2D Spherical Coordinates

Rotations from \ to dimensions:

	1	2
1	$\left[\begin{array}{cc} 1& 0\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc} \cos\left( \phi\right)& -{\sin\left( \phi\right)}\\ \sin\left( \phi\right)& \cos\left( \phi\right)\end{array}\right]$
2	$\left[\begin{array}{cc} \cos\left( \phi\right)& \sin\left( \phi\right)\\ -{\sin\left( \phi\right)}& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{cc} 1& 0\\ 0& 1\end{array}\right]$

traditional

${{{{R_{1,2}}}\left( {\phi_1}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{cc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_1}\right)}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{cc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} \cos\left( {\phi_1}\right)\\ \sin\left( {\phi_1}\right)\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{cc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_1}\right)}}}\\ {{r}} {{\cos\left( {\phi_1}\right)}}\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {r}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$ | = $\left[\begin{array}{cc} \cos\left( {\phi_1}\right)& -{{{r}} {{\sin\left( {\phi_1}\right)}}}\\ \sin\left( {\phi_1}\right)& {{r}} {{\cos\left( {\phi_1}\right)}}\end{array}\right]$ = $r$

my alternative

3D Spherical Coordinates

Rotations from \ to dimensions:

	1	2	3
1	$\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc} \cos\left( \phi\right)& -{\sin\left( \phi\right)}& 0\\ \sin\left( \phi\right)& \cos\left( \phi\right)& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc} \cos\left( \phi\right)& 0& \sin\left( \phi\right)\\ 0& 1& 0\\ -{\sin\left( \phi\right)}& 0& \cos\left( \phi\right)\end{array}\right]$
2	$\left[\begin{array}{ccc} \cos\left( \phi\right)& \sin\left( \phi\right)& 0\\ -{\sin\left( \phi\right)}& \cos\left( \phi\right)& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc} 1& 0& 0\\ 0& \cos\left( \phi\right)& -{\sin\left( \phi\right)}\\ 0& \sin\left( \phi\right)& \cos\left( \phi\right)\end{array}\right]$
3	$\left[\begin{array}{ccc} \cos\left( \phi\right)& 0& -{\sin\left( \phi\right)}\\ 0& 1& 0\\ \sin\left( \phi\right)& 0& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{ccc} 1& 0& 0\\ 0& \cos\left( \phi\right)& \sin\left( \phi\right)\\ 0& -{\sin\left( \phi\right)}& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

traditional

${{{{R_{1,2}}}\left( {\phi_2}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& -{\sin\left( {\phi_2}\right)}& 0\\ \sin\left( {\phi_2}\right)& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( -{{\phi_1}}\right)& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{r}} {{\cos\left( {\phi_1}\right)}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_2}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& -{\sin\left( {\phi_2}\right)}& 0\\ \sin\left( {\phi_2}\right)& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( -{{\phi_1}}\right)& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}\\ \cos\left( {\phi_1}\right)\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_2}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& -{\sin\left( {\phi_2}\right)}& 0\\ \sin\left( {\phi_2}\right)& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( -{{\phi_1}}\right)& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}\\ -{{{r}} {{\sin\left( {\phi_1}\right)}}}\end{array}\right]}$
${{{{\hat{e}_{\phi_2}}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{{{R_{1,2}}}\left( {\phi_2}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& -{\sin\left( {\phi_2}\right)}& 0\\ \sin\left( {\phi_2}\right)& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( -{{\phi_1}}\right)& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ {{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ 0\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {r}$
${{|\hat{e}_{\phi_2}|}} = {{{r}} {{\sin\left( {\phi_1}\right)}}}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$, ${\hat{e}_{\phi_2}}$ | = $\left[\begin{array}{ccc} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}& {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}& -{{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}& {{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}& {{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ \cos\left( {\phi_1}\right)& -{{{r}} {{\sin\left( {\phi_1}\right)}}}& 0\end{array}\right]$ = ${{{r}^{2}}} {{\sin\left( {\phi_1}\right)}}$

