# Talk:Implicit function

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Field: Analysis

## Discussion

I added the "Formula for two variables" section a while back. Now I see that the triple product rule article essentially derives the same thing. Should I remove it and link to that article? --Spoon! 23:41, 31 August 2006 (UTC)

Sorry i'm anon i should make an account. Anyway, i think that the heavy use of the graph of functions (implicit & explicit) on this page is not very mathematically precise. I do understand that the material is being presented in a somewhat laymans terms, but the graph of a function is not the same as the function itself. I don't have time to write an example of how i would change part of the page, i'll tryto solve to come back when i do have time.

I really think that the "Implicit function theorem" part should be separated (I guess separated again...), since it is very useful for mathematicians and is an important theorem. Implicitely-defined function, however, is not a deep mathematical concept, and, as mentioned above, is treated very lightly and not at all mathematically. yuliya 23:32, 26 March 2007 (UTC)

At the bottom of the page, it says "...then there exists ... and a differentiable function g: ..." Can't we change that to "continuously differentiable." In fact, f being C^r (r > 0) will imply g is (Munkres, Analysis on Manifolds). Dchudz 16:24, 3 May 2007 (UTC)

I also think g should be continuously differentiable but only have my lecture notes as a reference. Would someone like to do that? --GSpeight 13:32, 9 May 2007 (UTC)

I would like to once again say that "Implicit function theorem" is an extremely important mathematical theorem that should have its own article. Yuliya 22:06, 2 September 2007 (UTC)

Suggestion: the example 3 under the section implicit differntation can be done using explicit differentiation, if you differentiate with respect to y rather than x(giving dx/dy = 3y^2 - 1) and then invert the equation (giving dy/dx = 1 /(3y^2 - 1) ), wouldln't a better example be to differntiate "x^2 + xy + y^2 = 0 " ? —Preceding unsigned comment added by 79.70.172.108 (talk) 21:45, 31 May 2008 (UTC)

== R is still not defined== I saw some discussion talking about what is R in R(x, f(x)) = 0, but it still doesn't say what R is! daviddoria (talk) 01:17, 8 December 2008 (UTC)

I added a brief definition. siℓℓy rabbit (talk) 02:00, 8 December 2008 (UTC)

## Splitting Implicit function theorem back out.

The implicit function theorem doesn't really fit here, and deserves its own page. Dfeuer 17:14, 7 October 2007 (UTC)

I tend to agree. However I think the theroem is important enough to deserve a short overview section as in WP:SUMMARY. --Salix alba (talk) 22:48, 15 October 2007 (UTC)

## Definition

There is no clear definition in this article of "implicit function". While the term "implicit function" seems to have some currency, it seems that the essential concept is not "implicit function" but "implicit function definition", or perhaps "implicit relation definition". I think it might be reasonable to clarify that "implicit function" is an informal concept, rather than a precisely defined mathematical object. It might then make sense to split implicit differentiation into its own article, or possibly to merge it into Implicit function theorem. Dfeuer 17:35, 7 October 2007 (UTC)

I agree with Dfeuer, but I would use stronger terms. The phrase "implicit function" seems here to be an informal way of saying either "relation, used to implicitly define (various, possibly local) functions", or "actual function, coming from an implicit definition". I would say that vague, informal definitions that do not concur with careful mathematical practice are not appropriate on Wikipedia. I would propose reworking the article significantly, to eliminate the vagueness, and renaming it to "Implicitly defined function" or "Implicit definition of functions". -- Spireguy (talk) 02:42, 21 April 2008 (UTC)

Unless someone objects, pretty soon I'm going to rework this article significantly, and probably move it, based on the comments above. Please comment if you disagree. -- Spireguy (talk) 22:34, 30 April 2008 (UTC)

