# Reference request: calculus of variations

I am searching for a good book to self-study calculus of variations.

It should be fairly complete; build up gradually from the very basics; offer detailed explanations; have some emphasis on applications of variational methods.

• For math? Physics? Have you taken undergraduate real analysis/ODEs/ ... ? – Tyler Dec 22 '14 at 13:39
• @Tyler I study math, but I'm also interested in applications to mathematical physics. Yes, I took undergraduate real analysis and I'm now taking ODEs (however, a self-contained book with some revision of the necessary material would be good). – Dal Dec 22 '14 at 13:50
• A book that was used a couple of times for a course mainly taken by graduate mathematics students (and some graduate physics students, and usually at least one undergraduate mathematics student each time it was offered) at the university I attended in the late 1970s is Gelfand/Fomin's Calculus of Variations. Also during this time a popular book for the physics graduate students was Weinstock's Calculus of Variations: With Applications to Physics and Engineering. – Dave L. Renfro Dec 22 '14 at 17:51
• I second @DaveL.Renfro advice. Get Gelfand/Fomin. – Artem Dec 22 '14 at 21:37
• @user62029 Do you need more references? – Kugelblitz Mar 10 '15 at 10:32

Well there are a huge amount of book suggestions regarding the calculus of variations gather-able from these links: https://mathoverflow.net/questions/46319/beginners-text-on-calculus-of-variations ; Introductory text for calculus of variations .

I will list links to sources considered to be the best:

1. Gelfand and Fomin's "Calculus of variations" http://www.amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/0486414485 . It has many advantages: It is cheap (so if you buy it and don't like it, it's not a big deal); It is written by good mathematicians, that are broad enough to see connections with many different areas; It has useful exercises, and they're reasonable and with an eye on applications; It has an appendix on Optimal Control.
2. http://www2.math.uu.se/~gunnar/varcalc.pdf
3. https://www.cs.iastate.edu/~cs577/handouts/variations.pdf
4. http://www.amazon.com/Lectures-Calculus-Variations-Optimal-Publishing/dp/0821826905
5. Robert Weinstock's Calculus of Variations: with Applications to Physics and Engineering http://www.amazon.com/Calculus-Variations-Applications-Physics-Engineering/dp/0486630692/ref=pd_cp_b_2

I believe suggestions 1 and 5 will do the trick; I'm sure they'll be helpful.

• I came here to post Gelfand and Weinstock. These are the best books to use with Gelfand suiting a more pure mathematical approach and Weinstock taking the entirely applied math/physics approach. – Tony S.F. Mar 8 '15 at 3:17
• ^Exactly sir @TonyS.F. – Kugelblitz Mar 8 '15 at 3:18
• Also @TonyS.F. Could you give me suggestions for studying contour integration? I'm a tenth grader, so graduate texts are not very comfortable to learn from (I could, but I'd have to devote too much time..) Any excellent undergrad text source or pdf would be appreciated. – Kugelblitz Mar 8 '15 at 3:21
• I do not know enough on the subject to tell you a good source but I can say when I took complex variables, which covered contour integration, as an undergraduate we used Brown and Churchill – Tony S.F. Mar 9 '15 at 3:06
• Seen that before; A tad too condensed for me sir @TonyS.F. I need a more expository textbook I guess... – Kugelblitz Mar 9 '15 at 3:09

http://www.amazon.com/The-Calculus-Variations-N-I-Akhiezer/dp/3718648059

This book may be worth checking out. I began learning from it and found it pretty thorough. You might want to have exposure to basic real analysis (convergence, etc...).