my alternative

${{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{ccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}\\ 0& 1& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ {{r}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{ccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}\\ 0& 1& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}\\ {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ \sin\left( {\phi_2}\right)\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{ccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}\\ 0& 1& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_2}}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{\left[\begin{array}{ccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0\\ 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}\\ 0& 1& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{r}} {{\cos\left( {\phi_2}\right)}}\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {{{r}} {{\cos\left( {\phi_2}\right)}}}$
${{|\hat{e}_{\phi_2}|}} = {r}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$, ${\hat{e}_{\phi_2}}$ | = $\left[\begin{array}{ccc} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}& -{{{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}& -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}}\\ {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}& {{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}& -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ \sin\left( {\phi_2}\right)& 0& {{r}} {{\cos\left( {\phi_2}\right)}}\end{array}\right]$ = ${{{r}^{2}}} {{\cos\left( {\phi_2}\right)}}$

4D Spherical Coordinates

Rotations from \ to dimensions:

	1	2	3	4
1	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} \cos\left( \phi\right)& -{\sin\left( \phi\right)}& 0& 0\\ \sin\left( \phi\right)& \cos\left( \phi\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} \cos\left( \phi\right)& 0& \sin\left( \phi\right)& 0\\ 0& 1& 0& 0\\ -{\sin\left( \phi\right)}& 0& \cos\left( \phi\right)& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} \cos\left( \phi\right)& 0& 0& -{\sin\left( \phi\right)}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sin\left( \phi\right)& 0& 0& \cos\left( \phi\right)\end{array}\right]$
2	$\left[\begin{array}{cccc} \cos\left( \phi\right)& \sin\left( \phi\right)& 0& 0\\ -{\sin\left( \phi\right)}& \cos\left( \phi\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \cos\left( \phi\right)& -{\sin\left( \phi\right)}& 0\\ 0& \sin\left( \phi\right)& \cos\left( \phi\right)& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \cos\left( \phi\right)& 0& \sin\left( \phi\right)\\ 0& 0& 1& 0\\ 0& -{\sin\left( \phi\right)}& 0& \cos\left( \phi\right)\end{array}\right]$
3	$\left[\begin{array}{cccc} \cos\left( \phi\right)& 0& -{\sin\left( \phi\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( \phi\right)& 0& \cos\left( \phi\right)& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \cos\left( \phi\right)& \sin\left( \phi\right)& 0\\ 0& -{\sin\left( \phi\right)}& \cos\left( \phi\right)& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos\left( \phi\right)& -{\sin\left( \phi\right)}\\ 0& 0& \sin\left( \phi\right)& \cos\left( \phi\right)\end{array}\right]$
4	$\left[\begin{array}{cccc} \cos\left( \phi\right)& 0& 0& \sin\left( \phi\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( \phi\right)}& 0& 0& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \cos\left( \phi\right)& 0& -{\sin\left( \phi\right)}\\ 0& 0& 1& 0\\ 0& \sin\left( \phi\right)& 0& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos\left( \phi\right)& \sin\left( \phi\right)\\ 0& 0& -{\sin\left( \phi\right)}& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

