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{title2=$\sum_i x_i\partial_i f=kf$}

= Euler homogeneous function theorem
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{synonym}

Let $f$ be a <differentiable function> on an open domain stable under positive rescaling, with $f(\lambda x)=\lambda^kf(x)$ for $\lambda>0$. The <chain rule> differentiates this <homogeneous function> identity at $\lambda=1$ to give
$$
\sum_ix_i\partial_i f(x)=k f(x).
$$
If $f$ is twice differentiable, differentiating once more gives $\sum_i x_i\partial_i\partial_jf=(k-1)\partial_j f$. In degree one its <Hessian matrix> therefore annihilates the radial vector $x$. This is the identity used in the <no-scale identity from degree-one homogeneity>. For an <extensive quantity>, the degree-one formula also supplies the thermodynamic identity behind the <Gibbs-Duhem equation>.


Euler theorem for homogeneous functions

{title2=$f(\lambda x)=\lambda^k f(x)$}
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A <function> is homogeneous of degree $k$ when rescaling its arguments by a common <scalar> $\lambda$ rescales its value by $\lambda^k$, whenever the arguments and that power are defined. For integer $k$ over a <vector space>, one can require every nonzero scalar; a <homogeneous polynomial> gives an example. On a domain stable under positive real rescaling, the positive-scaling version allows real $k$. Degree-one <positive homogeneity> is the convention used for <norms> and many <convex functions>. The <Euler theorem for homogeneous functions> turns this scaling law into a differential identity for a <differentiable function>.


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 Incoming links: Homogeneous function