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Tenemos:

  1. \bar{r} := \frac{r}{r+a}
  2. \Delta := \frac{(1-\bar{r})^4}{a^2} \partial_{\bar{r}\bar{r}} + \frac{(1-\bar{r})^4}{a^2}\frac{2}{\bar{r}} \partial_{\bar{r}}
  3. \Delta \Theta_X = 6 \pi [ \frac{(1-\bar{r})^2}{a} \partial_{\bar{r}} S^*_{\bar{r}} + \frac{1 - \bar{r}}{a} \frac{2}{\bar{r}} S^*_{\bar{r}} + \frac{1-\bar{r}}{a} \frac{\cot \theta}{\bar{r}} S^*_{\theta}]
  4. \Delta X^{\bar{r}} = 8 \pi S^*_{\bar{r}} - \frac{1}{3} \frac{(1-\bar{r})^2}{a} \partial_{\bar{r}} \Theta_X
  5. \hat{A}^{\bar{r}\bar{r}} = \frac{4}{3}\frac{(1-\bar{r})^2}{a} \partial_{\bar{r}} X^{\bar{r}} - \frac{2}{3}2\frac{1-\bar{r}}{a}\frac{1}{\bar{r}} X^{\bar{r}}
  6. \Delta \Theta_\beta = \frac{3}{2}[\frac{(1-\bar{r})^4}{a^2} \partial_{\bar{r}\bar{r}}u + \frac{(1-\bar{r})^3(2-\bar{r})}{2a^2} \frac{4}{\bar{r}}u + \frac{(1-\bar{r})^2}{a^2} \frac{2}{\bar{r}^2}u]

con:

u:=\alpha \psi^{-6} \hat{A}^{\bar{r}\bar{r}}

y:

\{\frac{2 (\bar{r}_i - 1)^4 (\bar{r_i} - (\bar{r}_{i+1} - \bar{r}_i))}{a^2 (\bar{r}_i - \bar{r}_{i-1}) \bar{r_i} (\bar{r}_{i+1} - \bar{r}_{i-1} )},

\frac{(\bar{r}_i - 1)^2 (\frac{-2}{h_\theta^2 \bar{r}_i^2} + \frac{(\bar{r}_i - 1)^2 ((r_{i+1}-r_i)-(r_i-r_{i-1})-2)}{(r_{i+1}-r_i)(r_i - r_{i-1})} + \frac{-2 \csc^2 \theta_i}{h_\varphi^2 \bar{r}_i^2})}{a^2},

\frac{2 (\bar{r}_i - 1)^4 ((\bar{r}_{i} - \bar{r}_{i-1}) + \bar{r_i})}{a^2 (\bar{r}_{i+1} - \bar{r}_i) \bar{r}_i (\bar{r}_{i+1} - \bar{r}_{i-1} )} \}

psiAlpha

phiAlpha1

psiAlpha2

psiAlpha3

Como \Delta u = f \leftrightarrow u = \Delta^{-1} f entonces \mathcal{M}(u) = \mathcal{M}(\Delta^{-1} f) y

\mathcal{M}(\Delta^{-1}f) = \frac{1}{r} M(f) + \frac{1}{r^2}n_i D^i(f) + \frac{3}{2} \frac{1}{r^3} n_{\langle i} n_{j \rangle} Q^{ij}(f) + O(\frac{1}{r^4}) +

+ \Delta_0^{-1} \mathcal{M}(f)

con

M(f) = - \frac{1}{4 \pi} \int f,

D^i(f) = - \frac{1}{4 \pi} \int x^i f,

Q^{ij}(f) = - \frac{1}{4 \pi} \int x^i x^j f

y \mathcal{M}(f) = 0 si f es de soporte compacto.

 1.- \boxed{\Delta \Theta_X = \frac{3}{4} 8 \pi \mathcal{D}^i S_i^*} donde \Theta_X := \mathcal{D}_j X^j

En este caso, f_{\Theta_X} := \frac{3}{4} 8 \pi \mathcal{D}^i S_i^* y, por tanto, \mathcal{M}(f_{\Theta_X})=0. De esta manera, tenemos:

M(f_{\Theta_X}) =,

D^i(f_{\Theta_X}) = - \frac{1}{4 \pi} \int x^i \frac{3}{4} 8 \pi \mathcal{D}^i S_i^* = -\frac{3}{2} (\int \mathcal{D}^j(x^i S_j^*) d^3x' - \int S_j^* \mathcal{D}^j x^i d^3x'),

Q^{ij}(f_{\Theta_X}) = - \frac{1}{4 \pi} \int x^i x^j \frac{3}{4} 8 \pi \mathcal{D}^i S_i^*

\mathcal{M}(\Delta^{-1}f_{\Theta_X}) = + O()

2.- \boxed{\Delta X^i = 8 \pi f^{ij} S_j^* - \frac{1}{3} \mathcal{D}^i \Theta_X}

Ahora hacemos f_{X^i} := 8 \pi f^{ij} S_j^* - \frac{1}{3} \mathcal{D}^i \Theta_X

M(f_{X^i}) = - \frac{1}{4 \pi} \int 8 \pi f^{ij} S_j^* - \frac{1}{3} \mathcal{D}^i \Theta_X,