traditional

${{{{R_{1,2}}}\left( {\phi_3}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& -{\sin\left( {\phi_3}\right)}& 0& 0\\ \sin\left( {\phi_3}\right)& \cos\left( {\phi_3}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_2}}\right)& 0& -{\sin\left( -{{\phi_2}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( -{{\phi_2}}\right)& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_1}}\right)& 0& 0& \sin\left( -{{\phi_1}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( -{{\phi_1}}\right)}& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}}\\ {{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ -{{{r}} {{\cos\left( {\phi_1}\right)}}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_3}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& -{\sin\left( {\phi_3}\right)}& 0& 0\\ \sin\left( {\phi_3}\right)& \cos\left( {\phi_3}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_2}}\right)& 0& -{\sin\left( -{{\phi_2}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( -{{\phi_2}}\right)& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_1}}\right)& 0& 0& \sin\left( -{{\phi_1}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( -{{\phi_1}}\right)}& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}}\\ {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ -{\cos\left( {\phi_1}\right)}\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_3}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& -{\sin\left( {\phi_3}\right)}& 0& 0\\ \sin\left( {\phi_3}\right)& \cos\left( {\phi_3}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_2}}\right)& 0& -{\sin\left( -{{\phi_2}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( -{{\phi_2}}\right)& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_1}}\right)& 0& 0& \sin\left( -{{\phi_1}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( -{{\phi_1}}\right)}& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}\\ {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}\\ {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_1}\right)}}\end{array}\right]}$
${{{{\hat{e}_{\phi_2}}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{{{R_{1,2}}}\left( {\phi_3}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& -{\sin\left( {\phi_3}\right)}& 0& 0\\ \sin\left( {\phi_3}\right)& \cos\left( {\phi_3}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_2}}\right)& 0& -{\sin\left( -{{\phi_2}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( -{{\phi_2}}\right)& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_1}}\right)& 0& 0& \sin\left( -{{\phi_1}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( -{{\phi_1}}\right)}& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}\\ -{{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_3}}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{{{R_{1,2}}}\left( {\phi_3}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& -{\sin\left( {\phi_3}\right)}& 0& 0\\ \sin\left( {\phi_3}\right)& \cos\left( {\phi_3}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_2}}\right)& 0& -{\sin\left( -{{\phi_2}}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( -{{\phi_2}}\right)& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( -{{\phi_1}}\right)& 0& 0& \sin\left( -{{\phi_1}}\right)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -{\sin\left( -{{\phi_1}}\right)}& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}\\ 0\\ 0\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {r}$
${{|\hat{e}_{\phi_2}|}} = {{{r}} {{\sin\left( {\phi_1}\right)}}}$
${{|\hat{e}_{\phi_3}|}} = {{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$, ${\hat{e}_{\phi_2}}$, ${\hat{e}_{\phi_3}}$ | = $\left[\begin{array}{cccc} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}& -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}\\ {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}& {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}& -{{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}& 0\\ -{\cos\left( {\phi_1}\right)}& {{r}} {{\sin\left( {\phi_1}\right)}}& 0& 0\end{array}\right]$ = ${{{r}^{3}}} {{\sin\left( {\phi_2}\right)}} {{{\sin\left( {\phi_1}\right)}^{2}}}$

my alternative

${{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{cccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}\\ {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{r}} {{\sin\left( {\phi_3}\right)}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}}\\ {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}}\\ \sin\left( {\phi_3}\right)\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}\\ {{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}\\ 0\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_2}}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ -{{{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_3}}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{\left[\begin{array}{cccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0\\ 0& 1& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ {{r}} {{\cos\left( {\phi_3}\right)}}\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {{{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}}$
${{|\hat{e}_{\phi_2}|}} = {{{r}} {{\cos\left( {\phi_3}\right)}}}$
${{|\hat{e}_{\phi_3}|}} = {r}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$, ${\hat{e}_{\phi_2}}$, ${\hat{e}_{\phi_3}}$ | = $\left[\begin{array}{cccc} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}}& -{{{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}& -{{{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}& -{{{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}\\ {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}}& {{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}& -{{{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}& -{{{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}\\ {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}}& 0& {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}& -{{{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ \sin\left( {\phi_3}\right)& 0& 0& {{r}} {{\cos\left( {\phi_3}\right)}}\end{array}\right]$ = ${{{r}^{3}}} {{\cos\left( {\phi_2}\right)}} {{\left({{1} + {{{{\cos\left( {\phi_3}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}{-{{\cos\left( {\phi_3}\right)}^{2}}}{-{{\sin\left( {\phi_3}\right)}^{2}}} + {{\cos\left( {\phi_3}\right)}^{4}}}\right)}}$

5D Spherical Coordinates

Rotations from \ to dimensions:

