Original, rigorous, and lively, this text offers a concise approach to classical and contemporary topics in differential calculus. Based on courses conducted by the author at the Universite Pierre et Marie Curie, it encourages readers to pursue the subject in greater depth. The calculus is presented in a Banach space setting, covering: Vector fields One-parameter groups of diffeomorphisms The Morse-Palais lemma Differentiable submanifolds The treatment also examines applications to differential equations and the calculus of variables. For upper-level undergraduates and graduate students of analysis. AUTHOR: A. Avez was a noted French mathematician who received his PhD in 1954 from the Universite de Paris. Thereafter he taught at the Universite Pierre et Marie Curie, Paris, which was founded in 1971. His work Differential Calculus was reprinted 26 times between 1983 and 1995.