polar, coordinate

chart coordinates:

x^{\tilde{μ}} = {r, ϕ}

chart coordinate basis:

e_{\tilde{μ}} = {e_{\tilde{r}}, e_{\tilde{ϕ}}}

embedding coordinates:

u^{I} = {x, y}

embedding basis

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

flat metric:

{η_{I}}_{J} = \overset{I ↓ J \to}{[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]}

transform from basis to coordinate:

{\tilde{e}}_{A}^{a} = \overset{A ↓ a \to}{[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]}

transform from coorinate to basis:

{\tilde{e}}^{a}_{A} = \overset{a ↓ A \to}{[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]}

tensor index associated with coordinate

r

has operator

e_{r} (ζ) =

\frac{\partial ζ}{\partial r}

tensor index associated with coordinate

ϕ

has operator

e_{ϕ} (ζ) =

\frac{\partial ζ}{\partial ϕ}

chart in embedded coordinates:

u = \overset{I ↓}{[\begin{matrix} r \cos (ϕ) \\ r \sin (ϕ) \end{matrix}]}

basis operators applied to chart:

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

{e_{u}}^{I} = \overset{u ↓ I \to}{[\begin{array}{cc} \cos (ϕ) & \sin (ϕ) \\ - r \sin (ϕ) & r \cos (ϕ) \end{array}]}

{e^{u}}_{I} = \overset{u ↓ I \to}{[\begin{array}{cc} \cos (ϕ) & \sin (ϕ) \\ - \frac{1}{r} \sin (ϕ) & \frac{1}{r} \cos (ϕ) \end{array}]}

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

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

basis determinant:

d e t (e) = r

{c_{a}}_{b}^{c} = \overset{a ↓ [b ↓ c \to]}{[\begin{matrix} \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \\ \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \end{matrix}]}

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

{g_{u}}_{v} = \overset{u ↓ v \to}{[\begin{array}{cc} 1 & 0 \\ 0 & r^{2} \end{array}]}

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

metric determinant:

d e t (g) = r^{2}

{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 = \overset{a ↓ [b ↓ c \to]}{[\begin{matrix} \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \\ \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \end{matrix}]}

metric:

g = \overset{u ↓ v \to}{[\begin{array}{cc} 1 & 0 \\ 0 & r^{2} \end{array}]}

metric inverse:

g = \overset{a ↓ b \to}{[\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r^{2}} \end{array}]}

metric derivative:

\partial g = \overset{a ↓ [b ↓ c \to]}{[\begin{matrix} \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \\ \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 2 r & 0 \end{array}]} \end{matrix}]}

1st kind Christoffel:

Γ = \overset{a ↓ [b ↓ c \to]}{[\begin{matrix} \overset{b ↓ c \to}{[\begin{array}{cc} 0 & 0 \\ 0 & - r \end{array}]} \\ \overset{b ↓ c \to}{[\begin{array}{cc} 0 & r \\ r & 0 \end{array}]} \end{matrix}]}

connection coefficients / 2nd kind Christoffel:

Γ = \overset{a ↓ [b ↓ c \to]}{[\begin{matrix} [\begin{array}{cc} 0 & 0 \\ 0 & - r \end{array}] \\ [\begin{array}{cc} 0 & \frac{1}{r} \\ \frac{1}{r} & 0 \end{array}] \end{matrix}]}

connection coefficients derivative:

\partial Γ = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ - 1 & 0 \end{array}]} \\ \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ - \frac{1}{r^{2}} & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} - \frac{1}{r^{2}} & 0 \\ 0 & 0 \end{array}]} \end{array}]}

connection coefficients squared:

(Γ^{2}) = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & - 1 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ - 1 & 0 \end{array}]} \\ \overset{c ↓ d \to}{[\begin{array}{cc} 0 & \frac{1}{r^{2}} \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} \frac{1}{r^{2}} & 0 \\ 0 & - 1 \end{array}]} \end{array}]}

Riemann curvature,

♯ ♭ ♭ ♭

R = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \\ \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \end{array}]}

Riemann curvature,

♯ ♯ ♭ ♭

R = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] & [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] \\ [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] & [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] \end{array}]}

Ricci curvature,

♯ ♭

R = \overset{a ↓ b \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]}

Gaussian curvature:

0

trace-free Ricci,

♯ ♭

(R^{T F}) = \overset{a ↓ b \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]}

Einstein / trace-reversed Ricci curvature,

♯ ♭

G = \overset{a ↓ b \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]}

Schouten,

♯ ♭

P = \overset{a ↓ b \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]}

Weyl,

♯ ♯ ♭ ♭

C = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \\ \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \end{array}]}

Weyl,

♭ ♭ ♭ ♭

C = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] & [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] \\ [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] & [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}] \end{array}]}

Plebanski,

♯ ♯ ♭ ♭

P = \overset{a ↓ b \to [c ↓ d \to]}{[\begin{array}{cc} \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \\ \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} & \overset{c ↓ d \to}{[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]} \end{array}]}

divergence:

{A^{i}}_{, i} + {Γ^{i}}_{i}_{j} A^{j} = A^{\hat{r}} \cdot \frac{1}{r} + \frac{\partial A^{\hat{ϕ}}}{\partial ϕ} + \frac{\partial A^{\hat{r}}}{\partial r}

geodesic:

\overset{a ↓}{[\begin{matrix} \ddot{\hat{r}} \\ \ddot{\hat{ϕ}} \end{matrix}]} = \overset{a ↓}{[\begin{matrix} r {\dot{\hat{ϕ}}}^{2} \\ - 2 \dot{\hat{ϕ}} \cdot \dot{\hat{r}} \cdot \frac{1}{r} \end{matrix}]}

parallel propagators:

