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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]].\nThe field $F$ is conservative on $U$ if there is a function $\\phi \\colon U \\to \\mathbb{R}$ of [[class C^1|0TCBZMEYGN]] with $F = \\nabla \\phi$ on $U$, that is, $F_i = \\partial \\phi / \\partial x_i$ for $i = 1, \\dots, n$.\n\n## Remarks\n\nA function $\\phi$ with $\\nabla \\phi = F$ is a [[scalar potential|0FP4BVDAAQ]] of $F$. Physics texts often use $-\\phi$ instead.\nThe field $F(x, y) = (2 x y, x^2)$ is conservative on $\\mathbb{R}^2$, with $\\phi(x, y) = x^2 y$.\nThe field $F(x, y) = (-y, x)$ is not conservative on $\\mathbb{R}^2$. Its line integral around the unit circle is $2\\pi \\ne 0$, which contradicts [[path independence|0BFN21TF5F]].\nIf $F$ is conservative and of class $C^1$, then its [[curl is zero|0M2JRW1J0N]]. The converse holds on convex sets but [[fails in general|0GRM3BS0DD]].\n\n## Explanations\n\n### L1 human\n\nA conservative field is the slope field of some landscape $\\phi$.\nGoing around a loop in a landscape brings you back to the same height. So the total work of the field around the loop is zero.\n",
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