** HENRI CARTAN’S NOTEWORTHY **

**TEXT ***DIFFERENTIAL CALCULUS ON NORMED SPACES: FINALLY AVAILABLE AGAIN AFTER NEARLY 50 YEARS!*

*DIFFERENTIAL CALCULUS ON NORMED SPACES: FINALLY AVAILABLE AGAIN AFTER NEARLY 50 YEARS!*

**Createspace Pbk: $ 13.99 USD **

**Kindle E-Book Version: $9.99 USD **

*ACCORDINGLY,THIS BOOK HAS *BEEN RETITLED WITH NEW ADDITIONS FOR A MODERN AUDIENCE!

*ACCORDINGLY,THIS BOOK HAS*BEEN RETITLED WITH NEW ADDITIONS FOR A MODERN AUDIENCE!

**BEST OF ALL, A BRAND NEW PREFACE & COMPREHENSIVE SUPPLEMENTARY BIBILIOGRAPHY HAS BEEN WRITTEN BY KARO MAESTRO AKA THE MATHEMAGICIAN! **

** 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 **

**About** **Cartan's Cours de calcul différentiel**

** Cartan's unique text has been reissued after nearly a half a century. It's the 1st 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. Bourbaki-the remarkable school of abstraction which the author helped co-found and popularize in Europe! **

** ****The 1st text's ****republication in a cheap edition finally makes readily available again the English translation of both long separated halves of Cartan’s famous 1965-6 analysis course at The University of Paris. The 2nd half has been in print for over a decade as Differential Forms, published by Dover Books. Without the 1st half, it has been very**

**difficult for readers of that 2nd half to have the proper prerequisites. Consequently,this publication finally makes the entire course available again 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
& its sequel that they present careful calculus on Banach spaces.**

** Cartan's**** approach over normed instead of metric spaces has the main advantage of**** a unified theory of functions of one and several variables. 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. It 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: **

- ·
**L****inear 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!*

** Prerequisites For The Course **

* The prerequisites for this text are: *

* 1)**A rigorous first **course in calculus/ elementary analysis using the ɛ-δ definitions of **convergence and limits on the real line,*

* 2)A careful course in linear algebra on **abstract vector spaces with norms and linear **transformations as well as fluency with matrix **computations and*

* 3) A basic computational course in differential **equations. Also: *

* *

* 4) 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. *

** New Features Of The 2017 Edition **

** 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!**

** ORDER NOW!**

# TEXT PREVIEW:

# Preface

**Table of Contents**

** 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 2nd half**

**been available from Dover Books as**

*Differential Forms.*The 2nd**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**

**paces, 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. 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..................) **

# How Do I Get It?

**Click here to buy pbk directly from Createspace.**

**Click here to buy the Kindle E-Book. **