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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $F \\colon U \\to \\mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].\nA scalar potential of $F$ is a function $\\phi \\colon U \\to \\mathbb{R}$ of class $C^1$ with $\\nabla \\phi = F$ on $U$.\n\n## Remarks\n\nIn 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.\nIf $U$ is connected, two scalar potentials of $F$ differ by a constant. Indeed, their difference has zero gradient on $U$.\nOn 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|05R9NF2J49]].\nExample: the gravitational field $F(x) = -x / \\|x\\|^3$ on $\\mathbb{R}^3 \\setminus \\{0\\}$ has the scalar potential $\\phi(x) = 1 / \\|x\\|$.\n\n## Explanations\n\n### L1 human\n\nA scalar potential is a height function for a vector field.\nThe field points in the direction in which the potential grows fastest, and its length is the rate of growth.\n",
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