A potential function is a scalar function whose gradient represents a vector
field. If 
,
then 
is a scalar potential for 
. In mechanics and electrostatics the sign convention 
is commonly used, so the force
points toward decreasing potential 
.
On a connected domain, a potential function is unique up to an additive constant. A conservative field has a potential function and its line integral is independent of the path. Conversely, a continuously differentiable vector field with zero curl has a potential function on a simply connected domain.
Potential functions are harmonic where they satisfy Laplace's equation, as occurs away from sources for Newtonian and electrostatic potentials. In the presence of sources they instead satisfy a Poisson's equation. A vector potential is a different construction in which a vector field is represented as the curl of another vector field.
