This book is the first volume of a two-volume text on mathematics for engineering students in universities and polytechnics, for use in the second and subsequent years of a first degree course. The text is primadly designed to assist engineedng undergraduates and their teachers, but we hope it may also prove of value to students of other disciplines that employ mathematics as a tool, to mathematicians who are interested in applications of their subject, and as a reference book for practising engineers and others. Volume J covers mathematical topics which most engineedng students are required to study; Volume 2 deals with more advanced subjects which are often available as options in the later stages of an undergraduate course. The text is based on courses in mathematics given by the authors to the engineedng students of the University of Nottingham. These courses have evolved over the last sixteen years, and have been developed in close consultation with our fellow teachers in the engineering departments of the University. In preparing the text, we have kept in mind the constraints imposed by the normal three or four year undergraduate course, and we believe that the choice of matedal in the two volumes is realistic in that respect. For completeness, some topics are pursued a little further than an engineedng mathematics lecture course would normally take them, but all the material and examples should be within the grasp of a competent engineering undergraduate student.