Solution
= Solution
With $\beta=0$, integrating $\Phi_x=2\alpha x+\alpha^2y+\alpha z$ gives
$$
\Phi=\alpha x^2+\alpha^2xy+\alpha xz+K(y,z).
$$
The other two <partial derivatives> require $K_y=K_z=0$. The origin condition fixes the remaining constant to zero. Hence the <scalar potential> is
$$
\boxed{\Phi(x,y,z)=\alpha x^2+\alpha^2xy+\alpha xz}.
$$
Its value at $(1,1,1)$ gives $\alpha^2+2\alpha=1$, so
$$
\boxed{\alpha=-1\pm\sqrt2}.
$$