2D Spherical Coordinates

Rotations from \ to dimensions:

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

traditional

R_{1, 2} (ϕ_{1}) \hat{x}

[\begin{array}{cc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \end{matrix}]

[\begin{matrix} r \cos (ϕ_{1}) \\ r \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{1}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{cc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \end{matrix}]) = [\begin{matrix} \cos (ϕ_{1}) \\ \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{cc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{1}) \\ r \cos (ϕ_{1}) \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

| =

[\begin{array}{cc} \cos (ϕ_{1}) & - r \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & r \cos (ϕ_{1}) \end{array}]

r

my alternative

R_{1, 2} (ϕ_{1}) \hat{x}

[\begin{array}{cc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \end{matrix}]

[\begin{matrix} r \cos (ϕ_{1}) \\ r \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{1}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{cc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \end{matrix}]) = [\begin{matrix} \cos (ϕ_{1}) \\ \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{cc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{1}) \\ r \cos (ϕ_{1}) \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

| =

[\begin{array}{cc} \cos (ϕ_{1}) & - r \sin (ϕ_{1}) \\ \sin (ϕ_{1}) & r \cos (ϕ_{1}) \end{array}]

r

3D Spherical Coordinates

Rotations from \ to dimensions:

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

traditional

R_{1, 2} (ϕ_{2}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{1}) \hat{x}

[\begin{array}{ccc} \cos (ϕ_{2}) & - \sin (ϕ_{2}) & 0 \\ \sin (ϕ_{2}) & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccc} \cos (- ϕ_{1}) & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]

[\begin{matrix} r \cos (ϕ_{2}) \sin (ϕ_{1}) \\ r \sin (ϕ_{2}) \sin (ϕ_{1}) \\ r \cos (ϕ_{1}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{2}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{ccc} \cos (ϕ_{2}) & - \sin (ϕ_{2}) & 0 \\ \sin (ϕ_{2}) & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccc} \cos (- ϕ_{1}) & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} \sin (ϕ_{1}) \cos (ϕ_{2}) \\ \sin (ϕ_{1}) \sin (ϕ_{2}) \\ \cos (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{2}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{ccc} \cos (ϕ_{2}) & - \sin (ϕ_{2}) & 0 \\ \sin (ϕ_{2}) & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccc} \cos (- ϕ_{1}) & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} r \cos (ϕ_{2}) \cos (ϕ_{1}) \\ r \sin (ϕ_{2}) \cos (ϕ_{1}) \\ - r \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{2}} = \frac{\partial}{\partial ϕ_{2}} (R_{1, 2} (ϕ_{2}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{2}} ([\begin{array}{ccc} \cos (ϕ_{2}) & - \sin (ϕ_{2}) & 0 \\ \sin (ϕ_{2}) & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccc} \cos (- ϕ_{1}) & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{1}) \sin (ϕ_{2}) \\ r \cos (ϕ_{2}) \sin (ϕ_{1}) \\ 0 \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r

| {\hat{e}}_{ϕ_{2}} | = r \sin (ϕ_{1})

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

{\hat{e}}_{ϕ_{2}}

| =

[\begin{array}{ccc} \sin (ϕ_{1}) \cos (ϕ_{2}) & r \cos (ϕ_{2}) \cos (ϕ_{1}) & - r \sin (ϕ_{1}) \sin (ϕ_{2}) \\ \sin (ϕ_{1}) \sin (ϕ_{2}) & r \sin (ϕ_{2}) \cos (ϕ_{1}) & r \cos (ϕ_{2}) \sin (ϕ_{1}) \\ \cos (ϕ_{1}) & - r \sin (ϕ_{1}) & 0 \end{array}]

r^{2} \sin (ϕ_{1})

my alternative

R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) \hat{x}

[\begin{array}{ccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) \\ 0 & 1 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]

