This book evolved from a course at our university for beginning graduate stu­ dents in mathematics-particularly students who intended to specialize in ap­ plied mathematics. The content of the course made it attractive to other math­ ematics students and to graduate students from other disciplines such as en­ gineering, physics, and computer science. Since the course was designed for two semesters duration, many topics could be included and dealt with in de­ tail. Chapters 1 through 6 reflect roughly the actual nature of the course, as it was taught over a number of years. The content of the course was dictated by a syllabus governing our preliminary Ph. D. examinations in the subject of ap­ plied mathematics. That syllabus, in turn, expressed a consensus of the faculty members involved in the applied mathematics program within our department. The text in its present manifestation is my interpretation of that syllabus: my colleagues are blameless for whatever flaws are present and for any inadvertent deviations from the syllabus. The book contains two additional chapters having important material not included in the course: Chapter 8, on measure and integration, is for the ben­ efit of readers who want a concise presentation of that subject, and Chapter 7 contains some topics closely allied, but peripheral, to the principal thrust of the course. This arrangement of the material deserves some explanation.