NOW, THE CLASSIC TEXTBOOK-
OLD SCHOOL ADVANCED CALCULUS
BY WILLIAM BENJAMIN FITE:
A LOST GEM ADVANCED
CALCULUS COURSE BOOK IS BACK
IN PRINT FOR PURCHASE AT A
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
“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)
About This Text
Specifically, Old School
Advanced Calculus is exactly what the
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 same for functions of
on Euclidean spaces.
In fact, this second
semester was considered the most
important one since this was 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
comprised virtually all the main topics
needed by both mathematics and
physical science majors using a
uniform terminology and level of
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
As a result,pure
mathematics students get exposed
to important physical and geometric
applications along with mathematical
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
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
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
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
Most noteworthy, a lengthy new
preface has been added 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:
• 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 just
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.
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!
Also For Sale At BCS:
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.
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
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