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  "body": "\n## Definition\n\nLet $a \u003c b$, and consider a second-order ordinary differential equation for $y$ on $[a, b]$.\nA *Neumann boundary condition* is a pair of conditions\n$$y'(a) = \\alpha, \\qquad y'(b) = \\beta,$$\nwith given real numbers $\\alpha, \\beta$.\nMore generally, let $\\Omega \\subseteq \\R^d$ be a bounded domain, that is, a connected open set, whose [[boundary|05D5EGMFYV]] $\\partial \\Omega$ is of class $C^1$.\nLet $\\nu(x)$ be the outward unit [[normal vector|0WQJJCCTC8]] at $x \\in \\partial \\Omega$, and let $g \\colon \\partial \\Omega \\to \\R$ be a given function.\nFor an equation for $u$ on $\\Omega$, the Neumann boundary condition is\n$$\\frac{\\partial u}{\\partial \\nu}(x) = \\nabla u(x) \\cdot \\nu(x) = g(x) \\quad \\text{for all } x \\in \\partial \\Omega,$$\nwhere $\\frac{\\partial u}{\\partial \\nu}$ is the [[directional derivative|06KWGHDN2J]] of $u$ along $\\nu$.\n\n## Remarks\n\nOn $[a, b]$ the outward normal derivatives are $-y'(a)$ and $y'(b)$, so the two forms agree up to the sign at $a$.\n\nExample: $y'' = 2$ on $[0, 1]$ with $y'(0) = 0$ and $y'(1) = 2$.\nThe function $y = x^2$ has $y'' = 2$, $y'(0) = 0$ and $y'(1) = 2$, so it solves this Neumann problem; so does $x^2 + c$ for every constant $c$.\n\nNon-example: $y(0) = 0$ together with $y'(1) = 0$ is not a Neumann condition, since the first condition prescribes a value.\n\nFor $\\Delta u = f$ on $\\Omega$ with Neumann data $g$, the divergence theorem gives the necessary condition $\\int_\\Omega f = \\int_{\\partial \\Omega} g$.\nSolutions, when they exist, are unique only up to an additive constant.\n",
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