Talk:Change of variables (PDE)

Ryan Reich's Complaint

Ryan Reich says:

This is a how-to manual, thus inappropriate content on Wikipedia. The example (and lack of any indication of a theory which it exemplifies) exhibits the subject as being primarily phenomenological, and the quote from Steele confirms the intent of the article as being instructive rather than expository.

Ryan Reich: Can you explain just what "instructive rather than expository" means? Michael Hardy (talk) 04:33, 18 July 2008 (UTC)

It was just a continuation of what I'd been saying. The article seems to be concerned with instructing the reader in how to perform a change of variables. It's the sort of material I would expect to see one day in a calculus class, and the article being centered around it means that the author is treating PDEs like a problem in symbolic manipulation, illustrating the heuristics and the phenomena without giving them a mathematical context. Change of variables is hardly specific to PDEs but the title suggests that there's something special about them. Lack of exposition means that the article reads like the solution to a homework problem, giving no larger significance to the subject either in mathematics or in the literature, or indicating that there's anything more to it than guess 'n check. Ryan Reich (talk) 15:49, 18 July 2008 (UTC)

Hi Ryan,

The theory is in section 2. If you don't think the theory is well-presented, please improve the exposition.

Change of variable for PDE is not much discussed but it is an essential technique. Change of variable for integral equations is discussed in Integration by substitution but this doesn't really give you much help.

Also, can you direct me to WP:Phenomenological?

Thanks, Erxnmedia (talk) 13:49, 18 July 2008 (UTC)

What's wrong with the word "phenomenological"?

The stuff in section 2 is hardly "theory": it simply says that change of variables can be done, which is not the subject of either the example or the quote. As I said above to Michael Hardy, there is no theory of what significance the substitutions have or any claim that there is a theory, other than the abstract nonsense in section 2, which has no connection with the example. The fact that you say that it is an "essential technique" means to me that you think of this article precisely as a how-to guide, which, unlike WP:Phenomenological, is covered by a policy. It may be useful, but the place for such useful information is not here (it is, actually, at Wikibooks). Ryan Reich (talk) 15:49, 18 July 2008 (UTC)

Hi Ryan,

Do you also want to delete the Integration by substitution article? It also has examples, and discusses a technique which you may consider is so obvious it doesn't even need to be discussed.

Also do you think that no symbolic computation method needs to be discussed?

Since you are more educated on this topic, if you agree Integration by substitution should not be deleted, do you think that you could edit the article so that

• It meets your standards
• Includes an example
• Properly presents the theory

It is true that Wikibooks has a section on this, but it was not a useful discussion from my point of view, in that it was:

• Less concise
• Did not have an example which isolated the main elements of the technique
• Did not discuss the technique in general

Also you say that the theory section is abstract nonsense. However, the following information is true about change of variable technique for PDEs, if what I said is nonsense, please tell me how to convey the following information:

• It's about differential equations (hence by a slight abstraction, about differential operators)
• It is generally helpful for there to be a bijection between the old set of variables and the new one, or else one has to
  • State the domain of applicability of the correspondence (where again it needs to be a bijection), and/or
  • Enumerate the (finite list) of exceptions (poles) where the otherwise-bijection fails (and say why you don't care about those exceptions)
• If you don't have the bijection then the solution to your reduced-form equation will not map in a useful way back to a solution of the original equation

Thanks, Erxnmedia (talk) 19:52, 18 July 2008 (UTC)

"Has examples" is hardly a flaw in a math article. The problem is when the article is the examples, as is the case here. Integration by substitution concerns itself not only with the technique of substitution, but with the general principle (this principle, by the way, is the first Theorem in the section on multiple variables). Granted, the beginning of the article has a certain formal similarity to this one in that it concerns a technique, but it's not about how to do the techique, it's about what the technique accomplishes; the examples illustrate this as well. In this article, on the other hand, you continually show that you are most concerned with documenting the use of the technique itself: a fully-worked example without any theoretical implications, and in the section purporting to be on the theoretical underpinnings, you have actually added two bullet points about technical issues arising in the application of the technique. Aside from that, the contents of the second section are similar to the statement of integration by substitution, which is not bad, but also not necessarily a reason to keep the article. After all, there is no article on how to multiply two numbers in base 13, although there is an article on multiplication of integers. Ryan Reich (talk) 21:21, 18 July 2008 (UTC)

Hi Ryan,

My intentions in creating the article are not relevant to the topic yourself, and are not a valid basis for criticism.

