Potential of a conservative vector field ($X=\nabla \Phi$)

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 $V(\varphi )$ that denotes the derivative-free energy density of a scalar field.

For a continuously differentiable curl-free vector field $F$ on a domain star-shaped about the origin, define $\Phi$ by the displayed integral. Differentiation and ${\partial }_i{F}_j={\partial }_j{F}_i$ give ${\partial }_i\Phi ={\int }_0^1[F_i(tx)+t{x}_j{\partial }_j{F}_i(tx)]dt={\int }_0^1{\partial }_t[t{F}_i(tx)]dt=F_i(x)$. Thus $F=\nabla \Phi$. To adopt the sign convention $F=-\nabla \varphi$, take $\varphi =-\Phi$. The star-shaped-domain hypothesis ensures every segment used in the construction lies in the domain; curl-free fields need not have global potentials on arbitrary domains.