Introduction to Calculus and Classical Analysis
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Beschreibung
This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This second edition includes corrections as well as some additional material.
Some features of the text:
* The text is completely self-contained and starts with the real number axioms;
* the integral is defined as the area under the graph, while the area is defined for every subset of the plane;
* there is a heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero;
* there are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more;
* traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals;
* there are 366 problems.
About the first edition:
This is a very intriguing, decidedly unusual, and very satisfying treatment of calculus and introductory analysis. It's full of quirky little approaches to standard topics that make one wonder over and over again, "Why is it never done like this?"
John Allen Paulos, author of Innumeracy and A Mathematician Reads the Newspaper
Involving rigorous analysis, computational dexterity, and a breadth of applications, this book is ideal for undergraduate majors. For this second edition, the author has corrected errors, rewritten large portions of the text, and has introduced new topics.
This is the second edition of an undergraduate one-variable analysis text. Apart from correcting errors and rewriting several sections, material has been added, notably in Chapter 1 and Chapter 4. A noteworthy addition is a re- variable computation of the radius of convergence of the Bernoulli series using the root test (Chapter 5). What follows is the preface from the ?rst edition. For undergraduate students, the transition from calculus to analysis is often disorienting and mysterious. What happened to the beautiful calculus formulas?Wheredid -? andopensetscomefrom?Itisnotuntillaterthatone integrates these seemingly distinct points of view. When teaching “advanced calculus”, I always had a di?cult time answering these questions. Now,everymathematicianknowsthatanalysisarosenaturallyintheni- teenthcenturyoutofthecalculusoftheprevioustwocenturies.Believingthat it was possible to write a book re?ecting, explicitly, this organic growth, I set outtodoso. I chose several of the jewels of classical eighteenth and nineteenth century analysisandinsertedthemattheendofthebook,insertedtheaxiomsforreals at the beginning, and ?lled in the middle with (and only with) the material necessaryforclarityandlogical completeness.Intheprocess,everylittle piece of one-variable calculus assumed its proper place, and theory and application were interwoven throughout.
Interesting applications, many of which are not usually found in advanced calculus books
Class tested and contains a wide variety of challenging exercises
this text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. The book contains many remarkable features, including:
- complete avoidance of /epsilon-/delta arguments by instead using sequences
- definition of the integral as the area under the graph, while area is defined for EVERY subset of the plane
- complete avoidance of complex numbers
- heavy emphasis on computational problems
- applications from many parts of analysis
- 344 problems with solutions in the back of the book
- interesting applications, many of which are not usually found in advanced calculus books.
For this second edition, the author has corrected errors and rewritten large portions of the text. In addition, the author has introduced new topics, such as a combinatorial proof that the radius of convergence of the Bernoulli series is 2p.
Autor*in
Omar Hijab
Themen in »Introduction to Calculus and Classical Analysis«
calculus convergence differential equation integral integration real number