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 Incoming links: Second-order Euler identity for homogeneous functions

The scaling relation says that

z

n

. Fix

(x, y)

and differentiate the identity with respect to the positive parameter

λ

. The chain rule gives

x z_{x} (λ x, λ y) + y z_{y} (λ x, λ y) = n λ^{n - 1} z (x, y) .

Setting

λ = 1

x z_{x} + y z_{y} = n z .

Differentiate once more with respect to

λ

, using equality of the mixed partial derivatives for the assumed twice continuously differentiable function:

x^{2} z_{xx} (λ x, λ y) + 2 x y z_{x y} (λ x, λ y) + y^{2} z_{yy} (λ x, λ y) = n (n - 1) λ^{n - 2} z (x, y) .

λ = 1

x^{2} z_{xx} + 2 x y z_{x y} + y^{2} z_{yy} = n (n - 1) z .

n (n - 1)

, rather than

n^{2}