Find the curl of $\mathrm a \times ( \mathrm b \times \mathrm r)$ I'm trying to find the curl of $a \times (b \times r)$ where $a$, $b$ are constant vectors and $r = (x,y,z)$.
I've worked through the problem to achieve the following answer:
$$(\vec a \cdot \vec r) (\nabla \times \vec b) + \nabla (\vec a \cdot \vec r) \times \vec b - (\vec a \cdot \vec b) (\nabla \times \vec r) - \nabla (\vec a \cdot \vec b) \times \vec r$$
I know that the final answer is, $a \times b$, however I'm unsure of the steps in between to get there. Help would be greatly appreciated.
 A: \begin{align*}
  \mathbf{a} \times (\mathbf{b} \times \mathbf{r}) &=
  (\mathbf{a \cdot r}) \mathbf{b}-(\mathbf{a \cdot b}) \mathbf{r} \\[5pt]
  \nabla \times [\mathbf{a} \times (\mathbf{b} \times \mathbf{r})] &=
  \nabla \times [(\mathbf{a \cdot r}) \mathbf{b}]-
  (\mathbf{a \cdot b}) \nabla \times \mathbf{r} \\[5pt]
  &=
  \begin{vmatrix}
    \mathbf{e}_{x} & \mathbf{e}_{y} & \mathbf{e}_{z} \\
    \partial_{x} & \partial_{y} & \partial_{z} \\
    (\mathbf{a \cdot r}) b_x &
    (\mathbf{a \cdot r}) b_y &
    (\mathbf{a \cdot r}) b_z \\
  \end{vmatrix} - \mathbf{0} \\[5pt]
  &= \sum_{xyz} \mathbf{e}_{x}
     \left \{ \frac{\partial}{\partial y} [(\mathbf{a\cdot r}) b_{z}]-
     \frac{\partial}{\partial z} [(\mathbf{a\cdot r}) b_{y}] \right \} \\[5pt]
  &= \sum_{xyz} \, \mathbf{e}_{x} (a_{y} b_{z}-a_{z} b_{y}) \\[5pt]
  &= \mathbf{a\times b}
\end{align*}
A: To solve this kind of problems usually I use levi-civita symbol.
$(a\times b)_{k}=\varepsilon_{ijk}a_{i}b_{j}$
In this case:
First, you have to notice any differential operation over a constant vector is zero. Therefore, $(\nabla\times[\vec{a}\times(\vec{b}\times\vec{r})])_{s}=\varepsilon_{pqs}\partial_{p}[\vec{a}\times(\vec{b}\times\vec{r})]_{q}=\varepsilon_{pqs}\partial_{p}(\varepsilon_{ijq}a_{i}(\vec{b}\times\vec{r})_{j})=\varepsilon_{pqs}\partial_{p}(\varepsilon_{ijq}a_{i}\varepsilon_{nmj}b_{n}x_{m})=\varepsilon_{pqs}\partial_{p}(\varepsilon_{jqi}\varepsilon_{jnm}a_{i}b_{n}x_{m})=\varepsilon_{pqs}\partial_{p}([\delta_{qn}\delta_{im}-\delta_{qm}\delta_{in}]a_{i}b_{n}x_{m})=\varepsilon_{pqs}([\delta_{qn}\delta_{im}-\delta_{qm}\delta_{in}]a_{i}b_{n}\delta_{mp})=\varepsilon_{pqs}([\delta_{qn}\delta_{ip}-\delta_{qp}\delta_{in}]a_{i}b_{n})=\varepsilon_{pqs}(a_{p}b_{q}-0)=\vec{a}\times\vec{b}$
