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EulerHomogeneity

# EulerHomogeneity - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Euler’s Theorem for Homogeneous Functions KC Border Let f : R n + R . We say that f is homogeneous of degree k if for all x R n + and all λ > 0, f ( λx ) = λ k f ( x ) . 1 Euler’s theorem Let f : R n + R be continuous, and also differentiable on R n ++ . Then f is homogeneous of degree k if and only if for all x R n ++ , kf ( x ) = n i =1 D i f ( x ) x i . ( * ) Proof : (= ) Suppose f is homogeneous of degree k . Fix x R n ++ , and define the function g : [0 , ) R (depending on x ) by g ( λ ) = f ( λx ) - λ k f ( x ) , and note that for all λ 0, g ( λ ) = 0 . Therefore g ( λ ) = 0 for all λ > 0. But by the chain rule, g ( λ ) = n i =1 D i f ( λx ) x i - k - 1 f ( x ) . Evaluate this at λ = 1 to obtain ( * ). ( =) Suppose kf ( x ) = n i =1 D i f ( x ) x i for all x R n ++ . Fix any x 0 and again define g : [0 , ) R (depending on x ) by g ( λ ) = f ( λx ) - λ k f ( x ) 1

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KC Border Euler’s Theorem for Homogeneous Functions 2 and note that g (1) = 0. Then g ( λ ) = n i =1 D i f ( λx ) x i - k - 1 f ( x ) = λ - 1 n i =1 D i f ( λx ) λx i - k - 1 f ( x ) =
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EulerHomogeneity - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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