D^i(f_{X^i}) = - \frac{1}{4 \pi} \int x^i (8 \pi f^{ij} S_j^* - \frac{1}{3} \mathcal{D}^i \Theta_X),

Q^{ij}(f_{X^i}) = - \frac{1}{4 \pi} \int x^i x^j (8 \pi f^{ij} S_j^* - \frac{1}{3} \mathcal{D}^i \Theta_X)

\mathcal{M}(\Delta^{-1}f_{X^i}) = + O()

3.- \boxed{\Delta \psi = -2 \pi E^* \psi^{-1} - \frac{1}{8} ( f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij}) \psi^{-7}}

En esta ocasión, f_\psi := -2 \pi E^* \psi^{-1} - \frac{1}{8} ( f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij}) \psi^{-7}

M(f_\psi) = - \frac{1}{4 \pi} \int -2 \pi E^* \psi^{-1} - \frac{1}{8} ( f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij}) \psi^{-7},

D^i(f_\psi) = - \frac{1}{4 \pi} \int x^i (-2 \pi E^* \psi^{-1} - \frac{1}{8} ( f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij}) \psi^{-7}),

Q^{ij}(f_\psi) = - \frac{1}{4 \pi} \int x^i x^j (-2 \pi E^* \psi^{-1} - \frac{1}{8} ( f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij}) \psi^{-7})

\mathcal{M}(\Delta^{-1}f_\psi) = + O()

4.- \boxed{ \Delta (\alpha \psi) = \big( 2 \pi (E^* + 2S^*) \psi^{-2} + \frac{7}{8} (f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij} ) \psi^{-8} \big) (\alpha \psi) }

Definimos f_{\alpha \psi}:=\big( 2 \pi (E^* + 2S^*) \psi^{-2} + \frac{7}{8} (f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij} ) \psi^{-8} \big) (\alpha \psi)

M(f_{\alpha \psi}) = - \frac{1}{4 \pi} \int \big( 2 \pi (E^* + 2S^*) \psi^{-2} + \frac{7}{8} (f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij} ) \psi^{-8} \big) (\alpha \psi),

D^i(f_{\alpha \psi}) = - \frac{1}{4 \pi} \int x^i (\big( 2 \pi (E^* + 2S^*) \psi^{-2} + \frac{7}{8} (f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij} ) \psi^{-8} \big) (\alpha \psi)),

Q^{ij}(f_{\alpha \psi}) = - \frac{1}{4 \pi} \int x^i x^j (\big( 2 \pi (E^* + 2S^*) \psi^{-2} + \frac{7}{8} (f_{il} f_{jm} \hat{A}^{lm} \hat{A}^{ij} ) \psi^{-8} \big) (\alpha \psi))

\mathcal{M}(\Delta^{-1} f_{\alpha \psi}) = + O()

5.- \boxed{\Delta \Theta_\beta = \frac{3}{4}\mathcal{D}^i (\mathcal{D}_j(2\alpha \psi^{-6}\hat{A}^{ij})) } con \Theta_\beta := \mathcal{D}_i \beta^i

Para esta ecuación, f_{\Theta_\beta}:=\frac{3}{4}\mathcal{D}^i (\mathcal{D}_j(2\alpha \psi^{-6}\hat{A}^{ij}))

M(f_{\Theta_\beta}) = - \frac{1}{4 \pi} \int \frac{3}{4}\mathcal{D}^i (\mathcal{D}_j(2\alpha \psi^{-6}\hat{A}^{ij})),

D^i(f_{\Theta_\beta}) = - \frac{1}{4 \pi} \int x^i \frac{3}{4}\mathcal{D}^i (\mathcal{D}_j(2\alpha \psi^{-6}\hat{A}^{ij})),

Q^{ij}(f_{\Theta_\beta}) = - \frac{1}{4 \pi} \int x^i x^j \frac{3}{4}\mathcal{D}^i (\mathcal{D}_j(2\alpha \psi^{-6}\hat{A}^{ij}))

\mathcal{M}(\Delta^{-1}f_{\Theta_\beta}) = + O()

6.- \boxed{\Delta \beta^i = \mathcal{D}_j(2\alpha\psi^{-6}\hat{A}^{ij})-\frac{1}{3}\mathcal{D}^i \Theta_\beta}

Finalmente, tenemos f_{\beta^i}:=\mathcal{D}_j(2\alpha\psi^{-6}\hat{A}^{ij})-\frac{1}{3}\mathcal{D}^i \Theta_\beta

M(f_{\beta^i}) = - \frac{1}{4 \pi} \int \mathcal{D}_j(2\alpha\psi^{-6}\hat{A}^{ij})-\frac{1}{3}\mathcal{D}^i \Theta_\beta,

D^i(f_{\beta^i}) = - \frac{1}{4 \pi} \int x^i (\mathcal{D}_j(2\alpha\psi^{-6}\hat{A}^{ij})-\frac{1}{3}\mathcal{D}^i \Theta_\beta),

Q^{ij}(f_{\beta^i}) = - \frac{1}{4 \pi} \int x^i x^j (\mathcal{D}_j(2\alpha\psi^{-6}\hat{A}^{ij})-\frac{1}{3}\mathcal{D}^i \Theta_\beta)

\mathcal{M}(\Delta^{-1}f_{\beta^i}) = + O()

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junio 2017
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