详细计算，条理清晰：\nabla \times \left ( \nabla \Phi \right ) =？

根据矢量分析理论，可以将上式展开为：
$$
\nabla \times \left ( \nabla \Phi \right ) = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \
\frac{\partial \Phi}{\partial x} & \frac{\partial \Phi}{\partial y} & \frac{\partial \Phi}{\partial z}
\end{vmatrix}
$$
对上式进行计算，可以得到：
$$
\nabla \times \left ( \nabla \Phi \right ) = \left ( \frac{\partial^2 \Phi}{\partial y \partial z} - \frac{\partial^2 \Phi}{\partial z \partial y} \right ) \hat{i} + \left ( \frac{\partial^2 \Phi}{\partial z \partial x} - \frac{\partial^2 \Phi}{\partial x \partial z} \right ) \hat{j} + \left ( \frac{\partial^2 \Phi}{\partial x \partial y} - \frac{\partial^2 \Phi}{\partial y \partial x} \right ) \hat{k}
$$
因此，$\nabla \times \left ( \nabla \Phi \right )$ 的结果为：
$$
\nabla \times \left ( \nabla \Phi \right ) = \left ( \frac{\partial^2 \Phi}{\partial y \partial z} - \frac{\partial^2 \Phi}{\partial z \partial y} \right ) \hat{i} + \left ( \frac{\partial^2 \Phi}{\partial z \partial x} - \frac{\partial^2 \Phi}{\partial x \partial z} \right ) \hat{j} + \left ( \frac{\partial^2 \Phi}{\partial x \partial y} - \frac{\partial^2 \Phi}{\partial y \partial x} \right ) \hat{k}
$$