Curl of cross product of two vectors: $\operatorname{curl}(r\times a)$

Question

I cannot find mistake in any method hand written is mine, can someone point out the mistake book image my solution

Answer 1

Your error is that $ \nabla \cdot a \neq 0$ but $\nabla \cdot a = a_1 \partial_1 + a_2 \partial_2 + a_3 \partial_3$. Therefore, the first term reads

$(\nabla \cdot \vec{a}) \vec{r} = a_1 \cdot \partial_1(x) \cdot \vec{i} + a_2 \cdot \partial_2(y) \cdot\vec{j} + a_3 \cdot \partial_3(z) \cdot\vec{k} = \vec{a}$,

as (due to my notation) $\partial_1(x) := \partial_x(x) = 1$ and so on.

Therefore, as the calculation of the second term was correct, we obtain $(\nabla \cdot \vec{a})\vec{r} - (\nabla \cdot \vec{r})\vec{a} = \vec{a} - 3 \vec{a} = -2 \vec{a}$