	1	2	3	4	5
1	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& -{\sin\left( \phi\right)}& 0& 0& 0\\ \sin\left( \phi\right)& \cos\left( \phi\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& 0& \sin\left( \phi\right)& 0& 0\\ 0& 1& 0& 0& 0\\ -{\sin\left( \phi\right)}& 0& \cos\left( \phi\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& 0& 0& -{\sin\left( \phi\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( \phi\right)& 0& 0& \cos\left( \phi\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& 0& 0& 0& \sin\left( \phi\right)\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ -{\sin\left( \phi\right)}& 0& 0& 0& \cos\left( \phi\right)\end{array}\right]$
2	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& \sin\left( \phi\right)& 0& 0& 0\\ -{\sin\left( \phi\right)}& \cos\left( \phi\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \cos\left( \phi\right)& -{\sin\left( \phi\right)}& 0& 0\\ 0& \sin\left( \phi\right)& \cos\left( \phi\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \cos\left( \phi\right)& 0& \sin\left( \phi\right)& 0\\ 0& 0& 1& 0& 0\\ 0& -{\sin\left( \phi\right)}& 0& \cos\left( \phi\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \cos\left( \phi\right)& 0& 0& -{\sin\left( \phi\right)}\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& \sin\left( \phi\right)& 0& 0& \cos\left( \phi\right)\end{array}\right]$
3	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& 0& -{\sin\left( \phi\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( \phi\right)& 0& \cos\left( \phi\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \cos\left( \phi\right)& \sin\left( \phi\right)& 0& 0\\ 0& -{\sin\left( \phi\right)}& \cos\left( \phi\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& \cos\left( \phi\right)& -{\sin\left( \phi\right)}& 0\\ 0& 0& \sin\left( \phi\right)& \cos\left( \phi\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& \cos\left( \phi\right)& 0& \sin\left( \phi\right)\\ 0& 0& 0& 1& 0\\ 0& 0& -{\sin\left( \phi\right)}& 0& \cos\left( \phi\right)\end{array}\right]$
4	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& 0& 0& \sin\left( \phi\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( \phi\right)}& 0& 0& \cos\left( \phi\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \cos\left( \phi\right)& 0& -{\sin\left( \phi\right)}& 0\\ 0& 0& 1& 0& 0\\ 0& \sin\left( \phi\right)& 0& \cos\left( \phi\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& \cos\left( \phi\right)& \sin\left( \phi\right)& 0\\ 0& 0& -{\sin\left( \phi\right)}& \cos\left( \phi\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& \cos\left( \phi\right)& -{\sin\left( \phi\right)}\\ 0& 0& 0& \sin\left( \phi\right)& \cos\left( \phi\right)\end{array}\right]$
5	$\left[\begin{array}{ccccc} \cos\left( \phi\right)& 0& 0& 0& -{\sin\left( \phi\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( \phi\right)& 0& 0& 0& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& \cos\left( \phi\right)& 0& 0& \sin\left( \phi\right)\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& -{\sin\left( \phi\right)}& 0& 0& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& \cos\left( \phi\right)& 0& -{\sin\left( \phi\right)}\\ 0& 0& 0& 1& 0\\ 0& 0& \sin\left( \phi\right)& 0& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& \cos\left( \phi\right)& \sin\left( \phi\right)\\ 0& 0& 0& -{\sin\left( \phi\right)}& \cos\left( \phi\right)\end{array}\right]$	$\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

