= Solution
{c}
For a twice continuously differentiable <scalar potential>, mixed second <partial derivatives> commute. Taking the spatial <curl> of the equation from part (i) gives
$$
\boxed{\nabla\times\ddot{\mathbf x}
=-\nabla\times\nabla\xi=\mathbf0.}
$$
For example, its first component is $-\partial_2\partial_3\xi+\partial_3\partial_2\xi=0$. The <curl> acts on the spatial representation of the <material acceleration>, not directly on its expression in reference coordinates.
This is a necessary local condition. It is sufficient locally, and sufficient globally on a <simply connected domain> under the usual smoothness assumptions. On a domain with holes, a global <scalar potential> additionally requires every closed-loop <line integral> of the <material acceleration> to vanish. The homogeneous motion in part (c) admits an explicit global <scalar potential>, so this topological qualification causes no obstruction there.
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