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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be an open set, let $f \\colon U \\to \\mathbb{R}^m$, and let $a \\in U$.\nThe function $f$ is differentiable at $a$ when there is a linear map $L \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ such that\n$$\\lim_{h \\to 0} \\frac{\\|f(a + h) - f(a) - L h\\|}{\\|h\\|} = 0.$$\nHere the limit is over $h \\ne 0$ with $a + h \\in U$, and $\\|\\cdot\\|$ is the Euclidean norm.\nThe map $L$ is then unique. It is the derivative $Df(a)$, also called the [[total derivative|0FNKFT8H6E]] of $f$ at $a$.\n\n## Remarks\n\nThis is the [[Fréchet derivative|08EVNYZF42]] for the spaces $\\mathbb{R}^n$ and $\\mathbb{R}^m$, where each linear map is continuous.\nA function that is differentiable at $a$ is continuous at $a$, and all its [[directional derivatives|06KWGHDN2J]] exist at $a$.\nThe converse fails: a function can have all partial derivatives at a point and not be differentiable there, see [[this counterexample|0TNZC55X3J]].\nFor $n = m = 1$, the definition is the usual one, with $L h = f'(a) h$.\n\n## Explanations\n\n### L1 human\n\nA differentiable function looks flat when you zoom in on its graph near the point.\nFor $f(x, y) = x^2 + y^2$ at $(1, 1)$, the change is $f(1 + h_1, 1 + h_2) - f(1, 1) = 2 h_1 + 2 h_2 + h_1^2 + h_2^2$.\nThe linear part is $L h = 2 h_1 + 2 h_2$, and the rest $h_1^2 + h_2^2 = \\|h\\|^2$ is small compared with $\\|h\\|$.\n",
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