Calculus (Calculus 4th edition by Michael Spivak)

Excelente texto para estudiar matemáticas en primer año de pregrado.

The prose in this book is perhaps the best writing on a technical subject I have ever seen. Calculus--a dry, boring, and mechanical subject for most other authors--is here presented as a beautiful and interesting intellectual achievement worthy of study on its own right. Even if you aren't a math for math's sake sort of person, it is always worthy of attention when an expert like Spivak explains what their world is like with such clarity and passion. And if any book may convince you that the people who say "math is beautiful" aren't nuts, it's this one.Spivak manages to deliver both an intuitive picture of a concept and the full mathematical rigor in a brilliant and playful style. He will often give a provisional definition of a tough concept to aid understanding first, but importantly and in contrast to more "accessible" math books, he signals very clearly that he is being intentionally imprecise. He then moves towards rigor by explaining exactly the way in which he has been imprecise, clearly driving the motivation for a more rigorous definition. The overall effect is that you rarely feel very lost and when he ultimately gives you the full picture, it often feels like an inevitability. A favorite example of this sort of style is at the start of Chapter 20: "The irrationality of e was so easy to prove that in this optional chapter we will attempt a more difficult feat, and prove that the number e is not merely irrational, but actually much worse. Just how a number might be even worse than irrational is suggested by a slight rewording of definitions..."Another impressive aspect of the book is the layout, where every relevant figure is only a glance away in the margin or directly inline with the text. It is the same style used in the Feynman lectures and Edward Tufte's books, and it is executed at its highest level here. Clear care went into the placement of each symbol in each equation and each figure.The exercises are quite hard, but there is a full solutions manual available for self-study (how I am working through the book). I will admit that I needed to bail out of this book at the very beginning, never having been exposed to doing proofs at this level before (formulaic high school geometry "proofs" don't count for much here). I used Velleman's "How To Prove It" and the first few chapters of Apostol's Calculus Volume I to get up to speed. Both these books are also recommended, and Apostol, in particular, gives an excellent and rigorous but more gentle on-ramp for the sort of thinking asked of you in Spivak Part I. In the long run, however, I think Spivak edges out Apostol for self-study because of the solutions manual.I picked up this book when I found that after 3 years of doing calculus in high school and college, I had forgotten most of it within a few years. I realized that while I could do the mechanics, I never really understood calculus in the first place. This book is probably a bit of overkill for just patching understanding, but I now have a much deeper appreciation and understanding of the mathematical way of thinking. It's not an easy book, but it is a wonderful one that will pay back dividends for hard work.But you don't have to do all the hard work just to appreciate what Spivak has done here. If you have an interest in good writing, this book is worth a look even if you aren't interested in learning the subject. I take special pleasure in reading great writing on any topic, and this book is up there with the best writing anywhere.

The Calculus fourth edition by Spivak sat on my desk for a while. When I first purchased it, I tried some of the beginning problems, then I felt that they were a little more rigorous and abstract, meaning more mastery of mathematics for me to do before I can do the exercises. So, there it remained for a couple of years. Until suddenly a few weeks ago, when I was working in Lebesgue Integration, that I needed to bridge the gap from Calculus to Analysis to Integration and Measure theory. I found a chapter devoted to Riemann-Darboux Integration in Spivak's "Calculus," with a picture in particular of inf and sup of a function in a particular interval. Nowhere have I found an explanation for Lim Sup and Lim Inf that satisfied definitively what I need to grasp for advanced mathematics. I began to understand Lim Sup and Lim Inf by inferring that they are the intersection of the union and the union of the intersection respectively of arbitrary sets where the puzzling thing is how to write the notation correctly to discuss these two concepts without resorting to pictures to describe what I am talking about? I must say the section on Riemann integration is superior to many other texts, and this is where I found my understanding of Lim Sup and Lim Inf increased, Spivak did not make the connection explicitly for me. But it was with most mathematical books, what is stated in print, I have to make plausible inferences or interpretations to squeeze more out of what is symbolically stated. It is the usual case that when looking for Lim Sup and Lim Inf, I would assume to look in introductory analysis textbooks and not in a Calculus textbook like Spivak's. Thus, he does present concepts further along the journey of a mathematician's training, addressed to so fundamental ideas as to enlighten them. Using highly developed concepts to redirect our attention to the fundamental ideas we started out with that were unanswered or seemed unsatisfying, is the approach found here. It gives the vague ideas more definitive and detailed accounts of what is going on in the assumptions being made when these ideas were first introduced. For Example, the supremum of a sequence is applied to functions and also the infemum of a sequence applied to functions here, is used to clarify the Riemann Sum both Upper and Lower Sums beautifully.Now, I would say that I am trying now to comprehend the notations used in Measure theory which is written in Set theoretic notation, with a lot of Lim Inf and Lim Sup, in other words Spivak provided me with an answer to my vexing fundamental question as a key to the higher and more advanced mathematics of Integration and Measure theory. I am thankful and grateful for this key.