Schaum's Outline of Theory and Problems of Advanced Calculus

I have taken extensive coursework in undergraduate calculus, advanced calculus, and a 2-semester real analysis course with Rudin's 'principles.' This book was incredibly helpful to me, because it provides some examples of material that Rudin covers, and these examples helped me gain a better understanding of the material (there are 1370 solved problems!). There is nothing in this book about topology per se, but it does cover just about everything else: numbers, sequences, limits/continuity, derivatives, integrals, partial derivatives, vector geometry, multiple integrals, line integrals, infinite series/products, improper integrals, Fourier series & integrals, and gamma/beta functions. This book shines in the plethora of material covered, and because it focuses on solving difficult problems as opposed to routine ones. Another thing is that this book DOES include most of the answers to the supplementary problems - answers are only omitted, for example, when the question says to prove something (prove that the series converges, etc). My analysis professor *loved* improper integrals, so the chapter on that was especially helpful. This text also includes Dirichlet's test for series, which is left out of Rudin. There are a few minor typos that are easily spotted, but for $20 these mistakes can be overlooked. Overall a great reference with many helpful solved problems: A+

Whilst as some reviewers have pointed out, the proofs offered are not always the most fundamental and rigorous available, this aspect of the text makes it surprisingly digestible. As an example, the last chapter (on complex variables) seems almost an afterthought. However the text in combination with the solved problems will have the reader performing integration using residues etc in no time. This area of calculus has many weighty tomes devoted to it, offering full and rigorous proofs of each theorem, but to gain a similar working facility to that described above one needs to trawl through almost an entire book, a much more time-consuming exercise. The book works similarly for many other areas of calculus. This I found very rewarding, as I was able to go from clueless to reasonably proficient at problem solving in a short time, which encouraged me to keep learning. The more advanced treatments are also much easier to digest once you have some familiarity and competence in the area. The only real let-down are the typos, which seem to be more concentrated in some chapters rather than others, however they are generally easily spotted and accounted for. The book is therefore perfect for scientists and engineers, as with a minimum of fuss it teaches the reader everything they need to know to perform calculations, and makes a good introductory text for aspiring mathematicians. At this price, you'd be crazy to not have it lying around somewhere.

This book has great sentimental value for me! I love this book!!! It's got everything you need to get through multivariable calculus course and/or vector calculus & Fourier Series. It doesn't really go into any analysis stuff though; it's all concrete, so you can look at this to get the 'rough & ready' versions of theorems/proofs & then find the more rigorous & abstract versions somewhere else, or later on in your studies. It's also a good reference since it covers a lot more than what you might do in a course, & could be helpful for more than one course. It has so many solved problems & theorems that might not be done elsewhere. Get this if you're in maybe 2nd-year math & are going to carry on in math, science or engineering.

These books are a cheap suplement for courses or whatever you desire, and in general, especially the ones written by the late Murray Spiegel, are quite excellent. These books are certainly not intended for courses as they cannot go into extreme depth. However, if they are assigned, the student is assumed to have enough sense to venture elsewhere (the library maybe) and find supplemental books for learning in depth stuff. Thats all part of learning. This book (now in an updated version with some really weird title which makes it worth buying alone! ) is quite excellent. Don't go and get it expecting to learn a course from. But it is good for a) a supplement for a course. b) a supplement for another course... it has lots of solved problems which oftentimes help in other courses and c) to refresh yourself on things you may have forgotten. I'd also recommend you get Spiegel's Mathematical Handbook. It is absolutely indespensible. It is just a fantastic thing that everyone should have. When you are born, they should cut your umbilical cord and hand you a copy of Spiegel's Mathematical Handbook and you will have no problems ever in life. Get both. Get a whole lot of Schaums because they are so damn cheap and useful.