An article on Change of variables in PDE is in the same category as an article on multiplication in general (over integers, real numbers, matrices, etc. -- any domain on which differentiation can be defined). My intent in providing the article was to define the topic (as in multiplication), provide an illustrative example, and a general definition.

Following your criticisms, you can improve the article by:

• Discussing the general principle
• Saying what the technique accomplishes
• Giving an example which you feel better illustrates what the technique accomplishes

Thanks, Erxnmedia (talk) 22:08, 18 July 2008 (UTC)

I'm going to say more about how this subject is not the same as Integration by substitution. The significance of substitution in integrals is that the formula brings derivatives into the computation of integrals. Granted, that's what the fundamental theorem of calculus does, but substitution is in general more than just the fundamental theorem (in the multivariable case, the two are distinct; in the single variable case, as the article shows, you can derive substitution from the fundamental theorem). Thus, although substitution is for many an "integration technique", in the large sense it is more than just a device for changing the symbols being manipulated, but is in fact a bridge between these two fundamental operations of analysis. In the context of calculus on manifolds, it expresses a certain functorial behavior of integration with respect to smooth maps, another important property. Then there's the probabilistic connection, which has similar philosophical import. One might summarize all this by saying that the substitution rule expresses a local/global connection (differentiation being local, integration being global), which in many parts of math is among the most important connections to make, local being easier to deal with and global being the object of interest.

Now consider the potential of the subject of change of variables in a PDE. Everything written in the article thus far affirms that change of variables is, in essence, an application of the chain rule to this particular situation. As you have been pointing out, it is the case that one has to be careful to which changes of variable one applies the chain rule (they should be everywhere smooth and invertible, of course), but even including this, there are basically two pieces to this theory: what one might call "change of domain", where one transfers functions on an open set U to another open set V via a diffeomorphism between U and V; and the chain rule, applied to relate the effects of a differential operator on a function before and after changing its domain. Beyond this, I cannot see a mathematical significance comparable to the one above. In my opinion, beyond giving an overblown exposition of the facts I've just stated, the only room left in the subject is to give examples (which are, of course, the technique in question), which makes this topic essentially one of how-to: given a principle which is so directly reducible to others which are more general and discussed elsewhere, the only question is how to make it useful. The only way to escape this fate is if the technique itself has a theory; for example, one might say that the long exact sequences of algebraic topology are a computation technique, but there is much theory that goes into what they are, which objects to apply them to, and how they relate to the topological category. If there is anything like a general principle for choosing a change of variables that is not simply an assembly of heuristics, it would be wondeful and would totally justify this article.

I don't see such a principle claimed here, and the way you've written the article and have been talking, it doesn't seem to me that you know of one. You've been getting defensive about the relevance of your intentions, but there is one way that they are relevant: in deciding whether this article has potential for improvement. You wrote the article, so you delineated the general outline, and as I see it, you intended to write and did write a how-to article. As I've explained, I think the subject of change of variables for a differential equation is a dead end as far as theory goes, which means that your intentions have written the article into a corner in which it is vulnerable to deletion. You're trying to enlist me to improve the article according to my criticisms, but if I really did that, it would then become redundant with a discussion of the chain rule and get deleted or, at best, merged. Besides, the sword of cooperation cuts both ways: you said you don't like the Wikibooks treatment of this subject; you can improve it.

