场（向量场）：X:U→Rn\mathbf{X}:U\rightarrow\mathbb{R}^nX:U→Rn 梯度场：Δf=gradf=(∂f∂x,∂f∂y,∂f∂z)T\Delta f=\text{grad} f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})^TΔf=gradf=(∂x∂f,∂y∂f,∂z∂f)T 000 阶微分形式的外微分 旋度场：curlX=rotX=(∂R∂y−∂Q∂z,∂P∂z−∂R∂x,∂Q∂x−∂P∂y)\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})curlX=rotX=(∂y∂R−∂z∂Q,∂z∂P−∂x∂R,∂x∂Q−∂y∂P) 111 阶微分形式的外微分 散度：divX=∂P∂x+∂Q∂z+∂R∂z\text{div} \mathbf{X}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial z}+\frac{\partial R}{\partial z}divX=∂x∂P+∂z∂Q+∂z∂R 222 阶微分形式的外微分 curl(∇f)=0\text{curl}({\nabla f})=0curl(∇f)=0 div(curlX)=0\text{div}({\text{curl} \mathbf{X}})=0div(curlX)=0