La salida ahora para un tensor dos veces contravariante en la base ortonormal queda:

CovDerTen2CarCom2,

Para primera ecuación:

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

definimos como antes

V^i := \mathcal{D}_j \alpha \psi^{-6} \hat{A}^{ij},

de manera que la ecuación original la reescribimos como

\Delta \Theta_\beta = \frac{3}{2} \mathcal{D}_i V^i,

De esta manera, en nuestras coordenadas obtenemos:

\Delta \Theta_\beta = \frac{3}{2} \mathcal{D}_i V^i = \frac{3}{2} (\mathcal{D}_{\bar{x}} V^{\bar{x}} + \mathcal{D}_{\bar{y}} V^{\bar{y}} + \mathcal{D}_{\bar{z}} V^{\bar{z}}) =

= \frac{3}{2} (\frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} V^{\bar{x}}+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} V^{\bar{y}} + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} V^{\bar{z}}),

donde

V^{\bar{x}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ),

V^{\bar{y}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ),

V^{\bar{z}} = \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) + \mathcal{D}_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) + \mathcal{D}_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ),

que desarrollando las covariantes quedan:

V^{\bar{x}} = \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ),

V^{\bar{y}} = \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ),

V^{\bar{z}} = \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) + \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ).

Por tanto, combiando todo, tenemos:

\Delta \Theta_\beta =

= \frac{3 + 3cos(\pi \bar{x})}{2 a \pi} \partial_{\bar{x}} \big [ \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}} ) +

+ \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}} ) \big ] +

+ \frac{3 + 3 \cos (\pi \bar{y})}{2b \pi} \partial_{\bar{y}} \big [ \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}} ) +

+ \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}} ) \big ] +

+ \frac{3 + 3 \cos(\pi \bar{z})}{2c \pi} \partial_{\bar{z}} \big [ \frac{1 + \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

\frac{1 + \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}} ) +

+ \frac{1 + \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} ( \alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}} ) \big ],

Finalmente, las ecuaciones:

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

son:

\Delta \beta^{\bar{x}} = 2 \big [ \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) + \mathcal{D}_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}}) + \mathcal{D}_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}}) \big ] - \frac{1}{3} \mathcal{D}_{\bar{x}} \Theta_\beta,

\Delta \beta^{\bar{y}} = 2 \big [ \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) + \mathcal{D}_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}}) + \mathcal{D}_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}}) \big ] - \frac{1}{3} \mathcal{D}_{\bar{y}} \Theta_\beta,

\Delta \beta^{\bar{z}} = 2 \big [ \mathcal{D}_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) + \mathcal{D}_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}}) + \mathcal{D}_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}}) \big ] - \frac{1}{3} \mathcal{D}_{\bar{z}} \Theta_\beta,

y al sustituir las derivadas covariantes:

\Delta \beta^{\bar{x}} = \frac{2 + 2 \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{x}}) +

+ \frac{2 + 2 \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{y}}) +

+ \frac{2 + 2 \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{x} \bar{z}}) - \frac{1 + \cos(\pi \bar{x})}{3a \pi} \partial_{\bar{x}} \Theta_\beta,

\Delta \beta^{\bar{y}} = \frac{2 + 2 \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{x}}) +

+ \frac{2 + 2 \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{y}}) +

+ \frac{2 + 2 \cos(\pi \bar{z})}{c} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{y} \bar{z}}) - \frac{1 + \cos(\pi \bar{y})}{3b \pi} \partial_{\bar{y}} \Theta_\beta,

\Delta \beta^{\bar{z}} = \frac{2 + 2 \cos(\pi \bar{x})}{a \pi} \partial_{\bar{x}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{x}}) +

+ \frac{2 + 2 \cos(\pi \bar{y})}{b \pi} \partial_{\bar{y}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{y}}) +

+ \frac{2 + 2 \cos(\pi \bar{z})}{c \pi} \partial_{\bar{z}} (\alpha \psi^{-6} \hat{A}^{\bar{z} \bar{z}}) - \frac{1 + \cos(\pi \bar{z})}{3c \pi} \partial_{\bar{z}} \Theta_\beta.

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