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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer, let $k \\ge 1$ be an integer or $k = \\infty$, and let $I \\subseteq \\R$ be an [[interval|0TK8Y3J5RW]] with more than one point.\nLet $\\gamma \\colon I \\to \\R^n$, $\\gamma(t) = (\\gamma_1(t), \\dots, \\gamma_n(t))$, be a [[parametrised curve|0JF1XXGWB4]].\nThe curve $\\gamma$ is a *differentiable curve of class $C^k$* if each coordinate function $\\gamma_i$ is of class $C^k$ (see [[smoothness classes|WK5HPW5Y24]]) on $I$.\nThis means that $\\gamma_i$ is $k$ times differentiable and its derivatives up to order $k$ are continuous.\nAt an end point of $I$ that belongs to $I$, the derivatives are one-sided.\nA *smooth curve* is a differentiable curve of class $C^\\infty$.\n\n## Remarks\n\nExample: $\\gamma(t) = (t, t^2)$, $t \\in \\R$, is a smooth curve, since both coordinates are polynomials.\nNon-example: $\\gamma(t) = (t, \\abs{t})$, $t \\in \\R$, is a continuous curve but not of class $C^1$, because the second coordinate has no derivative at $t = 0$.\nA differentiable curve can still have a cusp in its image: $t \\mapsto (t^2, t^3)$ is smooth, but its velocity vanishes at $t = 0$.\nCurves whose velocity never vanishes are the [[regular curves|06JMR2XWXW]].\n",
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