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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).
\end{displaymath} (1)

Then define $x'\equiv xt$ and $y'\equiv yt$. Then
$\displaystyle nt^{n-1} f(x,y)$ $\textstyle =$ $\displaystyle {\partial f\over \partial x'}{\partial x'\over \partial t}
+ {\partial f\over \partial y'}{\partial y'\over \partial t}$  
  $\textstyle =$ $\displaystyle x{\partial f\over\partial x'}+y{\partial f\over\partial y'}
= x{\partial f\over\partial (xt)} +y{\partial f\over\partial (yt)}.$ (2)

Let $t=1$, then
x{\partial f\over \partial x}+y{\partial f\over \partial y}=nf(x,y).
\end{displaymath} (3)

This can be generalized to an arbitrary number of variables
x_i{\partial f\over \partial x_i} = nf({\bf x}),
\end{displaymath} (4)

where Einstein Summation has been used.

© 1996-9 Eric W. Weisstein