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  "body": "\n## Definition\n\nLet $U$ be an open subset of the [[complex plane|0CTE7A31XQ]], identified with $\\mathbb{R}^2$, and let $z_0 \\in U$.\nLet $f : U \\to \\mathbb{C}$ be differentiable at $z_0$ as a map from an open subset of $\\mathbb{R}^2$ to $\\mathbb{R}^2$, with Jacobian matrix $J$ at $z_0$.\nThe map $f$ is conformal at $z_0$ if $J = \\lambda Q$ for a real number $\\lambda \u003e 0$ and a rotation matrix $Q$. A rotation matrix is an orthogonal $2 \\times 2$ matrix with determinant $1$.\nThe map $f$ is conformal on $U$ if it is conformal at each point of $U$.\n\n## Remarks\n\nEquivalently, $f$ is conformal at $z_0$ when its real derivative at $z_0$ is the map $h \\mapsto a h$ for a complex number $a \\neq 0$.\nA map that is conformal at $z_0$ preserves the angles between curves through $z_0$, and their orientation.\nSome authors also allow orientation-reversing maps, such as $z \\mapsto \\bar{z}$, whose Jacobian is a positive number times an orthogonal matrix.\nOther authors use \"conformal map\" for an injective holomorphic map on an open set.\n\n## Explanations\n\n### L1 human\n\nA conformal map can stretch and turn a small figure, but it does not distort its shape.\nIf two curves cross at an angle of $30$ degrees, then their images cross at the same angle, in the same direction.\nFor example, $z \\mapsto 2z$ and $z \\mapsto iz$ are conformal at each point.\n",
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  "name": "Conformal map",
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