= Solution
The three components of the <curl> cancel pairwise:
$$
\nabla\times\mathbf F=(1-1,1-1,1-1)=\mathbf0.
$$
The domain is all of $\mathbb R^3$, which is <simply connected>, so this is consistent with a <conservative vector field>. To construct its <scalar potential>, integrate $\Phi_x=y+z$ to obtain $\Phi=x(y+z)+h(y,z)$. Matching $\Phi_y=x+z$ gives $h_y=z$, hence $h=yz+k(z)$. Finally $\Phi_z=x+y$ forces $k'(z)=0$. Consequently
$$
\boxed{\Phi(x,y,z)=xy+xz+yz+C,\qquad\mathbf F=\nabla\Phi.}
$$
Every <scalar potential> differs by a constant on this connected domain. The endpoint values differ by one, agreeing with the <line integral> in part (a).
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