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  "body": "\n## Summary\n\nThe first variation of a [[functional|1EQM0DJ4NC]] $J$ at $y$ in the direction $\\eta$ is the rate of change of $J(y + t\\eta)$ at $t = 0$.\nIt is the analogue for functionals of the [[directional derivative|06KWGHDN2J]].\nAt a minimiser it vanishes in every admissible direction, and for [[integral functionals|JRQZ803Q5Y]] this condition leads to the [[Euler–Lagrange equation|PS0VV149ET]].\n\n## Explanations\n\n### L1 human\n\nPush a function $y$ a little in the direction of another function $\\eta$, and watch how the value of the functional changes.\nThe first variation is the speed of that change at the start.\nAt the bottom of a valley every direction is flat to first order, so at a minimiser the first variation is zero.\nFor $J(y) = \\int_0^1 y^2 \\, dx$, the first variation at $y$ in the direction $\\eta$ is $2 \\int_0^1 y \\eta \\, dx$.\n",
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