### Vector analysis

Using the above expressions
$\mathsf{rot}\mathsf{rot}\mathbf{A}=\sum_{ijk}\varepsilon_{ijk} \frac{\hat{\mathbf{x_i}}}{h_jh_k} \frac{\partial}{\partial x_j}\left(\sum_{lm} \varepsilon_{klm} \frac{h_k}{h_lh_m} \frac{\partial h_mA_m}{\partial x_l}\right). ^$
Let is notice now that $\varepsilon_{ijk} \varepsilon_{klm}=\varepsilon_{ijk} \varepsilon_{lmk}= (\delta_{il}\delta_{jm} -\delta_{im}\delta_{jl})$, provided $i\ne j\ne k\ne i$.
Substituting this into the above relation we get
$\mathsf{rot}\mathsf{rot}\mathbf{A}=\sum_{ijk, i\ne j\ne k\ne i} \frac{\hat{\mathbf{x_i}}}{h_jh_k} \frac{\partial }{\partial x_j}\left[\frac{h_k}{h_ih_j}\left(\frac{\partial h_jA_j}{\partial x_i} - \frac{\partial h_iA_i}{\partial x_j}\right)\right]. ^$
In spherical coordinates $x_1=r$, $x_2=\theta$, $x_3=\varphi$, $h_1=1$, $h_2=r$, $h_3=r\sin\theta$, and
$\mathsf{rot}\mathsf{rot}\mathbf{A}=\frac{\hat{\mathbf{r}}}{r^2\sin\theta}\left\{\frac{\partial }{\partial \theta} \left[\sin\theta\left(\frac{\partial rA_\theta}{\partial r} -\frac{\partial A_r}{\partial \theta}\right) \right]\right.\\ +\left. \frac{\partial }{\partial \varphi} \left[\sin\theta \left(\frac{\partial r\sin\theta A_\phi}{\partial r} -\frac{\partial A_r}{\partial \varphi}\right)\right] \right\}\\ +\frac{\hat{\mathbf{\theta}}}{r\sin\theta} \left\{\frac{\partial }{\partial \varphi}\left[ \frac{1}{r^2\sin\theta}\left( \frac{\partial r\sin\theta A_\varphi}{\partial \theta} -\frac{\partial rA_\theta}{\partial \varphi}\right)\right]\right.\\ +\left.\frac{\partial}{\partial r}\left[\sin\theta\left(\frac{\partial A_r}{\partial \theta} -\frac{\partial rA_\theta}{\partial r}\right)\right]\right\}\\ + \frac{\hat{\mathbf{\varphi}}}{r}\left\{\frac{\partial }{\partial r} \left[\frac{1}{\sin\theta} \left(\frac{\partial A_r}{\partial \varphi} -\frac{\partial r\sin\theta A_\varphi}{\partial r}\right) \right]\right.\\ +\left.\frac{\partial }{\partial \theta}\left[\frac{1}{r^2\sin\theta}\left(\frac{\partial rA_\theta}{\partial \varphi} -\frac{\partial r\sin\theta A_\varphi}{\partial \theta}\right)\right]\right\} ^$