[\begin{matrix} r \cos (ϕ_{1}) \cos (ϕ_{2}) \\ r \sin (ϕ_{1}) \cos (ϕ_{2}) \\ r \sin (ϕ_{2}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{ccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) \\ 0 & 1 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} \cos (ϕ_{2}) \cos (ϕ_{1}) \\ \cos (ϕ_{2}) \sin (ϕ_{1}) \\ \sin (ϕ_{2}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{ccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) \\ 0 & 1 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \cos (ϕ_{2}) \sin (ϕ_{1}) \\ r \cos (ϕ_{1}) \cos (ϕ_{2}) \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{2}} = \frac{\partial}{\partial ϕ_{2}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) \hat{x}) = \frac{\partial}{\partial ϕ_{2}} ([\begin{array}{ccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) \\ 0 & 1 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{2}) \cos (ϕ_{1}) \\ - r \sin (ϕ_{2}) \sin (ϕ_{1}) \\ r \cos (ϕ_{2}) \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r \cos (ϕ_{2})

| {\hat{e}}_{ϕ_{2}} | = r

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

{\hat{e}}_{ϕ_{2}}

| =

[\begin{array}{ccc} \cos (ϕ_{2}) \cos (ϕ_{1}) & - r \cos (ϕ_{2}) \sin (ϕ_{1}) & - r \sin (ϕ_{2}) \cos (ϕ_{1}) \\ \cos (ϕ_{2}) \sin (ϕ_{1}) & r \cos (ϕ_{1}) \cos (ϕ_{2}) & - r \sin (ϕ_{2}) \sin (ϕ_{1}) \\ \sin (ϕ_{2}) & 0 & r \cos (ϕ_{2}) \end{array}]

r^{2} \cos (ϕ_{2})

4D Spherical Coordinates

Rotations from \ to dimensions:

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

traditional

R_{1, 2} (ϕ_{3}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{2}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{1}) \hat{x}

[\begin{array}{cccc} \cos (ϕ_{3}) & - \sin (ϕ_{3}) & 0 & 0 \\ \sin (ϕ_{3}) & \cos (ϕ_{3}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (- ϕ_{2}) & 0 & - \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (- ϕ_{2}) & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{cccc} \cos (- ϕ_{1}) & 0 & 0 & \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (- ϕ_{1}) & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]

[\begin{matrix} r \sin (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ r \sin (ϕ_{2}) \sin (ϕ_{1}) \sin (ϕ_{3}) \\ r \cos (ϕ_{2}) \sin (ϕ_{1}) \\ - r \cos (ϕ_{1}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{3}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{2}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{cccc} \cos (ϕ_{3}) & - \sin (ϕ_{3}) & 0 & 0 \\ \sin (ϕ_{3}) & \cos (ϕ_{3}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (- ϕ_{2}) & 0 & - \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (- ϕ_{2}) & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{cccc} \cos (- ϕ_{1}) & 0 & 0 & \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (- ϕ_{1}) & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{3}) \\ \sin (ϕ_{1}) \sin (ϕ_{2}) \sin (ϕ_{3}) \\ \sin (ϕ_{1}) \cos (ϕ_{2}) \\ - \cos (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{3}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{2}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{cccc} \cos (ϕ_{3}) & - \sin (ϕ_{3}) & 0 & 0 \\ \sin (ϕ_{3}) & \cos (ϕ_{3}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (- ϕ_{2}) & 0 & - \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (- ϕ_{2}) & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{cccc} \cos (- ϕ_{1}) & 0 & 0 & \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (- ϕ_{1}) & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} r \cos (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \\ r \sin (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \\ r \cos (ϕ_{2}) \cos (ϕ_{1}) \\ r \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{2}} = \frac{\partial}{\partial ϕ_{2}} (R_{1, 2} (ϕ_{3}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{2}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{2}} ([\begin{array}{cccc} \cos (ϕ_{3}) & - \sin (ϕ_{3}) & 0 & 0 \\ \sin (ϕ_{3}) & \cos (ϕ_{3}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (- ϕ_{2}) & 0 & - \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (- ϕ_{2}) & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{cccc} \cos (- ϕ_{1}) & 0 & 0 & \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (- ϕ_{1}) & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} r \cos (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \\ r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \\ - r \sin (ϕ_{1}) \sin (ϕ_{2}) \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{3}} = \frac{\partial}{\partial ϕ_{3}} (R_{1, 2} (ϕ_{3}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{2}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{3}} ([\begin{array}{cccc} \cos (ϕ_{3}) & - \sin (ϕ_{3}) & 0 & 0 \\ \sin (ϕ_{3}) & \cos (ϕ_{3}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (- ϕ_{2}) & 0 & - \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (- ϕ_{2}) & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{cccc} \cos (- ϕ_{1}) & 0 & 0 & \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sin (- ϕ_{1}) & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{1}) \\ r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \\ 0 \\ 0 \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r

