## NOW, **THE CLASSIC TEXTBOOK-**

*OLD SCHOOL** **ADVANCED CALCULUS *** **

*OLD SCHOOL*

**BY WILLIAM BENJAMIN FITE:**

** A LOST GEM **ADVANCED

## CALCULUS COURSE BOOK IS BACK

## IN PRINT FOR PURCHASE AT A

## WONDERFULLY AFFORDABLE

## PRICE!

**MOREOVER, THE NEW EDITION CONTAINS A LENGTHY **HISTORICAL PREFACE ON THE HISTORY OF ADVANCED

## CALCULUS CLASSES & A SHORT, WELL-CHOSEN SUPPLEMENTARY

## BIBLIOGRAPHY ON CLASSICAL ADVANCED CALCULUS TEXTS FOR THE PURPOSE OF DIRECTING THE USER IN FURTHER STUDY!

**ALSO,BOTH ADDITIONS ARE **

**AUTHORITATIVELY & LUCIDLY **

**WRITTEN BY **BCS FOUNDER AND

## CEO KARO MAESTRO AKA THE

## MATHEMAGICIAN!

** A Review:**

** “As the author states in his preface;**

*….this book has been written to supply an introductory course in **mathematical analysis for those who are looking forward to **specializing in mathematics. *

## To the reviewer it seems that the stated purpose has been attained in a satisfactory manner. Assuming

## familiarity with the working rules and simpler applications of the calculus, the first part of the book discusses the

## meanings of the fundamental concepts of the derivative and integral together with proofs of the fundamental theorems. The selection of subject matter in the book conforms with that which has now become somewhat standardized for a course in advanced calculus……….. There is sufficient material in the book for a year’s work. The style is clear, and the treatment is perhaps as rigorous as

## is possible for the average student at this stage of his development.”

## – L.L. Smail, *The American Mathematical Monthly*, Vol. 45, No. 7 (Aug. – Sep.,**1938)**

*The American Mathematical Monthly*, Vol. 45, No. 7 (Aug. – Sep.,

** About This Text **

** Specifically, *** Old School *

*Old School*

*Advanced **Calculus *** is ****exactly what the **

*Advanced*

**title says it is: A full ****year **course in

## advanced calculus the way it was

## offered at all American universities

## for most of the 20th century.

** In particular, Fite’s book has been **

**reissued in the ****hope that it brings the **

**advanced ****calculus course as it was **

**taught for ****nearly half a century **

**back ****into the ****consciousness of **

**mathematics and ****physical **

**science ****students and ****educators with **

**this comprehensive, ****long-out-of-print text. **

** In truth, the traditional American university ****advanced calculus syllabus slowly ****emerged over decades and reached ****more or less final form in the early ****1930’s. Subsequently, it was a standard mandatory ****undergraduate sequence until the early ****1970’s. Eventually,for various reasons,the course**** was then sundered into various “analysis for ****mathematicians” and “analysis ****for physical science students” courses. Therefore, mathematics and science students afterwards had very different post-calculus requirements then previous generations. **** **

