Los cálculos para cilíndricas, esféricas, esféricas compactificadas y cartesianas compactificadas ya están términados. Para hacer una pequeña comprobación de que son correctos, vamos a calcular, en cada caso, como quedaría la divergencia de un campo vectorial \mathcal{D}_i X^i utilizando las derivadas covariantes encontrados en los enlaces anteriores y compararla con el resultado que obtendriamos utilizando la fórmula para la divergencia en coordenadas curvilineas q^i:

\nabla \cdot \mathbf{X} = \frac{1}{\Pi_j h_j} \frac{\partial}{\partial q^i} (X^i \Pi_{j \neq i} h_j).

\{ \partial_r, \partial_{\theta}, \partial_z \}

\mathcal{D}_i X^i = \mathcal{D}_r X^r + \mathcal{D}_{\theta} X^{\theta} + \mathcal{D}_{z} X^{z} = \partial_r X^r + \partial_{\theta} X^{\theta} + \frac{1}{r} X^{\theta} + \partial_z X^z,

\{ \partial_r, \frac{1}{r} \partial_{\theta}, \partial_z \}

\mathcal{D}_i X^i = \partial_r X^r + \frac{1}{r} \partial_{\theta} X^{\theta} + \frac{1}{r} X^{\theta} + \partial_z X^z,

y con la fórmula para curvilineas obtenemos:

divCyl

\{ \partial_r, \partial_{\theta}, \partial_{\varphi} \}

\mathcal{D}_i X^i = \partial_r X^r + \frac{2}{r} X^r + \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \partial_{\varphi} X^{\varphi},

\{ \partial_r, \frac{1}{r} \partial_{\theta}, \frac{\csc \theta}{r} \partial_{\varphi} \}

\mathcal{D}_i X^i = \frac{1}{r} \big [ r \partial_r X^r + 2 X^r + \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \csc \theta \partial_{\varphi} X^{\varphi} \big ],

y con la fórmula para curvilineas obtenemos:

divSph

\{ \partial_{\bar{r}}, \partial_{\theta}, \partial_{\varphi} \}

\mathcal{D}_i X^i = \partial_{\bar{r}} X^r + \frac{1+\bar{r}}{1-\bar{r}} \frac{2}{\bar{r}} X^r + \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \partial_{\varphi} X^{\varphi},

\{ \frac{(1-\bar{r})^2}{a} \partial_{\bar{r}}, \frac{1-\bar{r}}{a \bar{r}} \partial_{\theta}, \frac{1-\bar{r}}{a \bar{r}} \csc \theta \partial_{\varphi} \}

\mathcal{D}_i X^i = \frac{1 - \bar{r}}{a \bar{r}} \big [ (\bar{r} - \bar{r}^2) \partial_{\bar{r}} X^r + 2 X^r + \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \csc \theta \partial_{\varphi} X^{\varphi} \big ],

y con la fórmula para curvilineas obtenemos:

divSphCom1

\{ \partial_{\bar{r}}, \partial_{\theta}, \partial_{\varphi} \}

\mathcal{D}_i X^i = \partial_{\bar{r}} X^r + \frac{4 \bar{r}}{(1-\bar{r})^2 \, \mbox{\scriptsize arctanh} \, \bar{r}} X^r + \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \partial_{\varphi} X^{\varphi},

\{ \frac{1-\bar{r}^2}{a} \partial_{\bar{r}}, \frac{1}{a \, \mbox{\scriptsize arctanh} \, \bar{r} } \partial_{\theta}, \frac{\csc \theta}{a \, \mbox{\scriptsize arctanh} \, \bar{r}} \partial_{\varphi} \}

\mathcal{D}_i X^i = \frac{1}{a \, \mbox{\scriptsize arctanh} \, \bar{r}} \big [ (1-\bar{r}^2) \, \mbox{arctanh} \, \bar{r} \partial_{\bar{r}} X^r + 2 X^r +

