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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field, let $X \\subseteq k^n$ be an [[affine variety|JERVQT4C12]] with [[ideal|HH6VEVM6Z0]] $I(X)$, and let $p \\in X$.\nFor $f \\in k[x_1, \\dots, x_n]$, its differential at $p$ is the linear form\n$$d_p f(v) = \\sum_{i=1}^{n} \\frac{\\partial f}{\\partial x_i}(p) \\, v_i, \\qquad v \\in k^n,$$\nwith the formal [[partial derivatives|09EGC6F8PT]].\nThe *Zariski tangent space* of $X$ at $p$ is the linear subspace\n$$T_p X = \\set{v \\in k^n : d_p f(v) = 0 \\text{ for all } f \\in I(X)}.$$\n\n## Remarks\n\nIf $I(X) = (f_1, \\dots, f_r)$, then $T_p X$ is the kernel of the [[Jacobian matrix|047NDTEPCV]] $\\left(\\frac{\\partial f_j}{\\partial x_i}(p)\\right)$, because $d_p(gf) = g(p) \\, d_p f$ when $f(p) = 0$.\n\nIntrinsic form: let $\\mathfrak m_p$ be the [[maximal ideal|0PV74TR5FP]] of the [[coordinate ring|G9WB27VWP7]] $k[X]$ of functions vanishing at $p$.\nThen $T_p X$ is isomorphic to the dual [[vector space|000AYPA9AJ]] of $\\mathfrak m_p / \\mathfrak m_p^2$.\nThe same space is the dual of the [[cotangent space|Z1CGB483J6]] of the [[local ring|MZ02BKPHJA]] $k[X]_{\\mathfrak m_p}$, so $T_p X$ only depends on a neighbourhood of $p$ and is defined for every variety.\n\nExample: for the parabola $X = V(y - x^2)$ at $p = (a, a^2)$, $d_p(y - x^2) = -2a \\, v_1 + v_2$, so $T_p X$ is the line $v_2 = 2a v_1$, of dimension $1$.\n\nExample: for the cusp $X = V(y^2 - x^3)$ at the origin, the polynomial $y^2 - x^3$ generates $I(X)$ and both of its partial derivatives vanish at $(0, 0)$, so $T_{(0,0)} X = k^2$ has dimension $2 \u003e \\dim X = 1$.\n\nOne always has $\\dim T_p X \\ge \\dim X$.\n",
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