This book was initially based on a short course of twenty lectures, given to second year students at University College Dublin during the autumn of 1992. Later, two chapters on integration theory were added to improve the balance between differential and integral calculus. The students had completed a one year course on differential and integral calculus for real valued functions of one real variable-this is the prerequisite for reading this book-and this course was designed as an introduction to the calculus of several variables. My initial motivation for writing this book was to provide my own stu dents with a friendly set of notes that they could read in their entirety. As the book took shape, I realized that I was addressing the subject in a manner somewhat different to the standard texts on several variable cal culus. It is difficult to explain precisely why this occurred. Nevertheless, an attempted explanation may also help you, the reader, in your approach and I will try to give a partial one.