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  "body": "\n## Definition\n\nLet $r \\colon D \\to \\mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is regular at the point $(u_0, v_0)$ of $D$, and let $p = r(u_0, v_0)$.\nThe tangent plane at $p$ is the plane $\\{ p + s \\, r_u(u_0, v_0) + t \\, r_v(u_0, v_0) : s, t \\in \\mathbb{R} \\}$.\nThe vector $N(u_0, v_0) = r_u(u_0, v_0) \\times r_v(u_0, v_0)$ is the normal vector of $r$ at $(u_0, v_0)$, and $n = N / \\|N\\|$ is the unit normal vector.\n\n## Remarks\n\nThe vector $N$ is perpendicular to $r_u$ and $r_v$, so it is perpendicular to the tangent plane. The other unit normal at $p$ is $-n$.\nFor the graph $z = f(x, y)$, the normal is $(-f_x, -f_y, 1)$, which points upward.\nIf the surface is the level set $g(x, y, z) = c$ of a $C^1$ function $g$ with $\\nabla g(p) \\ne 0$, then the [[gradient|0NS2QYVC4Y]] $\\nabla g(p)$ is a normal vector at $p$.\nFor the unit sphere with $r(\\varphi, \\theta) = (\\sin \\varphi \\cos \\theta, \\sin \\varphi \\sin \\theta, \\cos \\varphi)$, one gets $r_\\varphi \\times r_\\theta = \\sin \\varphi \\, r(\\varphi, \\theta)$, which points outward for $0 \u003c \\varphi \u003c \\pi$.\n\n## Explanations\n\n### L1 human\n\nThe tangent plane is the flat plane that touches the surface at a point, and the normal vector sticks straight out of it.\nThe cross product of the two coordinate tangent vectors gives a normal at once.\n",
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