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  "body": "\n## Definition\n\nLet $S \\subseteq \\mathbb{R}^n$.\nA vector field on $S$ is a [[vector-valued function|018GTRS7A4]] $F \\colon S \\to \\mathbb{R}^n$, $F(x) = (F_1(x), \\dots, F_n(x))$.\nThe vector field is continuous, or of class $C^k$, if each component $F_i$ is [[continuous|0S9EK5MQPE]], or of [[class C^k|0TCBZMEYGN]], on $S$.\n\n## Remarks\n\nOne pictures $F$ by an arrow $F(x)$ drawn with its tail at the point $x$.\nFor a differentiable function $f \\colon U \\to \\mathbb{R}$ on an open set $U$, the gradient $\\nabla f$ is a vector field on $U$. Such fields are gradient fields.\nThe field $F(x, y) = (-y, x)$ on $\\mathbb{R}^2$ describes a counterclockwise rotation about the origin. The field $F(x) = x$ points away from the origin.\nIn $\\mathbb{R}^3$ one often writes $F = P \\, \\mathbf{i} + Q \\, \\mathbf{j} + R \\, \\mathbf{k}$, with $P$, $Q$, $R$ the components of $F$.\n\n## Explanations\n\n### L1 human\n\nA vector field attaches an arrow to each point.\nThink of the wind on a weather map: at each place there is a wind direction and a wind speed.\n",
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