Free Ebook Basic Calculus: From Archimedes to Newton to its Role in Science (Textbooks in Mathematical Sciences), by Alexander J. Hahn
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Basic Calculus: From Archimedes to Newton to its Role in Science (Textbooks in Mathematical Sciences), by Alexander J. Hahn
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This introductory calculus text was developed by the author through his teaching of an honors calculus course at Notre Dame. The book develops calculus, as well as the necessary trigonometry and analytic geometry, from witin the relevant historical context, and yet it is not a textbook in the history of mathematics as such. The notation is modern, and the material is selected to cover the basics of the subject. Special emphasis is placed on pedagogy throughout. Whhile emphasizing the broad applications of the subject, emphasis is placed on the mathematical content of the subject.
- Sales Rank: #1275183 in Books
- Published on: 1998-07-17
- Original language: English
- Number of items: 1
- Dimensions: 1.24" h x 8.56" w x 10.05" l,
- Binding: Hardcover
- 545 pages
Most helpful customer reviews
10 of 12 people found the following review helpful.
Finally, a calculus text for the mature student!
By Ken Christenson
I've been wanting to learn calculus for several years now, but at 50 years old I need reasons to stuff even one new idea into an already crowded and reasonably well balanced head. I keep getting distracted looking for context: answers to the questions which always pop up in an older brain trying to make things FIT! And in the standard texts and even books for laymen I haven't been getting my questions satisfactorily answered. Until now.
Thank you Alexander J. Hahn for CONTEXT, CONTEXT, CONTEXT.
13 of 14 people found the following review helpful.
This book succeeds in its important goals.
By A Customer
If you want a sound understanding and appreciation of basic calculus, have a look at Alexander Hahn's "Basic Calculus". An author presenting calculus could seek to impart (A) Proficiency in its techniques, (B) Justification of its methods, (C) Understanding of its principles or (D) Appreciation of its significance in the history of mathematics and science. Many pre-engineering calculus texts do a good job of drilling proficiency and building an appreciation of calculus by showing its utility in scientific applications. It is, however, easy to walk away from a pre-engineering calculus course without an intuitive grasp of the subject, or any sense of why calculus is important in the history of thought. There are also many mathematically oriented calculus texts that present the logical structure of calculus very well. These books stress precise concepts and proofs that were actually developed after calculus had revolutionized the physical sciences. Since the precise justification was developed after the fact, it is clear that one can understand the concepts without understanding the justification. Unfortunately many of formal calculus texts also show that it is possible to understand the justification (follow the proofs) without an understanding of the subject matter. History of mathematics books can convey an appreciation of why calculus was a great intellectual achievement, but they often tell the reader very little about the mathematics itself. There are many fewer books that attempt to convey an understanding of calculus and an appreciation of its significance in the history of mathematics and in science. Hahn's book addresses this neglected area.
Hahn's book has two parts, "From Archimedes to Newton" and "Calculus and the Sciences". The first part focuses on building an understanding of basic concepts and appreciation of calculus as a watershed in the history of mathematics. Hahn starts with Greek geometry and shows its application to astronomy. This serves as a review of some pre-calculus mathematics, as a historical preamble, and as an introduction to a set of problems that become a challenge, motivation, and later an important triumph of calculus. The first chapters paint a vivid "hands-on" picture of an intellectual world in which geometry was completely separate from the theory of numbers and algebra, and where astronomy and mechanics were thought to be literally as different as heaven and earth. These chapters are typical. They teach some mathematical ideas, show why they represent important advances inside mathematics, and illustrate major applications that make them important outside mathematics.
The narrative continues with advances of the late Hellenistic world: trigonometry, conic sections, and mechanics. Hahn then skips nearly all of the middle ages, which though beloved of historians, in the history of mathematics were truly dark ages. He takes up the story again with the early scientific revolution, Copernicus, Galileo, Kepler, Descartes. This discussion shows how the gap between geometry and algebra began to close with the introduction of analytic geometry, and introduces the work on the problems of tangents and quadrature that set the stage for calculus.
