spherical, log-radial, coordinate

chart coordinates:

x^{\tilde{μ}} = {ρ, θ, ϕ}

chart coordinate basis:

e_{\tilde{μ}} = {e_{\tilde{ρ}}, e_{\tilde{θ}}, e_{\tilde{ϕ}}}

embedding coordinates:

u^{I} = {x, y, z}

embedding basis

e_{I} = {e_{x}, e_{y}, e_{z}}

transform from basis to coordinate:

{\tilde{e}}_{ρ}^{ρ} = 1

;

{\tilde{e}}_{θ}^{θ} = 1

;

{\tilde{e}}_{ϕ}^{ϕ} = 1

transform from coorinate to basis:

{\tilde{e}}^{ρ}_{ρ} = 1

;

{\tilde{e}}^{θ}_{θ} = 1

;

{\tilde{e}}^{ϕ}_{ϕ} = 1

tensor index associated with coordinate

ρ

is index

ρ

with operator

e_{ρ} (ζ) =

\frac{\partial ζ}{\partial ρ}

tensor index associated with coordinate

θ

is index

θ

with operator

e_{θ} (ζ) =

\frac{\partial ζ}{\partial θ}

tensor index associated with coordinate

ϕ

is index

ϕ

with operator

e_{ϕ} (ζ) =

\frac{\partial ζ}{\partial ϕ}

flat metric:

{η_{x}}_{x} = 1

;

{η_{y}}_{y} = 1

;

{η_{z}}_{z} = 1

chart in embedded coordinates:

u^{x} = \frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})} s i n (θ) c o s (ϕ)

;

u^{y} = \frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})} s i n (θ) s i n (ϕ)

;

u^{z} = \frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})} c o s (θ)

basis operators applied to chart:

{e_{u}}^{I} = {u^{I}}_{, u}

{e_{ρ}}^{x} = \frac{A c o s h (\frac{1}{w} (ρ)) s i n (θ) c o s (ϕ)}{w s i n h (\frac{1}{w})}

;

{e_{ρ}}^{y} = \frac{A c o s h (\frac{1}{w} (ρ)) s i n (θ) s i n (ϕ)}{w s i n h (\frac{1}{w})}

;

{e_{ρ}}^{z} = \frac{A c o s h (\frac{1}{w} (ρ)) c o s (θ)}{w s i n h (\frac{1}{w})}

;

{e_{θ}}^{x} = \frac{A c o s (θ) s i n h (\frac{1}{w} (ρ)) c o s (ϕ)}{s i n h (\frac{1}{w})}

;

{e_{θ}}^{y} = \frac{A c o s (θ) s i n h (\frac{1}{w} (ρ)) s i n (ϕ)}{s i n h (\frac{1}{w})}

;

{e_{θ}}^{z} = \frac{- A s i n h (\frac{1}{w} (ρ)) s i n (θ)}{s i n h (\frac{1}{w})}

;

{e_{ϕ}}^{x} = \frac{- A s i n (ϕ) s i n (θ) s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})}

;

{e_{ϕ}}^{y} = \frac{A s i n (θ) s i n h (\frac{1}{w} (ρ)) c o s (ϕ)}{s i n h (\frac{1}{w})}

{e^{ρ}}_{x} = \frac{w s i n (θ) c o s (ϕ) s i n h (\frac{1}{w})}{A c o s h (\frac{1}{w} (ρ))}

;

{e^{ρ}}_{y} = \frac{w s i n (θ) s i n (ϕ) s i n h (\frac{1}{w})}{A c o s h (\frac{1}{w} (ρ))}

;

{e^{ρ}}_{z} = \frac{w c o s (θ) s i n h (\frac{1}{w})}{A c o s h (\frac{1}{w} (ρ))}

;

{e^{θ}}_{x} = \frac{c o s (θ) c o s (ϕ) s i n h (\frac{1}{w})}{A s i n h (\frac{1}{w} (ρ))}

;

{e^{θ}}_{y} = \frac{s i n h (\frac{1}{w}) c o s (θ) s i n (ϕ)}{A s i n h (\frac{1}{w} (ρ))}

;

{e^{θ}}_{z} = \frac{- s i n h (\frac{1}{w}) s i n (θ)}{A s i n h (\frac{1}{w} (ρ))}

;

{e^{ϕ}}_{x} = \frac{- s i n (ϕ) s i n h (\frac{1}{w})}{A s i n h (\frac{1}{w} (ρ)) s i n (θ)}

;

{e^{ϕ}}_{y} = \frac{s i n h (\frac{1}{w}) c o s (ϕ)}{A s i n h (\frac{1}{w} (ρ)) s i n (θ)}

{e_{u}}^{I} {e^{v}}_{I} = \overset{u ↓ v \to}{[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]}

{e_{u}}^{I} {e^{u}}_{J} = \overset{I ↓ J \to}{[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]}

basis determinant:

d e t (e) = \frac{A^{3} {s i n h (\frac{1}{w} (ρ))}^{2} c o s h (\frac{1}{w} (ρ)) s i n (θ)}{w {s i n h (\frac{1}{w})}^{3}}

{c_{a}}_{b}^{c} = 0

{g_{u}}_{v} = {e_{u}}^{I} {e_{v}}^{J} {η_{I}}_{J}

{g_{ρ}}_{ρ} = \frac{A^{2} {c o s h (\frac{1}{w} (ρ))}^{2}}{w^{2} {s i n h (\frac{1}{w})}^{2}}

;

{g_{θ}}_{θ} = \frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}{{s i n h (\frac{1}{w})}^{2}}

;

{g_{ϕ}}_{ϕ} = \frac{- A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} (- 1 + {c o s (θ)}^{2})}{{s i n h (\frac{1}{w})}^{2}}

{g_{u}}_{v} = {e_{u}}^{I} {e_{v}}^{J} {η_{I}}_{J}

metric determinant:

d e t (g) = \frac{- A^{6} {c o s h (\frac{1}{w} (ρ))}^{2} {s i n h (\frac{1}{w} (ρ))}^{4} (- 1 + {c o s (θ)}^{2})}{w^{2} {s i n h (\frac{1}{w})}^{6}}

{Γ_{a}}_{b}_{c} = \frac{1}{2} ({g_{a}}_{b}_{, c} + {g_{a}}_{c}_{, b} - {g_{b}}_{c}_{, a} + {c_{a}}_{b}_{c} + {c_{a}}_{c}_{b} - {c_{c}}_{b}_{a})

commutation coefficients:

{c_{a}}_{b}^{c} = 0

metric:

{g_{ρ}}_{ρ} = \frac{A^{2} {c o s h (\frac{1}{w} (ρ))}^{2}}{w^{2} {s i n h (\frac{1}{w})}^{2}}

;

{g_{θ}}_{θ} = \frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}{{s i n h (\frac{1}{w})}^{2}}

;

{g_{ϕ}}_{ϕ} = \frac{- A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} (- 1 + {c o s (θ)}^{2})}{{s i n h (\frac{1}{w})}^{2}}

metric inverse:

{g^{ρ}}^{ρ} = \frac{w^{2} {s i n h (\frac{1}{w})}^{2}}{A^{2} {c o s h (\frac{1}{w} (ρ))}^{2}}

;

{g^{θ}}^{θ} = \frac{{s i n h (\frac{1}{w})}^{2}}{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{g^{ϕ}}^{ϕ} = \frac{{s i n h (\frac{1}{w})}^{2}}{- A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} (- 1 + {c o s (θ)}^{2})}

metric derivative:

{\partial g}_{ρ}_{ρ}_{ρ} = \frac{2 A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ))}{w^{3} {s i n h (\frac{1}{w})}^{2}}

;

{\partial g}_{θ}_{θ}_{ρ} = \frac{2 A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ))}{w {s i n h (\frac{1}{w})}^{2}}