| {\hat{e}}_{ϕ_{2}} | = r \sin (ϕ_{1})

| {\hat{e}}_{ϕ_{3}} | = r \sin (ϕ_{1}) \sin (ϕ_{2})

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

{\hat{e}}_{ϕ_{2}}

{\hat{e}}_{ϕ_{3}}

| =

[\begin{array}{cccc} \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{3}) & r \cos (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) & r \cos (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) & - r \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{1}) \\ \sin (ϕ_{1}) \sin (ϕ_{2}) \sin (ϕ_{3}) & r \sin (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) & r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) & r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \\ \sin (ϕ_{1}) \cos (ϕ_{2}) & r \cos (ϕ_{2}) \cos (ϕ_{1}) & - r \sin (ϕ_{1}) \sin (ϕ_{2}) & 0 \\ - \cos (ϕ_{1}) & r \sin (ϕ_{1}) & 0 & 0 \end{array}]

r^{3} \sin (ϕ_{2}) {\sin (ϕ_{1})}^{2}

my alternative

R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) \hat{x}

[\begin{array}{cccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]

[\begin{matrix} r \cos (ϕ_{3}) \cos (ϕ_{2}) \cos (ϕ_{1}) \\ r \cos (ϕ_{3}) \cos (ϕ_{2}) \sin (ϕ_{1}) \\ r \sin (ϕ_{2}) \cos (ϕ_{3}) \\ r \sin (ϕ_{3}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{cccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} \cos (ϕ_{2}) \cos (ϕ_{3}) \cos (ϕ_{1}) \\ \cos (ϕ_{2}) \cos (ϕ_{3}) \sin (ϕ_{1}) \\ \cos (ϕ_{3}) \sin (ϕ_{2}) \\ \sin (ϕ_{3}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{cccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \cos (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \\ r \cos (ϕ_{1}) \cos (ϕ_{2}) \cos (ϕ_{3}) \\ 0 \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{2}} = \frac{\partial}{\partial ϕ_{2}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) \hat{x}) = \frac{\partial}{\partial ϕ_{2}} ([\begin{array}{cccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \cos (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \\ - r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \\ r \cos (ϕ_{2}) \cos (ϕ_{3}) \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{3}} = \frac{\partial}{\partial ϕ_{3}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) \hat{x}) = \frac{\partial}{\partial ϕ_{3}} ([\begin{array}{cccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{3}) \cos (ϕ_{1}) \cos (ϕ_{2}) \\ - r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \\ - r \sin (ϕ_{3}) \sin (ϕ_{2}) \\ r \cos (ϕ_{3}) \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r \cos (ϕ_{2}) \cos (ϕ_{3})

| {\hat{e}}_{ϕ_{2}} | = r \cos (ϕ_{3})

