Homogeneous Functions Part 2

Summary

The video explains how to identify nonhomogeneous functions and introduces Euler’s theorem for homogeneous functions.

Nonhomogeneous function example

Consider [ f(x)=\frac{2x}{x+3}. ] Scaling (x) by 2 gives [ f(2x)=\frac{4x}{2x+3}. ] This is not equal to the original function multiplied by a consistent power of 2, so (f) is not homogeneous.

Euler’s theorem for homogeneous functions

If (f(x_1,x_2,\ldots)) is homogeneous of degree (k), Euler’s theorem states [ k f=\sum_i x_i\frac{\partial f}{\partial x_i}. ] That is, multiply each variable by the partial derivative of the function with respect to that variable, then add the products. The result equals (k) times the function.

For example, let (f(x)=7x^2), which has degree (k=2). Since (f’(x)=14x), [ x f’(x)=14x^2=2(7x^2)=2f(x). ] This confirms Euler’s theorem for this function.

Linearly homogeneous functions

When (k=1), Euler’s theorem becomes [ f=\sum_i x_i\frac{\partial f}{\partial x_i}. ] A homogeneous function of degree 1 is called linearly homogeneous.

For example, consider (f(x,y)=7x+2y). Scaling both variables by 2 gives [ f(2x,2y)=14x+4y=2(7x+2y)=2f(x,y), ] so the function is homogeneous of degree 1. Its partial derivatives are (f_x=7) and (f_y=2), and [ x f_x+y f_y=7x+2y=f(x,y). ] This verifies Euler’s theorem for the example.

Speaker

The video features one instructor or narrator speaking in a voiceover and noting that the audio was broken. No other speakers or sources are identified.