• 场（向量场）：$\mathbf{X}:U\rightarrow\mathbb{R}^n$
• 梯度场：$\Delta f=\text{grad} f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})^T$
  • $0$ 阶微分形式的外微分
• 旋度场：$\text{curl} \mathbf{X} = \text{rot} \mathbf{X} = (\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})$
  • $1$ 阶微分形式的外微分
• 散度：$\text{div} \mathbf{X}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial z}+\frac{\partial R}{\partial z}$
  • $2$ 阶微分形式的外微分
• $\text{curl}({\nabla f})=0$
• $\text{div}({\text{curl} \mathbf{X}})=0$