Proof: There is no function $f\in {C}^{2}({\mathbb{R}}^{\mathbb{3}})$ with gradient $\mathrm{\nabla}f(x,y,z)=(yz,xz,x{y}^{2})$

How can one show that there is no function, which is a continuously partially derivable function $f\in {C}^{2}({\mathbb{R}}^{\mathbb{3}})$ with this gradient $\mathrm{\nabla}f(x,y,z)=(yz,xz,x{y}^{2})$.

I thought about using the Hessian matrix since one has to calculate all second partial derivatives of f there.

Since only the gradient is given, can I calculate the antiderivatives first:

$yz=xyz$

$xz=xyz$

$x{y}^{2}=x{y}^{2}z$

Now I want to calculate the antiderivatives of the antiderivatives:

$xyz=\frac{yz{x}^{2}}{2}$

$xyz=\frac{yz{x}^{2}}{2}$

$x{y}^{2}z={\displaystyle \frac{{y}^{2}z{x}^{2}}{2}}$

I didn't calculate the antiderivatives of the partial derivatives and I don't even know if that way is correct...