This book originated from a one-semester course on introductory engineering mathematics taught at MIT over the past ten years primarily to first-year graduate students in engineering. While all students in my class have gone through standard calculus and ordinary differential equations in their undergraduate years, many still feel more awe than confidence and enthusiasm toward applied mathematics. Upon entering graduate school they need a quick and friendly exposure to the elementary techniques of partial differential equations for studying other advanced subjects and the existing literature, and for analyzing original problems. For them a popular first step is to take a course in advanced calculus, which is usually taught to large classes. To cater to a large audience with diverse backgrounds, an author or instructor tends to concentrate on mathematical principles and techniques. Applications to physics and engineering are often kept at an elementary level so that little effort is needed to set up the examples before, or interpret them after, finding the solutions. In some branches of engineering, students get further exposure to and practice in theoretical analysis in many other courses in their own fields. However, in other branches such reinforcements are less emphasized; all too often practical problems are dealt with by tentative arguments undeservingly called the Engineering Approach.

In engineering endeavors rooted in physical sciences, deep understanding and precise analysis cannot usually be achieved without the help of mathematics.