The book differs from other introduction to mathematical methodsat this level in several important areas.First, it does not follow the usual presentation of a description ofthe theory followed by examples and exercises. Rather we use examplesto introduce the theory. This approach is not new; it goes backto the methods by which the scribes of Ancient Babylon learnedmathematics: by example problems and, to judge from the numbersof surviving cuneiform tablets, lots of them!Second, to help the reader digest the text, it is broken up intoquite short sections (often a page or so) followed by exercises. It maybe tempting to skip the exercises, (especially if one is used to doingonly a selection of “end-of-chapter” problems) on the grounds thatone can get through the book more quickly that way. This is true, inthe same sense that watching a film speeded up x8 will get to the endmore quickly, but it will be without much understanding of the plot.There are optional additional exercises at the end of each chapter;the ones in the text represent the minimum we think you need.Third, however, we have tried to avoid too many “plug-andchug”exercises, that is, exercises which you solve by following thetext but substituting some different numbers (“pattern matching”).These are useful to reinforce memory, but they are not very usefulto develop or test understanding. Rather, we have tried to make theexercises diagnostic in the sense that they do test understanding,that is, they test the ability to use what has been learned in a slightlydifferent context. An instructor can therefore use these to targetsupport for students.Finally, and related to the previous point, while we hope it is perfectly possible to use this book for self-study, it was not designed for that purpose. It is intended for use as a course text. In this regard it might be useful to say a little about the background to the writing of the book - especially if you are intrigued to know why there are so many authors. We would also like to thank contributions to various versions of the text from Paul Abel, Mike Dampier, Andrew King, and Tim Yeoman.About forty years ago, it was agreed that our conventionalpresentation of mathematical methods for our physics students -lectures, marked homework,and examinations - was not as effective as we might have hoped. So, instead of spending lecture time going through theory and exercises on the board, we produced a text as, in effect, the lecture notes, and refocused class time on weekly workshops and small group tutorials. Lectures were restricted to a weekly one-hour introduction to the topics for that week. We made the examinations harder, by requiring passes separately in the major topics(calculus of one variable, many variables, linear algebra, differential equations, vector calculus) and the pass rates soared. The initial text has been refined over the years (hence the number of authors) and this book is another, more outward-facing version, which we are pleased to have the opportunity to share with you.