Cartan H. Differential Calculus

Kershaw Publishing Company, 1971. — 161 p. — ISBN: 0-395-12033-0.Differential Calculus provides an introduction to some of the most beautiful parts of classical analysis in a modern setting, where the reader is assumed to have some familiarity with the real and complex number fields and with linear algebra at a level which is by and large covered in sophomore level mathematics courses. Often in the course of studying this book one is impressed by the masterful hand of H. Gartan both in the general presentation of the subject matter and in the details of the proofs.
The book is divided into two chapters. The first develops the differential calculus in Banach spaces. After an introductory section providing the requisite background on Banach spaces, the derivative is defined, and proofs are given of the two basic theorems—the mean value theorem and the inverse function. The chapter proceeds with the introduction and study of higher order derivatives and a proof of Taylor's formula. It closes with a study of local MAXIMA and minima including both necessary and sufficient conditions for the existence of such minima.
The second chapter is devoted to differential equations. Existence and uniqueness theorems for ordinary differential equations are proved. Applications of this material to linear equations and to obtaining various properties of solutions of differential equations are then given. Finally the relation between partial differential equations of the first order and ordinary differential equations is discussed.
Differential Calculus could be used for a semester junior calculus course modernizing the classical advanced calculus of the junior year. A second way of using this book would be to follow its use with Gartan's companion volume, Differential Forms, for a full year course. This would be an analysis course having a geometric flavor, and providing an excellent background for further mathematical study particularly in such areas as the Theory of Lie Groups, Differential Geometry, Differentiable Manifolds, or Differential Topology.