Calculus Without Limits by John C. Sparks
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[edit] Review by Bruce R. Gilson (Rockville, MD United States)
An unusual approach This book is a very atypical introduction to calculus, which some people may like and others will hate. It clearly shows the orientation of its author, who is an engineer and not a mathematician.
Calculus can be presented in two approaches, one using infinitesimals (the way Leibniz thought of it) and one using limits (closer to Newton's picture, though he too would use infinitesimals when it suited him). Originally, either mode of thinking was applied intuitively, but mathematicians could not define the infinitesimal approach in a rigorous manner, and in the 19th century the subject was rigorously developed using limits. Eventually a rigorous approach was developed using infinitesimals by Robinson in the 20th century, but most mathematicians came to prefer the limit approach, because it was the first one developed with a degree of rigor to suit the mathematical community. But scientists and engineers developed an intuitive way of thinking about calculus, based primarily on the infinitesimal approach. So a typical scientist would come to calculus by means of a freshman course that presented the subject using limits, with only a slight reduction of the rigor to a level appropriate to college freshmen, and only in his physics and chemistry courses be presented with calculus as a problem-solving tool using a more intuitive approach based on infinitesimals. The author of this book seems to feel that it is better to present an intuitive approach to calculus, forgetting about rigor (he was once, he states, exposed to calculus in a course that was overly rigorous, even beyond the level usually met in freshman calculus classes!) and it is this which he does in this book.
The author starts off with an intuitive proof of the Pythagorean theorem, which works quite well. He does follow this with an equally intuitive proof of a geometric fallacy, which should clearly demonstrate the limits of intuitive thinking! But it seems that despite this, he intends to go on purely intuitively. The resulting book is one that mathematicians will find infuriating, but scientists and engineers will see as reflecting the way calculus is actually used in their disciplines. And so, I think, what you will think of this book will depend on your own orientation.
Before concluding I should say that one serious defect of this book, I would say, is the total absence of an index. In addition, I think that the title is somewhat misleading; the author uses the limit concept more than he implies. A better title would be "Calculus by an intuitive approach." It is the intuitive approach, even more than the fact that he uses infinitesimals rather than limits for many of his arguments, that defines the uniqueness of this book.
Rating: 10
[edit] Review E. Jennings Taylor
Perspective from a Practicing Technologist I have a PhD in Materials Science and have been "practicing" scientific and engineering R&D for about 25 years. As a consequence, I'm fairly astute regarding calculus matters.
I was intrigued about this book because of its' approach to calculus with minimal emphasis on the limit theorem and because my son is entering high school mathematics.
I found the book to be very well laid out and the progression of subject matter to be very logical. Most importantly, the book provided real world examples for "using" the mathematical concepts introduced. One of my favorite was the example of transporting a ladder around a corner. While one would not use calculus in such a situation, these sort of examples, really analogies, provide an excellent perspective for the reader.
Finally, the book read almost like a novel. It was enjoyable and left one to anticipate the next chapter.
I think this book is an enjoyable read and suggest it for 1) someone like myself with the challenge of helping bring relevance to their young students, 2) a student currently being introduced to calculus and 3) a former student seeking a refresher.
Rating: 10
[edit] Review by Karl D. Hahn
Review from the author of www.karlscalculus.org
Only someone who has fallen in love with calculus can know its poetry. Just as any musician can tell you how music is far deeper than mere spots on a score sheet, John C. Sparks leads you to understand how calculus is far deeper than mere cryptic symbols on a blackboard.
The sad truth is that too many calculus classes and texts teach students only the rules by which you manipulate the cryptic symbols. But those cryptic symbols, like all symbols, stand for something greater. Spots on a score sheet stand for melodic sounds, and mathematical symbols stand for beautiful ideas. It is these ideas that Calculus Without Limits teaches.
Without grasping the ideas and concepts of calculus, a student is left only with the grind of applying pencil lead to paper in this or that prescribed manner and has thoroughly missed the point. So it's hardly a surprise that so many calculus students are frustrated. But for a student who reads John C. Sparks' explanations of how the symbols assemble themselves into something meaningful, the symbology becomes just a tool -- as it should have been all along. The real knowledge and beauty behind the symbols glimmer through. And for a student who makes the special effort (and all mathematics learning requires special effort, even from those who love it) to follow that light where it leads -- that student too might very well fall in love and know the poetry.
Rating: 10
[edit] Review by John Gabriel
While Sparks has tried to explain basic calculus without limits, he really adds nothing to existing books. Calculus can truly be used without limits. In fact, limits did not exist when calculus was invented. Prof. Doverman in his free book gives some nice examples that are 100% rigorous and yet include no limits at all. Although Doverman does not have all the answers, his book is worth reading. It is a step in the right direction.
Sparks' book is wishy, lacks rigour and is somewhat disappointing. Poetry has no place in mathematics, only cold, hard facts. Unfortunately, Sparks' book falls short in many places because it indirectly uses the exact methods used in the process of finding limits. It is still 'Calculus with limits', not 'calculus without limits'.
John Gabriel's web pages: http://mathphile.blogspot.com http://www.geocities.com/john_gabriel