traditional

${{{{R_{1,2}}}\left( {\phi_4}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_3}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{5,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{5,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& -{\sin\left( {\phi_4}\right)}& 0& 0& 0\\ \sin\left( {\phi_4}\right)& \cos\left( {\phi_4}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_3}}\right)& 0& -{\sin\left( -{{\phi_3}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( -{{\phi_3}}\right)& 0& \cos\left( -{{\phi_3}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_2}}\right)& 0& 0& \sin\left( -{{\phi_2}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( -{{\phi_2}}\right)}& 0& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_1}}\right)& 0& 0& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}}\\ -{{{r}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{r}} {{\cos\left( {\phi_1}\right)}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_4}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_3}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{5,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{5,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& -{\sin\left( {\phi_4}\right)}& 0& 0& 0\\ \sin\left( {\phi_4}\right)& \cos\left( {\phi_4}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_3}}\right)& 0& -{\sin\left( -{{\phi_3}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( -{{\phi_3}}\right)& 0& \cos\left( -{{\phi_3}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_2}}\right)& 0& 0& \sin\left( -{{\phi_2}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( -{{\phi_2}}\right)}& 0& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_1}}\right)& 0& 0& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}\\ -{{{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ \cos\left( {\phi_1}\right)\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_4}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_3}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{5,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{5,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& -{\sin\left( {\phi_4}\right)}& 0& 0& 0\\ \sin\left( {\phi_4}\right)& \cos\left( {\phi_4}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_3}}\right)& 0& -{\sin\left( -{{\phi_3}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( -{{\phi_3}}\right)& 0& \cos\left( -{{\phi_3}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_2}}\right)& 0& 0& \sin\left( -{{\phi_2}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( -{{\phi_2}}\right)}& 0& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_1}}\right)& 0& 0& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}}\\ {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}\\ -{{{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_1}\right)}}}\end{array}\right]}$
${{{{\hat{e}_{\phi_2}}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{{{R_{1,2}}}\left( {\phi_4}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_3}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{5,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{5,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& -{\sin\left( {\phi_4}\right)}& 0& 0& 0\\ \sin\left( {\phi_4}\right)& \cos\left( {\phi_4}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_3}}\right)& 0& -{\sin\left( -{{\phi_3}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( -{{\phi_3}}\right)& 0& \cos\left( -{{\phi_3}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_2}}\right)& 0& 0& \sin\left( -{{\phi_2}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( -{{\phi_2}}\right)}& 0& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_1}}\right)& 0& 0& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}}\\ {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_3}}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{{{R_{1,2}}}\left( {\phi_4}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_3}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{5,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{5,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& -{\sin\left( {\phi_4}\right)}& 0& 0& 0\\ \sin\left( {\phi_4}\right)& \cos\left( {\phi_4}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_3}}\right)& 0& -{\sin\left( -{{\phi_3}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( -{{\phi_3}}\right)& 0& \cos\left( -{{\phi_3}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_2}}\right)& 0& 0& \sin\left( -{{\phi_2}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( -{{\phi_2}}\right)}& 0& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_1}}\right)& 0& 0& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}}\\ -{{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}}}\\ 0\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_4}}} = {{\frac{\partial}{\partial {\phi_4}}}\left({{{{{R_{1,2}}}\left( {\phi_4}\right)}} {{{{R_{3,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{3,1}}}\left( -{{\phi_3}}\right)}} {{{{R_{4,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{4,1}}}\left( -{{\phi_2}}\right)}} {{{{R_{5,1}}}\left( {{\frac{1}{2}} {π}}\right)}} {{{{R_{5,1}}}\left( -{{\phi_1}}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_4}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& -{\sin\left( {\phi_4}\right)}& 0& 0& 0\\ \sin\left( {\phi_4}\right)& \cos\left( {\phi_4}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_3}}\right)& 0& -{\sin\left( -{{\phi_3}}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( -{{\phi_3}}\right)& 0& \cos\left( -{{\phi_3}}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& \sin\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( {{\frac{1}{2}} {π}}\right)}& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_2}}\right)& 0& 0& \sin\left( -{{\phi_2}}\right)& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{\sin\left( -{{\phi_2}}\right)}& 0& 0& \cos\left( -{{\phi_2}}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& -{\sin\left( {{\frac{1}{2}} {π}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {{\frac{1}{2}} {π}}\right)& 0& 0& 0& \cos\left( {{\frac{1}{2}} {π}}\right)\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( -{{\phi_1}}\right)& 0& 0& 0& -{\sin\left( -{{\phi_1}}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( -{{\phi_1}}\right)& 0& 0& 0& \cos\left( -{{\phi_1}}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}\\ 0\\ 0\\ 0\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {r}$
${{|\hat{e}_{\phi_2}|}} = {{{r}} {{\sin\left( {\phi_1}\right)}}}$
${{|\hat{e}_{\phi_3}|}} = {{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}}}$
${{|\hat{e}_{\phi_4}|}} = {{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}}}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$, ${\hat{e}_{\phi_2}}$, ${\hat{e}_{\phi_3}}$, ${\hat{e}_{\phi_4}}$ | = $\left[\begin{array}{ccccc} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}& -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}}\\ {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_1}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}}& {{r}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}}& {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}& -{{{r}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}}}& 0\\ -{{{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}}& -{{{r}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_2}\right)}}}& {{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}}& 0& 0\\ \cos\left( {\phi_1}\right)& -{{{r}} {{\sin\left( {\phi_1}\right)}}}& 0& 0& 0\end{array}\right]$ = ${{\sin\left( {\phi_3}\right)}} {{\sin\left( {\phi_1}\right)}} {{\left({{-{{{{r}^{4}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{4}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_1}\right)}^{2}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_1}\right)}^{2}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\sin\left( {\phi_2}\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_2}\right)}^{2}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_2}\right)}^{2}}} {{{\sin\left( {\phi_1}\right)}^{2}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\sin\left( {\phi_2}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{4}}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{4}}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{4}}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{4}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_2}\right)}^{4}}} {{{\cos\left( {\phi_1}\right)}^{4}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{2}}} {{{\sin\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_2}\right)}^{2}}} {{{\cos\left( {\phi_1}\right)}^{4}}}}}\right)}}$

