Talk:Homogeneous function

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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)

The gradient is not even mentioned. Why not simply state the theorem in sum-of-derivative form? Even if a beginner digs through these comments, most likely looking up "gradient" will confuse them unnecessarily. — Preceding unsigned comment added by 140.112.177.67 (talk) 15:01, 21 January 2019 (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: $f\colon \mathbb {R} ^{2}\to \mathbb {R}$, $k=1$, with $f(x,y)=x$ (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 $k$. 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 $\alpha =0$? Clearly the definition of homogeneous of degree $k$ for $k<0$ has to be modified so that the condition holds for all $\alpha \neq 0$, and I've just changed this too. Mark (talk) 16:31, 11 February 2008 (UTC)

Notation

Notation such as

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

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

It's therefore better to write

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

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

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 <math> 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 <math> tag construction. 184.17.182.96 (talk) 06:26, 21 December 2012 (UTC)

Keep it simple please

People who look up homogeneous function may not necessarily understand what "ƒ : V → W is a function between two vector spaces over a field F" means; Likewise people who know what a Banach space are not likely to wonder "what the heck is a homogeneous function" and look it up in Wikipedia. Do not scare people away from math please. And foremost, be mindful Wikipedia is basically an encyclopedia; it's meant for ordinary people to look up stuff :) --Sahir 08:41, 8 December 2015 (UTC)

Typo in the introduction?

In the second line, should it say $f(ax,y)=a^{k}f(x,y)$ rather than $f(ax,ty)=a^{k}f(x,y)$? It looks like a typo but I don't know enough about the subject to edit. Mrdouglasweathers (talk) 12:35, 1 November 2016 (UTC)

This makes sense. I fixed it. --Erel Segal (talk) 13:16, 1 November 2016 (UTC)

degree 0 example?

How about an example of homogeneity of degree 0? Btyner (talk) 21:48, 1 September 2017 (UTC)

functions defined on positive cones or rays - and absolute homogeneity

The former is covered by the introduction, but that's it; later on, the square root is given as example, without discussing that it is not a function on a vector space. Absolute homogeneity is defined only later, but - being a defining property of norms - it could very well go into the introduction. (And: Zero removed? What about "[...] e.g. making matrix rank homogeneous of degree zero, once one omits the exceptional s=0".) — Preceding unsigned comment added by 193.90.163.192 (talk) 09:17, 6 December 2021 (UTC)