;

{\partial g}_{ϕ}_{ϕ}_{ρ} = \frac{- 2 A^{2} s i n h (\frac{1}{w} (ρ)) c o s h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{w {s i n h (\frac{1}{w})}^{2}}

;

{\partial g}_{ϕ}_{ϕ}_{θ} = \frac{2 A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ) c o s (θ)}{{s i n h (\frac{1}{w})}^{2}}

1st kind Christoffel:

{Γ_{ρ}}_{ρ}_{ρ} = \frac{A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ))}{w^{3} {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{ρ}}_{θ}_{θ} = \frac{- A^{2} s i n h (\frac{1}{w} (ρ)) c o s h (\frac{1}{w} (ρ))}{w {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{ρ}}_{ϕ}_{ϕ} = \frac{A^{2} s i n h (\frac{1}{w} (ρ)) c o s h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{w {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{θ}}_{ρ}_{θ} = \frac{A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ))}{w {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{θ}}_{θ}_{ρ} = \frac{A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ))}{w {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{θ}}_{ϕ}_{ϕ} = \frac{- A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ) c o s (θ)}{{s i n h (\frac{1}{w})}^{2}}

;

{Γ_{ϕ}}_{ρ}_{ϕ} = \frac{- A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{w {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{ϕ}}_{θ}_{ϕ} = \frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ) c o s (θ)}{{s i n h (\frac{1}{w})}^{2}}

;

{Γ_{ϕ}}_{ϕ}_{ρ} = \frac{- A^{2} c o s h (\frac{1}{w} (ρ)) s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{w {s i n h (\frac{1}{w})}^{2}}

;

{Γ_{ϕ}}_{ϕ}_{θ} = \frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ) c o s (θ)}{{s i n h (\frac{1}{w})}^{2}}

connection coefficients / 2nd kind Christoffel:

{Γ^{ρ}}_{ρ}_{ρ} = \frac{s i n h (\frac{1}{w} (ρ))}{w c o s h (\frac{1}{w} (ρ))}

;

{Γ^{ρ}}_{θ}_{θ} = \frac{- w s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))}

;

{Γ^{ρ}}_{ϕ}_{ϕ} = \frac{w s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{c o s h (\frac{1}{w} (ρ))}

;

{Γ^{θ}}_{ρ}_{θ} = \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))}

;

{Γ^{θ}}_{θ}_{ρ} = \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))}

;

{Γ^{θ}}_{ϕ}_{ϕ} = - s i n (θ) c o s (θ)

;

{Γ^{ϕ}}_{ρ}_{ϕ} = \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))}

;

{Γ^{ϕ}}_{θ}_{ϕ} = \frac{c o s (θ)}{s i n (θ)}

;

{Γ^{ϕ}}_{ϕ}_{ρ} = \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))}

;

{Γ^{ϕ}}_{ϕ}_{θ} = \frac{c o s (θ)}{s i n (θ)}

connection coefficients derivative:

{\partial Γ}^{ρ}_{ρ}_{ρ}_{ρ} = \frac{(c o s h (\frac{1}{w} (ρ)) - s i n h (\frac{1}{w} (ρ))) (c o s h (\frac{1}{w} (ρ)) + s i n h (\frac{1}{w} (ρ)))}{w^{2} {c o s h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{ρ}_{θ}_{θ}_{ρ} = \frac{- ({c o s h (\frac{1}{w} (ρ))}^{2} - {s i n h (\frac{1}{w} (ρ))}^{2})}{{c o s h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{ρ}_{ϕ}_{ϕ}_{ρ} = \frac{- {c o s h (\frac{1}{w} (ρ))}^{2} + {s i n h (\frac{1}{w} (ρ))}^{2} + {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} - {c o s (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}{{c o s h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{ρ}_{ϕ}_{ϕ}_{θ} = \frac{- 2 w c o s (θ) s i n (θ) s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))}

;

{\partial Γ}^{θ}_{ρ}_{θ}_{ρ} = \frac{(s i n h (\frac{1}{w} (ρ)) - c o s h (\frac{1}{w} (ρ))) (s i n h (\frac{1}{w} (ρ)) + c o s h (\frac{1}{w} (ρ)))}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{θ}_{θ}_{ρ}_{ρ} = \frac{(s i n h (\frac{1}{w} (ρ)) - c o s h (\frac{1}{w} (ρ))) (s i n h (\frac{1}{w} (ρ)) + c o s h (\frac{1}{w} (ρ)))}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{θ}_{ϕ}_{ϕ}_{θ} = (s i n (θ) - c o s (θ)) (s i n (θ) + c o s (θ))

;

{\partial Γ}^{ϕ}_{ρ}_{ϕ}_{ρ} = \frac{(s i n h (\frac{1}{w} (ρ)) - c o s h (\frac{1}{w} (ρ))) (s i n h (\frac{1}{w} (ρ)) + c o s h (\frac{1}{w} (ρ)))}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{ϕ}_{θ}_{ϕ}_{θ} = - \frac{1}{1 - {c o s (θ)}^{2}}

;

{\partial Γ}^{ϕ}_{ϕ}_{ρ}_{ρ} = \frac{(s i n h (\frac{1}{w} (ρ)) - c o s h (\frac{1}{w} (ρ))) (s i n h (\frac{1}{w} (ρ)) + c o s h (\frac{1}{w} (ρ)))}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{\partial Γ}^{ϕ}_{ϕ}_{θ}_{θ} = - \frac{1}{1 - {c o s (θ)}^{2}}

connection coefficients squared:

{(Γ^{2})}^{ρ}_{ρ}_{ρ}_{ρ} = \frac{{s i n h (\frac{1}{w} (ρ))}^{2}}{w^{2} {c o s h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{ρ}_{ρ}_{θ}_{θ} = - 1

;

{(Γ^{2})}^{ρ}_{ρ}_{ϕ}_{ϕ} = - (1 - {c o s (θ)}^{2})

;

{(Γ^{2})}^{ρ}_{θ}_{ρ}_{θ} = \frac{- {s i n h (\frac{1}{w} (ρ))}^{2}}{{c o s h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{ρ}_{θ}_{θ}_{ρ} = - 1

;

{(Γ^{2})}^{ρ}_{θ}_{ϕ}_{ϕ} = \frac{- w s i n h (\frac{1}{w} (ρ)) s i n (θ) c o s (θ)}{c o s h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{ρ}_{ϕ}_{ρ}_{ϕ} = \frac{{s i n h (\frac{1}{w} (ρ))}^{2} (- 1 + {c o s (θ)}^{2})}{{c o s h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{ρ}_{ϕ}_{θ}_{ϕ} = \frac{w s i n (θ) s i n h (\frac{1}{w} (ρ)) c o s (θ)}{c o s h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{ρ}_{ϕ}_{ϕ}_{ρ} = - (1 - {c o s (θ)}^{2})

;

{(Γ^{2})}^{ρ}_{ϕ}_{ϕ}_{θ} = \frac{- w s i n h (\frac{1}{w} (ρ)) s i n (θ) c o s (θ)}{c o s h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{θ}_{ρ}_{ρ}_{θ} = \frac{{c o s h (\frac{1}{w} (ρ))}^{2}}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{θ}_{ρ}_{θ}_{ρ} = \frac{1}{w^{2}}

;

{(Γ^{2})}^{θ}_{ρ}_{ϕ}_{ϕ} = \frac{- c o s h (\frac{1}{w} (ρ)) s i n (θ) c o s (θ)}{w s i n h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{θ}_{θ}_{ρ}_{ρ} = \frac{{c o s h (\frac{1}{w} (ρ))}^{2}}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{θ}_{θ}_{θ}_{θ} = - 1

;

{(Γ^{2})}^{θ}_{θ}_{ϕ}_{ϕ} = - {c o s (θ)}^{2}

;