| {\hat{e}}_{ϕ_{3}} | = r

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

{\hat{e}}_{ϕ_{2}}

{\hat{e}}_{ϕ_{3}}

| =

[\begin{array}{cccc} \cos (ϕ_{2}) \cos (ϕ_{3}) \cos (ϕ_{1}) & - r \cos (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) & - r \cos (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) & - r \sin (ϕ_{3}) \cos (ϕ_{1}) \cos (ϕ_{2}) \\ \cos (ϕ_{2}) \cos (ϕ_{3}) \sin (ϕ_{1}) & r \cos (ϕ_{1}) \cos (ϕ_{2}) \cos (ϕ_{3}) & - r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) & - r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \\ \cos (ϕ_{3}) \sin (ϕ_{2}) & 0 & r \cos (ϕ_{2}) \cos (ϕ_{3}) & - r \sin (ϕ_{3}) \sin (ϕ_{2}) \\ \sin (ϕ_{3}) & 0 & 0 & r \cos (ϕ_{3}) \end{array}]

r^{3} \cos (ϕ_{2}) (1 + {\cos (ϕ_{3})}^{2} {\sin (ϕ_{3})}^{2} - {\cos (ϕ_{3})}^{2} - {\sin (ϕ_{3})}^{2} + {\cos (ϕ_{3})}^{4})

5D Spherical Coordinates

Rotations from \ to dimensions:

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

traditional

R_{1, 2} (ϕ_{4}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{3}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{2}) R_{5, 1} (\frac{1}{2} π) R_{5, 1} (- ϕ_{1}) \hat{x}

[\begin{array}{ccccc} \cos (ϕ_{4}) & - \sin (ϕ_{4}) & 0 & 0 & 0 \\ \sin (ϕ_{4}) & \cos (ϕ_{4}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{3}) & 0 & - \sin (- ϕ_{3}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (- ϕ_{3}) & 0 & \cos (- ϕ_{3}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{2}) & 0 & 0 & \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (- ϕ_{2}) & 0 & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{1}) & 0 & 0 & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]

[\begin{matrix} r \sin (ϕ_{2}) \sin (ϕ_{3}) \cos (ϕ_{4}) \sin (ϕ_{1}) \\ r \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{4}) \sin (ϕ_{1}) \\ r \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{3}) \\ - r \cos (ϕ_{2}) \sin (ϕ_{1}) \\ r \cos (ϕ_{1}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{4}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{3}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{2}) R_{5, 1} (\frac{1}{2} π) R_{5, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{ccccc} \cos (ϕ_{4}) & - \sin (ϕ_{4}) & 0 & 0 & 0 \\ \sin (ϕ_{4}) & \cos (ϕ_{4}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{3}) & 0 & - \sin (- ϕ_{3}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (- ϕ_{3}) & 0 & \cos (- ϕ_{3}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{2}) & 0 & 0 & \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (- ϕ_{2}) & 0 & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{1}) & 0 & 0 & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} \sin (ϕ_{2}) \sin (ϕ_{3}) \cos (ϕ_{4}) \sin (ϕ_{1}) \\ \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{4}) \sin (ϕ_{1}) \\ \sin (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ - \cos (ϕ_{2}) \sin (ϕ_{1}) \\ \cos (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{4}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{3}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{2}) R_{5, 1} (\frac{1}{2} π) R_{5, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{ccccc} \cos (ϕ_{4}) & - \sin (ϕ_{4}) & 0 & 0 & 0 \\ \sin (ϕ_{4}) & \cos (ϕ_{4}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{3}) & 0 & - \sin (- ϕ_{3}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (- ϕ_{3}) & 0 & \cos (- ϕ_{3}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{2}) & 0 & 0 & \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (- ϕ_{2}) & 0 & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{1}) & 0 & 0 & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} r \sin (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{4}) \\ r \sin (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \sin (ϕ_{4}) \\ r \cos (ϕ_{3}) \sin (ϕ_{2}) \cos (ϕ_{1}) \\ - r \cos (ϕ_{1}) \cos (ϕ_{2}) \\ - r \sin (ϕ_{1}) \end{matrix}]

