Message-ID: <20041018174715.12976.qmail@plover.com>
Subject: Book review: Differential Equations / Simmons
Organization: Plover Systems
Date: Mon, 18 Oct 2004 13:47:15 -0400


On a whim, I picked up the following book at a rummage sale yesterday:

        Differential Equations with Applications and Historical Notes.
        George F. Simmons.

It is really excellent.  It's clear and it not only makes the subject
interesting but deeply fascinating.  By page 30 for example, he has
treated falling objects with air resistance, and shown how to calulate
terminal veolcities, he has shown how to solve the brachistochrone
problem.

It's fun even to read the exercises.  Each one gets me excited to try
to find out the answer.  Here are some examples, all from chapter 1:

        Consider a bead at the highest part of a circle in a vertical
        plane, and let that point be joined to any lower point on the
        circle by a straight wire, If the bead slides down the wire
        without friction, show that it will reach the circle in the
        sane time regardless of the position of the lower point.

        A chain 4 feet long starts with 1 foot hanging over the edge
        of a table.  Neglect friction, and find the time for the chain
        to slide off the table.

        A smooth football having the shape of a prolate spheroid 12
        inches long and 6 inches thick is lying outdoors in a
        rainstorm.  Find the paths along which the water will run down
        its sides.

        The clepsydra, or ancient water clock, was a bowl from which
        water was allowed to escape through a small hole in the
        bottom.  It was often used in Greek and Roman courts to time
        the speeches of lawyers, in order to keep them from talking
        too much.  Find the shape it should have if the water level is
        to fall at a constant rate.

This one is my favorite so far:

        A destroyer is hunting a submarine in a dense fog.  The fog
        lifts for a moment, discloses the submarine on the surface 3
        miles away, and immediately descends.  The speed of the
        destroyer is twice that of the submarine, and it is known that
        the latter will at once dive and depart at full speed in a
        straight course of unknown direction.  What path should the
        destroyer follow to be certain of passing directly over the
        submarine?  Hint: Establish a polar coordinate system with the
        origin at the point where the submarine was sighted.

I wouldn't even have realized that there *was* such a path, and I had
to ponder for a while to persuade myself that there was.

The differential equations class I took as a youth was disappointing,
because it seemed like little more than a bag of tricks that would
work for a few equations, leaving the vast majority of interesting
problems insoluble.  If your experience of differential equations was
similar, you might enjoy this book as much as I am.