{(Γ^{2})}^{θ}_{ϕ}_{ρ}_{ϕ} = \frac{- c o s h (\frac{1}{w} (ρ)) c o s (θ) s i n (θ)}{w s i n h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{θ}_{ϕ}_{θ}_{ϕ} = - (1 - {c o s (θ)}^{2})

;

{(Γ^{2})}^{θ}_{ϕ}_{ϕ}_{ρ} = \frac{- c o s h (\frac{1}{w} (ρ)) s i n (θ) c o s (θ)}{w s i n h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{θ}_{ϕ}_{ϕ}_{θ} = - {c o s (θ)}^{2}

;

{(Γ^{2})}^{ϕ}_{ρ}_{ρ}_{ϕ} = \frac{{c o s h (\frac{1}{w} (ρ))}^{2}}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{ϕ}_{ρ}_{θ}_{ϕ} = \frac{c o s h (\frac{1}{w} (ρ)) c o s (θ)}{w s i n h (\frac{1}{w} (ρ)) s i n (θ)}

;

{(Γ^{2})}^{ϕ}_{ρ}_{ϕ}_{ρ} = \frac{1}{w^{2}}

;

{(Γ^{2})}^{ϕ}_{ρ}_{ϕ}_{θ} = \frac{c o s (θ) c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ)) s i n (θ)}

;

{(Γ^{2})}^{ϕ}_{θ}_{ρ}_{ϕ} = \frac{c o s (θ) c o s h (\frac{1}{w} (ρ))}{w s i n (θ) s i n h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{ϕ}_{θ}_{θ}_{ϕ} = \frac{{c o s (θ)}^{2}}{1 - {c o s (θ)}^{2}}

;

{(Γ^{2})}^{ϕ}_{θ}_{ϕ}_{ρ} = \frac{c o s (θ) c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ)) s i n (θ)}

;

{(Γ^{2})}^{ϕ}_{θ}_{ϕ}_{θ} = - 1

;

{(Γ^{2})}^{ϕ}_{ϕ}_{ρ}_{ρ} = \frac{{c o s h (\frac{1}{w} (ρ))}^{2}}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}}

;

{(Γ^{2})}^{ϕ}_{ϕ}_{ρ}_{θ} = \frac{c o s (θ) c o s h (\frac{1}{w} (ρ))}{w s i n (θ) s i n h (\frac{1}{w} (ρ))}

;

{(Γ^{2})}^{ϕ}_{ϕ}_{θ}_{ρ} = \frac{c o s h (\frac{1}{w} (ρ)) c o s (θ)}{w s i n h (\frac{1}{w} (ρ)) s i n (θ)}

;

{(Γ^{2})}^{ϕ}_{ϕ}_{θ}_{θ} = \frac{{c o s (θ)}^{2}}{1 - {c o s (θ)}^{2}}

;

{(Γ^{2})}^{ϕ}_{ϕ}_{ϕ}_{ϕ} = - 1

Riemann curvature,

♯ ♭ ♭ ♭

{R^{a}}_{b}_{c}_{d} = 0

Riemann curvature,

♯ ♯ ♭ ♭

{R^{a}}^{b}_{c}_{d} = 0

Ricci curvature,

♯ ♭

{R^{a}}_{b} = 0

Gaussian curvature:

0

trace-free Ricci,

♯ ♭

{(R^{T F})}^{a}_{b} = 0

Einstein / trace-reversed Ricci curvature,

♯ ♭

{G^{a}}_{b} = 0

Schouten,

♯ ♭

{P^{a}}_{b} = 0

Weyl,

♯ ♯ ♭ ♭

{C^{a}}^{b}_{c}_{d} = 0

Weyl,

♭ ♭ ♭ ♭

{C_{a}}_{b}_{c}_{d} = 0

Plebanski,

♯ ♯ ♭ ♭

{P^{a}}^{b}_{c}_{d} = 0

divergence:

{A^{i}}_{, i} + {Γ^{i}}_{i}_{j} A^{j} = \frac{\partial A^{ρ}}{\partial ρ} + \frac{\partial A^{θ}}{\partial θ} + \frac{\partial A^{ϕ}}{\partial ϕ} + A^{ρ} \cdot s i n h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{c o s h (ρ \cdot \frac{1}{w})} + 2 A^{ρ} \cdot c o s h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{s i n h (ρ \cdot \frac{1}{w})} + A^{θ} \cdot c o s (θ) \frac{1}{s i n (θ)}

geodesic:

\overset{a ↓}{[\begin{matrix} \ddot{ρ} \\ \ddot{θ} \\ \ddot{ϕ} \end{matrix}]} = \overset{a ↓}{[\begin{matrix} {\dot{ϕ}}^{2} w^{2} s i n h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{c o s h (ρ \cdot \frac{1}{w})} + - 1 {\dot{ϕ}}^{2} w^{2} {c o s (θ)}^{2} s i n h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{c o s h (ρ \cdot \frac{1}{w})} + - 1 {\dot{ρ}}^{2} s i n h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{c o s h (ρ \cdot \frac{1}{w})} + {\dot{θ}}^{2} w^{2} s i n h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{c o s h (ρ \cdot \frac{1}{w})} \\ - 2 \dot{ρ} \cdot \dot{θ} \cdot c o s h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{s i n h (ρ \cdot \frac{1}{w})} + w {\dot{ϕ}}^{2} s i n h (ρ \cdot \frac{1}{w}) c o s (θ) s i n (θ) \frac{1}{w} \frac{1}{s i n h (ρ \cdot \frac{1}{w})} \\ - 2 \dot{ϕ} \cdot \dot{ρ} \cdot s i n (θ) c o s h (ρ \cdot \frac{1}{w}) \frac{1}{w} \frac{1}{s i n (θ)} \frac{1}{s i n h (ρ \cdot \frac{1}{w})} + - 2 \dot{ϕ} \cdot \dot{θ} \cdot w s i n h (ρ \cdot \frac{1}{w}) c o s (θ) \frac{1}{w} \frac{1}{s i n (θ)} \frac{1}{s i n h (ρ \cdot \frac{1}{w})} \end{matrix}]}

parallel propagators:

[Γ_{ρ}] = [\begin{matrix} \frac{s i n h (\frac{1}{w} (ρ))}{w c o s h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 \\ 0 & 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} \end{matrix}]

\int_{ρ_{L}}^{ρ_{R}} d ρ ([\begin{matrix} \frac{s i n h (\frac{1}{w} (ρ))}{w c o s h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 \\ 0 & 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} \end{matrix}])

[\begin{matrix} l o g (\frac{c o s h (\frac{1}{w} (ρ_{R}))}{c o s h (\frac{1}{w} (ρ_{L}))}) & 0 & 0 \\ 0 & l o g (\frac{| s i n h (\frac{1}{w} (ρ_{R})) |}{| s i n h (\frac{1}{w} (ρ_{L})) |}) & 0 \\ 0 & 0 & l o g (\frac{| s i n h (\frac{1}{w} (ρ_{R})) |}{| s i n h (\frac{1}{w} (ρ_{L})) |}) \end{matrix}]

P_{ρ}

e^{(- (\int_{ρ_{L}}^{ρ_{R}} d ρ ([\begin{matrix} \frac{s i n h (\frac{1}{w} (ρ))}{w c o s h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 \\ 0 & 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} \end{matrix}])))}

[\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}]

{P_{ρ}}^{- 1}

e^{(\int_{ρ_{L}}^{ρ_{R}} d ρ ([\begin{matrix} \frac{s i n h (\frac{1}{w} (ρ))}{w c o s h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 \\ 0 & 0 & \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} \end{matrix}]))}

[\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{R}))}{c o s h (\frac{1}{w} (ρ_{L}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{R})) |}{| s i n h (\frac{1}{w} (ρ_{L})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{R})) |}{| s i n h (\frac{1}{w} (ρ_{L})) |} \end{matrix}]

