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Euler identity for a differentiable degree-one homogeneous function

Statement

Let $C \subseteq R^{d}$ be open and $f : C \to R$ be $C^{1}$ . Suppose that for every $x \in C$ there is a nondegenerate interval $J_{x} \subset (0, \infty)$ containing $1$ such that $λ x \in C$ and $f (λ x) = λ f (x)$ for every $λ \in J_{x}$ . Here nondegenerate means that $J_{x}$ contains a point different from $1$ ; $1$ may be an endpoint. Then

$f (x) = D f (x) [x] = \sum_{j = 1}^{d} x_{j} \partial_{j} f (x) (x \in C) .$

The radial identity need only hold on one side of $1$ : openness of $C$ and $C^{1}$ regularity supply an ambient derivative, and the interval makes $1$ a nonisolated point at which that derivative is determined by the radial identity. This uses the same function on the same domain; it asserts nothing about a family of functions on different tagged domains.

Facts & Assumptions

Given: The open domain $C$ , the $C^{1}$ function $f$ , and for each $x \in C$ the stated radial homogeneity on a nondegenerate interval containing $λ = 1$ .

[F1]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : If $f$ and $g$ are totally differentiable at the relevant points, then $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ .

Proof

technique · direct

1.1givenF1construct

Fix $x \in C$ . Openness of $C$ and continuity of $λ \mapsto λ x$ give an open interval $K_{x} \subset (0, \infty)$ around $1$ with $λ x \in C$ for $λ \in K_{x}$ . Define $g : K_{x} \to R$ by $g (λ) = f (λ x)$ . By [F1], $g^{'} (1) = D f (x) [x]$ . This ambient derivative exists even if $1$ is an endpoint of the interval $J_{x}$ on which homogeneity is assumed.

2.1givenstep 1.1algebra∎

Since $J_{x}$ is a nondegenerate interval containing $1$ , it contains points $λ \neq 1$ arbitrarily close to $1$ , also lying in $K_{x}$ . For these points, homogeneity and $g (1) = f (x)$ give $(g (λ) - g (1)) / (λ - 1) = f (x)$ . The ambient derivative in step 1.1 exists, so its limit along this possibly one-sided set is $g^{'} (1) = f (x)$ . Therefore $f (x) = D f (x) [x]$ , and writing the total derivative in Euclidean coordinates gives $D f (x) [x] = \sum_{j = 1}^{d} x_{j} \partial_{j} f (x)$ .

Depends on

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

Dependency tree · 2 levels

Sources

Daniel Arovas, Thermodynamics and Statistical Mechanics, §2.7