Damn you Euler, with your complicated yet important results

01Oct06

Euler’s Homogeneous Function Theorem — from Wolfram MathWorld:

F is homogenous of degree n, so f(tx,ty)=(t^n)f(x,y)

Differentiate with repect to t:

Hint: Define x’=xt andy’=yt

Then

n(t^[n-1])f(x,y)=df/dx’ dx’/dt + df/dy’ dy’/dt
= df/dx’ x + df/dy’ y

This has not been obvious to me for some time. Glad I looked it up.

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