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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^nf(x,y).
(1)

Then define x^'=xt and y^'=yt. Then

nt^(n-1)f(x,y)=(partialf)/(partialx^')(partialx^')/(partialt)+(partialf)/(partialy^')(partialy^')/(partialt)
(2)
=x(partialf)/(partialx^')+y(partialf)/(partialy^')
(3)
=x(partialf)/(partial(xt))+y(partialf)/(partial(yt)).
(4)

Let t=1, then

x(partialf)/(partialx)+y(partialf)/(partialy)=nf(x,y).
(5)

This can be generalized to an arbitrary number of variables

x_i(partialf)/(partialx_i)=nf(x),
(6)

where Einstein summation has been used.

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Cite this as:

Weisstein, Eric W. "Euler's Homogeneous Function Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html

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