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  "body": "\n## Proof\n\nWe restrict $f$ to a line through $a$ and use the one-variable theorem.\n\n1. Let $U$, $f$ and $a$ be as in the statement. Assume first that $f$ has a [[local maximum|04CPG9K1C4]] at $a$.\n2. There is $\\varepsilon \u003e 0$ with $f(x) \\le f(a)$ for all $x \\in U$ with $\\norm{x - a} \u003c \\varepsilon$. Since $U$ is open, we may shrink $\\varepsilon$ so that all $x$ with $\\norm{x - a} \u003c \\varepsilon$ lie in $U$.\n3. Let $i$ be an index for which $\\partial f / \\partial x_i (a)$ exists. Let $e_i$ be the $i$-th standard basis vector, and define $\\varphi(t) = f(a + t e_i)$ on the open interval $I = (-\\varepsilon, \\varepsilon)$.\n4. For $t \\in I$ we have $\\norm{(a + t e_i) - a} = \\abs{t} \u003c \\varepsilon$, so $\\varphi(t) \\le f(a) = \\varphi(0)$. Thus $\\varphi$ has a local maximum at $0$.\n5. By the definition of the [[partial derivative|05AXG3QRKR]], $\\varphi$ is differentiable at $0$ with $\\varphi'(0) = \\partial f / \\partial x_i (a)$.\n6. By [[Fermat's theorem on stationary points|0PCVBRV9VJ]], $\\varphi'(0) = 0$. So $\\partial f / \\partial x_i (a) = 0$.\n7. If $f$ has a local minimum at $a$, then $-f$ has a local maximum at $a$, and $\\partial (-f)/\\partial x_i (a) = -\\partial f / \\partial x_i (a)$. So step 6 applied to $-f$ gives again $\\partial f / \\partial x_i (a) = 0$.\n8. Now let $f$ be [[differentiable|0135XZZQ8M]] at $a$. Then every partial derivative exists at $a$, and $Df(a) e_i = \\partial f / \\partial x_i (a)$, since $Df(a)$ has the [[Jacobian matrix|0YF0VXKXGB]] of partial derivatives.\n9. By steps 6 and 7, all these partial derivatives are $0$. So the [[gradient|0NS2QYVC4Y]] $\\nabla f(a)$, the vector of partial derivatives, is $0$.\n10. The linear map $Df(a)$ vanishes on each basis vector $e_i$, so it is the zero linear map.\n\n## Idea\n\nAlong each coordinate line through $a$, the function still has a local extremum at $a$. So the one-variable Fermat theorem makes each partial derivative zero. Openness of $U$ is used to make the line segment around $a$ lie in the domain on both sides.\n",
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