电动力学数学基础
$\nabla$
  记$\boldsymbol{g_i}$为协变基矢量，$\boldsymbol{g^i}$为逆变基矢量，$\varphi$为标量场，$\boldsymbol{u,v}$为矢量场，$\boldsymbol{T}$为张量场，则梯度、散度、旋度可表示如下如下
$$\nabla\boldsymbol{T}=\boldsymbol{g^i}\partial_i\boldsymbol{T}\qquad\boldsymbol{T}\nabla=\partial_i\boldsymbol{T}\boldsymbol{g^i}\\
\nabla\cdot\boldsymbol{T}=\boldsymbol{g^i}\cdot\partial_i\boldsymbol{T}\qquad\boldsymbol{T}\cdot\nabla=\partial_i\boldsymbol{T}\cdot\boldsymbol{g^i}\\
\nabla\times\boldsymbol{T}=\boldsymbol{g^i}\times\partial_i\boldsymbol{T}\qquad\boldsymbol{T}\times\nabla=\partial_i\boldsymbol{T}\times\boldsymbol{g^i}$$
  算子之间常用的等式有
$$\nabla(\varphi\boldsymbol{T})=\nabla\varphi\boldsymbol{T}+\varphi\nabla\boldsymbol{T}\\
\nabla\cdot(\varphi\boldsymbol{T})=\nabla\varphi\cdot\boldsymbol{T}+\varphi\nabla\cdot\boldsymbol{T}\\
\nabla\times(\varphi\boldsymbol{T})=\nabla\varphi\times\boldsymbol{T}+\varphi\nabla\times\boldsymbol{T}\\
\nabla(\boldsymbol{u}\cdot\boldsymbol{v})=\nabla\boldsymbol{u}\cdot\boldsymbol{v}+\nabla\boldsymbol{v}\cdot\boldsymbol{u}\\
\nabla(\boldsymbol{u}\cdot\boldsymbol{v})=\boldsymbol{u}\times(\nabla\times\boldsymbol{v})+\boldsymbol{v}\times(\nabla\times\boldsymbol{u})+\boldsymbol{u}\cdot(\nabla\boldsymbol{v})+\boldsymbol{v}\cdot(\nabla\boldsymbol{u})\\
\nabla\times\boldsymbol{u}\times \boldsymbol{v}=(\boldsymbol{u}\nabla-\nabla\boldsymbol{u})\cdot\boldsymbol{v}\\
\nabla\times(\boldsymbol{u}\times \boldsymbol{v})=\boldsymbol{u}(\nabla\cdot\boldsymbol{v})-\boldsymbol{v}(\nabla\cdot\boldsymbol{u})-\boldsymbol{u}\cdot(\nabla\boldsymbol{v})+\boldsymbol{v}\cdot(\nabla\boldsymbol{u})\\
\nabla\times\nabla\varphi=0\\
\nabla\cdot(\nabla\times\boldsymbol{u})=0$$