A few years ago, I was one of the examiners for the first year mathematics examination for Oxford undergraduates studying physics or engineering. I wanted to set a question which started as follows: ‘A friend of yours has just been to his first lecture on the vector operators grad, div and curl. What would you tell him to help him appreciate their significance?’ One of the other examiners, a mathematician, commented that this was a silly question since the students merely had to write down the standard definitions, as these determined all their properties.

This exchange of views convinced me that mathematicians have a very different approach to mathematics from that of scientists. It also persuaded me that, in order to enable science students to feel fully at home with the mathematics they need, at first for their courses and later for applying it effectively to scientific problems, it was necessary to supplement the conventional mathematical presentations with one that stressed explanations which are more meaningful to scientists.

This book aims to do just that. It is based on tutorials that I have been giving in Oxford for several years to physics students as part of their first year mathematics course. For their weekly tutorials, students come either singly or in pairs to see their tutor.