[Γ_{θ}] = [\begin{matrix} 0 & \frac{- w s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & 0 & \frac{c o s (θ)}{s i n (θ)} \end{matrix}]

\int_{θ_{L}}^{θ_{R}} d θ ([\begin{matrix} 0 & \frac{- w s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & 0 & \frac{c o s (θ)}{s i n (θ)} \end{matrix}])

[\begin{matrix} 0 & \frac{- w s i n h (\frac{1}{w} (ρ)) (θ_{R} - θ_{L})}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{- c o s h (\frac{1}{w} (ρ)) (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & 0 & l o g (\frac{| s i n (θ_{R}) |}{| s i n (θ_{L}) |}) \end{matrix}]

P_{θ}

e^{(- (\int_{θ_{L}}^{θ_{R}} d θ ([\begin{matrix} 0 & \frac{- w s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & 0 & \frac{c o s (θ)}{s i n (θ)} \end{matrix}])))}

[\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ)) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}]

{P_{θ}}^{- 1}

e^{(\int_{θ_{L}}^{θ_{R}} d θ ([\begin{matrix} 0 & \frac{- w s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & 0 & 0 \\ 0 & 0 & \frac{c o s (θ)}{s i n (θ)} \end{matrix}]))}

[\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{w s i n h (\frac{1}{w} (ρ)) s i n (θ_{L} - θ_{R})}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ)) s i n (θ_{L} - θ_{R})}{- w s i n h (\frac{1}{w} (ρ))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{R}) |}{| s i n (θ_{L}) |} \end{matrix}]

[Γ_{ϕ}] = [\begin{matrix} 0 & 0 & \frac{w s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{c o s h (\frac{1}{w} (ρ))} \\ 0 & 0 & - c o s (θ) s i n (θ) \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & \frac{c o s (θ)}{s i n (θ)} & 0 \end{matrix}]

\int_{ϕ_{L}}^{ϕ_{R}} d ϕ ([\begin{matrix} 0 & 0 & \frac{w s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{c o s h (\frac{1}{w} (ρ))} \\ 0 & 0 & - c o s (θ) s i n (θ) \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & \frac{c o s (θ)}{s i n (θ)} & 0 \end{matrix}])

[\begin{matrix} 0 & 0 & \frac{- w s i n h (\frac{1}{w} (ρ)) (ϕ_{L} \cdot {c o s (θ)}^{2} - ϕ_{L} - ϕ_{R} \cdot {c o s (θ)}^{2} + ϕ_{R})}{c o s h (\frac{1}{w} (ρ))} \\ 0 & 0 & - s i n (θ) c o s (θ) (ϕ_{R} - ϕ_{L}) \\ \frac{- c o s h (\frac{1}{w} (ρ)) (ϕ_{L} - ϕ_{R})}{w s i n h (\frac{1}{w} (ρ))} & \frac{- c o s (θ) (ϕ_{L} - ϕ_{R})}{s i n (θ)} & 0 \end{matrix}]

P_{ϕ}

e^{(- (\int_{ϕ_{L}}^{ϕ_{R}} d ϕ ([\begin{matrix} 0 & 0 & \frac{w s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{c o s h (\frac{1}{w} (ρ))} \\ 0 & 0 & - c o s (θ) s i n (θ) \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & \frac{c o s (θ)}{s i n (θ)} & 0 \end{matrix}])))}

[\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ)}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ) c o s (θ) s i n h (\frac{1}{w} (ρ))}{2 c o s h (\frac{1}{w} (ρ))} & \frac{w s i n h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ)}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ)}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ)}^{2} {s i n (θ)}^{2} + {ϕ_{L}}^{3} {c o s (θ)}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ)}^{2} - {ϕ_{L}}^{3} {s i n (θ)}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ)}^{2} - {ϕ_{R}}^{3} {c o s (θ)}^{4} - {ϕ_{R}}^{3} {s i n (θ)}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{3} {s i n (θ)}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ) c o s (θ) c o s h (\frac{1}{w} (ρ))}{2 w s i n h (\frac{1}{w} (ρ))} & \frac{1}{2} ({c o s (θ)}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ) c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ))} & \frac{- c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ)} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})

{P_{ϕ}}^{- 1}

e^{(\int_{ϕ_{L}}^{ϕ_{R}} d ϕ ([\begin{matrix} 0 & 0 & \frac{w s i n h (\frac{1}{w} (ρ)) (- 1 + {c o s (θ)}^{2})}{c o s h (\frac{1}{w} (ρ))} \\ 0 & 0 & - c o s (θ) s i n (θ) \\ \frac{c o s h (\frac{1}{w} (ρ))}{w s i n h (\frac{1}{w} (ρ))} & \frac{c o s (θ)}{s i n (θ)} & 0 \end{matrix}]))}

[\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ)}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ) c o s (θ) s i n h (\frac{1}{w} (ρ))}{2 c o s h (\frac{1}{w} (ρ))} & \frac{- w s i n h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ)}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ)}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ)}^{2} {s i n (θ)}^{2} + {ϕ_{L}}^{3} {c o s (θ)}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ)}^{2} - {ϕ_{L}}^{3} {s i n (θ)}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ)}^{2} - {ϕ_{R}}^{3} {c o s (θ)}^{4} - {ϕ_{R}}^{3} {s i n (θ)}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{3} {s i n (θ)}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ))} \\ \frac{(2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ) c o s (θ) c o s h (\frac{1}{w} (ρ))}{2 w s i n h (\frac{1}{w} (ρ))} & \frac{1}{2} ({c o s (θ)}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (- c o s (θ) s i n (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} + {ϕ_{L}}^{3} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})) \\ \frac{c o s h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ))} & \frac{c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ)} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})

propagator commutation:

[

P_{ρ}

P_{θ}

] =

[\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}] [\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{L}))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ_{L}))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}] - [\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{R}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ_{R})) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ_{R}))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}] [\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}]

[\begin{matrix} 0 & \frac{w s i n (θ_{L} - θ_{R}) (| s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) - s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) |)}{c o s h (\frac{1}{w} (ρ_{R})) | s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L} - θ_{R}) (| s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) - s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) |)}{w | s i n h (\frac{1}{w} (ρ_{R})) | s i n h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

[

P_{ρ}

P_{ϕ}

] =

[\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}] ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ_{L}) c o s (θ_{L}) s i n h (\frac{1}{w} (ρ_{L}))}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{w s i n h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ_{L}))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ_{L}) c o s (θ_{L}) c o s h (\frac{1}{w} (ρ_{L}))}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{1}{2} ({c o s (θ_{L})}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ_{L}) c o s (θ_{L}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{- c o s (θ_{L}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ_{L})} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) - ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ_{L}) c o s (θ_{L}) s i n h (\frac{1}{w} (ρ_{R}))}{2 c o s h (\frac{1}{w} (ρ_{R}))} & \frac{w s i n h (\frac{1}{w} (ρ_{R})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ_{R}))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ_{L}) c o s (θ_{L}) c o s h (\frac{1}{w} (ρ_{R}))}{2 w s i n h (\frac{1}{w} (ρ_{R}))} & \frac{1}{2} ({c o s (θ_{L})}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ_{L}) c o s (θ_{L}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ_{R})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ_{R}))} & \frac{- c o s (θ_{L}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ_{L})} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) [\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}]

