Useful Identities

The following identities are useful:

$${\bf A} = A^{\,r}\,{\cal J}\,\nabla\theta\times\nabla\varphi + A^{\,\thet... ...bla\varphi\times\nabla r + A^{\,\varphi}\,{\cal J}\,\nabla r\times\nabla\theta,$$ (14.4)

$${\bf A} = A_r\,\nabla r+A_\theta\,\nabla\theta+A_\varphi\,\nabla\varphi,$$ (14.5)

$${\bf A}\cdot{\bf B} = A_r\,B^{\,r}+A_\theta \,B^{\,\theta}+A_\varphi\,B^{\,\varphi}=A^{\,r}\,B_r+A^{\,\theta}\,B_{\theta}+A^{\,\varphi}\,B_\varphi,$$ (14.6)

$$({\bf A}\times {\bf B})_r = {\cal J}\,(A^{\,\theta}\,B^{\,\varphi}-A^{\,\varphi}\,B^{\,\theta}),$$ (14.7)

$$({\bf A}\times {\bf B})_\theta = {\cal J}\,(A^{\,\varphi}\,B^{\,r}-A^{\,r}\,B^{\,\varphi}),$$ (14.8)

$$({\bf A}\times {\bf B})_\varphi = {\cal J}\,(A^{\,r}\,B^{\,\theta}-A^{\,\theta}\,B^{\,r}),$$ (14.9)

$${\cal J}\,({\bf A}\times {\bf B})^{\,r} = A_{\theta}\,B_{\varphi}-A_{\varphi}\,B_{\theta},$$ (14.10)

$${\cal J}\,({\bf A}\times {\bf B})^{\,\theta} = A_{\varphi}\,B_{r}-A_{r}\,B_{\varphi},$$ (14.11)

$${\cal J}\,({\bf A}\times {\bf B})^{\,\varphi} =A_{r}\,B_{\theta}-A_{\theta}\,B_{r},$$ (14.12)

$${\cal J}\,\nabla \cdot{\bf A} = \frac{\partial\,({\cal J}\,A^{\,r})}{\partial r} + \frac{\parti... ...\partial \theta}+ \frac{\partial\,({\cal J}\,A^{\,\varphi})}{\partial \varphi},$$ (14.13)

$${\cal J}\,(\nabla\times {\bf A})^{\,r} = \frac{\partial A_\varphi}{\partial \theta} -\frac{\partial A_\theta}{\partial \varphi},$$ (14.14)

$${\cal J}\,(\nabla\times {\bf A})^{\,\theta} = \frac{\partial A_r}{\partial \varphi} -\frac{\partial A_\varphi}{\partial r},$$ (14.15)

$${\cal J}\,(\nabla\times {\bf A})^{\,\varphi} = \frac{\partial A_\theta}{\partial r} -\frac{\partial A_r}{\partial \theta}.$$ (14.16)

Here, ${\bf A}$ and ${\bf B}$ are arbitrary vector fields. Moreover, subscript/superscript $r$, $\theta$, $\varphi$ refer to covariant/contravariant components of a vector in the $r$, $\theta$, $\varphi$ coordinate system.