Leonhard Euler proved an interesting theorem to the effect that

if F(λx₁, λx₂, …, λx_m) = λⁿF(x₁, x₂, …, x_m)
then
Σ_i=1^mx_i∂F/∂x_i = nF(x₁, x₂, …, x_m)

This can be represented more succinctly by letting X represent the variables {x₁, x₂, …, x_m} as an m-dimensional column vector. A function such that

F(λX) = λⁿF(X)

is said to be homogeneous of degree n.

Euler's Theorem for Homogeneous Functions is then

(∂F/∂X)·X = nF(X)

where (∂F/∂X) is the partial derivatives of F with respect to the x_i represented as a row vector.

Exponentially-Weighted Homogeneous Functions

In physics there arise functions of the form

V(X) = exp(−A·X)F(X)

where F(X) is a homogeneous function and A·X is Σa_ix_i.

For such functions

(∂F/∂X)·X = (n−A·X)V(X)

Proof

Differentiate both sides of
V(λX) = exp(−λA·X)λⁿF(X)
with respect to λ

The result is
(∂V/∂λX)·X = nλ^n-1exp(−λA·X)F(X) − (A·X)λⁿexp(−λA·X)F(X)

Now set λ equal to 1 to get
(∂V/∂X)·X = nV(X) − (A·X)V(x) = (n−A·X)V(X)

As an example, consider the Yukawa potential

V(r) = −H*exp(−r/r₀)/r

Here m=1, F(r)=−H/r and A=(1/r₀). The function F(r) is homogeneous of degree −1.

By the theorem above

r∂V/∂r = (−1 − r/r₀)V = −(1+ r/r₀)V