Chain law with functions of several variables

Question

A function 'f' of two variables is said to be homogeneous of degree 'n' if f(tx,ty) = t^n*f(x,y) whenever t > 0.

 

How can I show that such a function 'f' satisfies the equation: x*(partial derivative of f with respect to x) + y*(partial derivative of f with respect to y) = n * f.

Answer 1

Let F(t)=f(tx,ty). Denote by f₁(x,y) the partial derivative of f with respect to x, and f₂(x,y) the partial derivative with respect to y. Since F(t)=t^nf(x,y). Taking derivative with respect to t using the Chain rule, we have

dF/dt=(∂f/∂(tx))(d(tx)/dt)+(∂f/∂(ty))(d(ty)/dt)=f₁(tx,ty)x+f₂(tx,ty)y.

On the other hand, dF/dt=nt^n-1f(tx,ty). Thus we have

f₁(tx,ty)x+f₂(tx,ty)y = nt^n-1f(tx,ty), Now let t=1.