[\begin{matrix} [\begin{matrix} 0 & \frac{w c o s (θ_{L}) s i n (θ_{L}) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})))}{2 {c o s h (\frac{1}{w} (ρ_{R}))}^{2}} & \frac{w (9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 2 {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 2 {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 2 {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 2 {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})))}{6 {c o s h (\frac{1}{w} (ρ_{R}))}^{2}} \\ \frac{- (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R}))) c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L}) c o s (θ_{L})}{2 w s i n h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ \frac{- c o s h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})))}{6 w s i n h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \end{matrix}] & 0 & 0 \\ 0 & [\begin{matrix} 0 & \frac{- w s i n (θ_{L}) | s i n h (\frac{1}{w} (ρ_{L})) | (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L}))) c o s (θ_{L})}{2 c o s h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) | s i n h (\frac{1}{w} (ρ_{R})) |} & \frac{- w | s i n h (\frac{1}{w} (ρ_{L})) | (6 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 2 {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 2 {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 2 {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 2 {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})))}{6 c o s h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) | s i n h (\frac{1}{w} (ρ_{R})) |} \\ \frac{- (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R}))) | s i n h (\frac{1}{w} (ρ_{L})) | c o s (θ_{L}) s i n (θ_{L})}{2 w s i n h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) |} & 0 & 0 \\ \frac{- | s i n h (\frac{1}{w} (ρ_{L})) | (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})))}{6 w s i n h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) |} & 0 & 0 \end{matrix}] & 0 \\ 0 & 0 & [\begin{matrix} 0 & \frac{- w s i n (θ_{L}) | s i n h (\frac{1}{w} (ρ_{L})) | (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L}))) c o s (θ_{L})}{2 c o s h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) | s i n h (\frac{1}{w} (ρ_{R})) |} & \frac{- w | s i n h (\frac{1}{w} (ρ_{L})) | (6 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 2 {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 2 {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 2 {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 2 {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})))}{6 c o s h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) | s i n h (\frac{1}{w} (ρ_{R})) |} \\ \frac{- (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{R}}^{2} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R}))) | s i n h (\frac{1}{w} (ρ_{L})) | c o s (θ_{L}) s i n (θ_{L})}{2 w s i n h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) |} & 0 & 0 \\ \frac{- | s i n h (\frac{1}{w} (ρ_{L})) | (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 6 ϕ_{L} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) + {ϕ_{L}}^{3} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) - {ϕ_{L}}^{3} s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{L})) s i n h (\frac{1}{w} (ρ_{R})) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) + {ϕ_{R}}^{3} c o s h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) - 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{L})) c o s h (\frac{1}{w} (ρ_{R})) + 6 ϕ_{R} \cdot s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})) - {ϕ_{R}}^{3} s i n h (\frac{1}{w} (ρ_{R})) c o s h (\frac{1}{w} (ρ_{L})))}{6 w s i n h (\frac{1}{w} (ρ_{R})) s i n h (\frac{1}{w} (ρ_{L})) | s i n h (\frac{1}{w} (ρ_{R})) |} & 0 & 0 \end{matrix}] \end{matrix}]

[

P_{θ}

P_{ϕ}

] =

[\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{L}))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ_{L}))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}] ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ_{L}) c o s (θ_{L}) s i n h (\frac{1}{w} (ρ_{L}))}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{w s i n h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ_{L}))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ_{L}) c o s (θ_{L}) c o s h (\frac{1}{w} (ρ_{L}))}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{1}{2} ({c o s (θ_{L})}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ_{L}) c o s (θ_{L}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{- c o s (θ_{L}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ_{L})} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) - ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ_{R}) c o s (θ_{R}) s i n h (\frac{1}{w} (ρ_{L}))}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{w s i n h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{R})}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{R})}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{R})}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{R})}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{3} {c o s (θ_{R})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} {c o s (θ_{R})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ_{L}))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ_{R}) c o s (θ_{R}) c o s h (\frac{1}{w} (ρ_{L}))}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{1}{2} ({c o s (θ_{R})}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ_{R}) c o s (θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{- c o s (θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ_{R})} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) [\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{L}))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ_{L}))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}]