** Thus,the classical AC syllabus was **

**composed of a**** first semester of single **

**variable ****calculus “done right” with **

**a ****rigorous presentation of the **

**various ****limits of functions on the ****real **

**line and their important ****applications.**

**Furthermore, the second semester **

**did ****the s****ame for ****functions of **

**several ****variables **

**on ****Euclidean spaces.**** **

**In fact, this ****second **

**semester was considered ****the most **

**important one since this w****as where **

**both mathematics and ****physical science **

**majors would ****learn functions of **

**several variables ****and its applications. **

**Indeed, there was not yet a standard **

**course ****later ****students ****would recognize **

**as a “Calculus 3/Multivariable **

**Calculus” ****course. ****Therefore, you ****had **

**to take ****advanced calculus to begin to **

**learn ****about functions in ****Rn. **

**Nevertheless****, ****all students in the **

**hard ****sciences ****needed to learn calculus **

**of ****several variables in order to **

**progress ****beyond the beginning level. **

**For this ****reason, the performance of **

**students in ****the classical AC course was **

**critical for ****future performance. **

** Specifically, the author does a **

**terrific job of ****combining a careful **

**“epsilon-delta” ****presentation of **

**calculus of ****one and ****several variables **

**with many ****applications to **

**classical ****physics, ****differential **

**equations and geometry. **

** The main advantage of ****the original **

**advanced calculus course,exemplified **

**by Fite, is a ****unified presentation of **

**mathematical analysis of one and **

**several ****variables. **

** Consequently,it **

**comprised ****virtually all the main topics **

**needed by ****both mathematics and **

**physical science ****majors using a **

**uniform ****terminology and level of **

**rigor. **

** Therefore, even if each semester**

**was taught by a different faculty **

**member, they were both bound by**

**more or less the same syllabus. **

**Consequently ,this format greatly**

**restricted dramatic divergence in their**

**respective course content. **

**Furthermore, when the **

**subject ****selection, notation **

**and rigor level is consistent ****like it is **

**with books like Fine’s, then **

**throughout the entire course a balance **

**that ****benefits all involved is achieved **

**and maintained.**** For example, both **

**physics and mathematics students **

**learn the basic structure of classical **

**mechanics identically. **

** As a result,pure **

**mathematics ****students get exposed **

**to ****important ****physical and geometric **

**applications ****along with m****athematical **

**rigor.**

** ****On the other hand, physics and **

**engineering ****students ****get ****exposed to **

**pure ****mathematics and the **

**abstract ****minimalist ****deductive **

**skills it builds in**

**them that will be invaluable ****when they **

**begin research.**

** Meanwhile, Fite’ ****s book requires **

**only high ****school ****algebra and geometry **

**as well as ****a year- ****long basic (non-**

**rigorous) ****single **** variable calculus **

**course as ****prerequisites. So that a **

**course based on ****Fite will give both the **

**beginning ****mathematics major and the **

**serious ****physical/social science major a **

**thorough grounding in **

**classical ****analysis. **

**In addition, its’ many **

**applications will assist greatly**

**in preparation for further research in **

**real variables, mathematical**

**physics or additional fields which **

**require some analysis background**

**beyond calculus. **

## More importantly, because of the

## low prerequisites, Fite is quite

## versatile. Therefore, it can be used for

## a number of different courses, either

## a standard classical advanced

## calculus course, an honors calculus

## course for strong freshman or

## independent reading by students or

## professors in other sciences who need

## to learn advanced calculus beyond

## calculus.

** **

** Most noteworthy, a lengthy new **

**preface has been a****dded by Karo **

**Maestro explaining the history of the **

**advanced calculus course in America. **

**In fact, Fite’s book was one ****of the first **

**standard such texts. Therefore, this **

**preface provides appropriate context. **

** Also,he has ****added a recommended **

**reading section ****reviewing many of the **

**other standard ****classical analysis texts **

**for ****additional reading.**

** Some Topics In Fite’s ***Old School Advanced Calculus*:

*Old School Advanced Calculus*:

** • An unusual “semi-axiomatic” presentation of the real numbers via Dedekind cuts of rationals **

** • limits and differentiability properties (derivatives, differentials, partial derivatives, etc.) of real valued functions of one and several variables including the Mean Value, Implicit And Inverse Function Theorems in R2 and R3 **

**• A careful presentation of the Riemann integral in one variable in terms of Darboux upper and lower sums and many standard applications of it, such as solids of revolution, areas under curves and trapezoidal approximation **

** • Techniques and formulas for indefinite integrals of all the standard functions of calculus**

** In addition, some important integrals that you usually don’t see in more modern analysis books, such as elliptic and Abelian integrals.**

**•Taylor’s formula and higher order derivatives with applications to approximation and differential equations **

**•Improper integrals, their convergence conditions and a detailed presentation of the gamma function with applications**

**•Multiple integrals, iterated integrals, line integrals and Green’s Theorem, change of variables and curvilinear coordinate systems as well as a number of physical applications **

**•An unusually comprehensive treatment of infinite series including unusual topics such as double series, quasi-uniform convergence and a detailed discussion of Weierstrass’ example of a continuous function with no derivative at any point in its domain, the theory and applications of power series, trigonometric series with emphasis on Fourier and orthogonal series and some of their applications to partial differential equations**

