which may be written more succinctly as:

Here, U is the unknown function, k a property of the material and h a source term. Some exact solutions are available, but not commonly in cases of practical interest. Even when solutions exist, they tend to be complicated and must be evaluated numerically for graphical presentation in digestible form.

Finite element analysis (FEA) is a numerical approach in which the system under study is divided into hundreds of regions, each one giving rise to one or more equations. The main task is to solve all these hundreds or thousands of simultaneous equations, which was impracticable before the days of transistorized computers.

Hundreds of textbooks on finite element analysis exist today, and they all present the various mathematical methods in detail. They assume that the reader will make calculations on simple systems by hand and then write FORTRAN code or other, provided in an appendix, in order to handle more complex situations.

Some twenty years ago, the choice was between writing one's own programs or abstaining completely from FEA. Today, expert software saves students from worrying about programming strategy, formatting, indices and graphics. The tools of FEA are now on anyone's desk, since the average personal computer is quite adequate for solving many such problems.

Programs for FEA calculations on a PC have been available for many years but they have generally been of a "black box" type. Now programs exist which permit you to enter the equations and the boundary conditions required by your mathematical model, solve them automatically, and present the results graphically in a variety of ways. There are two general software packages of similar structure and usage. These products are PDEase2D, available from Macsyma Inc., 20 Academy Street, Arlington, MA 02174, and FlexPDE49 from PDE Solutions, Inc., PO Box 549, Sunol, CA 94586.

One of the most attractive applications of this software is in university education. An extremely large range of problems involving classical fields may easily be studied in detail, and with realistic boundary conditions. Non-linear equations and boundary conditions as well as space-dependent materials properties are no longer serious limitations. The number of cases for study is virtually unlimited, and by solving problems under various conditions the student may develop an intuitive feeling for how fields behave.

No one would even consider taking courses in Object Oriented Programming, with applications to advanced string, windows and mouse handling, before using a modern word processor. We all leave these chores to specialists, who spend years learning the tricks of their profession and then devote more years to produce the software we use every day.

Numerical algorithms and programming really have little to do with the concepts of physics and if months or even years may be gained by skipping these items, then there is no reason to hesitate. Every scientist has to make a choice about what to learn and what to leave alone. No one can pretend to master all of physics, not even all of classical physics. The real choice is between additional ignorance within the discipline of physics or within some adjacent field.

On the other hand, no scientist would be content to use a numerical toolbox, such as PDEase2D or FlexPDE, without knowing at least the principles of operation. Some details are even essential to the formulation of boundary conditions. In an attempt to include a general description of the method and at the same time de-emphasize its importance to the reader, I have included an appendix on this subject.

The main purpose of this volume is to illustrate how fields change when you modify the geometry, use different materials and change the boundary conditions. It should be regarded as a companion to textbooks in the various branches of physics, rather than a textbook on its own. Even if you have not purchased the corresponding software package, you can use this book for information on fields and about the potential value of FEA.