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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $f \\colon U \\to \\mathbb{R}$ be twice differentiable on $U$.\nThen\n$$\\Delta f = \\frac{\\partial^2 f}{\\partial x_1^2} + \\frac{\\partial^2 f}{\\partial x_2^2} + \\dots + \\frac{\\partial^2 f}{\\partial x_n^2}.$$\n\n## Remarks\n\nProof: $\\nabla f = (\\partial f / \\partial x_1, \\dots, \\partial f / \\partial x_n)$, and the [[divergence|02NSMQVQWK]] adds the derivative of the $i$-th component with respect to $x_i$.\nSo $\\Delta f$ is the trace of the [[Hessian matrix|0SCSQWXRA4]] of $f$. The trace does not change under a rotation of coordinates, so neither does $\\Delta f$.\nExamples: $\\Delta (x^2 + y^2) = 4$, $\\Delta (x^2 - y^2) = 0$ and $\\Delta \\|x\\|^2 = 2n$ on $\\mathbb{R}^n$.\nIn polar coordinates in the plane, $\\Delta f = f_{rr} + \\tfrac{1}{r} f_r + \\tfrac{1}{r^2} f_{\\theta\\theta}$ for $r \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nIn Cartesian coordinates, the Laplacian adds up the second derivatives of $f$ in the $n$ coordinate directions.\nOnly the unmixed second partial derivatives appear, not mixed ones like $f_{xy}$.\n",
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