= Solution
{c}
The space $\mathbb R^3$ is <simply connected>, and the <electric field> has zero <curl>. Hence it is a <conservative vector field>. More explicitly, <Stokes theorem> gives zero <circulation> around any closed loop, since such a loop bounds a spanning <surface> in $\mathbb R^3$. Therefore the <line integral> between two fixed endpoints is independent of the chosen path.
Fix a reference point $\mathbf r_0$ and define the <electric potential>
$$
\varphi(\mathbf r)=-\int_{\mathbf r_0}^{\mathbf r}\mathbf E\cdot d\boldsymbol\ell+C.
$$
Taking a short straight terminal segment in each coordinate direction shows $\partial_j\varphi=-E_j$. Consequently
$$
\boxed{\mathbf E=-\nabla\varphi.}
$$
The <electric potential> is determined up to an additive constant; the later condition at infinity fixes that constant for the finite <charge distribution> in part (b).
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