{\hat{e}}_{ϕ_{2}} = \frac{\partial}{\partial ϕ_{2}} (R_{1, 2} (ϕ_{4}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{3}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{2}) R_{5, 1} (\frac{1}{2} π) R_{5, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{2}} ([\begin{array}{ccccc} \cos (ϕ_{4}) & - \sin (ϕ_{4}) & 0 & 0 & 0 \\ \sin (ϕ_{4}) & \cos (ϕ_{4}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{3}) & 0 & - \sin (- ϕ_{3}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (- ϕ_{3}) & 0 & \cos (- ϕ_{3}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{2}) & 0 & 0 & \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (- ϕ_{2}) & 0 & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{1}) & 0 & 0 & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \cos (ϕ_{4}) \\ r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \sin (ϕ_{4}) \\ r \cos (ϕ_{3}) \cos (ϕ_{2}) \sin (ϕ_{1}) \\ r \sin (ϕ_{2}) \sin (ϕ_{1}) \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{3}} = \frac{\partial}{\partial ϕ_{3}} (R_{1, 2} (ϕ_{4}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{3}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{2}) R_{5, 1} (\frac{1}{2} π) R_{5, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{3}} ([\begin{array}{ccccc} \cos (ϕ_{4}) & - \sin (ϕ_{4}) & 0 & 0 & 0 \\ \sin (ϕ_{4}) & \cos (ϕ_{4}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{3}) & 0 & - \sin (- ϕ_{3}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (- ϕ_{3}) & 0 & \cos (- ϕ_{3}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{2}) & 0 & 0 & \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (- ϕ_{2}) & 0 & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{1}) & 0 & 0 & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{4}) \\ r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \sin (ϕ_{4}) \\ - r \sin (ϕ_{1}) \sin (ϕ_{3}) \sin (ϕ_{2}) \\ 0 \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{4}} = \frac{\partial}{\partial ϕ_{4}} (R_{1, 2} (ϕ_{4}) R_{3, 1} (\frac{1}{2} π) R_{3, 1} (- ϕ_{3}) R_{4, 1} (\frac{1}{2} π) R_{4, 1} (- ϕ_{2}) R_{5, 1} (\frac{1}{2} π) R_{5, 1} (- ϕ_{1}) \hat{x}) = \frac{\partial}{\partial ϕ_{4}} ([\begin{array}{ccccc} \cos (ϕ_{4}) & - \sin (ϕ_{4}) & 0 & 0 & 0 \\ \sin (ϕ_{4}) & \cos (ϕ_{4}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & - \sin (\frac{1}{2} π) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (\frac{1}{2} π) & 0 & \cos (\frac{1}{2} π) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{3}) & 0 & - \sin (- ϕ_{3}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (- ϕ_{3}) & 0 & \cos (- ϕ_{3}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & \sin (\frac{1}{2} π) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (\frac{1}{2} π) & 0 & 0 & \cos (\frac{1}{2} π) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{2}) & 0 & 0 & \sin (- ϕ_{2}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ - \sin (- ϕ_{2}) & 0 & 0 & \cos (- ϕ_{2}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (\frac{1}{2} π) & 0 & 0 & 0 & - \sin (\frac{1}{2} π) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (\frac{1}{2} π) & 0 & 0 & 0 & \cos (\frac{1}{2} π) \end{array}] [\begin{array}{ccccc} \cos (- ϕ_{1}) & 0 & 0 & 0 & - \sin (- ϕ_{1}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (- ϕ_{1}) & 0 & 0 & 0 & \cos (- ϕ_{1}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{4}) \sin (ϕ_{1}) \\ r \sin (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{4}) \\ 0 \\ 0 \\ 0 \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r

| {\hat{e}}_{ϕ_{2}} | = r \sin (ϕ_{1})

| {\hat{e}}_{ϕ_{3}} | = r \sin (ϕ_{1}) \sin (ϕ_{2})

| {\hat{e}}_{ϕ_{4}} | = r \sin (ϕ_{2}) \sin (ϕ_{1}) \sin (ϕ_{3})