[\begin{matrix} [\begin{matrix} \frac{1}{2} (- c o s (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})) & \frac{w c o s (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}))}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{- w c o s (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{4} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{R})}^{2} - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{R})}^{2} - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{R})}^{2} + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{R})}^{2} + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {c o s (θ_{R})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} {c o s (θ_{R})}^{2} + 3 {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{R})}^{2})}{6 c o s h (\frac{1}{w} (ρ_{L}))} \\ \frac{c o s (θ_{L} - θ_{R}) c o s h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}))}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{1}{2} (c o s (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})) & \frac{1}{6} (c o s (θ_{L} - θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{L}}^{3} s i n (θ_{R}) c o s (θ_{R}) + 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{L}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) + 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{R}}^{3} s i n (θ_{L}) c o s (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{R}))) \\ 0 & \frac{- c o s (θ_{L} - θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{R}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{L}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{R}) - {ϕ_{L}}^{3} s i n (θ_{L}) c o s (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{R}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{L}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{R}) + 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{L}) - {ϕ_{R}}^{3} s i n (θ_{R}) c o s (θ_{L}))}{6 s i n (θ_{R}) s i n (θ_{L})} & 0 \end{matrix}] & [\begin{matrix} \frac{w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{- w^{2} {s i n h (\frac{1}{w} (ρ_{L}))}^{2} s i n (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}))}{2 {c o s h (\frac{1}{w} (ρ_{L}))}^{2}} & \frac{w^{2} {s i n h (\frac{1}{w} (ρ_{L}))}^{2} s i n (θ_{L} - θ_{R}) (9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{4} + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} - 3 {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{R})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{4} - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{4} - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + 3 {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} - {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} {c o s (θ_{R})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {c o s (θ_{R})}^{4})}{6 {c o s h (\frac{1}{w} (ρ_{L}))}^{2}} \\ \frac{1}{2} (- s i n (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}))) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{3} s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{L}) - 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{L}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{3} s i n (θ_{L}) c o s (θ_{L}) + 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{R}))}{6 c o s h (\frac{1}{w} (ρ_{L}))} \\ 0 & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{L}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{R}) - {ϕ_{L}}^{3} s i n (θ_{R}) c o s (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{L}) + {ϕ_{L}}^{3} c o s (θ_{R}) s i n (θ_{L}) - 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{L}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{R}) - {ϕ_{R}}^{3} s i n (θ_{L}) c o s (θ_{R}) - 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{L}) + {ϕ_{R}}^{3} c o s (θ_{L}) s i n (θ_{R}) + 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{R}))}{6 s i n (θ_{L}) s i n (θ_{R}) c o s h (\frac{1}{w} (ρ_{L}))} & 0 \end{matrix}] & 0 \\ [\begin{matrix} \frac{- s i n (θ_{L} - θ_{R}) c o s h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{1}{2} (s i n (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}))) & \frac{1}{6} (s i n (θ_{L} - θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{4} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{R})}^{2} - 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} + 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} + 3 {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} - 6 ϕ_{L} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{3} {c o s (θ_{R})}^{4} - {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} {c o s (θ_{R})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} + {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{4} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{R})}^{2} + 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} - {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + 6 ϕ_{R} \cdot {c o s (θ_{R})}^{2} - 3 {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{R})}^{4} + {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - {ϕ_{R}}^{3} {s i n (θ_{R})}^{2})) \\ \frac{{c o s h (\frac{1}{w} (ρ_{L}))}^{2} s i n (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}))}{2 w^{2} {s i n h (\frac{1}{w} (ρ_{L}))}^{2}} & \frac{c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{c o s h (\frac{1}{w} (ρ_{L})) s i n (θ_{L} - θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{L}}^{3} s i n (θ_{R}) c o s (θ_{R}) + 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{L}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{3} s i n (θ_{L}) c o s (θ_{L}) + 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{L}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{R}))}{6 w s i n h (\frac{1}{w} (ρ_{L}))} \\ 0 & \frac{- s i n (θ_{L} - θ_{R}) c o s h (\frac{1}{w} (ρ_{L})) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{R}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{L}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{R}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{L}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{R}) + 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{L}) - {ϕ_{L}}^{3} s i n (θ_{L}) c o s (θ_{R}) - {ϕ_{R}}^{3} s i n (θ_{R}) c o s (θ_{L}))}{6 w s i n (θ_{L}) s i n (θ_{R}) s i n h (\frac{1}{w} (ρ_{L}))} & 0 \end{matrix}] & [\begin{matrix} \frac{1}{2} (- c o s (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})) & \frac{w c o s (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}))}{2 c o s h (\frac{1}{w} (ρ_{L}))} & \frac{- w c o s (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ_{L})) (9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{4} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{R})}^{2} - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{R})}^{2} - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{R})}^{2} + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{R})}^{2} + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {c o s (θ_{R})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} {c o s (θ_{R})}^{2} + 3 {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{R})}^{2})}{6 c o s h (\frac{1}{w} (ρ_{L}))} \\ \frac{c o s (θ_{L} - θ_{R}) c o s h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}))}{2 w s i n h (\frac{1}{w} (ρ_{L}))} & \frac{1}{2} (c o s (θ_{L} - θ_{R}) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})) & \frac{1}{6} (c o s (θ_{L} - θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{L}}^{3} s i n (θ_{R}) c o s (θ_{R}) + 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{L}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) + 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{R}}^{3} s i n (θ_{L}) c o s (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{R}))) \\ 0 & \frac{- c o s (θ_{L} - θ_{R}) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{R}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{L}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{R}) - {ϕ_{L}}^{3} s i n (θ_{L}) c o s (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{R}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{L}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{R}) + 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{L}) - {ϕ_{R}}^{3} s i n (θ_{R}) c o s (θ_{L}))}{6 s i n (θ_{R}) s i n (θ_{L})} & 0 \end{matrix}] & 0 \\ 0 & 0 & [\begin{matrix} \frac{- | s i n (θ_{L}) | (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})}{2 | s i n (θ_{R}) |} & \frac{w | s i n (θ_{L}) | s i n h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}))}{2 c o s h (\frac{1}{w} (ρ_{L})) | s i n (θ_{R}) |} & \frac{- w | s i n (θ_{L}) | s i n h (\frac{1}{w} (ρ_{L})) (9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{4} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{4} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ_{R})}^{2} - 6 ϕ_{L} \cdot {c o s (θ_{L})}^{2} + 6 ϕ_{L} \cdot {c o s (θ_{R})}^{2} - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{4} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{L})}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ_{R})}^{2} + 6 ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 6 ϕ_{R} \cdot {c o s (θ_{R})}^{2} + 3 {ϕ_{L}}^{3} {c o s (θ_{L})}^{2} - {ϕ_{L}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} + {ϕ_{L}}^{3} {s i n (θ_{L})}^{2} - {ϕ_{L}}^{3} {s i n (θ_{R})}^{2} + {ϕ_{L}}^{3} {c o s (θ_{R})}^{2} {s i n (θ_{R})}^{2} - 3 {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{4} - {ϕ_{R}}^{3} {c o s (θ_{R})}^{4} - {ϕ_{R}}^{3} {s i n (θ_{R})}^{2} {c o s (θ_{R})}^{2} + 3 {ϕ_{R}}^{3} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{3} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {c o s (θ_{L})}^{2} {s i n (θ_{L})}^{2} + {ϕ_{R}}^{3} {s i n (θ_{R})}^{2})}{6 c o s h (\frac{1}{w} (ρ_{L})) | s i n (θ_{R}) |} \\ \frac{| s i n (θ_{L}) | c o s h (\frac{1}{w} (ρ_{L})) (2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{L}) s i n (θ_{L}) - 2 ϕ_{L} \cdot ϕ_{R} \cdot c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{L}}^{2} s i n (θ_{L}) c o s (θ_{L}) - {ϕ_{R}}^{2} s i n (θ_{L}) c o s (θ_{L}) + {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) + {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}))}{2 w s i n h (\frac{1}{w} (ρ_{L})) | s i n (θ_{R}) |} & \frac{| s i n (θ_{L}) | (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{L})}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ_{R})}^{2} - {ϕ_{L}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{L}}^{2} {c o s (θ_{R})}^{2} - {ϕ_{R}}^{2} {c o s (θ_{L})}^{2} + {ϕ_{R}}^{2} {c o s (θ_{R})}^{2})}{2 | s i n (θ_{R}) |} & \frac{| s i n (θ_{L}) | (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{R}) - {ϕ_{L}}^{3} s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{L}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{L}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{R}) - {ϕ_{R}}^{3} s i n (θ_{L}) c o s (θ_{L}) + 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{R}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{R}))}{6 | s i n (θ_{R}) |} \\ 0 & \frac{- | s i n (θ_{L}) | (3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{L}) s i n (θ_{R}) - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} c o s (θ_{R}) s i n (θ_{L}) + 6 ϕ_{L} \cdot s i n (θ_{L}) c o s (θ_{R}) - 6 ϕ_{L} \cdot s i n (θ_{R}) c o s (θ_{L}) + {ϕ_{L}}^{3} c o s (θ_{L}) s i n (θ_{R}) - {ϕ_{L}}^{3} s i n (θ_{L}) c o s (θ_{R}) - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{L}) s i n (θ_{R}) + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} c o s (θ_{R}) s i n (θ_{L}) - 6 ϕ_{R} \cdot s i n (θ_{L}) c o s (θ_{R}) + {ϕ_{R}}^{3} c o s (θ_{R}) s i n (θ_{L}) + 6 ϕ_{R} \cdot s i n (θ_{R}) c o s (θ_{L}) - {ϕ_{R}}^{3} s i n (θ_{R}) c o s (θ_{L}))}{6 s i n (θ_{L}) s i n (θ_{R}) | s i n (θ_{R}) |} & 0 \end{matrix}] \end{matrix}]

propagator partials

\frac{\partial}{\partial ρ} ([\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}]) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

\frac{\partial}{\partial θ} ([\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}]) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

\frac{\partial}{\partial ϕ} ([\begin{matrix} \frac{c o s h (\frac{1}{w} (ρ_{L}))}{c o s h (\frac{1}{w} (ρ_{R}))} & 0 & 0 \\ 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} & 0 \\ 0 & 0 & \frac{| s i n h (\frac{1}{w} (ρ_{L})) |}{| s i n h (\frac{1}{w} (ρ_{R})) |} \end{matrix}]) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

\frac{\partial}{\partial ρ} ([\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ)) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}]) = [\begin{matrix} 0 & \frac{- s i n (θ_{L} - θ_{R}) ({c o s h (\frac{1}{w} (ρ))}^{2} - {s i n h (\frac{1}{w} (ρ))}^{2})}{{c o s h (\frac{1}{w} (ρ))}^{2}} & 0 \\ \frac{- s i n (θ_{L} - θ_{R}) ({c o s h (\frac{1}{w} (ρ))}^{2} - {s i n h (\frac{1}{w} (ρ))}^{2})}{w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

\frac{\partial}{\partial θ} ([\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ)) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}]) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

\frac{\partial}{\partial ϕ} ([\begin{matrix} c o s (θ_{L} - θ_{R}) & \frac{- w s i n (θ_{L} - θ_{R}) s i n h (\frac{1}{w} (ρ))}{c o s h (\frac{1}{w} (ρ))} & 0 \\ \frac{c o s h (\frac{1}{w} (ρ)) s i n (θ_{L} - θ_{R})}{w s i n h (\frac{1}{w} (ρ))} & c o s (θ_{L} - θ_{R}) & 0 \\ 0 & 0 & \frac{| s i n (θ_{L}) |}{| s i n (θ_{R}) |} \end{matrix}]) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

