# Talk:Homogeneous function

WikiProject Mathematics (Rated Start-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Mid Importance
Field:  Algebra

## [Untitled Section]

What the hell is x times gradient of f(x) supposed to mean, dot product?

It means that for a vector function f(x) that is homogenous of degree k, the dot production of a vector x and the gradient of f(x) evaluated at x will equal k * f(x). CodeLabMaster 12:12, 05 August 2007 (UTC)
Yes, as can be seen from the furmula under that one. I've added the dot and changed vector symbols to bold. mazi 18:04, 22 February 2006 (UTC)

## Not all homogeneous functions are differentiable

The article, before I changed it a moment ago, implied that all homogeneous functions are differentiable. Here's a counterexample: f: R^2 --> R, k=1, with f(x,y)= either x (if xy>0) or 0 (otherwise). --Steve 03:23, 6 August 2007 (UTC)

## Is the derivative theorem correct?

According to planetmath [1], the theorem about derivatives is not correct unless we replace "homogeneous" by "positive homogeneous" throughout. Their counterexample is wrong (I just submitted a correction on the site), but could that claim be correct? Does anyone have a reference, or a proof, or a proper counterexample?

Update: The person maintaining that planetmath page responded to my correction by taking away the counterexample but keeping the claim. Again, a reference, proof, or proper counterexample is needed to resolve this. --Steve 15:51, 5 October 2007 (UTC)

The result is correct for functions which are homogeneous of degree ${\displaystyle k}$. I've added the elementary proof of this result to the page (and merged "Other properties" with "Euler's theorem" as the proofs are very similar). Is the planetmath contributor worried about ${\displaystyle \alpha =0}$? Clearly the definition of homogeneous of degree ${\displaystyle k}$ for ${\displaystyle k<0}$ has to be modified so that the condition holds for all ${\displaystyle \alpha \neq 0}$, and I've just changed this too. Mark (talk) 16:31, 11 February 2008 (UTC)

## Notation

Notation such as

${\displaystyle {\frac {\partial f}{\partial x_{1}}}(\alpha \mathbf {x} )}$

can be confusing: Are we differentiating the expression with respect to the first component of ${\displaystyle \mathbf {x} }$ or do we mean the partial derivative of ${\displaystyle f}$ with respect to its first argument evaluated at the point ${\displaystyle \alpha \mathbf {x} }$?

It's therefore better to write

${\displaystyle {\frac {\partial f}{\partial x_{1}}}(\alpha \mathbf {y} )}$:

now it's clear that ${\displaystyle x_{1},\ldots ,x_{n}}$ are the arguments of ${\displaystyle f}$ and we are differentiating with respect to the first argument of ${\displaystyle f}$ evaluated at the point ${\displaystyle \alpha \mathbf {y} }$.

I've cleaned up the notation in my proof of Euler's theorem accordingly. 91.21.25.30 (talk) 21:50, 11 February 2008 (UTC)

I agree. Sorry about my incorrect edit to that effect earlier, thanks for reverting :-) --Steve (talk) 18:00, 11 February 2008 (UTC)

## the name

Euler's theorem? why the name, is he the 1st guy prove this? if yes, why don't we use his work as a reference? Jackzhp (talk) 17:29, 4 December 2008 (UTC)

Keep in mind the Euler lived in the 18th century and wrote mostly in Latin so not really a good reference for a modern audience. It might be worth adding the original work as a historical reference though.--RDBury (talk) 21:44, 18 April 2010 (UTC)

## restriction of k for nth degree homogeneity?

What are the restrictions on k? Must k be contained within the domain of the vector space, reals, or what?

Thanks, Jgreeter (talk) 06:51, 16 May 2011 (UTC)

For arbitrary fields, k should be an integer. For the reals, it makes sense to define this notion also for real numbers k. For example, the square root is homogeneous with k=1/2.
But it seems to me that usually k is an integer. --Aleph4 (talk) 16:49, 16 May 2011 (UTC)

## Note on change http://en.wikipedia.org/w/index.php?title=Homogeneous_function&diff=529079679&oldid=520543743

The Phi used in the equations, that is /varphi inside a [itex] tag, was rendered differently in my browser (mobile Safari) than the &phi used in the descriptions below (see Phi#Computing) -- so I switched the descriptions to use the clumsier but definitely-consistent [itex] tag construction. 184.17.182.96 (talk) 06:26, 21 December 2012 (UTC)