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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be an open set that is star-shaped with respect to a point $a \\in U$.\nThis means that the segment from $a$ to $x$ lies in $U$ for every $x \\in U$.\nLet $f \\colon U \\to \\R$ be a [[smooth function|0V33BNKZ9S]].\nThen there are smooth functions $g_1, \\dots, g_n \\colon U \\to \\R$ such that\n$$f(x) = f(a) + \\sum_{i=1}^{n} (x_i - a_i)\\, g_i(x) \\quad \\text{for all } x \\in U,$$\nand $g_i(a) = \\dfrac{\\partial f}{\\partial x_i}(a)$ for $i = 1, \\dots, n$, where $\\partial f / \\partial x_i$ is the [[partial derivative|09EGC6F8PT]].\n\n## Remarks\n\nThe proof is one line of calculus.\nFor $x \\in U$, the fundamental theorem of calculus and the chain rule give\n$$f(x) - f(a) = \\int_0^1 \\frac{d}{dt} f(a + t(x - a))\\, dt = \\sum_{i=1}^n (x_i - a_i) \\int_0^1 \\frac{\\partial f}{\\partial x_i}(a + t(x - a))\\, dt.$$\nSet $g_i(x) = \\int_0^1 \\frac{\\partial f}{\\partial x_i}(a + t(x - a))\\, dt$.\nDifferentiation under the integral sign shows that $g_i$ is smooth, and $g_i(a) = \\frac{\\partial f}{\\partial x_i}(a)$.\n\nExample: for $f(x) = x^3$ on $\\R$ and $a = 1$, one has $x^3 = 1 + (x - 1)(x^2 + x + 1)$, so $g_1(x) = x^2 + x + 1$ and $g_1(1) = 3 = f'(1)$.\n\nIf $f$ is only of class $C^k$ with $k \\ge 1$, the same formula gives functions $g_i$ of class $C^{k-1}$ only.\nFor example, $f(x) = x \\abs{x}$ is $C^1$ on $\\R$, and with $a = 0$ the formula gives $g_1(x) = \\abs{x}$, which is not differentiable at $0$.\n\n## Explanations\n\n### L1 human\n\nA smooth function that vanishes at a point can be divided by the coordinates of that point, and the quotients are still smooth.\nIn one variable it says: if $f(a) = 0$, then $f(x) = (x - a) g(x)$ with $g$ smooth.\nThis is the smooth analogue of the factor theorem for polynomials.\n\n### L2 human\n\nIn terms of [[germs|7SWSRGDT4A]], the lemma says that the maximal ideal $\\mathfrak{m}_a$ of the ring of smooth germs at $a$ is generated by the germs of $x_1 - a_1, \\dots, x_n - a_n$.\nApplying the lemma twice gives $f(x) = f(a) + \\sum_i \\frac{\\partial f}{\\partial x_i}(a)(x_i - a_i) + \\sum_{i,j} (x_i - a_i)(x_j - a_j) h_{ij}(x)$ with $h_{ij}$ smooth.\nThis is the key step in the proof that derivations of the ring of germs are exactly the tangent vectors, and in the proof of the Morse lemma.\n",
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