Problem: Let $v=(v_1,v_2,v_3)$ be smooth vector field. Show that $-lap v=curlcurl v- Div v$.
Let $curl v=w$. Then $$eex ea -lap v_1&=-(p_1p_1+p_2p_2+p_3p_3)v_1\ &=-p_1p_1v_1-p_1p_2v_2-p_1p_3v_3\ &quad -p_2(p_2v_1-p_1v_2)-p_3(p_3v_1-p_1v_3)\ &=-p_1(p_1v_1+p_2v_2+p_3v_3) +p_2w_3-p_3w_2\ &=-p_1Div v+(curl w)_1. eeaeeex$$ And similarly, $$eex ea -lap v_2&=-p_2Div v+(curl w)_2,\ -lap v_3&=-p_2Div v+(curl w)_3. eea eeex$$