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  "body": "\n## Summary\n\nA Neumann boundary condition prescribes the derivative of the unknown function in the direction normal to the boundary.\nFor an ordinary differential equation on $[a, b]$, it fixes $y'(a)$ and $y'(b)$.\nFor a partial differential equation on a domain, it fixes the outward normal derivative $\\frac{\\partial u}{\\partial \\nu}$ on the boundary, which is a flux.\nIt is one of the three classical conditions in a [[boundary value problem|TM8VQJWGHZ]], with the [[Dirichlet|EETVA6QYX4]] and Robin conditions.\nSee the [[definition|REH63MEBNC]].\n\n## Explanations\n\n### L1 human\n\nA Neumann condition says how steeply the solution leaves the boundary, not where it is.\nFor heat flow, it prescribes the heat flux through the boundary.\nA perfectly insulated wall lets no heat through, which is the condition $\\frac{\\partial u}{\\partial \\nu} = 0$.\nFor $y'' = 0$ on $[0, 1]$ with $y'(0) = y'(1) = 0$, every constant is a solution, so the solution is not unique.\n",
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