Calculus: A Modern Approach (Dover Books on Mathematics) by Karl Menger (PDF)

Description

One of the twentieth century’s most original mathematicians and thinkers, Karl Menger taught students of many backgrounds. In this, his radical revision of the traditional calculus text, he presents pure and applied calculus in a unified conceptual frame, offering a thorough understanding of theory as well as of the methodology underlying the use of calculus as a tool.The most outstanding feature of this text is the care with which it explains basic ideas, a feature that makes it equally suitable for beginners and experienced readers. The text begins with a “mini-calculus” which brings out the fundamental results without recourse to the notions of limit and continuity. The standard subject matter is then presented as a pure and unambiguous calculus of functions. The issues surrounding the applications of pure calculus to problems in the sciences are faced in a forthright manner by carefully analyzing the meaning of “variable quantity” and clarified by resuscitating Newton’s concept of fluents. The accompanying exercises are original, insightful and an integral part of the text. This Dover edition features a new Preface and Guide to Further Reading by Bert Schweizer and Abe Sklar.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I own this book. It dives straight into derivatives and continues on from there – a very steep dive indeed for anyone new to the subject. In my opinion it proceeds with excessively complex explanations of what should be easily comprehended concepts of calculus. It may, in my opinion, be suitable for the calculus-adept student or teacher but falls significantly short in any attempt to teach the subject to a novice. I do NOT recommend this book to anyone attempting to understand calculus on his or her own (i.e. beginners). If this method is considered a modern approach, then perhaps we should fall back to methods that have turned fewer newcomers off to the subject. Consider “Calculus Made Easy” by Silvanus P. Thompson and “Teach Yourself Calculus” by Hugh Neil instead.

⭐Disclaim: I’ve studied calculus several yeas ago.First, this book is not really “advanced” in contents, covering the standard topics of any undergraduate course in calculus. What differentiates it from other texts are several non common ideas in the use of the symbols that arguably result in a more consistent treatment if the subjects.Specifically interesting is the use of the normally implicit identity function (here named “j”) that upon judicious introduction brings a lot of clarity and consistency to the expressions. The author indicates this innovation is the equivalent (in calculus) to the ancient introduction of the symbol for zero in the arithmetic.Another topic where the author devotes several paragraphs is the meaning of the concept of “variable”, making explicit its different contexts of application. This is the most common pattern of the book: making explicit (and extending) several ideas that are usually taken for granted without enough reflection.The final result could be confusing for anybody used to traditional calculus books, but deserves attention from educators, math writers and mathematicians in general.

⭐If you are aiming to learn Calculus softly and thoroughly, then choose another book: Karl Menger’s is far probably a manifest on teaching calculus rather than a textbook. If you are trying to find out why can’t you understand more clearly the underlying concepts the “soft book” intends to clarify, then go to “Calculus: a Modern Approach”. If you teach Calculus and you’re aware that your students face difficulties, read Menger’s works. They are not conventional. As a matter of fact, Menger remakes the whole world in a few more days than the previous Creator, but this time in a little bit more consistent fashion, or at least guided by some criticism. Menger is either loved or hated. His detractors once managed to insert at the front matter in one of his works, disguised as a print error: Karl Menger, University of Dame Notre. Since to him everything should be done in a different and particular way, Notre Dame naturally should be written Dame Notre. Foes are foes, but “Calculus: a Modern Approach” should be a mandatory text for calculus teachers and be always kept in unending reeditions.

⭐An excellent book by the son of the founder of the Austrian School of Economics (Carl Menger.) Calculus in general needs to be approached from this angle, not the objectivist set theory way.

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