= Solution
On a connected domain, a <vector field> is a <conservative vector field> if and only if it is the <gradient> of a single-valued <scalar potential>, $\boxed{\mathbf H=\boldsymbol\nabla\phi}$. Equivalently, its <line integral> is path independent, or its integral around every closed curve is zero. For a continuously differentiable field on a <simply connected> domain, this is equivalent to $\boldsymbol\nabla\times\mathbf H=\mathbf0$. Without the domain hypothesis, vanishing <curl> alone is only a local test and is not the necessary-and-sufficient global statement.
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