+ \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \csc \theta \partial_{\varphi} X^{\varphi} \big ],

y con la fórmula para curvilineas obtenemos:

divSphCom2

\{ \partial_{\bar{r}}, \partial_{\theta}, \partial_{\varphi} \}

\partial_{\bar{r}} X^r + ( \pi \tan \frac{\pi \bar{r}}{2} + 2 \pi \csc (\pi \bar{r} ) ) X^r + \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \partial_{\varphi} X^{\varphi},

\{ \frac{1 + \cos ( \pi \bar{r})}{a \pi} \partial_{\bar{r}} , \frac{ \cot \frac{ \pi \bar{r} }{2} }{a} \partial_{\theta} , \frac{ \cot \frac{ \pi \bar{r} }{2} }{a} \csc \theta \partial_{\varphi} \}

\mathcal{D}_i X^i = \frac{ \cot \frac{ \pi \bar{r} }{2} }{a} \big [ \frac{1 + \cos ( \pi \bar{r} ) }{\bar{r}} \tan \frac{\pi \bar{r}}{2} \partial_{ \bar{r} } X^r + 2 X^r +

+ \partial_{\theta} X^{\theta} + \cot \theta X^{\theta} + \csc \theta \partial_{\varphi} X^{\varphi} \big ],

y con la fórmula para curvilineas obtenemos:

divSphCom3

\{ \partial_{\bar{x}} , \partial_{\bar{y}}, \partial_{\bar{z}} \}

\mathcal{D}_i X^i = \partial_{\bar{x}} X^{\bar{x}} + \frac{2 \bar{x}}{1-\bar{x}^2} X^{\bar{x}} + \partial_{\bar{y}} X^{\bar{y}} + \frac{2 \bar{y}}{1-\bar{y}^2} X^{\bar{y}} + \partial_{\bar{z}} X^{\bar{z}} + \frac{2 \bar{z}}{1-\bar{z}^2} X^{\bar{z}},

\{ \frac{|\bar{x}^2 -1|}{a} \partial_{\bar{x}} , \frac{|\bar{y}^2 -1|}{b} \partial_{\bar{y}}, \frac{|\bar{z}^2 -1|}{c} \partial_{\bar{z}} \}

\mathcal{D}_i X^i = \frac{|\bar{x}^2 -1|}{a} \partial_{\bar{x}} X^{\bar{x}} + \frac{|\bar{y}^2 - 1|}{b} \partial_{\bar{y}} X^{\bar{y}} + \frac{|\bar{z}^2 - 1|}{c} \partial_{\bar{z}} X^{\bar{z}},

y con la fórmula para curvilineas obtenemos:

divCarCom1

\{ \partial_{\bar{x}} , \partial_{\bar{y}}, \partial_{\bar{z}} \}

\mathcal{D}_i X^i = \partial_{\bar{x}} X^{\bar{x}} + \pi \tan \frac{\pi \bar{x}}{2} X^{\bar{x}} + \partial_{\bar{y}} X^{\bar{y}} + \pi \tan \frac{\pi \bar{y}}{2} X^{\bar{y}} + \partial_{\bar{z}} X^{\bar{z}} + \pi \tan \frac{\pi \bar{z}}{2} X^{\bar{z}}

\{ \frac{1 + \cos ( \pi \bar{x} ) }{a \pi} \partial_{\bar{x}} , \frac{1 + \cos ( \pi \bar{y} ) }{b \pi} \partial_{\bar{y}}, \frac{1 + \cos ( \pi \bar{z} ) }{c \pi} \partial_{\bar{z}} \}

\mathcal{D}_i X^i = \frac{1 + \cos ( \pi \bar{x} ) }{a \pi} \partial_{\bar{x}} X^{\bar{x}} + \frac{1 + \cos ( \pi \bar{y} ) }{b \pi} \partial_{\bar{y}} X^{\bar{y}} + \frac{1 + \cos ( \pi \bar{z} ) }{c \pi} \partial_{\bar{z}} X^{\bar{z}},

y con la fórmula para curvilineas obtenemos:

divCarCom2

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