via Lie bracket:
${{{{ \epsilon} _i} _j} _k} = {\overset{i\downarrow[{j\downarrow k\rightarrow}]}{\left[ \begin{matrix} \overset{j\downarrow k\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{matrix} \right]} \\ \overset{j\downarrow k\rightarrow}{\left[ \begin{matrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{matrix} \right]} \\ \overset{j\downarrow k\rightarrow}{\left[ \begin{matrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
flat metric:
${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
tetrad:
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[ \begin{matrix} -{\frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} & \cos\left( \psi\right) & \frac{\sin\left( \psi\right)}{\sin\left( \theta\right)} \\ \frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)} & \sin\left( \psi\right) & -{\frac{\cos\left( \psi\right)}{\sin\left( \theta\right)}} \\ 1 & 0 & 0\end{matrix} \right]}}$
inverse tetrad by inverting the coefficients of the tetrad:
${{{ e} ^u} _I} = {\overset{u\downarrow I\rightarrow}{\left[ \begin{matrix} 0 & \cos\left( \psi\right) & {{\sin\left( \theta\right)}} {{\sin\left( \psi\right)}} \\ 0 & \sin\left( \psi\right) & -{{{\sin\left( \theta\right)}} {{\cos\left( \psi\right)}}} \\ 1 & 0 & \cos\left( \theta\right)\end{matrix} \right]}}$
verify that the two are orthogonoal:
${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
coordinate metric:
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} \frac{{1} + {{{\cos\left( \theta\right)}^{2}} - {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} & -{\frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} \\ -{\frac{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} & \frac{{{1} - {{\cos\left( \theta\right)}^{2}}} + {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{\sin\left( \theta\right)}^{2}} & \frac{{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}}{\sin\left( \theta\right)} \\ -{\frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} & \frac{{{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}}{\sin\left( \theta\right)} & 1\end{matrix} \right]}}$
${{{ g} ^u} ^v} = {{{{{ e} ^u} _I}} {{{{ e} ^v} _J}} {{{{ \eta} ^I} ^J}}}$
${{{ g} ^u} ^v} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} {{1} - {{\cos\left( \theta\right)}^{2}}} + {{{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} & {{{\cos\left( \theta\right)}^{2}}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}} & {{\sin\left( \psi\right)}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} \\ {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}} & {1} - {{{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} & -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\cos\left( \psi\right)}}} \\ {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}} & -{{{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \psi\right)}}} & {1} + {{\cos\left( \theta\right)}^{2}}\end{matrix} \right]}}$
${{{{ c} _a} _b} ^c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & -{1} \\ 0 & 1 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -{1} & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{ c} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ \frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)} & -{\frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} & -{1} \\ -{\frac{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} & \frac{{{1} - {{\cos\left( \theta\right)}^{2}}} + {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{\sin\left( \theta\right)}^{2}} & \frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{\frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} & \frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)} & 1 \\ 0 & 0 & 0 \\ \frac{{{-{1}} - {{\cos\left( \theta\right)}^{2}}} + {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{\sin\left( \theta\right)}^{2}} & \frac{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}} & \frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \frac{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}} & \frac{{-{1}} + {{{\cos\left( \theta\right)}^{2}} - {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} \\ \frac{{1} + {{{\cos\left( \theta\right)}^{2}} - {{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} & -{\frac{{{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{{{\cos\left( \theta\right)}^{3}}} {{{\cos\left( \psi\right)}^{3}}}} + {{{{{{\cos\left( \psi\right)}^{3}}} {{\cos\left( \theta\right)}}} - {{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}} - {{{{\cos\left( \theta\right)}^{3}}} {{\cos\left( \psi\right)}}}}}\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & \frac{{{4}} {{\cos\left( \theta\right)}} {{\left({{-{1}} + {{{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{\cos\left( \psi\right)}^{2}}}\right)}} {{\sin\left( \psi\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & \frac{{{4}} {{{\cos\left( \theta\right)}^{2}}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}} \\ \frac{{{2}} {{\cos\left( \theta\right)}} {{\left({{{{2}} {{{\cos\left( \psi\right)}^{2}}}} + {{{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{\cos\left( \theta\right)}^{2}}}}\right)}} {{\sin\left( \psi\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & \frac{{{2}} {{\left({{{-{{{2}} {{{\cos\left( \psi\right)}^{3}}} {{{\cos\left( \theta\right)}^{3}}}}} - {{{2}} {{{\cos\left( \psi\right)}^{3}}} {{\cos\left( \theta\right)}}}} + {{{2}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}} + {{{{\cos\left( \theta\right)}^{3}}} {{\cos\left( \psi\right)}}}}\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & \frac{{{2}} {{\left({{-{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}}}} + {{\cos\left( \theta\right)}^{2}}}\right)}}}{{\sin\left( \theta\right)}^{2}} \\ \frac{{{\left({{1} + {{\cos\left( \theta\right)}^{2}}}\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}} & \frac{{{1} - {{\cos\left( \psi\right)}^{2}}} - {{{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \frac{{{2}} {{\cos\left( \theta\right)}} {{\left({{{{2}} {{{\cos\left( \psi\right)}^{2}}}} + {{{{2}} {{{\cos\left( \psi\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{\cos\left( \theta\right)}^{2}}}}\right)}} {{\sin\left( \psi\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & \frac{{{2}} {{\left({{{-{{{2}} {{{\cos\left( \psi\right)}^{3}}} {{{\cos\left( \theta\right)}^{3}}}}} - {{{2}} {{{\cos\left( \psi\right)}^{3}}} {{\cos\left( \theta\right)}}}} + {{{2}} {{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}} + {{{{\cos\left( \theta\right)}^{3}}} {{\cos\left( \psi\right)}}}}\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & \frac{{{2}} {{\left({{-{{{2}} {{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}}}} + {{\cos\left( \theta\right)}^{2}}}\right)}}}{{\sin\left( \theta\right)}^{2}} \\ \frac{{{4}} {{\left({{{-{{{{\cos\left( \psi\right)}^{3}}} {{{\cos\left( \theta\right)}^{3}}}}} - {{{{\cos\left( \psi\right)}^{3}}} {{\cos\left( \theta\right)}}}} + {{{{\cos\left( \theta\right)}^{3}}} {{\cos\left( \psi\right)}}}}\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}} & -{\frac{{{4}} {{{\cos\left( \psi\right)}^{2}}} {{\sin\left( \psi\right)}} {{\cos\left( \theta\right)}} {{\left({{1} + {{\cos\left( \theta\right)}^{2}}}\right)}}}{{{\sin\left( \theta\right)}} {{{\sin\left( \theta\right)}^{2}}}}} & -{\frac{{{4}} {{{\cos\left( \theta\right)}^{2}}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} \\ \frac{{-{{\cos\left( \psi\right)}^{2}}} + {{{\cos\left( \theta\right)}^{2}} - {{{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{\left({{1} + {{\cos\left( \theta\right)}^{2}}}\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} & -{\frac{{{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}{\sin\left( \theta\right)}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \frac{{{\left({{1} + {{\cos\left( \theta\right)}^{2}}}\right)}} {{\sin\left( \psi\right)}} {{\cos\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}} & \frac{{{1} - {{\cos\left( \psi\right)}^{2}}} - {{{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{\cos\left( \psi\right)}} {{\cos\left( \theta\right)}}}{\sin\left( \theta\right)}} \\ \frac{{-{{\cos\left( \psi\right)}^{2}}} + {{{\cos\left( \theta\right)}^{2}} - {{{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \psi\right)}^{2}}}}}}{{\sin\left( \theta\right)}^{2}} & -{\frac{{{\left({{1} + {{\cos\left( \theta\right)}^{2}}}\right)}} {{\cos\left( \psi\right)}} {{\sin\left( \psi\right)}}}{{\sin\left( \theta\right)}^{2}}} & -{\frac{{{\cos\left( \theta\right)}} {{\sin\left( \psi\right)}}}{\sin\left( \theta\right)}} \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{1}{4} & 0 \\ -{\frac{1}{4}} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{1}{4} \\ 0 & 0 & 0 \\ -{\frac{1}{4}} & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\frac{1}{4}} & 0 \\ \frac{1}{4} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & \frac{1}{4} \\ 0 & -{\frac{1}{4}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{1}{4}} \\ 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & -{\frac{1}{4}} \\ 0 & \frac{1}{4} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$