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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open.\nA function $u \\colon U \\to \\mathbb{R}$ is harmonic on $U$ if $u$ is of [[class C^2|0TCBZMEYGN]] on $U$ and its [[Laplacian|0XQ2ZVN1PZ]] vanishes on $U$:\n$$\\Delta u = \\frac{\\partial^2 u}{\\partial x_1^2} + \\dots + \\frac{\\partial^2 u}{\\partial x_n^2} = 0.$$\n\n## Remarks\n\nEach affine function is harmonic. In the plane, $x^2 - y^2$, $x y$ and $e^x \\sin y$ are harmonic, and so is $\\ln(x^2 + y^2)$ on $\\mathbb{R}^2 \\setminus \\{0\\}$.\nOn $\\mathbb{R}^3 \\setminus \\{0\\}$, $1 / \\|x\\|$ is harmonic. It is the potential of a unit point charge at the origin.\nThe real and imaginary parts of a holomorphic function of $z = x + i y$ are harmonic functions of $(x, y)$.\nMean value property: let the closed ball of centre $a$ and radius $r$ lie in $U$. Then $u(a)$ is the average of $u$ on the sphere of radius $r$ about $a$, and also on the ball.\n\n## Explanations\n\n### L1 human\n\nA harmonic function has no local bumps or dips: its value at each point is the average of its values around the point.\nA stretched rubber sheet held at fixed heights on its rim has a harmonic height function inside.\n",
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