Talk:Implicit function theorem
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In view of the importance of this theorem, this article seems quite unadequate... but better than nothing! MFH 05:50, 10 Mar 2005 (UTC)
- That's what the "edit this page" link is for ;) Dysprosia 07:19, 10 Mar 2005 (UTC)
By reversible matrix, we mean invertible, right? Smmurphy(Talk) 01:37, 25 October 2006 (UTC)
- I agree, this is an important theorem, and the tiny section on the theorem in the article Implicit function is not very clear. As far as the merge is concerned, I think that there is no useful information on the aforementioned page that needs to be added to this one. TSchellhous 21:47, 12 December 2006 (UTC)
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[edit] Split from Implicit function
This is a major theorem, and deserves its own article. Dfeuer 17:09, 7 October 2007 (UTC)
[edit] Relationship with Inverse function theorem
What is the relationship between the implicit function theorem and the inverse function theorem? That should probably be mentioned in both articles Dfeuer 17:10, 7 October 2007 (UTC)
- The intended link might be that both theorems require the jacobian to be invertible, but it's still unclear to me. :-/ EverGreg (talk) 09:27, 15 May 2008 (UTC)
[edit] Proof
Does anyone have a (GDFL or public domain) proof available? Dfeuer 17:15, 7 October 2007 (UTC)
[edit] Example
In the example of the inverse function theorem, shouldn't the left part of (Df)(a,b) have -1's instead of 1's? Arthena(talk) 19:48, 21 January 2008 (UTC)
Also, the statement that one cannot implicitly create a differentiable function near (1,0) and (-1,0) seems pretty false. Sure you can't write an expression of y in terms of x, but clearly you can write x in terms of y. This theorem would be fairly weak if it couldn't fully characterize one of the most basic shapes in geometry. jordan (talk) 23:33, 7 May 2008 (UTC)
[edit] Relation
Is the term "relation" and the respecitve link, appropriate in this context? —Preceding unsigned comment added by Stdazi (talk • contribs) 22:06, 26 January 2008 (UTC)
- The link does not seem right. Arthena(talk) 23:08, 26 January 2008 (UTC)
- I've removed the respective link temporarily... Stdazi (talk) 11:05, 27 January 2008 (UTC)
- I've put it back :) A "relation" is like a "multi-valued function". For example, the (x,y) co-oridinates of a circle are a relation, even though we can't write y as a function of x because there are two values. That is the type of relation defined in relation (mathematics), and the type meant here. The intro to relation (mathematics) is misleading, so it now link straight to the definition. LachlanA (talk) 19:07, 19 February 2008 (UTC)
- I've removed the respective link temporarily... Stdazi (talk) 11:05, 27 January 2008 (UTC)
[edit] Curve genus
The question answeared by the implicit function theorem "can we write f(x,y)=0 as y=g(x)?" is also addressed by a theorem about so-called Curve genus [1]. An implicit function can be parameterized using rational polynoms iff curve genus equals zero. This is elaborated on in [2]. The curve genus is also used in the Riemann curve theorem [3] but I can't tell if this is subsumed with a similar "genus" in the Riemann–Roch theorem. EverGreg (talk) 09:23, 15 May 2008 (UTC)
- This is not correct. If a curve can be parameterized by rational polynomials, then it is an algebraic curve, so f(x,y) is a polynomial. But the implicit function theorem applies to f which are not polynomial. f need not even be analytic, nor does it have to be smooth, only C1. For instance, if you took a continuous everywhere non-differentiable function and called its antiderivative g, then the relation g(y) - x (that is, the graph of g turned a quarter circle) is a perfectly good f, and the implicit function theorem applies.
- The Riemann-Roch theorem is usually stated using the geometric genus of a smooth plane curve. This is the same thing as the arithmetic genus, which is what appears in the "Riemann curve theorem" link you cited. (I've never heard that theorem given a name before, but I'm used to hearing it stated as the birational invariance of the arithmetic genus, not as a theorem about plane curves.) Ozob (talk) 23:00, 15 May 2008 (UTC)
- Yes, it was not correct to say that "the curve theorem" and the implicit function theorem address the same issue. I think the motivation for stating the "curve theorem" for plane curves is that the Plücker Characteristics [4] can be determined by inspecting a computer generated plot of f(x,y)=0 where f is a polynomial. This will then tell you whether or not f can be parameterized. But now that I've learned that this can be determined from the jacobian determinant, the curve theorem strikes me as a somewhat roundabout approach. :-/ EverGreg (talk) 10:54, 16 May 2008 (UTC)
