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The Plan

In the study of any special issue in mathematics, two things may be of importance, namely, subjects and order of them. I just wrote down a plan to study the theory of partial differential equations in a treatise manner. My program is as follows, considering the subjects and their order based on the prerequisites and logical relationships between them:

  1. Logic

  2. Topology

  3. Linear Algebra

  4. Mathematical Analysis

  5. Real Analysis

  6. Functional Analysis

  7. Calculus of Variations

  8. Classical Theory of Ordinary Differential Equations (ODEs)

  9. Classical Theory of Partial Differential Equations (PDEs)

Questions

  1. I am not sure that I have chosen the proper order for self-studying and also there may be things that I even don't know about to cover. So please tell me subjects that I should cover based on what you think and the ideal order of learning them.

  2. It will be really nice if you suggest just one reading in each subject that you prefer is the best for a theoretical study.

  3. How much time does it take to do the whole thing? 1, 2, 3 or 4 years?

Notification

Any other plan or thought you have is welcome. Please write whatever you think as answers not comments! :)

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    $\begingroup$ Welcome to the site! Could you please include the prerequisite knowledge on which to start the sequence and the ultimate goal of the activity. $\endgroup$
    – quid ♦
    Nov 18, 2015 at 9:58
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    $\begingroup$ Maybe latter. But I am not a PDE person. But give it some time. This site is not as fast paced as math.se for example. There is a good chance somebody else will answer. $\endgroup$
    – quid ♦
    Nov 18, 2015 at 21:10
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    $\begingroup$ I feel I cannot give an answer to this but here are some remarks:it may be possible to study things in this way, but if you are passionate about PDEs I would really recommend another approach. The thing you propose might be very systematic but especially if you are self-studying you may run out of steam quite quickly. Especially to start with logic seems not really needed. sure some logic and set-theory is needed everywhere, but you do not need to read a book on it beforehand. Then topology, more or less the same; knowing some topology is good, but even better is to know what for you need it. $\endgroup$
    – quid ♦
    Nov 21, 2015 at 1:30
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    $\begingroup$ Linear Algebra that's a good subject to get started with, real analysis too (given your background). What I would actually propose you to do is get some PDE book and look at it. I am not a PDE person at all but I did study it at some point, for some specific see if you can get Fritz John "Partial Differential Equations" and try to read a bit and see what you are missing as background. Or some other math-y Introductory PDE book. $\endgroup$
    – quid ♦
    Nov 21, 2015 at 1:30
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    $\begingroup$ Also you need to define what "treatise manner" means. $\endgroup$
    – guest
    Apr 8, 2018 at 1:07

2 Answers 2

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Despite the reputation that mathematics has for "logical order" and such, that assertion is misleading about how to study it, and how to understand it, I think. For one thing, it is not the case that there is a "true natural logical order", but only that things are order-able, in various ways, depending on tastes/prejudices. And, then, still, I doubt that any one of these logical/linear orderings really serves anyone well.

A fundamental symptom of a failure of linearizing to accurately reflect the sense of mathematics is that examples, counter-examples, motivations, and applications (within mathematics) cannot (in my experience) be made to naturally fit into any such scheme. Artificially, sure, we can refuse to mention motivations and examples, but that's terrible, not helpful.

Also, possibly contrary to various myths, studying "logic" is not a prerequisite for much of anything. For that matter, studying from a book/course about "how to do proofs" is a very peculiar thing, often making artificial mountains out of intuitive molehills... leaving the reader with no sense of why we need "proof" of trivial things, nor how to prove non-trivial things.

So, altogether, don't just read one thing at a time. Go back and forth between. And don't feel obliged to do all the exercises, or finish each chapter in every detail before moving on. Keep in mind that textbook ordering is invariably a choice, and is inescapably artificial. Nothing sacred, and not even necessarily/reliably helpful to the reader.

Sure, the topics you've mentioned are good... don't think of them as restricting you!

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  • $\begingroup$ (+1) Thank you Paul. :) $\endgroup$
    – Hosein Rahnama
    Jul 1, 2018 at 8:30
  • $\begingroup$ +1 for making artificial mountains out of intuitive molehills $\endgroup$
    – Dave L Renfro
    Jul 1, 2018 at 10:56
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Complementing Paul Garrett's advice in his answer, you might want to try a book with minimal prerequisites to get more of a feel for PDEs before going into more depth. One option could be Borthwick's Introduction to Partial Differential equations:

From the publisher's website:

This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise. Within each section the author creates a narrative that answers the five questions:

  • What is the scientific problem we are trying to understand?
  • How do we model that with PDE?
  • What techniques can we use to analyze the PDE?
  • How do these techniques apply to this equation?
  • What information or insight did we obtain by developing and analyzing the PDE?

The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.

If you do wish to follow a learning path of sorts, then you might try something along the lines of the following: calculus (single variable and multivariable/vector), linear algebra, real analysis, complex analysis, general topology (or at least metric spaces), measure theory & integration, and functional analysis. However, no set-in-stone ordering is intended, as some subjects can be studied in parallel or you can jump back and forth in grasshopper fashion. Along the way, picking up a little knowledge of ODEs would make sense, but no need to make a big detour just to get started on PDEs.

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  • $\begingroup$ (+1) Thanks for the reference. :) $\endgroup$
    – Hosein Rahnama
    Jul 1, 2018 at 8:31
  • $\begingroup$ For what it's worth, the elementary PDE courses I've had used the first half of The Analysis and Solution of Partial Differential Equations by Robert L. Street [taken during Fall 1976] (1973; the book is stronger than average on modeling aspects) and the first two-thirds of Partial Differential Equations of Mathematical Physic [taken during Fall 1981] (1980 2nd edition; the book is easier to look up things in than Street's book, and the 4 chapters prior to Fourier series and (continued) $\endgroup$
    – Dave L Renfro
    Jul 1, 2018 at 11:09
  • $\begingroup$ separation of variables, eigenvalue problems, and boundary value problems were (to me, at least) much more difficult than this later material), and all but the complex variables stuff in A First Course in Partial Differential Equations by H. F. Weinberger [taken Fall 1989] (1965; I've briefly discussed this book in comments here and here). $\endgroup$
    – Dave L Renfro
    Jul 1, 2018 at 11:15

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