The mathematical solution of a physical problem usually involves the

following steps:

1 The construction Of a mathematical model. At this step we must care-

fully define the variables involved and select a consistent system of

units. Tile actual mathematical equations describing the physical

problem are usually mathematical realizations of physical laws or

imposed constraints. Frequently, some simplifying assumptions are

made at this stage in order to make the model simple enough to solve.

Solution of the mathematical equations obtained in step 1. Ideally, we

would like to be able to find exact solutions of the equations, but in

many cases only approximate solutions are available. In this case,

theoretical support for the existence and uniqueness of the solution

would be reassuring.

Interpretation of the solution of the mathematical problem in terms of

the original physical situation. It is important that the mathematical

results be consistent with physical intuition and laboratory evidence.

If they are not, then the construction of the model must be critically

reexamined.

This book will be concerned mostly with step 2. We will begin by con-

sidering briefly a few simple mathematical models of physical situations which will

motivate the topics to be covered in this book. Solutions to these models will be studied

later.