\frac{\partial}{\partial ρ} ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ)}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ) c o s (θ) s i n h (\frac{1}{w} (ρ))}{2 c o s h (\frac{1}{w} (ρ))} & \frac{w s i n h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ)}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ)}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ)}^{2} {s i n (θ)}^{2} + {ϕ_{L}}^{3} {c o s (θ)}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ)}^{2} - {ϕ_{L}}^{3} {s i n (θ)}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ)}^{2} - {ϕ_{R}}^{3} {c o s (θ)}^{4} - {ϕ_{R}}^{3} {s i n (θ)}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{3} {s i n (θ)}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ) c o s (θ) c o s h (\frac{1}{w} (ρ))}{2 w s i n h (\frac{1}{w} (ρ))} & \frac{1}{2} ({c o s (θ)}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ) c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ))} & \frac{- c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ)} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) = [\begin{matrix} 0 & \frac{c o s (θ) s i n (θ) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} - {ϕ_{L}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - {ϕ_{R}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} + {ϕ_{L}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + {ϕ_{R}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2})}{2 {c o s h (\frac{1}{w} (ρ))}^{2}} & \frac{- (9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} {c o s (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} {c o s (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - 6 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + 3 {ϕ_{L}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} - 6 ϕ_{L} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} + 6 ϕ_{L} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} - 3 {ϕ_{L}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} + 6 ϕ_{L} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} - 2 {ϕ_{L}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} - {ϕ_{L}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} + {ϕ_{L}}^{3} {s i n (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - {ϕ_{L}}^{3} {c o s (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} {s i n (θ)}^{2} + {ϕ_{L}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} {s i n (θ)}^{2} {c o s (θ)}^{2} - {ϕ_{L}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} {s i n (θ)}^{2} - 6 ϕ_{L} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} + 2 {ϕ_{L}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} + {ϕ_{L}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} - 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} {c o s (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} {c o s (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + 6 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + 6 ϕ_{R} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} - 3 {ϕ_{R}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} - 6 ϕ_{R} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} + 3 {ϕ_{R}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{2} - 6 ϕ_{R} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} + 2 {ϕ_{R}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} + {ϕ_{R}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} + {ϕ_{R}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} {s i n (θ)}^{2} {c o s (θ)}^{2} - {ϕ_{R}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} {s i n (θ)}^{2} + 6 ϕ_{R} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} - 2 {ϕ_{R}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} - {ϕ_{R}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} {c o s (θ)}^{4} + {ϕ_{R}}^{3} {s i n (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - {ϕ_{R}}^{3} {c o s (θ)}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} {s i n (θ)}^{2})}{6 {c o s h (\frac{1}{w} (ρ))}^{2}} \\ \frac{- c o s (θ) s i n (θ) (2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} - 2 ϕ_{L} \cdot ϕ_{R} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} + {ϕ_{L}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} + {ϕ_{R}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - {ϕ_{L}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - {ϕ_{R}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2})}{2 w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}} & 0 & 0 \\ \frac{3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - 6 ϕ_{L} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} + {ϕ_{L}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} - {ϕ_{L}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2} + 6 ϕ_{L} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s h (\frac{1}{w} (ρ))}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n h (\frac{1}{w} (ρ))}^{2} - {ϕ_{R}}^{3} {c o s h (\frac{1}{w} (ρ))}^{2} + 6 ϕ_{R} \cdot {c o s h (\frac{1}{w} (ρ))}^{2} - 6 ϕ_{R} \cdot {s i n h (\frac{1}{w} (ρ))}^{2} + {ϕ_{R}}^{3} {s i n h (\frac{1}{w} (ρ))}^{2}}{6 w^{2} {s i n h (\frac{1}{w} (ρ))}^{2}} & 0 & 0 \end{matrix}]

\frac{\partial}{\partial θ} ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ)}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ) c o s (θ) s i n h (\frac{1}{w} (ρ))}{2 c o s h (\frac{1}{w} (ρ))} & \frac{w s i n h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ)}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ)}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ)}^{2} {s i n (θ)}^{2} + {ϕ_{L}}^{3} {c o s (θ)}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ)}^{2} - {ϕ_{L}}^{3} {s i n (θ)}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ)}^{2} - {ϕ_{R}}^{3} {c o s (θ)}^{4} - {ϕ_{R}}^{3} {s i n (θ)}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{3} {s i n (θ)}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ) c o s (θ) c o s h (\frac{1}{w} (ρ))}{2 w s i n h (\frac{1}{w} (ρ))} & \frac{1}{2} ({c o s (θ)}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ) c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ))} & \frac{- c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ)} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) = [\begin{matrix} s i n (θ) c o s (θ) (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) & \frac{w s i n h (\frac{1}{w} (ρ)) (4 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - 2 ϕ_{L} \cdot ϕ_{R} - 2 {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{L}}^{2} - 2 {ϕ_{R}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2})}{2 c o s h (\frac{1}{w} (ρ))} & \frac{w (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} - {ϕ_{R}}^{3}) s i n (θ) s i n h (\frac{1}{w} (ρ)) c o s (θ)}{3 c o s h (\frac{1}{w} (ρ))} \\ \frac{c o s h (\frac{1}{w} (ρ)) (4 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - 2 ϕ_{L} \cdot ϕ_{R} - 2 {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{L}}^{2} - 2 {ϕ_{R}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2})}{2 w s i n h (\frac{1}{w} (ρ))} & - s i n (θ) c o s (θ) (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) & \frac{1}{6} (- ({ϕ_{L}}^{3} - 6 ϕ_{L} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} - 2 {ϕ_{L}}^{3} {c o s (θ)}^{2} + 12 ϕ_{L} \cdot {c o s (θ)}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} - 12 ϕ_{R} \cdot {c o s (θ)}^{2} + 2 {ϕ_{R}}^{3} {c o s (θ)}^{2})) \\ 0 & \frac{3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} + {ϕ_{L}}^{3} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} - {ϕ_{R}}^{3} + 6 ϕ_{R}}{6 (1 - {c o s (θ)}^{2})} & 0 \end{matrix}]

\frac{\partial}{\partial ϕ} ([\begin{matrix} \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot {c o s (θ)}^{2} - {ϕ_{L}}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} {c o s (θ)}^{2})) & \frac{w (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2}) s i n (θ) c o s (θ) s i n h (\frac{1}{w} (ρ))}{2 c o s h (\frac{1}{w} (ρ))} & \frac{w s i n h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{4} - 9 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} + 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} - 3 ϕ_{L} \cdot {ϕ_{R}}^{2} {s i n (θ)}^{2} + 6 ϕ_{L} \cdot {ϕ_{R}}^{2} + 6 ϕ_{L} \cdot {c o s (θ)}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{4} + 9 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {c o s (θ)}^{2} {s i n (θ)}^{2} + 3 ϕ_{R} \cdot {ϕ_{L}}^{2} {s i n (θ)}^{2} - 6 ϕ_{R} \cdot {ϕ_{L}}^{2} - 6 ϕ_{R} \cdot {c o s (θ)}^{2} + 6 ϕ_{R} + {ϕ_{L}}^{3} {c o s (θ)}^{2} {s i n (θ)}^{2} + {ϕ_{L}}^{3} {c o s (θ)}^{4} - 3 {ϕ_{L}}^{3} {c o s (θ)}^{2} - {ϕ_{L}}^{3} {s i n (θ)}^{2} + 2 {ϕ_{L}}^{3} + 3 {ϕ_{R}}^{3} {c o s (θ)}^{2} - {ϕ_{R}}^{3} {c o s (θ)}^{4} - {ϕ_{R}}^{3} {s i n (θ)}^{2} {c o s (θ)}^{2} + {ϕ_{R}}^{3} {s i n (θ)}^{2} - 2 {ϕ_{R}}^{3})}{6 c o s h (\frac{1}{w} (ρ))} \\ \frac{- {(ϕ_{L} - ϕ_{R})}^{2} s i n (θ) c o s (θ) c o s h (\frac{1}{w} (ρ))}{2 w s i n h (\frac{1}{w} (ρ))} & \frac{1}{2} ({c o s (θ)}^{2} (2 ϕ_{L} \cdot ϕ_{R} - {ϕ_{L}}^{2} - {ϕ_{R}}^{2})) & \frac{1}{6} (s i n (θ) c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3} + {ϕ_{L}}^{3})) \\ \frac{- c o s h (\frac{1}{w} (ρ)) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 w s i n h (\frac{1}{w} (ρ))} & \frac{- c o s (θ) (3 ϕ_{L} \cdot {ϕ_{R}}^{2} + {ϕ_{L}}^{3} - 6 ϕ_{L} - 3 ϕ_{R} \cdot {ϕ_{L}}^{2} + 6 ϕ_{R} - {ϕ_{R}}^{3})}{6 s i n (θ)} & \frac{1}{2} (- ({ϕ_{L}}^{2} - 2 ϕ_{L} \cdot ϕ_{R} + {ϕ_{R}}^{2})) \end{matrix}] + O (A^{4})) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

volume element:

{(\frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})})}^{2} s i n (θ)

volume integral:

(\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (\frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ)}{{s i n h (\frac{1}{w})}^{2}}))) Δ ϕ

finite volume (0,0)-form:

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + Δ t \cdot (\frac{1}{V (x_{C})} (- (\int_{ϕ_{L}}^{ϕ_{R}} d ϕ (\int_{θ_{L}}^{θ_{R}} d θ (J (ρ_{R}) \cdot {e_{ρ}}^{\bar{ρ}} (ρ_{R}) \cdot F^{ρ} (ρ_{R}) - J (ρ_{L}) \cdot {e_{ρ}}^{\bar{ρ}} (ρ_{L}) \cdot F^{ρ} (ρ_{L})))) - (\int_{ϕ_{L}}^{ϕ_{R}} d ϕ (\int_{ρ_{L}}^{ρ_{R}} d ρ (J (θ_{R}) \cdot {e_{θ}}^{\bar{θ}} (θ_{R}) \cdot F^{θ} (θ_{R}) - J (θ_{L}) \cdot {e_{θ}}^{\bar{θ}} (θ_{L}) \cdot F^{θ} (θ_{L})))) - (\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (J (ϕ_{R}) \cdot {e_{ϕ}}^{\bar{ϕ}} (ϕ_{R}) \cdot F^{ϕ} (ϕ_{R}) - J (ϕ_{L}) \cdot {e_{ϕ}}^{\bar{ϕ}} (ϕ_{L}) \cdot F^{ϕ} (ϕ_{L}))))) + S (x_{C}))

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + Δ t \cdot (\frac{1}{(\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (\frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ)}{{s i n h (\frac{1}{w})}^{2}}))) Δ ϕ} (- (\int_{ϕ_{L}}^{ϕ_{R}} d ϕ (\int_{θ_{L}}^{θ_{R}} d θ ({(\frac{A s i n h (\frac{1}{w} (ρ_{R}))}{s i n h (\frac{1}{w})})}^{2} s i n (θ) F^{ρ} (ρ_{R}) - {(\frac{A s i n h (\frac{1}{w} (ρ_{L}))}{s i n h (\frac{1}{w})})}^{2} s i n (θ) F^{ρ} (ρ_{L})))) - (\int_{ϕ_{L}}^{ϕ_{R}} d ϕ (\int_{ρ_{L}}^{ρ_{R}} d ρ ({(\frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})})}^{2} s i n (θ_{R}) F^{θ} (θ_{R}) - {(\frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})})}^{2} s i n (θ_{L}) F^{θ} (θ_{L})))) - (\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ ({(\frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})})}^{2} s i n (θ) F^{ϕ} (ϕ_{R}) - {(\frac{A s i n h (\frac{1}{w} (ρ))}{s i n h (\frac{1}{w})})}^{2} s i n (θ) F^{ϕ} (ϕ_{L}))))) + S (x_{C}))

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + Δ t \cdot (\frac{1}{(\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (\frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ)}{{s i n h (\frac{1}{w})}^{2}}))) Δ ϕ} (- \frac{- A^{2} (F^{ρ} (ρ_{L}) \cdot {s i n h (\frac{1}{w} (ρ_{L}))}^{2} c o s (θ_{L}) - F^{ρ} (ρ_{L}) \cdot {s i n h (\frac{1}{w} (ρ_{L}))}^{2} c o s (θ_{R}) - F^{ρ} (ρ_{R}) \cdot {s i n h (\frac{1}{w} (ρ_{R}))}^{2} c o s (θ_{L}) + F^{ρ} (ρ_{R}) \cdot {s i n h (\frac{1}{w} (ρ_{R}))}^{2} c o s (θ_{R}))}{{s i n h (\frac{1}{w})}^{2}} Δ ϕ - (\int_{ρ_{L}}^{ρ_{R}} d ρ (\frac{A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} (F^{θ} (θ_{R}) \cdot s i n (θ_{R}) - F^{θ} (θ_{L}) \cdot s i n (θ_{L}))}{{s i n h (\frac{1}{w})}^{2}})) Δ ϕ - (\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (\frac{- A^{2} {s i n h (\frac{1}{w} (ρ))}^{2} s i n (θ) (F^{ϕ} (ϕ_{L}) - F^{ϕ} (ϕ_{R}))}{{s i n h (\frac{1}{w})}^{2}})))) + S (x_{C}))

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + S (x_{C}) \cdot Δ t + - 1 Δ t \cdot (\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (- 1 A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) F^{ϕ} (ϕ_{L}) \cdot \frac{1}{{s i n h (\frac{1}{w})}^{2}} - - 1 A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) F^{ϕ} (ϕ_{R}) \cdot \frac{1}{{s i n h (\frac{1}{w})}^{2}}))) \frac{1}{Δ ϕ} \frac{1}{\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) \frac{1}{{s i n h (\frac{1}{w})}^{2}}))} + - 1 Δ t \cdot (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} F^{θ} (θ_{R}) \cdot s i n (θ_{R}) \frac{1}{{s i n h (\frac{1}{w})}^{2}} - A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} F^{θ} (θ_{L}) \cdot s i n (θ_{L}) \frac{1}{{s i n h (\frac{1}{w})}^{2}})) \frac{1}{\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) \frac{1}{{s i n h (\frac{1}{w})}^{2}}))} + F^{ρ} (ρ_{L}) \cdot Δ t \cdot A^{2} {s i n h (ρ_{L} \cdot \frac{1}{w})}^{2} c o s (θ_{L}) \frac{1}{{s i n h (\frac{1}{w})}^{2}} \frac{1}{\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) \frac{1}{{s i n h (\frac{1}{w})}^{2}}))} + - 1 F^{ρ} (ρ_{L}) \cdot Δ t \cdot A^{2} {s i n h (ρ_{L} \cdot \frac{1}{w})}^{2} c o s (θ_{R}) \frac{1}{{s i n h (\frac{1}{w})}^{2}} \frac{1}{\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) \frac{1}{{s i n h (\frac{1}{w})}^{2}}))} + - 1 F^{ρ} (ρ_{R}) \cdot Δ t \cdot A^{2} {s i n h (ρ_{R} \cdot \frac{1}{w})}^{2} c o s (θ_{L}) \frac{1}{{s i n h (\frac{1}{w})}^{2}} \frac{1}{\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) \frac{1}{{s i n h (\frac{1}{w})}^{2}}))} + F^{ρ} (ρ_{R}) \cdot Δ t \cdot A^{2} {s i n h (ρ_{R} \cdot \frac{1}{w})}^{2} c o s (θ_{R}) \frac{1}{{s i n h (\frac{1}{w})}^{2}} \frac{1}{\int_{θ_{L}}^{θ_{R}} d θ (\int_{ρ_{L}}^{ρ_{R}} d ρ (A^{2} {s i n h (ρ \cdot \frac{1}{w})}^{2} s i n (θ) \frac{1}{{s i n h (\frac{1}{w})}^{2}}))}