I object rather strongly to any proposal to move the present article to another one. The term "implicit function" is well-established in mathematics to refer to a function, possibly multiply-valued, defined by an implicit relation. See, for instance, the Springer EOM entry L.D. Kudryavtsev (2001), "Implicit function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Basically any good calculus textbook (Apostol, Stewart, etc.) treats implicit functions as bona fide objects of study, although they also do not go to great lengths to be precise. Any attempt to redefine the term would probably be a neologism, which is expressly to be avoided on Wikipedia. This article could be cleaned up considerably, but I really don't think it needs to be totally reinvented. I am going to post over at WT:MATH to see what others have to say about this. silly rabbit (talk) 22:58, 30 April 2008 (UTC)
Thanks for the response. The intention was not to redefine the term "implicit function", but to remove it, since heretofore the article did not justify its use of the term. You have now provided a reference for the usage---let's put that into the article, no?
Given that reference, I feel less need to do the reworking that I intended, but I would still say that the usage is not very standard. I wouldn't say that Apostol and Stewart (and Shifrin, Williamson/Trotter, Corwin/Szczarba---I just checked) "do not go to great lengths to be precise." They are precise; they just don't use the term "implicit function" to describe a relation defined by an equation. There isn't much need to do so: that's what the word "relation" is for. Instead, to avoid confusion, they (and most modern mathematics texts, as far as I know) reserve the word "function" for an actual function. So in the current standard terminology, the implicit function theorem is a theorem for deciding when an equation, which always defines a relation, actually (locally) defines a (smooth) function. It's not a theorem about "implicit functions", but a theorem about implicitly defined (actual) functions.
It may be that the term "implicit function" in this sense is more commonly used than I think. The reference just provided shows that it has some currency. It would be good to get a consensus on just how common the term is, to see whether the article should be changed or not. If it's an obscure or obsolete usage, then I would still say that change is needed. If it's not, let's put in the precise definition presumably given by the EOM reference mentioned by Silly rabbit. But by all means, let's hear from other people. -- Spireguy (talk) 02:58, 1 May 2008 (UTC)
Yeah, the concept can be defined formally, you just need a function h from a cartesian product of sets XxY to a set Z and an element z of Z. (It can be restated in any category with finite products actually.) Then, you obtain a so-called implicit relation R between X and Y: xRy iff h(x,y)=z. If this relation is a function, then it's called an implicit function. Generally, in Topology, an implicit relation is functional only locally (in the neighborhood of a point (a,b) in XxY satisfying f(a,b)=z). It's what says the eom reference. But sometimes, some mathematicians call implicit function what is an implicit relation. Cenarium (talk) 04:34, 1 May 2008 (UTC)
I would say that the term "implicit function" generally refers to an actual function satisfying an implicit relation. However, as the article indicates, there is often a tendency not to worry about whether an implicit relation defines a bona fide function. Indeed, in most introductory calculus courses, the term "implicit function" is used to mean a solution of an implicit relation, irrespective of whether it defines a function, but with the intimation that it is, at least locally. In more advanced mathematics, this lack of precision is justified by the implicit function theorem. I think the article at present, despite its imperfection, does attempt to be clear about the potential dual usage of the term "implicit function". In the first, more formal sense, it means a true function satisfying an implicit relation. In the second sense, it means a multiple-valued function; that is, a function with several branches. This latter definition could be made more precise, if necessary, using some ideas from topology: an implicit function is the sheaf of germs of solutions y of the equation R(x,y)=0 in which y is a function of x. This is the approach taken by Lars Ahlfors in his treatment of global analytic functions, and in particular algebraic functions (one of the most important classes of — multiply-defined — implicit functions). silly rabbit (talk) 12:42, 1 May 2008 (UTC)
First, let me say that I appreciate the thoughtful responses, and I will try to respond in kind. Let me clarify what I'm still worried about. My main concerns here are (1) usage and (2) logic. (1) The EOM reference shows that the exact usage "implicit function" does exist, but I think there is still a valid discussion to be had about how widely the term is used. In particular, I cannot recall seeing an introductory calculus text use the term "implicit function." (In addition to the ones mentioned above, I just checked Spivak and Rudin.) They instead speak of (bona fide) functions being implicity defined by an equation. Please let me know a specific reference otherwise if there is one.
(2) Look at the exact wording currently in the article:

Formally, a function f:XY is said to be an implicit function if it satisfies the equation

R(x,f(x)) = 0

for all xX.