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

{\hat{e}}_{ϕ_{2}}

{\hat{e}}_{ϕ_{3}}

{\hat{e}}_{ϕ_{4}}

| =

[\begin{array}{ccccc} \sin (ϕ_{2}) \sin (ϕ_{3}) \cos (ϕ_{4}) \sin (ϕ_{1}) & r \sin (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{4}) & r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \cos (ϕ_{4}) & r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{4}) & - r \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{4}) \sin (ϕ_{1}) \\ \sin (ϕ_{2}) \sin (ϕ_{3}) \sin (ϕ_{4}) \sin (ϕ_{1}) & r \sin (ϕ_{3}) \cos (ϕ_{1}) \sin (ϕ_{2}) \sin (ϕ_{4}) & r \sin (ϕ_{3}) \sin (ϕ_{1}) \cos (ϕ_{2}) \sin (ϕ_{4}) & r \cos (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \sin (ϕ_{4}) & r \sin (ϕ_{3}) \sin (ϕ_{1}) \sin (ϕ_{2}) \cos (ϕ_{4}) \\ \sin (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) & r \cos (ϕ_{3}) \sin (ϕ_{2}) \cos (ϕ_{1}) & r \cos (ϕ_{3}) \cos (ϕ_{2}) \sin (ϕ_{1}) & - r \sin (ϕ_{1}) \sin (ϕ_{3}) \sin (ϕ_{2}) & 0 \\ - \cos (ϕ_{2}) \sin (ϕ_{1}) & - r \cos (ϕ_{1}) \cos (ϕ_{2}) & r \sin (ϕ_{2}) \sin (ϕ_{1}) & 0 & 0 \\ \cos (ϕ_{1}) & - r \sin (ϕ_{1}) & 0 & 0 & 0 \end{array}]

\sin (ϕ_{3}) \sin (ϕ_{1}) (- r^{4} {\sin (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{4} + r^{4} {\cos (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{2} - r^{4} {\cos (ϕ_{1})}^{2} {\sin (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{2} + r^{4} {\sin (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{2} - r^{4} {\cos (ϕ_{1})}^{2} {\sin (ϕ_{1})}^{2} {\sin (ϕ_{2})}^{2} - r^{4} {\sin (ϕ_{2})}^{2} {\sin (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{2} + r^{4} {\cos (ϕ_{2})}^{2} {\sin (ϕ_{1})}^{2} + r^{4} {\sin (ϕ_{1})}^{2} {\sin (ϕ_{2})}^{2} - r^{4} {\cos (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{4} - r^{4} {\sin (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{4} - r^{4} {\cos (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{4} + r^{4} {\sin (ϕ_{1})}^{2} {\cos (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{4} + r^{4} {\cos (ϕ_{2})}^{4} {\cos (ϕ_{1})}^{4} + r^{4} {\sin (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{2} {\sin (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{2} - r^{4} {\sin (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{2} {\cos (ϕ_{2})}^{2} + r^{4} {\sin (ϕ_{2})}^{2} {\cos (ϕ_{2})}^{2} {\cos (ϕ_{1})}^{4})

my alternative

R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) R_{5, 1} (ϕ_{4}) \hat{x}

[\begin{array}{ccccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{4}) & 0 & 0 & 0 & - \sin (ϕ_{4}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (ϕ_{4}) & 0 & 0 & 0 & \cos (ϕ_{4}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]

[\begin{matrix} r \cos (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) \\ r \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ r \sin (ϕ_{2}) \cos (ϕ_{3}) \cos (ϕ_{4}) \\ r \sin (ϕ_{3}) \cos (ϕ_{4}) \\ r \sin (ϕ_{4}) \end{matrix}]

{\hat{e}}_{r} = \frac{\partial}{\partial r} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) R_{5, 1} (ϕ_{4}) \hat{x}) = \frac{\partial}{\partial r} ([\begin{array}{ccccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{4}) & 0 & 0 & 0 & - \sin (ϕ_{4}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (ϕ_{4}) & 0 & 0 & 0 & \cos (ϕ_{4}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} \cos (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) \\ \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ \cos (ϕ_{3}) \sin (ϕ_{2}) \cos (ϕ_{4}) \\ \cos (ϕ_{4}) \sin (ϕ_{3}) \\ \sin (ϕ_{4}) \end{matrix}]

