Euler's theorem statement with sin u example

Euler's theorem

If f(x, y) is a homogeneous function of degree n (f(tx, ty) = tⁿ f(x, y)) with continuous first partial derivatives, then

x ∂f/∂x + y ∂f/∂y = n f.

Proof of the given result

Given sin u = (√x − √y)/(√x + √y).

Let z = sin u = (x^(1/2) − y^(1/2))/(x^(1/2) + y^(1/2)). Replace x by tx and y by ty: the factor t^(1/2) cancels from numerator and denominator, so z(tx, ty) = t⁰ z. Thus z is homogeneous of degree n = 0.

By Euler's theorem:

x ∂z/∂x + y ∂z/∂y = 0 · z = 0

Now ∂z/∂x = cos u ∂u/∂x and ∂z/∂y = cos u ∂u/∂y, so

cos u ( x ∂u/∂x + y ∂u/∂y ) = 0
x ∂u/∂x + y ∂u/∂y = 0      (cos u ≠ 0)

Hence proved. ∎