my alternative

${{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{{{R_{5,1}}}\left( {\phi_4}\right)}} {{\hat{x}}}$ = ${{\left[\begin{array}{ccccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& 0& 0& 0& -{\sin\left( {\phi_4}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {\phi_4}\right)& 0& 0& 0& \cos\left( {\phi_4}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}$ = $\left[\begin{array}{c} {{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{r}} {{\sin\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{r}} {{\sin\left( {\phi_4}\right)}}\end{array}\right]$
${{{{\hat{e}_{r}}} = {{\frac{\partial}{\partial r}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{{{R_{5,1}}}\left( {\phi_4}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial r}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& 0& 0& 0& -{\sin\left( {\phi_4}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {\phi_4}\right)& 0& 0& 0& \cos\left( {\phi_4}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}\\ {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}\\ {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_3}\right)}}\\ \sin\left( {\phi_4}\right)\end{array}\right]}$
${{{{\hat{e}_{\phi_1}}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{{{R_{5,1}}}\left( {\phi_4}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_1}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& 0& 0& 0& -{\sin\left( {\phi_4}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {\phi_4}\right)& 0& 0& 0& \cos\left( {\phi_4}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_1}\right)}}\\ 0\\ 0\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_2}}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{{{R_{5,1}}}\left( {\phi_4}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_2}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& 0& 0& 0& -{\sin\left( {\phi_4}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {\phi_4}\right)& 0& 0& 0& \cos\left( {\phi_4}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ {{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_2}\right)}}\\ 0\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_3}}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{{{R_{5,1}}}\left( {\phi_4}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_3}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& 0& 0& 0& -{\sin\left( {\phi_4}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {\phi_4}\right)& 0& 0& 0& \cos\left( {\phi_4}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}}}\\ -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_3}\right)}}}\\ {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}}\\ 0\end{array}\right]}$
${{{{\hat{e}_{\phi_4}}} = {{\frac{\partial}{\partial {\phi_4}}}\left({{{{{R_{1,2}}}\left( {\phi_1}\right)}} {{{{R_{3,1}}}\left( {\phi_2}\right)}} {{{{R_{1,4}}}\left( {\phi_3}\right)}} {{{{R_{5,1}}}\left( {\phi_4}\right)}} {{\hat{x}}}}\right)}} = {{\frac{\partial}{\partial {\phi_4}}}\left({{{\left[\begin{array}{ccccc} \cos\left( {\phi_1}\right)& -{\sin\left( {\phi_1}\right)}& 0& 0& 0\\ \sin\left( {\phi_1}\right)& \cos\left( {\phi_1}\right)& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_2}\right)& 0& -{\sin\left( {\phi_2}\right)}& 0& 0\\ 0& 1& 0& 0& 0\\ \sin\left( {\phi_2}\right)& 0& \cos\left( {\phi_2}\right)& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_3}\right)& 0& 0& -{\sin\left( {\phi_3}\right)}& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ \sin\left( {\phi_3}\right)& 0& 