Hi,
I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options:
I work on Machine Learning, and that where I intend to apply this.
Thanks!
Hi, I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options: I work on Machine Learning, and that where I intend to apply this. Thanks! 


I would personally recommend Gelfand and Fomin's "Calculus of variations". It has many advantages:
Overall, I think this is a good book to have anyways, you'll always want to have a look there even if you get a book that is concerned more directly with applications (although as I said, they already keep an eye on what those ideas are useful for outside pure math) 


Bruce van Brunt's The Calculus of Variations. 


Chris Bishop's book "Pattern Recognition and Machine Learning" has some stuff on applying variational methods to machine learning. You might have a look at that and follow his references. Also, to me, "A Primer on the Calculus of Variations and Optimal Control Theory" by Mike MestertonGibbons looks nice. You can get a sample of it (table of contents, the first few pages) at http://www.ams.org/bookstoregetitem/item=STML50 


Like most people above, I am not really sure what you are doing with this information. However, after you have looked at the continuous case, you might consider looking at the discrete calculus of variations. [1] (listed below) has a very nice chapter (chapter 8) on the discrete calculus of variations. [1] Kelley, W. & Peterson, A. (2001). Difference Equations: An Introduction with Applications (2nd Ed.). San Diego, CA: Academic Press. 


Thank you everyone for your recommendations and suggestions. As pointed out by some I was, perhaps, not very specific about my needs. I am working with ML and my need to understand the Calculus of Variations came up when I began looking at LDA (www.cs.princeton.edu/~blei/papers/BleiNgJordan2003.pdf). @PeterR  I own a copy of Bishop, but I needed something more comprehensive in way of being introduced to this subject. I feel the coverage of the topic in the book is highly tailored towards its applications in it. I would like to be in a position apply it on my own models sometime in the near future  so I want to build a good foundation for now. Your other suggestion seems promising from a cursory look  Thanks! @Deane Yang  "I know of very few places where the ideas or techniques of calculus of variations are used beyond the calculation of the first and second variations of the functional being optimized"  I pretty much suspect the same from what I have seen so far :) @sfilip  I have begun looking at Gelfland and Fomin and it does look promising. Thanks! And again, thanks a lot everyone for your suggestions! 

