Thinking About Equations || Several Instructive Examples

✍ Scribed by Bernstein, Matt A.; Friedman, William A.

• Publisher: John Wiley & Sons, Inc.
• Year: 2009
• Tongue: English
• Weight: 452 KB
• Edition: 1
• Category: Article
• ISBN: 0470186208
• DOI: 10.1002/9780470495032.ch8

✦ Synopsis

In the fi nal chapter, we present several instructive examples. In Section 8.1 , we present two examples that illustrate how a clever choice of coordinate system can simplify the equations needed to solve a problem. Then, in Section 8.2 , we discuss how insight can be gained when it is discovered that the solution to a problem has unanticipated properties. The solution might be simpler or more complicated than expected, or it might suggest unexpected symmetry properties. In Section 8.3 , we examine an indirect method of solving problems by starting with known solutions that apply to a broad class of problems. We illustrate this method with an example from the fi eld of electrostatics. Specifically, we discuss an algorithm that generates expressions guaranteed to satisfy the two -dimensional version of Laplace ' s equation, even though it is not known beforehand to which problem the generated solution will correspond. Finally, in Section 8.4 , we present examples from the fi elds of chaos theory and cosmology that illustrate the far -reaching ways in which equations can alter our understanding of the world.

8.1 CHOICE OF COORDINATE SYSTEM

Choosing the optimal coordinate system can greatly simplify fi nding the solution to a problem. Consider the simple problem of calculating the distance 214