**•classical differential geometry of curves and surfaces in R2 and R3 with applications to mechanics **

**•Brief but broad introductions to the calculus of variations and functions of a complex variable**

**…and much more! **

** As said above, the prerequisites for a **

**course based on Fite are very minimal. **

**In fact, those prerequisites are j****ust **

**high school algebra and geometry **

**as well as a non-rigorous single **

**variable calculus course. Therefore,it’s **

**also ideal for self study. ****Further, no **

**experience ****with rigorous mathematics **

**or proof ****techniques is necessary.**

**It’s quite important to understand why this is true in a historical context. When Fite wrote this book, it would be 20 years before independent “methods of proof” type courses were introduced. Previously, the advanced calculus course was where math students-both pure and applied-began to transition from plug and chug type courses to where rigorous proof was the order of the day.**

## ** As a result,while being certainly more careful then a current calculus course, Fite’s book is somewhat less so then a modern real analysis course such as one based on Apostol or Rudin would be. Hence, it would therefore be more accessible to a larger number of undergraduates then those books without sacrificing any rigor.**

** Which brings us to the major reason we’ve decided to reissue this text. The reforms that made the old AC course obsolete were done in response to the strengthening of mathematics requirements in 1960’s as a result of the Space Age. Subsequently,today’s undergrads have regressed at least to the pre-World War II preparation level. Consequently, the audience for the far more abstract analysis courses is now limited to the very strongest pure mathematics students. **

** ***Therefore, it is the belief of the publishers that a return to the original advanced calculus sequence-with appropriate modifications, of course-should be strongly considered.*

*Therefore, it is the belief of the publishers that a return to the original advanced calculus sequence-with appropriate modifications, of course-should be strongly considered.*

## As a result,the republication of this book at such an inexpensive price certainly encourages this rethinking of today’s course syllabus. At any rate, buy this book to help fuel this movement in academia!

**Above all, with this text now available and affordable,this course in advanced calculus can studied as originally intended for a new generation of mathematics & science students and teachers of analysis! Accordingly, it should become a standard text now for university students and teachers, either as a main text or as a supplement for self study!**

** ORDER NOW!**

** TEXT PREVIEW:**

# Also For Sale At BCS:

* Vector Analysis: A Supplement to Old School Advanced Calculus *

*Vector Analysis: A Supplement to Old School Advanced Calculus*

**Authored by Keith S. Miller With Additions by Karo Maestro **

## ** Certainly,this brief and inexpensive ****text intends to provide a modern introduction to vector analysis in R2 and R3. **

** Furthermore, it complements the very rigorous and wonderfully written presentation of classical analysis in our companion book, *** Old School Advanced **Calculus* by William Benjamin Fite.

*Old School Advanced*

*Calculus*by William Benjamin Fite.

**Granted that Fite’s book is otherwise **

**very ****comprehensive, his presentation **

**of ****functions of several variables is **

**rather ****archaic and purely analytic. **

**Nevertheless, vector ****analysis is now a **

**necessary ****part of ****the ****mathematical **

**training ****of both ****mathematics and **

**physical ****science ****students.Therefore, **

**the absence of ****vector valued calculus **

**in low ****dimensional Euclidean spaces **

**in Fite ****is a highly problematic void.**

** Simultaneously,the ****republishing of **

**this book by ****Miller with**** Fite **

**is ****specifically ****intended the rectify this **

**for ****both ****groups of students. **

**In ****addition, ****despite mostly classical **

**language, ****Miller carefully ****connects the **

**material to ****modern ****formulations so **

**he doesn’t ****alienate ****pure mathematics **

**majors. **

**For example,he ****carefully lays out **

**vector algebra in the first chapter **

**using the old 19th century “arrows” **

**language while simultaneously **

**detailing their algebraic structure as a **

**vector space over the real or complex **

**numbers. As a result, this keeps the **

**book’s intended audience **

**very general. Therefore, this invites not only mathematics majors,**

**but students of ****physics,engineering **