{\hat{e}}_{ϕ_{1}} = \frac{\partial}{\partial ϕ_{1}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) R_{5, 1} (ϕ_{4}) \hat{x}) = \frac{\partial}{\partial ϕ_{1}} ([\begin{array}{ccccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{4}) & 0 & 0 & 0 & - \sin (ϕ_{4}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (ϕ_{4}) & 0 & 0 & 0 & \cos (ϕ_{4}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ r \cos (ϕ_{2}) \cos (ϕ_{3}) \cos (ϕ_{4}) \cos (ϕ_{1}) \\ 0 \\ 0 \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{2}} = \frac{\partial}{\partial ϕ_{2}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) R_{5, 1} (ϕ_{4}) \hat{x}) = \frac{\partial}{\partial ϕ_{2}} ([\begin{array}{ccccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{4}) & 0 & 0 & 0 & - \sin (ϕ_{4}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (ϕ_{4}) & 0 & 0 & 0 & \cos (ϕ_{4}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \cos (ϕ_{4}) \sin (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) \\ - r \cos (ϕ_{4}) \sin (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ r \cos (ϕ_{4}) \cos (ϕ_{3}) \cos (ϕ_{2}) \\ 0 \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{3}} = \frac{\partial}{\partial ϕ_{3}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) R_{5, 1} (ϕ_{4}) \hat{x}) = \frac{\partial}{\partial ϕ_{3}} ([\begin{array}{ccccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{4}) & 0 & 0 & 0 & - \sin (ϕ_{4}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (ϕ_{4}) & 0 & 0 & 0 & \cos (ϕ_{4}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \cos (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \sin (ϕ_{3}) \\ - r \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \sin (ϕ_{3}) \\ - r \sin (ϕ_{2}) \cos (ϕ_{4}) \sin (ϕ_{3}) \\ r \cos (ϕ_{3}) \cos (ϕ_{4}) \\ 0 \end{matrix}]

{\hat{e}}_{ϕ_{4}} = \frac{\partial}{\partial ϕ_{4}} (R_{1, 2} (ϕ_{1}) R_{3, 1} (ϕ_{2}) R_{1, 4} (ϕ_{3}) R_{5, 1} (ϕ_{4}) \hat{x}) = \frac{\partial}{\partial ϕ_{4}} ([\begin{array}{ccccc} \cos (ϕ_{1}) & - \sin (ϕ_{1}) & 0 & 0 & 0 \\ \sin (ϕ_{1}) & \cos (ϕ_{1}) & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{2}) & 0 & - \sin (ϕ_{2}) & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \sin (ϕ_{2}) & 0 & \cos (ϕ_{2}) & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{3}) & 0 & 0 & - \sin (ϕ_{3}) & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sin (ϕ_{3}) & 0 & 0 & \cos (ϕ_{3}) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}] [\begin{array}{ccccc} \cos (ϕ_{4}) & 0 & 0 & 0 & - \sin (ϕ_{4}) \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \sin (ϕ_{4}) & 0 & 0 & 0 & \cos (ϕ_{4}) \end{array}] [\begin{matrix} r \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} - r \sin (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) \\ - r \sin (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ - r \sin (ϕ_{2}) \sin (ϕ_{4}) \cos (ϕ_{3}) \\ - r \sin (ϕ_{4}) \sin (ϕ_{3}) \\ r \cos (ϕ_{4}) \end{matrix}]

| {\hat{e}}_{r} | = 1

| {\hat{e}}_{ϕ_{1}} | = r \cos (ϕ_{2}) \cos (ϕ_{3}) \cos (ϕ_{4})

| {\hat{e}}_{ϕ_{2}} | = r \cos (ϕ_{3}) \cos (ϕ_{4})

| {\hat{e}}_{ϕ_{3}} | = r \cos (ϕ_{4})

