= Euler theorem for homogeneous functions
{c}
{title2=$\sum_i x_i\partial_i f=kf$}
= Euler homogeneous function theorem
{c}
{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>.
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