1 Topics to Review This chapter is self-contained and requires basic knowledge from A PREVIEW calenlus,such as partial deriva- tives.We make occasional refer- OF APPLICATIONS ences to topics from a first course in ordinary differential equations. AND TECHNIQUES These topics are found in Ap- pendix A and can be called on as needed. Looking Ahead... Everything should be made as simple as possible,but no simpler This chapter is intended to intro- -ALBERT EINSTEIN duce some of the basic notions occurring in the study of partial differential equations and bound- ary value problems.Section 1.1 introduces the notion of a par- tial differential equation and of fers some clementary but useful solution techniques.Section 1.2 focuses primarily on the develop- Partial differential equations arise in modeling numerous phenomena in sci- ment of a simple but central ex- ence and engineering.In this chapter we preview the ideas and techniques ample in partial differential equa- that will be studied in this book.Our goal is to convey something of the fa- tions,the vibrating string.The vor of partial differential equations and their use in applications rather than reader is taken to a point where to give a systematic development.In particular,by focusing on the vibration several of the major landmarks of of a stretched string,we will see how partial differential equations arise in the field may be surveyed.This modeling a physical phenomenon and how the interpretation of their solu- preview shows clearly the need tions helps us to understand this phenomenon.The chapter will culminate for the treatment of Fourier se- with a brief discussion of Fourier series that highlights their importance to ries of Chapter 2 and the other the development of the applications of partial differential equations. more systematic developments to follow