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  "body": "\n## Summary\n\nThe heat kernel is the solution of the [[heat equation|VP2HN86VAG]] that starts from a unit amount of heat concentrated at one point.\nOn $\\R^n$ it is the Gaussian $\\Phi(x, t) = (4 \\pi t)^{-n/2} e^{-\\abs{x}^2 / (4t)}$.\nEvery bounded continuous initial temperature evolves by convolution with it.\nOn domains and on Riemannian manifolds the heat kernel encodes the spectrum of the Laplacian.\nIt is a basic tool in geometric analysis and index theory.\n\n## Explanations\n\n### L1 human\n\nTouch a cold infinite metal plate with a hot needle for an instant and watch the heat spread.\nAt each later time the temperature is a bell-shaped curve centred at the touched point, getting wider and flatter as time goes on.\nThe total amount of heat stays the same, so the area under the bell is always one.\nAny initial temperature is a sum of many such touches, so its evolution is the matching sum of bells.\n",
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