That doesn't make logical sense. What is R in this definition? (I know what it is in the context of the article, but I'm talking just about the definition given.) Put another way, if we have a definition of the form "a function f:XY is said to be an implicit function if..." then I should be able to look at a function and say if it is implicit or not. But that doesn't make sense. "Implicitness" is not a property of a function, it is a property of how one defines a function. That's why I claim that the usage "implicitly defined function" is more correct than the usage "implicit function" (as well as being far more common).
As to the connection with multivalued functions, that term already exists, and has currency (although it's somewhat archaic in most contexts, since it conflicts with the preferred modern usage of "function", as pointed out by the article on that topic). So one doesn't need to have the term "implicit function" as a rough synonym for "multivalued function". It's true that most "multivalued functions" are defined implicitly. But why not just say that, instead of saying that they are "implicit functions"?
I feel like I am being very picky, but that's the idea, no? -- Spireguy (talk) 15:09, 1 May 2008 (UTC)
Not at all. I think I now better understand your concern. I'm worried now about going overboard in the lead, but perhaps the following formal definition is warranted:
Formally, if X, Y, and Z are three sets, and R : X×YZ is a function from the Cartesian product of X and Y into Z, then a function f:XY is said to be an implicit function if it satisfies the equation
R(x,f(x)) = c
for all xX, where c is a fixed element of Z.
I think we are making progress here. Perhaps the above can be worked on, and included in the lead? silly rabbit (talk) 15:24, 1 May 2008 (UTC)
So far the objections seem to be about 1) common usage 2) precision 3) better usage. As far as #1 goes, I object to your claim that "implicitly defined function" is far more common. For example, Google Scholar shows the usage of "implicit function" is about 200 times more common than "implicitly defined function". I can't recall the last time I've ever heard a mathematician even say "implicitly defined function". Of course, the majority of such mentions is in relation to implicit function theorems, but that is a major context in which such defined functions are discussed. #2 is not a big deal, as nicely pointed out by Oded. By the way, it's fine to make assertions that something is or is not "careful mathematical practice", but in reality, mathematical practice is often not "careful". Actually, mathematics is often discussed and thought about in "incorrect" ways, and terminology like "implicit function" is just a reflection of that. #3 really is irrelevant, no matter how incorrect "implicit function" may be, that is the common usage and it serves no purpose to create extra redirects. --C S (talk) 18:49, 1 May 2008 (UTC)
• It is not always necessary to be precise. We have an article titled complex analysis, but there is no mathematical definition of what complex analysis means, nor would there ever be one. You will not find any theorem proving that something is complex analysis, nor would it be reasonable to state a theorem saying that something is an implicit function. Implicit function is well established terminology. While we need to say what this means, there is no need for the definition to be precise mathematically. Just as the notion of number can have different meanings depending on context (positive, real, complex,...) so can the precise meaning of implicit function be context dependent. Oded (talk) 16:03, 1 May 2008 (UTC)
I think some progress has been made towards consensus in the last two posts. I agree now with Spireguy that the article really does lack a good definition of even the most basic implicit function (qua an actual function (mathematics)), and that there should be a way to work this into the article somehow. That said, I also do not feel that we should strive for a maximum of precision at the expense of clarity. silly rabbit (talk) 16:07, 1 May 2008 (UTC)

Here's my (Spireguy) proposed (draft) intro:

In mathematics, an implicit definition of a function is a way of using an equation to specify the values of the function. It contrasts with an explicit definition of a function. In this context, the term implicit function can refer to either the equation (i.e. the relation) or the resulting function.[1]

An explicit definition of a function f gives an algorithmic procedure, usually given by an algebraic formula, that produces the output (dependent variable) of f for a given input (independent variable); for example, f(x) = x2.

By contrast, given a function R of two variables, one can try to implicitly define a function f in the following way:

y = f(x) if and only if R(x,y) = 0.

We say in this case that f has been implicitly defined by the equation R(x,y)=0. In many cases, this is not possible without also restricting the domain or range of f; see the examples below for clarification.

1. ^ However the term "implicit function" is not currently popular usage, and it does not denote a special type of function; in fact, any function f can trivially be defined implicitly via the equation y-f(x) = 0.

