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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer and let $k$ be a real number.\nThe *homogeneity equation* of degree $k$ is the [[functional equation|SRAN4EKWS0]]\n$$f(tx) = t^k f(x) \\quad \\text{for all } t \u003e 0 \\text{ and } x \\in \\R^n \\setminus \\set{0},$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon \\R^n \\setminus \\set{0} \\to \\R$.\nA solution is called a *positively homogeneous function of degree $k$*.\n\n## Remarks\n\nExample: $f(x, y) = x^2 + xy$ is a solution with $k = 2$, since $f(tx, ty) = t^2 x^2 + t^2 xy = t^2 f(x, y)$.\nNon-example: $f(x) = x + 1$ on $\\R \\setminus \\set{0}$ is not a solution for any $k$, since $f(2) = 3$, $f(1) = 2$ would need $2^k = 3/2$, while $f(4) = 5$, $f(2) = 3$ would need $2^k = 5/3$.\nThe general solution is $f(x) = \\norm{x}^k g(x / \\norm{x})$, where $g$ is an arbitrary function on the unit sphere and $\\norm{\\cdot}$ is the Euclidean norm.\nFor differentiable $f$, Euler's homogeneous function theorem says that $f$ is a solution exactly when $\\sum_i x_i \\, \\partial f / \\partial x_i = k f$.\nOther versions require the identity for all nonzero real $t$ or for $t$ in a field, which gives a stronger notion of homogeneity.\n",
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