The Calculus 7

This is one of my favorite math textbooks. I'd say it's the best of its kind, that is, intended for a standard intro to calculus course.When you look at reviews of mathematics textbooks, you see that reviewers tend to break them down into two groups: the less rigorous texts more focused on intuitive exposition and applications, and the more rigorous texts focused on proofs (e.g. Spivak). Leithold is one of those rare few that should satisfy readers looking for either one. He gives you both a sound proof-based theoretical basis for "The Calculus" as well as a vast amount of practice working through problems.The result is a massive book, but well worth it. At each stage, he builds up the theory of calculus from the basic principles presented in Chapter 1, as you would expect from an analysis text. At the same time, he also provides lots of illustrations and exercises to build intuition. He provides enough explanation and worked examples that you are rarely expected to "figure things out for yourself" as some authors would have you do.TC7 is pretty diligent about assuring that you have the algebraic manipulation techniques at your disposal to work the problems, and if you don't, there is a nice 11-part appendix at the end of the book to bring you up to speed. (You still shouldn't expect to read this book without first learning precalculus, but the appendix is nice if you need a refresher or missed a couple of topics when you learned it the first time.) When I picked up this book, I hadn't done serious math in years and I was concerned that, although this wasn't my first look at calculus, I might have to go back and use a precalculus text to get back up to speed with solving equations; thanks to the appendix, I found that everything I needed was right here in TC7.TC7 is also crammed full of applications. There are entire sections of the book devoted to applications, including all of Chapter 6 which is mainly devoted to physics. There are so many exercises in this book that the instructor's solutions manual is nearly 1000 pages long. You couldn't possibly look through this book and then say that you don't know what math is good for.Despite the consistently rigorous presentation of calculus, this is not a text that focuses on teaching students to write their own proofs. TC7 is mainly focused on how to read proofs and how to solve calculus problems, not teaching original mathematical thinking. However, the problem sets do ask for some proofs, especially toward the end of the review questions for each chapter. For students to really tackle those questions, I think they would need some supplemental preparation (which I have myself, but a freshman in college probably would not).

In college we were using the TCWAG, the predecessor of TC7. TCWAG was like the bible of calculus classes in my university and I learned a lot from it. After more than a decade, I am going back to graduate school and I need to relearn my calculus and I decided to use TC7 instead of the TCWAG. I'll just come back to update this review once I am deep into this textbook.

THIS IS THE BEST CALCULUS TEXT BOOK I HAVE EVER USED AND IS THE BOOK IS LOOKED ALMOST AS NEW.

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I'm a junior math major at a college in the US and I'm using this book to learn what I should have learned in my earlier calculus classes. Most calc classes these days seriously skimp on the proofs - they either relegate them to an appendix of proofs, which is both discouraging and inconvenient for a new student, or they simply don't include them at all. Leithold does neither in this book.All of Leithold's proofs are clear, concise and rigorous. While there isn't any mention of metric spaces or the like - this isn't an analysis book, nor should it be - anyone who is looking to solidify their understanding of (lower-level) calculus should definitely pick up this book.A word of warning: I'm commenting on the third edition of this book (from 1976). While I would hope that they wouldn't change such a great text over the years, it's possible that the newer editions didn't retain the rigorous and clear nature of the earlier texts. I have no idea if it's different or not. If you want to be sure, go for one of the earlier editions - or see if you can snag a look at the new one from a friend or in a library.EDIT: I originally had an example theorem from the book here, but I can no longer find this exact statement in the text, so I've removed the reference to it from the review.This is the kind of theorem which wouldn't even be mentioned in most calculus texts. After all, the proofs that it is useful in are being omitted anyway, so why bother? But in THIS text, it is given explicitly and a full proof accompanies it. It's one of those theorems that you know intuitively to be true, but often hesitate to use in a proof on a test as you don't have a rigorous proof of it to back you up. But this is the kind of complete, rigorous approach you will find in Leithold. I imagine that if one studied Leithold's book extensively and then took a beginning Real Analysis class, one would still score a 50% on the final exam, even if one had no knowledge of metric spaces and the like. That's how good this book is. As another example, he proves all of the Riemann integration theorems from the definition, as opposed to just using geometric arguments.BUY IT! You will not be disappointed. :)[if this helps you to trust my opinion, I just received an A in my analysis class]P. S. This book (in the older edition at least) covers everything up until the theorems of Stokes & Green, etc, but does not go over those. It does go over line integrals though. However, it's not really a drawback - if you learn everything in this book, you'll be able to tackle a book like "Advanced Calculus" by Kaplan, which does go over those things, just fine. And in my opinion, most calc books are doing a disservice by tackling those advanced theorems while ignoring the proofs of the earlier theorems which prep you for more advanced material. Leithold's is the right approach.

I've been using my old edition of Leithold's Calculus since 1976 when I was a college freshman, and I've only recently updated to the latest edition. This latest edition holds up just as well as my old copy did 30 years ago. If you are attending a big university, you are about to learn that teaching undergraduates is far down on the agenda of many professors. Leithold's text can be your life line in such a situation. All proofs are clearly derived, there are many worked examples, and it will be useful to you through all three semesters of college Calculus. So even if this is not your assigned textbook, you might want to get a copy. Even twenty five years after I graduated, when I have a calculus question I can usually read the explanation in Leithold and get what I need. A good companion to this book is the Schaum's outline of Calculus. It too will take you through all three semesters of Calculus with plenty of solved problems.