Operador Laplaciano en coordenadas ortogonales compactificadas

Vamos a compactificar el operador Laplaciano en coordenadas esféricas y en coordenadas cartesianas:

$latex (bar{r}, theta, varphi) longrightarrow (r,theta,varphi) longrightarrow (x,y,z)$

$latex phi(bar{r},theta,varphi) = (frac{bar{r}}{bar{r}+a} sin theta cos varphi, frac{bar{r}}{bar{r}+a} sin theta sin varphi, frac{bar{r}}{bar{r}+a} cos theta)$

de manera que, con la formula ya vista en este post, nos queda:

$latex (bar{r}+a)^2 bigg [ frac{(bar{r}+a)^2}{a^2} frac{partial^2}{partial bar{r}^2} + frac{2}{bar{r}} frac{(bar{r}+a)^2}{a^2} frac{partial}{partial bar{r}} + frac{1}{bar{r}^2} frac{partial^2}{partial theta^2} + frac{cot theta}{bar{r}^2} frac{partial}{partial theta} + frac{csc^2 theta}{bar{r}^2} frac{partial^2}{partial varphi^2} bigg ] u = $

$latex = f(x(bar{r},theta,varphi),y(bar{r},theta,varphi),z(bar{r},theta,varphi))$.

De la misma manera:

$latex (bar{x}, bar{y}, bar{z}) longrightarrow (x,y,z)$

$latex phi(bar{x},bar{y},bar{z}) = (tanh frac{bar{x}}{a}, tanh frac{bar{y}}{b}, tanh frac{bar{z}}{c})$

de manera que:

$latex bigg [ a^2 cosh^4 frac{bar{x}}{a} frac{partial^2}{partial bar{x}^2} + 2 a cosh^3 frac{bar{x}}{a} sinh frac{bar{x}}{a} frac{partial}{partial bar{x}} + $

$latex + b^2 cosh^4 frac{bar{y}}{a} frac{partial^2}{partial bar{y}^2} + 2 b cosh^3 frac{bar{y}}{b} sinh frac{bar{y}}{b} frac{partial}{partial bar{y}} + $

$latex + c^2 cosh^4 frac{bar{z}}{c} frac{partial^2}{partial bar{z}^2} + 2 c cosh^3 frac{bar{z}}{c} sinh frac{bar{z}}{c} frac{partial}{partial bar{z}} bigg ] u(bar{x},bar{y},bar{z}) = f(x(bar{x}),y(bar{y}),z(bar{z})$.

O, con el otro cambio:

$latex (bar{x}, bar{y}, bar{z}) longrightarrow (x,y,z)$

$latex phi(bar{x},bar{y},bar{z}) = frac{2}{pi}(arctan frac{bar{x}}{a}, arctan frac{bar{y}}{b}, arctan frac{bar{z}}{c})$

de manera que:

$latex frac{pi^2}{2} bigg [frac{(bar{x}^2+a^2)^2}{2a^2} frac{partial^2}{partial bar{x}^2} + frac{bar{x}(bar{x}^2+a^2)}{a^2} frac{partial}{partial bar{x}} + $

$latex + frac{(bar{y}^2+b^2)^2}{2b^2} frac{partial^2}{partial bar{y}^2} + frac{bar{y}(bar{y}^2+b^2)}{b^2} frac{partial}{partial bar{y}} + $

$latex + frac{(bar{z}^2+c^2)^2}{2c^2} frac{partial^2}{partial bar{z}} + frac{bar{z}(bar{z}^2+c^2)}{c^2} frac{partial}{partial bar{z}} bigg ] u(bar{x},bar{y},bar{z}) = f(x(bar{x}),y(bar{y}),z(bar{z})$

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