Some graphical conventions are adopted: definitions are displayed over grey backgrounds, while statements appear on blue; examples are marked with a blue vertical bar at the side; exercises with solutions are boxed (e.g., 12. ). This second edition is enriched by two appendices, devoted to differential and integral calculus, respectively. Therein, the interested reader may find the rigorous explanation of many results that are merely stated without proof in the previous chapters, together with useful additional material. We completely omitted the proofs whose technical aspects prevail over the fundamental notions and ideas. These may be found in other, more detailed, texts, some of which are explicitly suggested to deepen relevant topics. All figures were created with MATLABTM and edited using the freely-available package psfrag. This volume originates from a textbook written in Italian, itself an expanded version of the lecture courses on Calculus we have taught over the years at the Politecnico di Torino. We owe much to many authors who wrote books on the subject: A. Bacciotti and F. Ricci, C. Pagani and S. Salsa, G. Gilardi to name a few. We have also found enduring inspiration in the Anglo-Saxon-flavoured books by T. Apostol and J. Stewart. Special thanks are due to Dr. Simon Chiossi, for the careful and effective work of translation. Finally, we wish to thank Francesca Bonadei – Executive Editor, Mathematics and Statistics, Springer Italia – for her encouragement and support in the preparation of this textbook.