Aug 17




Createspace Pbk: $ 13.99  USD 

Kindle E-Book Version: $9.99 USD 




Some Reviews: 


 "Written by the hand of a master in a nice and pleasant style, it is highly recommended to all those interested in the study of differential calculus......In conclusion, a wonderful geometrical (coordinate-free) introduction to the subject."-M.Puta (Timişoara), Zentralblatt MATH 

"……the book consists of two parts….. the first exposes the differential calculus in Banach spaces, and the ordinary differential equations. The second part develops, again on open sets in Banach spaces, the calculus of exterior forms, their integration in the finite dimensional case, and applications to variational calculus and to the differential geometry a la Eli Cartan of curves and surfaces in Euclidean space. In short, it is an excellent modem textbook on advanced analysis put in a non-standard form."- I. Weisman,  Zentralblatt MATH 

"Cartan’s work provides an excellent text for an undergraduate course in advanced calculus, but at the same time, it furnishes the reader with an excellent foundation for global and nonlinear algebra." – Mathematical Review 

Brilliantly successful.” – Bulletin de l’Association des Professeurs de Mathematiques   


“The presentation is precise and detailed, the style lucid and almost conversational……clearly, an outstanding text and work of reference.” –Annales  

This one-of-a-kind text,now reissued after nearly a half a century, is the first half of the English translation of Cartan’s famous Cours de calcul différentiel, given by the author at the University of Paris at the height of the Bourbaki movement in mathematics education-which the author helped co-found and popularize in Europe!  


 The first half of the course text’s republication in an inexpensive edition finally makes readily available again the English translations of both long separated halves of Cartan’s famous 1965-6 analysis course at The University of Paris: The second half has been in print for over a decade as Differential Forms, published by Dover Books. Without the first half, it has been very difficult for readers of that second half to be prepared with the proper prerequisites as Cartan originally intended.


At the time of its publication, this presentation represented the height of rigor and abstract in a calculus course for talented students. It’s like no other text on calculus/analysis at any level you’ll find anywhere.


What distinguishes Cartan’s course presented in this text and its sequel (more on that later) is that it gives a careful and abstract treatment of differential and integral calculus on Banach spaces, instead of as customary in analysis, metric or topological spaces.


The main advantage of this completely general approach over abstract metric spaces is it allows for a unified theory of functions of one and several variables that doesn’t have to be redone. For example, there is a single definition of a derivative as a linear transformation between subspaces of a Banach space. The needed vector space structure is already present and doesn’t have to be added ad hoc as in the case of metric spaces. Special cases, such as partial derivatives and Taylor’s theorem, are derived as needed.


As the title indicates, this text focuses solely on the foundations of differential calculus on Banach spaces. The integral calculus is developed in the sequel.

Some topics covered in the text:

  • ·      A review of basic linear algebra and topology in abstract normed spaces
  • ·       Multilinear continuous functions via the exterior product  
  • ·      The Frechet derivative as a linear transformation 
  • ·      The General Mean Value And Inverse Function Theorems 
  • ·      Taylor’s formula and higher order derivatives ·      General existence and uniqueness theorems for linear homogeneous and nonhomogeneous ordinary differential equations
  • ·      Connection between solution spaces for partial differential equations and systems of ordinary differential equations
  • ……and much more!

The prerequisites for this text are a rigorous first

course in calculus using the ɛ-δ definitions of

convergence and limits (or equivalently, a course in

one variable advanced calculus or elementary

analysis), a careful course in linear algebra on

abstract vector spaces with norms and linear

transformations as well as fluency with matrix

computations and a basic course in differential

equations. A knowledge of the computational aspects

of multivariable calculus will also be needed for some

parts of the book. The basic definitions of topology

(metric and topological spaces, open and closed sets,

etc.) will be needed as well. 

A new, detailed preface has been added by the publisher to provide historical context and educational perspective on the text’s approach and structure as well as the author’s intentions. Further, a detailed bibliography has been added comparing and contrasting Cartan’s book(s) to the current standard analysis texts at this level. He also offers detailed suggestions for how it/they can be used by modern students as either main text(s) or as inexpensive supplements for a standard text like Rudin.   


With both texts now available at very affordable prices, the entire course can now be easily obtained and studied as it was originally intended for a new generation of mathematics students and teachers of analysis! This classic and its sequel can now be used and studied easily and should become standard analysis texts now for university students and teachers!









Table of Contents

Preview page 30-41

                              And Don’t Forget....

The main purpose of me printing this new edition

of DC is so that both volumes of Cartan’s famous

course can now be available readily and cheaply to

English-speaking students and teachers of

mathematics! As I said, the second half volume

has been available from Dover Books as

Differential Forms for over a decade.  The second

half is just as clear, beautifully written and

informative as the first. The text’s focus is on

differential forms as tools in calculus on Banach

spaces. The main use of forms is in path integrals

in finite dimensional Banach spaces. As the

Frechet derivative generalizes and unifies all the

special cases of differentiation in Euclidean

spaces, abstract path integrals over differential

forms unify all the integrals of calculus via Stokes’

Theorem. These methods are then applied to

problems in classical differential geometry and the

calculus of variations. The section on differential

geometry is particularly important as it uses Eli

Cartan’s method of moving frames-something

usually not presented at a level suitable to

undergraduates. Since both books are available so

cheaply now, there’s no reason not to have both!



(Ok, enough advertising for someone else.God knows Dover can afford their own..................) 

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