Euler’s Theorem

Let z = f(x,y) be a homogeneous function in x and y of degree n, and having continuous partial derivatives, then

x(∂z/∂x+ y(∂z/∂z) = n z

Proof:

Since z is a homogeneous function of degree n, we can write

F(x,y) = xⁿ f(y/x)

Substituting (y/x) by t, we get, f(x,y) = xⁿ f(t)

Differentiating both the sides partially with respect to x and y respectively, we have,

 = ∂z/∂x= nx^n-1f(t) + xⁿ f ’ (t) (∂t/∂x)

 = nx^n-1f(y/x) + xⁿ f’ (y/x) (-y/x²) (∴ t =y/x)

 = nx^n-1f(y/x) – xⁿ y x ^n-2 , f’ (y/x)

Again, we have, ∂t/∂y= xn f ’ (t) (1/x)

= nx^n-1f ’ (y/x)

Multiplying ∂z/∂x by x and  ∂z/∂y by y and then adding their products we get,

x(∂z/∂x) +y(∂z/∂y) = n xⁿ f(y/x) = n f(x,y) =nz

Hence, x(∂z/∂x) + y(∂z/∂y) = nz Proved.