Euler's Homogeneous Function Theorem

Let $f(x,y)$ be a Homogeneous Function of order $n$ so that

$$f(tx,ty)=t^n f(x,y).$$ (1)

Then define $x'\equiv xt$ and $y'\equiv yt$. Then

$$nt^{n-1} f(x,y) = {\partial f\over \partial x'}{\partial x'\over \partial t} + {\partial f\over \partial y'}{\partial y'\over \partial t}$$

$$= x{\partial f\over\partial x'}+y{\partial f\over\partial y'} = x{\partial f\over\partial (xt)} +y{\partial f\over\partial (yt)}.$$ (2)

Let $t=1$, then

$$x{\partial f\over \partial x}+y{\partial f\over \partial y}=nf(x,y).$$ (3)

This can be generalized to an arbitrary number of variables

$$x_i{\partial f\over \partial x_i} = nf({\bf x}),$$ (4)

where Einstein Summation has been used.