= Solution
A <conservative force> has work depending only on the endpoints, so it can be written
$$
\boxed{\mathbf F=-\nabla U,\qquad W_{A\to B}=U(A)-U(B),\qquad
\oint\mathbf F\cdot d\mathbf r=0.}
$$
The arbitrary additive constant in <potential energy> has no effect on force. A smooth curl-free force is conservative on a simply connected domain; a hole or excluded singularity requires checking global circulation as well.
Uniform gravity has $U=mgz$, central gravity has $U=-GMm/r$ with zero at infinity, and a linear spring has $U=kx^2/2$ relative to equilibrium. If the potential is time independent and no other force does work, $K+U$ is conserved by the <work-energy theorem>. A time-dependent potential or nonconservative work can change mechanical energy; having an instantaneous gradient force alone is not sufficient for energy conservation. <University Physics Volume 1 by William Moebs and others>[Chapter8, conservative forces, potential energy and mechanical-energy conservation]{ref}
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