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  "body": "\n## Summary\n\nThe Gateaux derivative of a map between vector spaces at a point measures the rate of change of the map along each direction.\nIt generalizes the directional derivative of calculus to normed and locally convex spaces of any dimension.\nIt is weaker than the Fréchet derivative: a map can have a Gateaux derivative at a point without being continuous there.\nIt is the derivative used in the calculus of variations, where it gives the first variation of a functional.\n\n## Explanations\n\n### L1 human\n\nTo differentiate a map at a point $x$, move along a straight line $x + t h$ and look at the rate of change at $t = 0$.\nThe Gateaux derivative asks only for this rate along every line through $x$, one line at a time.\nFor $f(x) = \\norm{x}^2$ on a real Hilbert space, the rate along $h$ is $2\\ip{x}{h}$.\nSince nothing is asked about how the lines fit together, the notion is weaker than the usual derivative.\n",
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