[Γ_{r}] = [\begin{array}{cc} 0 & 0 \\ 0 & \frac{1}{r} \end{array}]

\int_{r_{L}}^{r_{R}} [\begin{array}{cc} 0 & 0 \\ 0 & \frac{1}{r} \end{array}] d r

[\begin{array}{cc} 0 & 0 \\ 0 & \log (\frac{1}{r_{L}} r_{R}) \end{array}]

P_{r}

\exp (- \int_{r_{L}}^{r_{R}} [\begin{array}{cc} 0 & 0 \\ 0 & \frac{1}{r} \end{array}] d r)

[\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r_{R}} r_{L} \end{array}]

{P_{r}}^{- 1}

\exp (\int_{r_{L}}^{r_{R}} [\begin{array}{cc} 0 & 0 \\ 0 & \frac{1}{r} \end{array}] d r)

[\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r_{L}} r_{R} \end{array}]

[Γ_{ϕ}] = [\begin{array}{cc} 0 & - r \\ \frac{1}{r} & 0 \end{array}]

\int_{ϕ_{L}}^{ϕ_{R}} [\begin{array}{cc} 0 & - r \\ \frac{1}{r} & 0 \end{array}] d ϕ

[\begin{array}{cc} 0 & r (ϕ_{L} - ϕ_{R}) \\ \frac{1}{r} (- ϕ_{L} + ϕ_{R}) & 0 \end{array}]

P_{ϕ}

\exp (- \int_{ϕ_{L}}^{ϕ_{R}} [\begin{array}{cc} 0 & - r \\ \frac{1}{r} & 0 \end{array}] d ϕ)

[\begin{array}{cc} \frac{\frac{1}{\exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} (- {ϕ_{L}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R})} & \frac{r ({ϕ_{L}}^{2} + {ϕ_{R}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{- ϕ_{L} + ϕ_{R} + ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 r \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}} & \frac{1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} \end{array}]

{P_{ϕ}}^{- 1}

\exp (\int_{ϕ_{L}}^{ϕ_{R}} [\begin{array}{cc} 0 & - r \\ \frac{1}{r} & 0 \end{array}] d ϕ)

[\begin{array}{cc} \frac{(- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}) (1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R})} & \frac{r (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + 2 ϕ_{L} \cdot ϕ_{R} - 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{ϕ_{L} - ϕ_{R} - ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 r \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}} & \frac{1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} \end{array}]

propagator commutation:

[

P_{r}

P_{ϕ}

] =

[\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r_{R}} r_{L} \end{array}] [\begin{array}{cc} \frac{\frac{1}{\exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} (- {ϕ_{L}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R})} & \frac{({ϕ_{L}}^{2} + {ϕ_{R}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})) r_{L}}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{- ϕ_{L} + ϕ_{R} + ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} r_{L}} & \frac{1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} \end{array}] - [\begin{array}{cc} \frac{\frac{1}{\exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} (- {ϕ_{L}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R})} & \frac{({ϕ_{L}}^{2} + {ϕ_{R}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})) r_{R}}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{- ϕ_{L} + ϕ_{R} + ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} r_{R}} & \frac{1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} \end{array}] [\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r_{R}} r_{L} \end{array}]

[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

propagator partials

\frac{\partial}{\partial r} ([\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r_{R}} r_{L} \end{array}]) = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

\frac{\partial}{\partial ϕ} ([\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{r_{R}} r_{L} \end{array}]) = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

\frac{\partial}{\partial r} ([\begin{array}{cc} \frac{\frac{1}{\exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} (- {ϕ_{L}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R})} & \frac{r ({ϕ_{L}}^{2} + {ϕ_{R}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{- ϕ_{L} + ϕ_{R} + ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 r \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}} & \frac{1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} \end{array}]) = [\begin{array}{cc} 0 & \frac{\frac{1}{\exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} ({ϕ_{L}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + {ϕ_{R}}^{2} - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{ϕ_{L} - ϕ_{R} - ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) r^{2} \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}} & 0 \end{array}]

\frac{\partial}{\partial ϕ} ([\begin{array}{cc} \frac{\frac{1}{\exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} (- {ϕ_{L}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) + 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 (- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R})} & \frac{r ({ϕ_{L}}^{2} + {ϕ_{R}}^{2} - {ϕ_{L}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - {ϕ_{R}}^{2} \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - 2 ϕ_{L} \cdot ϕ_{R} + 2 ϕ_{L} \cdot ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}))}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}} (ϕ_{L} - ϕ_{R})} \\ \frac{- ϕ_{L} + ϕ_{R} + ϕ_{L} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) - ϕ_{R} \cdot \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 r \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}) \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}}} & \frac{1 + \exp (2 \sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})}{2 \exp (\sqrt{- {ϕ_{L}}^{2} - {ϕ_{R}}^{2} + 2 ϕ_{L} \cdot ϕ_{R}})} \end{array}]) = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

volume element:

r

volume integral:

\frac{1}{2} Δ (r^{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})} 0 + S (x_{C}))

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + Δ t \cdot (\frac{1}{\frac{1}{2} Δ (r^{2}) Δ ϕ} 0 + S (x_{C}))

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + Δ t \cdot (\frac{1}{\frac{1}{2} Δ (r^{2}) Δ ϕ} 0 + S (x_{C}))

u (x_{C}, t_{R}) = u (x_{C}, t_{L}) + S (x_{C}) \cdot Δ t