The story of the birth of calculus concludes with Leibniz, Newton, and Newton's masterwork, "The Principia", which closes the gap between our understanding of terrestrial and celestial motion. The problem with presenting ideas in the context of their history is that concepts are rarely clear at their inception. This can make the historical presentation murky and unnecessarily pedantic. Hahn avoids pedantry by developing what might be called historical reconstructions of important ideas. For example, Hahn reconstructs both Leibniz's and Newton's presentation of the Fundamental theorem of calculus. His narrative captures the distinctive differences in the two men's approach but the presentation does not do without the conceptual clarity of the notion of mathematical functions, in the interests of historical purity. There is also a good detailed account of Newton's derivation of Kepler's Laws. In this case Hahn's reconstruction goes as far as using Newton's actual diagrams and following closely the steps of Newton's derivation. Students of history of science will find this part of the book very enlightening, and may perhaps even forgive Hahn the practical expedient of using some modern notation.
The second part of the book covers the growth and maturation of calculus, and focuses on building an appreciation of the diverse applications of calculus in the sciences. Hahn starts by covering some of the ideas of nineteenth century calculus that made its concepts precise and its justification sound. However, he considers these ideas only in so far as they help to clarify concepts and aid understanding. You will, for example, find an exposition of why the idea of differentiability is a good analysis of the intuitive idea of the smoothness of a function, but you will not find a proof of L'Hospital's Rule. The remainder of the text discusses applications. Each chapter introduces some new mathematics and then exploits it along with the basic concepts of calculus to illuminate some application area. The applications are essentially the "interesting stuff" that is left out of most purely mathematical texts. The reader will be able to "see" the calculus in the ordinary furnishings of our technical society: Telescopes, Suspension Bridges, Nuclear Reactors, E. coli, Banking, Rockets and Guns.
Part Two builds on the foundation laid in Part One to not only give the reader understanding and appreciation of calculus, but does it in a way that is consistently interesting and enjoyable.
13 of 15 people found the following review helpful.
A really different calculus textbook.
By A Customer
Calculus has traditionally been the window through which college students first glimpse the intimate relationship between mathematics and the physical sciences. Unfortunately, few calculus textbooks give more than an inkling of this relationship, perhaps because of their author's desire to present the mathematics in full detail and the lack of a common background of scientific knowledge among our students. Alexander Hahn's Basic Calculus is a delightful exception; here, calculus is developed as the thread uniting the solution of historically important scientific problems by the likes of Archimedes, Ptolemy, Leibniz, and Newton. The scientific applications are not only of historical interest; they include potassium-argon dating in geology, modeling the forces and the motion of a bullet in a gun barrel, examples of mathematics in microeconomics, and even a thoughtful discussion of cosmology. The scientific ideas are developed in far more detail than is customary in mathematics textbooks. Each is presented as a story---the problem is placed in a historical context, including vignettes of the people involved; then the student is led through the basic science and the associated mathematical analysis, using technically accurate data.
The exercises at the end of each chapter are unusual too. After a few routine skill-building problems, the remainder pose small scientific problems to extend the discussion in the text. Students who work through any one of the many scientific topics Hahn treats in this way, working the associated exercises, will feel a justified sense of accomplishment. And in the process they will gain an accurate sense of how calculus developed and how it continues to play a key role in science and engineering. This fall my college has been debating how to structure the science component of our general education requirements, and if our entering students had enough mathematical preparation to profit from a study of calculus, I could not think of a better way to introduce them to science and mathematics than through a course from Hahn's book. This book would certainly be my choice of text for an honors course. I would expect it to convert some students who had planned on another major field of study to change to science or mathematics, and those who did not change their plans would still have a lasting experience of the attraction of the scientific enterprise.
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