Ryan Reich (talk) 02:39, 19 July 2008 (UTC)

Hi Ryan,

A few points:

1. It is not a requirement in practice that "changes of variable ... should be everywhere smooth and invertible". In practice, the change of variable should be smooth and invertible in the region of interest for the solution of the problem at hand, except perhaps for a finite number of poles.
2. I am not "being defensive about my intentions". I am pointing out that, based on prior experience in other domains editing Wikipedia articles, that
   1. Intentions and motivations for taking an interest in a topic and writing about it are not a valid basis for criticism of what has been written. When is the last time you referreed a JAMS article and criticized an article for the motivations of the writer? Maybe an article on category theory because you think anybody who writes about category theory should move to New Mexico? How is it ever relevant in dicussing mathematics to question or discuss the motivations of the writer of some text about mathematics?
   2. In practice, 2 seconds after an article has been created, there will be 10 people coming at that article with 10 completely different motivations, no matter what the topic. This is the strength of Wikipedia, as in general, the cross-purposes eventually cancel themselves out modulo a small error term. In this sense, pragmatically speaking, motivations are irrelevant, so you're just wasting energy by discussing them.
   3. In this case, by classification of motivation, you are a deletionist and I am an inclusionist. Wikipedia doesn't favor either position, see Deletionism and inclusionism in Wikipedia.
3. You write some nice things about integration by substitution. Why don't you include them in that article? My guess is you won't because you don't think any kind of commentary or exposition belongs in a piece of writing about mathematics which is going to stay in one place -- I'm guessing that in your mind, commentary and exposition are only permitted in passing discussions or in a classroom, coffee lounge or advisor's office.
4. I picked the Black Scholes equation example because it is short and gives a dramatic simplification. However the example doesn't make clear that the context for change of variable (you are criticizing the text for lacking context) is Coordinate system transformation. Because in the Black Scholes case it is not clear that a coordinate system transformation is being done, and because I don't know (I'm not a professional mathematician) whether it is necessary for a change of variables to constitute a coordinate system transformation in order to validly apply the technique, I left that out.

I know I'm talking into air here because you want to delete this article and don't want to be "enlisted" to improve it, but, you know, I'm just saying.

Thanks, Erxnmedia (talk) 13:45, 19 July 2008 (UTC)

Theory

Roughly speaking, it seems that the method of change of variables is a composition of

• A coordinate system transformation
• The chain rule

Under the right preconditions, this composition constitutes an algorithm, behind which the theory is no more or less than the proof of correctness of that algorithm.

Similarly, the method of characteristics is an algorithm. The proof of correctness of the method of characteristics algorithm, together with the history of prior discussions of the method of characteristics, is given in a clear and rigorous mathematical exposition in this paper:

• Delgado, Manuel. "The Lagrange-Charpit Method", SIAM Review, Vol. 39, No. 2 (Jun., 1997), pp. 298-304, Society for Industrial and Applied Mathematics

What I would really love to see evolve in the theory section for the method of change of variable would be something very much in the spirit and form of the presentation provided by Delgado's article for method of characteristics.

Thanks, Erxnmedia (talk) 00:30, 23 July 2008 (UTC)

Theoretical Underpinnings/Encyclopedic Work

I am new to this discussion (which seems to have been resolved of late judging by the lack of continued discussion) but I think I can be of some assistance in cleaning up some of the issues presented. I actually encountered the page while doing a little review of the COV technique for PDEs. Here are a few thoughts.

Perhaps the theory section should come first, followed by the example. Maybe that is just my personal pedagogical preference?

A previous contributor accurately pointed at that this is largely an exercise in change of coordinates and chain rule. That is perfectly true, except for there is more to the story. COV for PDEs forms the basis for the so called "Method of Mappings", a powerful technique for solving Inverse Problems on a domain that changes in time. I can provide several references from Banks, Kojima et al from the early 1990s when this technique was developed. Would this help to further establish the article by providing some encyclopedic basis for the technique? My personal research will (hopefully) produce another powerful application of the technique in the next few months. I would not be comfortable placing such results on Wikipedia myself, but would be glad to submit considerations via the discussion page for someone else's consideration.

In the process of performing this research, perhaps I could also beef up the theoretical underpinnings of the technique?

Thoughts? Clayt85 (talk) 18:56, 5 June 2009 (UTC)