Scalar potential

Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F \colon U \to \mathbb{R}^n$ be a continuous vector field. A scalar potential of $F$ is a function $\phi \colon U \to \mathbb{R}$ of class $C^1$ with $\nabla \phi = F$ on $U$.

Remarks

In physics, a potential is often a function $V$ with $F = -\nabla V$, so that $V$ is a potential energy. The two conventions differ by a sign. If $U$ is connected, two scalar potentials of $F$ differ by a constant. Indeed, their difference has zero gradient on $U$. On a connected open set $U$ one finds a potential by fixing $p \in U$ and setting $\phi(x) = \int_\gamma F \cdot d\mathbf{r}$ for a path $\gamma$ from $p$ to $x$. This works when $F$ is conservative. Example: the gravitational field $F(x) = -x / \|x\|^3$ on $\mathbb{R}^3 \setminus \{0\}$ has the scalar potential $\phi(x) = 1 / \|x\|$.

Explanations

L1 human

A scalar potential is a height function for a vector field. The field points in the direction in which the potential grows fastest, and its length is the rate of growth.