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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^2$ be open, let $f \\colon U \\to \\mathbb{R}$, and let $(a, b) \\in U$ be a point at which $f$ is [[differentiable|0135XZZQ8M]].\nThe tangent plane to the [[graph|0JBJYT69ZV]] $z = f(x, y)$ at the point $(a, b, f(a, b))$ is the plane in $\\mathbb{R}^3$ with the equation\n$$z = f(a, b) + \\frac{\\partial f}{\\partial x}(a, b)\\,(x - a) + \\frac{\\partial f}{\\partial y}(a, b)\\,(y - b).$$\nMore generally, let $U \\subseteq \\mathbb{R}^n$ be open and let $f$ be differentiable at $a \\in U$. The tangent hyperplane to the graph at $(a, f(a))$ is the set of points $(x, z) \\in \\mathbb{R}^n \\times \\mathbb{R}$ with $z = f(a) + \\nabla f(a) \\cdot (x - a)$.\n\n## Remarks\n\nThe vector $\\big(\\partial f / \\partial x(a, b), \\, \\partial f / \\partial y(a, b), \\, -1\\big)$ is normal to the tangent plane.\nFor $f(x, y) = x^2 + y^2$ at $(1, 1)$, the tangent plane is $z = 2 + 2(x - 1) + 2(y - 1)$, that is, $z = 2x + 2y - 2$.\nThe tangent plane exists only where $f$ is differentiable. The existence of the [[partial derivatives|09EGC6F8PT]] alone gives a plane, but this plane need not touch the graph well.\n\n## Explanations\n\n### L1 human\n\nPut a flat sheet on top of a smooth hill so that it touches the hill at one point and does not tilt against it.\nThat sheet is the tangent plane. Near the point of contact, the hill and the sheet are almost the same.\n",
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