Euler’s theorem. Declared by Leonhard Euler on Homogeneous Functions, it is a characterization of homogeneous functions.
Summary
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- 1 Definition
- 2 Example 1
- 3 Euler’s First Theory
- 4 Example 2
- 5 Source
- 6 See also
Definition
The function f (x, y) is called the Homogeneous Function of degree n if for any real factor K the equality is verified
f (kx, ky) = k n f (x, y).
Example 1
Say if the given function is homogeneous and what is the degree of homogeneity.
z = f (x, y) = x 2 + xy – y 2
f (λx, λy) = (λx) 2 + (λx) (λy) – (λy) 2 = λ 2 x 2 + λ 2 xy – λ 2 y 2 = λ 2 (x 2 + xy – y 2 )
f (λx, λy) = λ 2 f (x, y)
Since the function z = f (x, y) meets the definition, we say that z is homogeneous of degree 2.
Euler’s First Theory
If z = f (x, y) is a homogeneous function of degree “n” and its first-order partial derivatives exist, then:
xf ´ x (x, y) + yf ´ y (x, y) = nf (x, y)
An integer rational function will be homogeneous, if all the terms of it are of the same degree.
Example 2
Prove whether the following function fulfills Euler’s theorem.
f (x, y) = x 2 – 2x 3 y 2 – y 5
f ´ x (x, y) = 5x 4 – 6x 2 y 2
f ´ y (x, y) = -4x 3 y – 5y 4
xf ´ x (x, y) + yf ´ y (x, y) = x (5x 4 – 6x 2 y 2 ) + y (-4x 3 y – 5y 4 ) = 5x 2 10x 3 y 2 – 5y 5 = 5 f (x, y)