| {\hat{e}}_{ϕ_{4}} | = r

{\hat{e}}_{r}

{\hat{e}}_{ϕ_{1}}

{\hat{e}}_{ϕ_{2}}

{\hat{e}}_{ϕ_{3}}

{\hat{e}}_{ϕ_{4}}

| =

[\begin{array}{ccccc} \cos (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) & - r \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) & - r \cos (ϕ_{4}) \sin (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) & - r \cos (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \sin (ϕ_{3}) & - r \sin (ϕ_{4}) \cos (ϕ_{2}) \cos (ϕ_{1}) \cos (ϕ_{3}) \\ \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) & r \cos (ϕ_{2}) \cos (ϕ_{3}) \cos (ϕ_{4}) \cos (ϕ_{1}) & - r \cos (ϕ_{4}) \sin (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) & - r \cos (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \sin (ϕ_{3}) & - r \sin (ϕ_{4}) \cos (ϕ_{2}) \sin (ϕ_{1}) \cos (ϕ_{3}) \\ \cos (ϕ_{3}) \sin (ϕ_{2}) \cos (ϕ_{4}) & 0 & r \cos (ϕ_{4}) \cos (ϕ_{3}) \cos (ϕ_{2}) & - r \sin (ϕ_{2}) \cos (ϕ_{4}) \sin (ϕ_{3}) & - r \sin (ϕ_{2}) \sin (ϕ_{4}) \cos (ϕ_{3}) \\ \cos (ϕ_{4}) \sin (ϕ_{3}) & 0 & 0 & r \cos (ϕ_{3}) \cos (ϕ_{4}) & - r \sin (ϕ_{4}) \sin (ϕ_{3}) \\ \sin (ϕ_{4}) & 0 & 0 & 0 & r \cos (ϕ_{4}) \end{array}]

\cos (ϕ_{2}) \cos (ϕ_{4}) (r^{4} - r^{4} {\cos (ϕ_{4})}^{2} + r^{4} {\cos (ϕ_{4})}^{4} + r^{4} {\cos (ϕ_{3})}^{4} - r^{4} {\cos (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{4} + r^{4} {\cos (ϕ_{3})}^{4} {\cos (ϕ_{4})}^{4} + r^{4} {\sin (ϕ_{3})}^{2} {\cos (ϕ_{3})}^{2} - r^{4} {\cos (ϕ_{3})}^{2} {\cos (ϕ_{4})}^{2} {\sin (ϕ_{3})}^{2} + r^{4} {\cos (ϕ_{3})}^{2} {\sin (ϕ_{3})}^{2} {\cos (ϕ_{4})}^{4} + r^{4} {\sin (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{2} - r^{4} {\sin (ϕ_{4})}^{2} {\cos (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{2} + r^{4} {\sin (ϕ_{4})}^{2} {\sin (ϕ_{3})}^{2} - r^{4} {\sin (ϕ_{4})}^{2} {\cos (ϕ_{4})}^{2} {\sin (ϕ_{3})}^{2} - r^{4} {\cos (ϕ_{3})}^{2} + r^{4} {\cos (ϕ_{3})}^{2} {\cos (ϕ_{4})}^{2} - r^{4} {\cos (ϕ_{3})}^{2} {\cos (ϕ_{4})}^{4} - r^{4} {\sin (ϕ_{3})}^{2} + r^{4} {\cos (ϕ_{4})}^{2} {\sin (ϕ_{3})}^{2} - r^{4} {\sin (ϕ_{3})}^{2} {\cos (ϕ_{4})}^{4} - r^{4} {\sin (ϕ_{4})}^{2} + r^{4} {\cos (ϕ_{4})}^{2} {\sin (ϕ_{4})}^{2} - r^{4} {\sin (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{4} + r^{4} {\sin (ϕ_{4})}^{2} {\cos (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{4} + r^{4} {\cos (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{2} {\sin (ϕ_{3})}^{2} {\sin (ϕ_{4})}^{2} - r^{4} {\sin (ϕ_{4})}^{2} {\cos (ϕ_{3})}^{2} {\sin (ϕ_{3})}^{2})