Home | | Maths 12th Std | Homogeneous Functions and Euler’s Theorem

Mathematics - Homogeneous Functions and Euler’s Theorem | 12th Maths : UNIT 8 : Differentials and Partial Derivatives

Chapter: 12th Maths : UNIT 8 : Differentials and Partial Derivatives

Homogeneous Functions and Euler’s Theorem

These types of functions are important in Ordinary differential equations.

Homogeneous Functions and Euler’s Theorem


Definition 8.12

(a) Let A = {( x, y ) | a < x < b, c < y < d } 2 , F : A , we say that F is a homogeneous function on A , if there exists a constant p such that F ( λ x, λ y ) = λ p F ( x , y) for all λ ∈ ℝ and sutitably restricted λ, x, y, such that (λ x , λ y) A . This constant p is called degree of F .

(b) Let B = {( x, y , z) | a < x < b , c < y < d, u < z < v} 3 , G : B , we say that G is a homogeneous function on B , if there exists a constant p such that G ( λ x, λ y , λ z ) = λ pG ( x, y , z) for all λ and sutitably restricted λ, x, y, z, such that (λ x , λ y, λz ) B . This constant p is called degree of G .

Note: Division by any variable may occur, to avoid division by zero, we say that λ, x, y, z are sutitably restricted real numbers.


These types of functions are important in Ordinary differential equations (Chapter 10). Let us consider some examples.

Consider F ( x, y) = x3 2 y3 + 5xy2 , (x , y) 2 . Then

F ( λ x , λ y) = (λ x ) 3 2(λ y)3 + 5(λ x )(λ y)2 = λ3 (x 3 2 y3 + 5xy2 )

and hence F is a homogeneous function of degree 3.

On the other hand,

G ( x, y) = ex2 + 3y2 is not a homogeneous function

because, G ( λ x , λ y) = e (λx)2  + 3(λ y)2 λpG ( x, y)

for any λ 1 and any p.

 

Example 8.21

Show that  is a homogeneous function of degree 1.

Solution

We compute


for all λ . So F is a homogeneous function of degree 1.

We state the following theorem of Leonard Euler on homogeneous functions.

 

Definition 8.13 (Euler)

Suppose that A = {( x, y ) | a < b, c < y < d } 2, F : A 2 . If F is having continuous partial derivatives and homogeneous on A , with degree p , then


Suppose that B = {( x, y , z) | a < x < b, c < y < d, u < z < v} 3 , F : B 3. If F is having continuous partial derivatives and homogeneous on B , with degree p , then


We omit the proof. The above theorem is also true for any homogeneous function of n variables; and is useful in certain calculations involving first order partial derivatives.

 

Example 8.22


Solution

Note that the function u is not homogeneous. So we cannot apply Euler’s Theorem for u.

However, note that  is homogeneous; because


 

Note:

Solving this problem by direct calculation will be possible; but will involve lengthy calculations.

Tags : Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Maths : UNIT 8 : Differentials and Partial Derivatives : Homogeneous Functions and Euler’s Theorem | Mathematics

Related Topics

12th Maths : UNIT 8 : Differentials and Partial Derivatives


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.