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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f = (f_1, \\dots, f_m) \\colon U \\to \\mathbb{R}^m$, let $a \\in U$, and let $L \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ be linear with components $L_1, \\dots, L_m$.\nThen $f$ is [[differentiable|0135XZZQ8M]] at $a$ with $Df(a) = L$ if and only if, for each $j \\in \\{1, \\dots, m\\}$, the component $f_j$ is differentiable at $a$ with $Df_j(a) = L_j$.\n\n## Remarks\n\nSo you can differentiate a vector-valued map one component at a time. The derivative of $f_j$ is row $j$ of the [[Jacobian matrix|047NDTEPCV]] of $f$.\nThe proof compares norms: each $|v_j| \\le \\|v\\|$, and $\\|v\\| \\le \\sqrt{m} \\max_j |v_j|$, for $v \\in \\mathbb{R}^m$.\nThis is the case of [[a map into a product|0QQMXFTX0A]] with $m$ factors equal to $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nThe map $f(t) = (\\cos t, \\sin t)$ has the derivative $h \\mapsto (-h \\sin t, \\, h \\cos t)$ at $t$.\nYou get it by differentiating $\\cos t$ and $\\sin t$ separately.\n",
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