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 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. 


One of my favourite books for calculus of variations is the following: "Introduction to the Calculus of Variations" by Bernard Dacorogna It is a great introductory book. It has plenty of solved problems to get familiar with the material. It starts with the basics on function spaces, and then introducing classical and direct methods. It then moves onto minimal surfaces and isoperimetric inequalities. It is a little expensive but it is worth it. 

