Homogeneous Function(redirected from Euler's theorem on homogeneous functions)
homogeneous function[‚hä·mə′jē·nē·əs ′fəŋk·shən]
a function of one or several variables that satisfies the following condition: when all independent variables of a function are simultaneously multiplied by the same (arbitrary) factor, the value of the function is multiplied by some power of this factor. In algebraic terms, a function f(x, y, …, u) is said to be homogeneous of degree n if for all values of x, y, …, u and for any λ
f (λx, λy, …, λu) = λnf(x, y, …, u)
For example, the functions x2 – 2y2, (x – y – 3z)/(z2 + xy), and are homogeneous of degree 2, –1, and 4/3, respectively. Functions homogeneous of degree n are characterized by Euler’s theorem that asserts that if the differential of each independent variable is replaced with the variable itself in the expression for the complete differential
then we obtain the function f(x, y, …, u) multiplied by the degree of homogeneity:
Homogeneous functions are frequently encountered in geometric formulas. In the equation x = f(a, b, …, l), where a, b, …, l are the lengths of segments expressed in terms of the same unit, f must be a homogeneous function (of degree 1, 2, or 3, depending on whether x signifies length, area, or volume). For example, in the formula for the volume of a truncated cone
V is a homogeneous function of degree 3 in h, R, and r.