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Unformatted text preview: 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 X 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 X i =1 D i f ( λx ) x i- kλ k- 1 f ( x ) . Evaluate this at λ = 1 to obtain ( * ). ( ⇐ =) Suppose kf ( x ) = n X 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...
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This note was uploaded on 06/20/2011 for the course OPR 201 taught by Professor Pp during the Spring '11 term at Thammasat University.
- Spring '11