I used to think that the best introduction to Calculus of Variations for scientists was Weinstock's book Calculus of Variations: with Applications to Physics and Engineering. I had discovered it as an undergrad student and, since then, I kept it at the top of my list. But, recently, I discovered Elsgolc's book and I must admit that it has to replace Weinstock's book at the top of the list.Weinstock's book is written at the right level for scientists - not highly abstract, but not imprecise for mathematicians. It contains all the right topics from calculus of variations and lots - really lots - of applications from science. However, I always felt that all these applications are more of distraction for someone who wants to learn just the topic of calculus of variations and then apply it in his/her own discipline. After all, the majority of the applications material in Weinstock's book can be found in physics and engineering books easily; and in these applications the calculus of variations part is only a small step to get a differential equation for the phenomenon under consideration. But the actual theory of the calculus of variations cannot be found so easily in the science books. Usually, these books devote a brief chapter to the topic of calculus of variations discussing only the main problem (which is often solved in a very unsatisfying way) and then state that other problems can be dealt similarly, essentially asking the reader to discover the remaining techniques on his/her own.My point in the previous paragraph is that scientists are in need to read a presentation of the various calculus of variations techniques in a crystal clear way, not read a copy of their mechanics text. And Elsgolc's book is exactly this: a careful, clean presentation of the theory without extremely long and unnecessary excursions to physics. It is written at the same level as Weinstock's book and it does contain simple examples to clarify the theory. One nice feature of the book is many two-column pages in which the author shows to the reader how the ideas of calculus of variations are similar to the ideas of the traditional calculus (of functions). The beginners will find this feature very valuable. The book is thinner than Weinstock's and yet it contains more topics than Weinstock's: Weinstock does not discuss extremals with cusps, neither he deals with sufficiency conditions for an extremum. The book does contain some quick applications (section 7 of chapter 1) to make the connection with science. The reader will find Weinstock's end-of-chapter problems more interesting and exciting but Elsgolc's are not bad. They are made to fit exactly the contents of the corresponding chapter. Most probably, beginners will find these problems more useful just because they are fine-tuned to the material.Finally I should say that, although I have presented the book as a good reading for scientists, it is also a good reading for mathematicians. In fact, mathematicians will be turned off by Weinstock's book. The extensive applications make the math scant. So, Elsgolc's book is the right choice.Overall, if you want to learn the topic, I recommend this book as first reading.