梯度，散度，旋度

问题等价于当
$$
nablatimesvec{v}=0,nablacdotvec{v}=0
$$
什么条件下$vec{v}=0$?
设$Omegasubsetmathbb{R}^3$区域。$partial Omega$为光滑曲面。$vec{v}$是$overline{Omega}$上的向量场
$$
nablatimesvec{v}=0,nablacdotvec{v}=0
$$
设$vec{v}=Pvec{i}+Qvec{j}+Rvec{k}$

$$
nablatimesvec{v}=0Rightarrow
leftlbrace
begin{aligned}
frac{partial R}{partial y}-frac{partial Q}{partial z}=0\
frac{partial P}{partial z}-frac{partial R}{partial x}=0\
frac{partial Q}{partial x}-frac{partial P}{partial y}=0
end{aligned}
right.
$$
$$
nablacdotvec{v}=0Rightarrow
frac{partial P}{partial x}+frac{partial Q}{partial y}+frac{partial R}{partial z}=0
$$
$$
frac{partial nablacdotvec{v}}{partial x}=Delta P=0
$$
$$
Delta P=Delta Q=Delta R=0
$$
命题：设$f,overline{Omega}rightarrowmathbb{R}$光滑函数$Delta f=0$且$f|_{partialOmega}equiv 0rightarrow fequiv 0$

证明
$$
nablacdot(fnabla f)=nabla fcdotnabla f+fnablacdotnabla f=|nabla f|^2+fnabla f=|nabla f|^2
$$
积分
$$int_Omega|nabla f|^2mathop{}mathrm{d}V=int_Omeganabla cdot(fnabla f)mathop{}mathrm{d}V=int_{partialOmega}(fnabla f)mathop{}mathrm{d}vec{S}=0rightarrow|nabla f|equiv 0$$
所以$f$是常值函数