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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^3$ be open, and let $F = (P, Q, R) \\colon U \\to \\mathbb{R}^3$ be a [[vector field|0N67NBKQ5T]] whose first [[partial derivatives|05AXG3QRKR]] exist on $U$.\nThe curl of $F$ is the vector field $\\operatorname{curl} F \\colon U \\to \\mathbb{R}^3$,\n$$\\operatorname{curl} F = \\nabla \\times F = \\left( \\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z}, \\ \\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x}, \\ \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right).$$\nA field with $\\operatorname{curl} F = 0$ on $U$ is irrotational, or curl-free.\n\n## Remarks\n\nAs a mnemonic, $\\nabla \\times F$ is the formal determinant of the matrix with rows $(\\mathbf{i}, \\mathbf{j}, \\mathbf{k})$, $(\\partial_x, \\partial_y, \\partial_z)$ and $(P, Q, R)$.\nFor $F(x, y, z) = (-y, x, 0)$, $\\operatorname{curl} F = (0, 0, 2)$. For $F(x, y, z) = (x, y, z)$, $\\operatorname{curl} F = 0$.\nFor a plane field $(P, Q)$, one uses the scalar curl $\\partial Q / \\partial x - \\partial P / \\partial y$, the third component of the curl of $(P, Q, 0)$. It appears in [[Green's theorem|0HR3N02317]].\nTake small loops orthogonal to a unit vector $n$. The component of $\\operatorname{curl} F$ along $n$ is the limit of the circulation of $F$ around these loops, divided by the enclosed area. This follows from [[Stokes' theorem|0AKRKVHW94]].\n\n## Explanations\n\n### L1 human\n\nPut a tiny paddle wheel into a fluid with velocity field $F$.\nIf it spins, the curl is not zero there. The curl points along the axis about which the wheel spins fastest.\n",
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