{
  "aliases": [
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^3$ be open, and let $F \\colon U \\to \\mathbb{R}^3$ be a vector field of class $C^1$ with $\\nabla \\times F = 0$ on $U$.\nLet $c_0, c_1 \\colon [0, 1] \\to U$ be piecewise smooth [[closed curves|0D032NSZKY]].\nSuppose that there is a piecewise smooth map $H \\colon [0, 1] \\times [0, 1] \\to U$ with $H(t, 0) = c_0(t)$, $H(t, 1) = c_1(t)$ and $H(0, s) = H(1, s)$ for all $s, t \\in [0, 1]$.\nThen\n$$\\oint_{c_0} F \\cdot d\\mathbf{r} = \\oint_{c_1} F \\cdot d\\mathbf{r}.$$\n\n## Remarks\n\nProof: apply [[Stokes' theorem|0AKRKVHW94]] to the surface $H([0, 1]^2)$. The curl term is zero, and the integrals along the curves $s \\mapsto H(0, s)$ and $s \\mapsto H(1, s)$ cancel.\nIf $c_1$ is a constant loop, then $\\oint_{c_0} F \\cdot d\\mathbf{r} = 0$.\nOn a [[simply connected|075C629XTH]] domain, each loop deforms to a point, and a smoothing argument gives a piecewise smooth $H$. So curl-free $C^1$ fields on such domains are [[conservative|05R9NF2J49]].\nThe condition $\\nabla \\times F = 0$ means that the 1-form $F_1 \\, dx + F_2 \\, dy + F_3 \\, dz$ is closed. This explains the name of the theorem.\n\n## Explanations\n\n### L1 human\n\nFor a field with no swirl, the work around a loop does not change when you slide and bend the loop continuously inside the domain.\nA loop that can shrink to a point therefore has zero work.\n",
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        "locator": "Helmholtz's theorem",
        "url": "https://en.wikipedia.org/wiki/Stokes%27_theorem"
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        "locator": "Conservative forces",
        "url": "https://en.wikipedia.org/wiki/Stokes%27_theorem"
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        "locator": "Vector field analogies",
        "url": "https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms"
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  "name": "Homotopy invariance of line integrals of closed forms",
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