证明△(uv)=u△v+v△u+2▽u•▽v 关于哈密顿算子的,矢量分析类的体,

△(uv)=(ið^2/ðx^2+jð^2/ðy^2+kð^2/ðz^2)(uv),而 ð^2/ðx^2=ð(u*ðv/ðx+vðu/ðx)/ðx=uð^2v/ðx^2+vð^2u/ðx^2+2ðu/ðx*ðv/ðx),同理ð^2/ðy^2=uð^2v/ðy^2+vð^2u/ðy^2+2ðu/ðy*ðv/ðy),ð^2/ðz^2=uð^2v/ðz^2+vð^2u/ðz^2+2ðu/ðz*ðv/ðz).所以△(uv)= u(ið^2v/ðx^2+jð^2v/ðy^2+kð^2v/ðz^2)+v(ið^2u/ðx^2+jð^2u/ðy^2+kð^2u/ðz^2)+2(iðu/ðx*ðv/ðx+jðu/ðy*ðv/ðy+kðu/ðz*ðv/ðz)=u△v+v△u+2(iðu/ðx+jðu/ðy+kðu/ðz)*(iðv/ðx+jðv/ðy+kðv/ðz)=u△v+v△u+2▽u•▽v.