= Potential of a conservative vector field
{title2=$X=\nabla\Phi$}
= Vector-field potential
{synonym}
A potential for a <conservative vector field> $X$ is a <function> with <scalar> values $\Phi$ whose <gradient> is $X$. Potentials differ by a constant on each connected component. The <fundamental theorem for line integrals> gives $\int_\gamma X\cdot d\mathbf x=\Phi(\gamma(1))-\Phi(\gamma(0))$. In mechanics a conservative force is conventionally written $-\nabla V$, so its potential energy is $V=-\Phi$. This spatial potential differs from the <scalar potential> $V(\phi)$ that denotes the derivative-free energy density of a <scalar field>.
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