0& \cos\left( {\phi_3}\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{ccccc} \cos\left( {\phi_4}\right)& 0& 0& 0& -{\sin\left( {\phi_4}\right)}\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \sin\left( {\phi_4}\right)& 0& 0& 0& \cos\left( {\phi_4}\right)\end{array}\right]}} {{\left[\begin{array}{c} r\\ 0\\ 0\\ 0\\ 0\end{array}\right]}}}\right)}} = {\left[\begin{array}{c} -{{{r}} {{\sin\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ -{{{r}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_3}\right)}}}\\ {{r}} {{\cos\left( {\phi_4}\right)}}\end{array}\right]}$
${{|\hat{e}_{r}|}} = {1}$
${{|\hat{e}_{\phi_1}|}} = {{{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}}}$
${{|\hat{e}_{\phi_2}|}} = {{{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}}}$
${{|\hat{e}_{\phi_3}|}} = {{{r}} {{\cos\left( {\phi_4}\right)}}}$
${{|\hat{e}_{\phi_4}|}} = {r}$
| ${\hat{e}_{r}}$, ${\hat{e}_{\phi_1}}$, ${\hat{e}_{\phi_2}}$, ${\hat{e}_{\phi_3}}$, ${\hat{e}_{\phi_4}}$ | = $\left[\begin{array}{ccccc} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}& -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}& -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}& -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}}}& -{{{r}} {{\sin\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}& {{r}} {{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_1}\right)}}& -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}& -{{{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\sin\left( {\phi_3}\right)}}}& -{{{r}} {{\sin\left( {\phi_4}\right)}} {{\cos\left( {\phi_2}\right)}} {{\sin\left( {\phi_1}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ {{\cos\left( {\phi_3}\right)}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}}& 0& {{r}} {{\cos\left( {\phi_4}\right)}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_2}\right)}}& -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_3}\right)}}}& -{{{r}} {{\sin\left( {\phi_2}\right)}} {{\sin\left( {\phi_4}\right)}} {{\cos\left( {\phi_3}\right)}}}\\ {{\cos\left( {\phi_4}\right)}} {{\sin\left( {\phi_3}\right)}}& 0& 0& {{r}} {{\cos\left( {\phi_3}\right)}} {{\cos\left( {\phi_4}\right)}}& -{{{r}} {{\sin\left( {\phi_4}\right)}} {{\sin\left( {\phi_3}\right)}}}\\ \sin\left( {\phi_4}\right)& 0& 0& 0& {{r}} {{\cos\left( {\phi_4}\right)}}\end{array}\right]$ = ${{\cos\left( {\phi_2}\right)}} {{\cos\left( {\phi_4}\right)}} {{\left({{{r}^{4}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_4}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_4}\right)}^{4}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{4}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{4}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{4}}} {{{\cos\left( {\phi_4}\right)}^{4}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_3}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{4}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( {\phi_3}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{4}}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_3}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{4}}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\sin\left( {\phi_4}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{4}}}}} + {{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{4}}}} + {{{{r}^{4}}} {{{\cos\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}} {{{\sin\left( {\phi_4}\right)}^{2}}}}{-{{{{r}^{4}}} {{{\sin\left( {\phi_4}\right)}^{2}}} {{{\cos\left( {\phi_3}\right)}^{2}}} {{{\sin\left( {\phi_3}\right)}^{2}}}}}}\right)}}$