Euler's homogeneous function theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 18/02/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Homogeneous_function


00:01:50 1 Examples
00:01:58 1.1 Example 1
00:03:28 1.2 Linear functions
00:04:54 1.3 Homogeneous polynomials
00:06:37 1.4 Min/max
00:09:56 1.5 Polarization
00:11:08 1.6 Rational functions
00:13:38 2 Non-examples
00:14:08 2.1 Logarithms
00:14:17 2.2 Affine functions
00:16:20 3 Positive homogeneity
00:16:50 4 Homogeneous distributions
00:23:03 5 Application to differential equations
00:26:12 6 See also
00:27:34 7 References



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SUMMARY
=======
In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
For example, a homogeneous function of two variables x and y is a real-valued function that satisfies the condition 
  
    
      
        f
        (
        α
        x
        ,
        α
        y
        )
        =
        
          α
          
            k
          
        
        f
        (
        x
        ,
        y
        )
      
    
    {\displaystyle f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)}
   for some constant 
  
    
      
        k
      
    
    {\displaystyle k}
   and all real numbers 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  .  The constant k is called the degree of homogeneity.
More generally, if ƒ : V → W is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if

for all nonzero α ∈ F and v ∈ V.  When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that (1) hold for all α  0.
Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry.  More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).