References for Methods of Applied Mathematics Course

[physics4fun]

Hi,
I am taking a methods of applied mathematics course in Spring 2010 semester. The previous semester proved to be a bit overwhelming and the professor used a textbook that I found inadequate in fully understanding the material. The course content will follow this syllabus with some additional changes:

.http://www-rohan.sdsu.edu/~rcarrete/teaching/M-342B/syllabus.html"

I have a basic problem with methods courses as they omit so much detail and try to cover too much material. The previous semester left me feeling like I had been given a very superficial understanding of the course content. I suppose that there is not enough time and this will always be a problem with this type of course. I would like to know if there are any other references that someone can recommend to cover these topics. Does anyone have specific references for the topics that will cover them in depth? I really do well with proof based explanations where I can see the development of theorem from definitions to result.

Thank you.

 

[Astronuc]

That certainly is an ambitious syllabus for a one semester course. I took one course in applied math that dealt with special functions, and this was after one course in complex variables and functions, and another separate course on partial differential equations. I don't know how one would do all this in a single course - except for 6-9 classroom hrs per week.

"Mathematical Methods in the Physical Sciences" by Boas, Wiley, 1983 is a classic which the professor referenced.

My texts for uni are 30+ years old.


A similar modern book is:
MATHEMATICAL METHODS FOR PHYSICISTS
http://www.elsevier.com/wps/find/bookdescription.cws_home/705838/description#toc
George Arfken, Miami University, Oxford, Ohio, U.S.A.
Hans Weber, University of Virginia, U.S.A.

1. Vector Analysis
2. Vector Analysis in Curved Coordinates and Tensors
3. Determinants and Matrices
4. Group Theory
5. Infinite Series
6. Functions of a Complex Variable I: Analytic Properties, Mapping
7. Functions of a Complex Variable II
8. The Gamma Function (Factorial Function)
9. Differential Equations
10. Sturm-Liouville Theory-Orthogonal Functions
11. Bessel Functions
12. Legendre Functions
13. More Special Functions
14. Fourier Series
15. Integral Transforms
16. Integral Equations
17. Calculus of Variations
18. Nonlinear Methods and Chaos
19. Probability


There are on-line course notes in DiffEQs, PDE, and probably the other topics. The Math & Science Learning Materials Math forums have some tutorials and links.

Calculus & Beyond Learning Materials
https://www.physicsforums.com/forumdisplay.php?f=179

There are a number of relatively inexpensive math books published by Dover.

 

[physics4fun]

Thank you for the recommendation about Dover books. I have the Boas book which I used last semester but I found it to be very lacking in details. Some people can learn this way; I can't. A lot of other students taking the class started using extra references. The professor even told me he relied upon students to find other resources because the book was short on details.

This was what I took last semester:

http://www-rohan.sdsu.edu/~antoniop/teaching/m342A/m342A_assignments.html"

I don't know what else to do other than keep looking for alternative sources and maybe other people will have suggestions. I read some of the reviews on Amazon.com about the book you mentioned and there are many reviews that give the same criticism that I have of the Boas book.

Regardless, thank you for the post as a starting point. I'll keep looking for resources.

 

[Studiot]

The book by Sadri Hassini (University of Illinois) gives a pretty modern treatment of your topics except the Chaos stuff. I think you will need another book for that.

Foundations of Mathematical Physics - Sadri Hassini - Prentice Hall.

If you can get hold of a copy Professor Riley's book is really good at explaining things, although a little old nowadays.

Mathematical Methods for the Physical Sciences

K F Riley - Cambridge University Press.

Vikas Publishing of New Delhi, India offers an absolutely mammoth tome

Mathematical Physics by B D Gupta.

An absolute mine of classical information and very cheap

Finally two chaos books

Chaos in Dynamical Systems by Edward Ott

and

Chaos and Non Linear Dynamics byRobert C Hilborn

should more than cover your syllabus in this area.

PS It is traditional for students of almost any technical discipline to arm themselves with one or more of the Schaum series. There's one for every eventuality.

 

[jasonRF]

Since you want more details than the standard "methods of ..." book, perhaps dedicated books are what you are looking for. Of course, working through multiple dedicated books in a semester isn't too practical. But you asked for "references".

You primarily need to learn about PDEs, series solutions of ODEs and special functions, phase-plane stuff, and complex analysis. So I would suggest:

Get the 2nd edition of the complex variables book by Saff and Snider. On amazon used copies go for like $5 plus shipping. It has a nice presentation of the fundamentals. Many other books are good as well.

For PDEs, I can think of a few of books. The 2nd or 3rd edition of Elementary Applied PDEs by Haberman is pretty good for applied math. Used copies are reasonable. I also like the Fourier Series and boundary value problems book by Churchill. Used copies of old editions are cheap, once again. A more old fashioned book (but one that is quite proof heavy) is A First Course in PDEs by Weinberger (a Dover book). It also covers complex analysis and integral transforms, but is too old fashioned to include delta functions and vector space ideas. It also includes solutions to all problems! I have spent many hours with this book and have learned a lot from it.

For series solutions of ODEs and Bessel functions you can consult many books on ODEs. Pick almost any book and you are fine. The assigned book is probably fine, although it may not include a rigorous proof. A classic reference for this (and other special functions) is "A course of modern analysis" by Whitaker and Watson, which can be found online for free (and legal!) . I wouldn't buy it unless you were so enchanted by the free version that you want a paper copy, but it does include proofs about the series solutions of ODEs. For a pure reference of special function properties/equations (no proofs) the classic is "handbook of mathematical functions" edited by abramowitz and stegun. Again this is legally free online, and I personally use it all the time (my paper-bound copy is worn out but I now use the electronic version a lot). Google will find it!

For phase-plane and nonlinear dynamics, a great book is the book by Strogatz "nonlinear dynamics and chaos" which is wonderful. I covet that book - I have only checked it out from the library at work but it was recalled before I could really work through it. this should be far down on the list of priorities, but it is really fun.

good luck, and be consoled by the fact that your quick overview course is really useful, even if it doesn't go into as much detail as you seem to want. You will know the most important ideas and techniques, and will have a good background and know where to look for more details if/when you need them. That is all most of us ever need.

Good luck,

jason

jason