Equivalence Of Second Differential of a 0 - Form and curl grad f

The statement that for a 0 -form  
\[\omega^0 , \: d (d \omega^0) =0\]
  is equivalent to the statement that for a function  
\[f, \: \mathbf{\nabla} \times (\mathbf{\nabla} f)=0\]
\[\omega^0 \: f\]
  are both twice differentiable.
\[\omega^0 =f(x_1,x_2,x_3)\]

\[\begin{equation} \begin{aligned} d(d \omega^) &= (\frac{\partial^2 d}{\partial x_2 \partial x_2} - \frac{\partial^2 d}{\partial x_1 \partial x_1}) dx_1 \wedge dx_2 + (\frac{\partial^2 d}{\partial x_3 \partial x_1} - \frac{\partial^2 d}{\partial x_1 \partial x_3}) dx_3 \wedge dx_1 \\ &+ (\frac{\partial^2 d}{\partial x_2 \partial x_3} - \frac{\partial^2 d}{\partial x_3 \partial x_2}) dx_2 \wedge dx_3 \\ &=0 \end{aligned} \end{equation}\]

On the other hand,
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \times (\mathbf{\nabla} f) &= (\frac{\partial}{\partial x_1}, (\frac{\partial}{\partial x_2},(\frac{\partial}{\partial x_3}) \times (\frac{\partial f}{\partial x_1}, (\frac{\partial f}{\partial x_2 \partial x_3}) \\ &=(\frac{\partial^2 f}{\partial x_2 \partial x_3}- \frac{\partial^2 f}{\partial x_3 \partial x_2} , \frac{\partial^2 f}{\partial x_3 \partial x_1}- \frac{\partial^2 f}{\partial x_1 \partial x_3}, \frac{\partial^2 f}{\partial x_1 \partial x_2}- \frac{\partial^2 f}{\partial x_2 \partial x_1} )\\ &=0 \end{aligned} \end{equation}\]

The components of  
\[\mathbf{\nabla} \times (\mathbf{\nabla} f)\]
  are the component functions of  
\[d ( d \omega^0 )\]
. The Theorem is proved.