I believe there are several problems with this, but let's start with why you decided to ignore my comments and continue to assert the non-prevalence of the term "implicit function". "Not currently popular usage"? Can you back this up? --C S (talk) 19:40, 1 May 2008 (UTC)
This draft confuses the usual, "explicit" definition of functions with algorithmic definitions. For example, the function on the real plane that takes (a,b) and returns 1 when a< b or returns 0 otherwise, where "<" is a well-ordering on the reals given by the axiom of choice, is certainly not algorithmic. But it would usually be considered an "explicit" definition (despite having an unconstructive component). --C S (talk) 20:02, 1 May 2008 (UTC)
Sorry, C S, I didn't see your comments, probably because I was sitting on my edit for a long time. So I wasn't ignoring them; very sorry to seem rude, not my intention at all. I'll respond to the particulars presently. -- Spireguy (talk) 20:05, 1 May 2008 (UTC)
OK, belatedly responding to C S's original comment.
1) I looked at the Google Scholar results. Indeed, the standalone phrase "implicit function" does get used; I see it only in applied mathematics articles, which may be why (as a pure mathematician) I had never seen this usage. Or, as I said before, maybe I just missed it. So I would say that the footnote, as I wrote it, is inappropriate. However the comment that an "implicit function" is not a particular kind of function is, I think, still important, since that would be a point of confusion. Also, it may still be significant that calculus texts (including advanced calculus) seem to studiously avoid the term.
2) I think the analogy to the vagueness of "complex analysis" is weak. We are talking about a specific mathematical object, not a whole field. Modern mathematical practice is to be careful and precise in defining the objects of study. And even if the notion is context-dependent, then still, in each context, there is a precise definition. (I really don't think context-dependence is the issue here, anyway. Even in the most basic context of the circle example, the current version is still imprecise.)
If the actual usage of the specific phrase "implicit function" is indeed vague, in some sense like "complex analysis" is vague, then the article should reflect that more correctly, and not pretend that there is some particular object called and "implicit function". As it stands, the article attempts to define the notion precisely, but comes out with something that's logically empty.
3) As I said in response to #1, the term does seem to have substantial usage, despite being avoided in all of the textbooks/pure math books that I have looked at (a dozen).
Response to C S's more recent comments: yes, the version I wrote used "algorithmic" as a substitute for "explicit"; that's one reason it was a draft. Not sure of the best way to describe "explicit", offhand.
Here's what I would like to see: the EOM definition of "implicit function" (not the whole entry, just the precise definition, or if they don't give one, the most accurate description of the concept). Since that's the only reference that people have produced that actually defines the concept, and this is Wikipedia, let's go with the reliable source. Without that, it's too easy to argue back and forth about what the term really means. Can someone transcribe that here? -- Spireguy (talk) 22:01, 1 May 2008 (UTC)
Regarding #2, again, I think when you state what "modern mathematical practice" is, I think you are mistaken. It is clear to me that "implicit function" as in the currrent context, imprecise as it is, does in fact fit well within modern mathematical practice. Let us use another example. How often have you heard the question, "Is the function well-defined?" As the question is stated, it is nonsensical. Obviously if it is a function it must be defined properly. If it is not a function, then the question presupposes it is one. Nonetheless, many mathematicians use this question and it is perfectly clear to them what it means. --C S (talk) 02:02, 2 May 2008 (UTC)
A good analogy, but remember, the intro we are discussing is giving a definition. Even in a book, paper, or talk which uses loose notions such as "is this function well-defined" (which are, as you note, ubiquitous), the definitions will (one hopes!) be precise. I would be fine with the article stating that "implicit function" is a loose term only---if it gave a clear idea of what this loose term meant---but one of my objections is that it seems to have it both ways. It gives a precise-looking definition, but one which is logically suspect. See my reply below as well.

