Potential, function and vector field

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In one paragraph

A function (a scalar field) gives one number at every point, like the height of the ground. A vector field gives an arrow at every point, like wind or the force on a particle. A potential is not a third kind of object: it is a function with a job, namely that its slope generates a vector field. If \phi is a potential, the arrows are its gradient, F=\nabla\phi (or F=-\nabla\phi in physics). The potential is the hidden landscape, and the field is the slope of that landscape.

The picture

Left: a hill drawn with elliptical dashed contour lines and blue arrows pointing uphill, crossing the contours at right angles. Right: orange arrows swirling around a centre, with a dotted loop; no hill produces this field. — Left: a potential (the hill, shown by its contour lines) and the vector field it generates (the arrows point uphill, at right angles to the contours). Right: a swirling field that no hill produces.

Think of a ball on a hillside. The height at each place is the function. The push on the ball, one arrow per place, is the vector field, and it always points along the steepest slope, at right angles to the contour lines. The hill is one number per point while the field is several, so a potential packs a whole field of arrows into a single function.

Why a potential is useful

Less to store. In n dimensions a vector field needs n numbers at every point, and a potential needs one.
Path independence. If a field comes from a potential, the total push along a route between two points depends only on the endpoints, because it equals the change in height. A long route and a short one give the same answer.
Special points. Where the arrows vanish the potential is flat, and the Hessian then says whether the point is a minimum, a saddle or a trough: see Jacobian and Hessian.

Not every vector field is a gradient

A swirling field cannot come from a hill: walking once around a loop, you would be pushed forward the whole way and finish with a net gain, but a hill gives back exactly what it takes. A field comes from a potential exactly when it has no swirl (it is curl-free, on a region with no holes). A flow circling a drain is the standard field with no potential.

In these notes

In a dually flat space there is a potential \psi(\theta), and the two coordinate systems are tied together by its gradient: \eta=\nabla\psi(\theta). The expectation parameters are the arrows, and the log-partition function \psi is the landscape.
The Hessian of \psi is the metric: see Jacobian and Hessian and distance, divergence and metric.
A Bregman divergence is built from the potential: it is the gap between \psi at one point and its tangent-line approximation taken from the other point: Chapter 1: dually flat structure.
Convexity and the transform: the potential \psi is convex, and its Legendre transform gives the dual potential: Convex function and the Legendre transform.

Takeaways

A potential is a function whose gradient is a vector field: the landscape behind the arrows.
The arrows point along the steepest slope, at right angles to the contour lines.
A field is a gradient only if it has no swirl; a circulating flow has no potential.
In dually flat geometry the potential \psi gives the other coordinates by its gradient and the metric by its Hessian.