Problem: Let $v=(v_1,v_2,v_3)$ be smooth vector field. Show that $-\lap v=\curl\curl v-\n \Div v$.
Let $\curl v=w$. Then $$\beex \bea -\lap v_1&=-(\p_1\p_1+\p_2\p_2+\p_3\p_3)v_1\\ &=-\p_1\p_1v_1-\p_1\p_2v_2-\p_1\p_3v_3\\ &\quad -\p_2(\p_2v_1-\p_1v_2)-\p_3(\p_3v_1-\p_1v_3)\\ &=-\p_1(\p_1v_1+\p_2v_2+\p_3v_3) +\p_2w_3-\p_3w_2\\ &=-\p_1\Div v+(\curl w)_1. \eea\eeex$$ And similarly, $$\beex \bea -\lap v_2&=-\p_2\Div v+(\curl w)_2,\\ -\lap v_3&=-\p_2\Div v+(\curl w)_3. \eea \eeex$$
原文：https://www.cnblogs.com/zhangzujin/p/9041050.html