## Sources

Please add sources for the article here:

Defines implicit functions as such. These are true functions in the mathematical sense. Quote: "A function f:EY given by an equation F(x,y)=z0, where F:X×YZ, xX, yY, z0Z, EX and X, Y and Z are certain sets, i.e. a function f:YX such that F(x,f(x))=z0 for any xE."
Signed by silly rabbit (talk) 23:38, 1 May 2008 (UTC)
Defines implicit functions in contrast to explicit functions, but only in the algebraic case. Specifically includes the possibility that functions may have multiple values.
Signed by silly rabbit (talk) 23:43, 1 May 2008 (UTC)

Sources establishing legitimacy of the term "Implicit function":

Section §10.2 is entitled "Implicit functions". Unfortunately, the text fails to define them properly.
Signed by silly rabbit (talk) 23:54, 1 May 2008 (UTC)
• Spivak, Michael (1965), Calculus on manifolds, Addison-Wesley (published 1995)
One of the sections in Chapter 2 is entitled "Implicit functions" (this is apparently not the case in subsequent editions of the book?).
Signed by silly rabbit (talk) 00:00, 2 May 2008 (UTC)
• Stewart, James (2006), Calculus: Concepts & Contexts (3rd ed.), Thomson Learning, Inc., ISBN 978-0-495-38491-5
The term "Implicit function" appears in the index, although the word does not appear in the actual text of the book. Instead it defines an "implicitly defined function" on page 233.
Signed by silly rabbit (talk) 23:52, 1 May 2008 (UTC)

Tertiary sources:

Defines implicit functions in such a way as to be actual functions (presumably).
Signed by silly rabbit (talk) 00:08, 2 May 2008 (UTC)

Other somewhat related resources:

Seems like an interesting resource for students in multivariable calculus, although lacking in a general definition of implicit functions. Here is a cite for their treatment of the implicit function theorem: http://www.ualberta.ca/MATH/gauss/fcm/calculus/multvrbl/basic/ImplctFnctns/implct_fnctn_thrm.htm

## Comments on sources

Thanks for the copious listing (and for transcribing the EOM reference---I couldn't get the link to work). I'm not sure about the Spivak reference. (When I mentioned Spivak, it was his Calculus, not Calculus on Manifolds, which I don't have next to me right now.) I concede the currency of the general term "implicit function." But note, still, how many of the references use it as a general term, but do not give it a specific definition. (The EOM being an exception, of course, and that's probably the best source to use.) That's not accidental; I think it's a reflection of how slippery the term is, and the fact that authors don't want confusion about the crucial term "function".

Anyway, I'm still not happy with the phrasing in the article, which characterizes an implicit function as a special kind of function. The logic still doesn't make sense. I'd like it to be clear that you cannot say "Let f be an implicit function..." or ask "Is f an implicit function"? (As opposed to, say, "f is a function implicitly defined by R(x,f(x))=0", which is clearly meaningful.) Maybe the modification necessary is more minor than I first thought, something like:

Formally, a function f:XY is said to be an implicit function (defined by the equation R(x,y)=0) if it satisfies the equation
R(x,f(x)) = 0
for all xX.

That brings R into the definition. An analogy: it doesn't make much sense to say "p is a characteristic polynomial", but it does make sense to say "p is the characteristic polynomial of the matrix A".

Also, a quick glance at the Google Scholar results seems to indicate that there are at least three usages: (1) the EOM usage; (2) the function R, not the function f derived from it; (3) the phrase "implicit function surface" comes up (as a synonym for "level set of R", I believe). If that holds up, it seems necessary to address at least the first two uses.

Yeah, I'm still being picky. I don't want to be annoying, but this is still bugging me. -- Spireguy (talk) 02:33, 2 May 2008 (UTC)

## Implicit differentiation

I believe this is a distinct topic that requires its own article. What do people think about a mergeout of this section? —Anonymous DissidentTalk 12:57, 26 May 2009 (UTC)

It does seem that the way the article is now implicit differentiation gets lost in the shuffle. I'm not sure that creating a new article is the best solution but there needs to be some changes.--RDBury (talk) 12:08, 25 September 2009 (UTC)
I dont know about a separate page, either, at least not until the existing section gets fleshed out. I can suggest some changes and additions to the first part. For example, the following sentence could be changed to clarify a bit: "It is impossible to express y explicitly as a function of x and dy/dx therefore this dy/dx cannot be found by explicit differentiation." (ie, moving the dy/dx term, striking 'this') I'm a bit rusty with explaining this in writing, so I hope the change leaves an accurate statement. -PrBeacon (talk) 02:45, 7 October 2010 (UTC)