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  "body": "\n## Summary\n\nThe shape derivative measures how a quantity that depends on a domain changes when the domain is deformed a little.\nOne moves each point $x$ of the domain to $x + \\theta(x)$ for a small vector field $\\theta$.\nThe shape derivative is the derivative of the quantity with respect to $\\theta$.\nFor smooth domains it usually depends only on the normal component of $\\theta$ on the boundary.\nThis fact is known as the Hadamard structure theorem.\nIt is the gradient that drives numerical methods of [[shape optimization|WWQJQXJECK]].\n\n## Explanations\n\n### L1 human\n\nPush the boundary of a region outward a little, in some places more than in others.\nThe area of the region grows, and to first order the growth is the outward push integrated along the boundary.\nThat first-order rate of change is the shape derivative of the area.\nFor other quantities, such as the drag on a wing, the shape derivative tells an engineer where to push the boundary to improve the design.\n",
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