div(rotA→)=∇⋅∇×A

根据向量分析的矢量恒等式，有：

$$
\begin{aligned}
\operatorname{div}(\operatorname{rot}\mathbf{A}) &= \nabla\cdot(\nabla\times\mathbf{A})\\
&=\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k}\\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\
\frac{\partial}{\partial y} & \frac{\partial}{\partial z} & \frac{\partial}{\partial x}\\
A_x & A_y & A_z\\
\end{vmatrix}\\
&=\left(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\right)+\left(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}\right)+\left(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\right)\\
&=\left(\frac{\partial^2 A_z}{\partial y\partial x}-\frac{\partial^2 A_y}{\partial z\partial x}\right)+\left(\frac{\partial^2 A_x}{\partial z\partial y}-\frac{\partial^2 A_z}{\partial x\partial y}\right)+\left(\frac{\partial^2 A_y}{\partial x\partial z}-\frac{\partial^2 A_x}{\partial y\partial z}\right)\\
&=\left(\frac{\partial^2 A_y}{\partial x\partial z}-\frac{\partial^2 A_z}{\partial x\partial y}\right)+\left(\frac{\partial^2 A_z}{\partial y\partial x}-\frac{\partial^2 A_x}{\partial y\partial z}\right)+\left(\frac{\partial^2 A_x}{\partial z\partial y}-\frac{\partial^2 A_y}{\partial z\partial x}\right)\\
&=\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k}\\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\
\frac{\partial}{\partial y} & \frac{\partial}{\partial z} & \frac{\partial}{\partial x}\\
A_y & A_z & A_x\\
\end{vmatrix}\\
&=\nabla\cdot(\nabla\times\mathbf{A}).
\end{aligned}
$$

因此，$\operatorname{div}(\operatorname{rot}\mathbf{A})=\nabla\cdot(\nabla\times\mathbf{A})$。