CHAPTER 5 - Differential Equations and Their Use in Science

Summary

The reader should be somewhat familiar with the concepts and the techniques of integral and differential calculus; yet knowledge of the theory of differential equations is not a prerequisite. What such equations are and how they must be treated will be explained (roughly but sufficiently for our purpose) later, when they naturally emerge from physical problems. It will turn out that differential equations are useful in science. We cannot understand how and why they are useful before we have used them.

SECTION 1. FIRST EXAMPLES

Rotating Fluid

One lump of sugar, or two? Cream? We have all observed a lady taking tea. What happens? The faster she stirs, the higher up the side of her cup the tea climbs. If she stirs too fast she spills it and ruins an afternoon. Her teacup contains a problem for her and a problem for us. Our problem is amenable to mathematical treatment: What is the surface shape of the rotating tea?

First consider a motionless liquid. We have all seen a glass of water when no one is kicking the table. Its surface looks flat, yet closer examination shows its surface to be not entirely horizontal; it curls up ever so slightly at the edges, due to surface tension. For water substitute mercury, and surface tension causes precisely the opposite effect, a curling down at the edges. A phenomenon distinctly visible in a mercury barometer.