How to Solve Applied Mathematical Problems
It is only possible to have a good understanding of applied mathematics and theoretical physics by spending a long time solving a wide variety of problems. However, most students find that after attending lectures or reading books on applied mathematics, they discover to their dismay that they are simply unable to solve many of the problems. This book attempts to show students of applied mathematics how problems can be solved. The subject of applied mathematics is extremely wide ranging, as can be seen from the contents list of this book which is far from being comprehensive in scope since otherwise the book would be enormously long.
One can try to formulate basic principles for solving problems in applied mathematics, and to a certain extent this can be done, but the best way is to study the solutions of a large representative selection of problems.
It is important to realize that solving problems in applied mathematics is strongly dependent on understanding, and is not just a matter of memory, although a knowledge of the relevant formulae plays a significant role.
Most of the selection of problems in this book belong to the standard repertoire and all have been given to mathematics and physics undergraduate students at the Queen's University of Belfast as examples or exercises at one time or another. As far as possible lengthy and very complicated problems have been avoided and the problems have been chosen to illustrate general principles. All the problems can be solved in closed analytical forms in terms of elementary functions or simple integrals.
The book starts with an introductory chapter in which there is a survey of the range of subjects which are covered in the subsequent chapters by means of a small representative selection of problems, together with some general principles for solving problems in applied mathematics.
The domain of applied mathematics and theoretical physics, or mathematical physics as it is often called, may be viewed as a set of nested Chinese boxes in which it is not possible to get at the inner boxes without first opening the outer ones.
Each chapter in this book opens a new box. The outermost box which we open in this book of elementary solved problems is labelled vector algebra. This subject is basic to much of applied mathematics since it is a three dimensional algebra which provides the formal basis for dealing with problems in space, for example geometry.
From here we proceed to kinematics or the study of the motion of points in space. Next we examine the dynamics of a particle which provides basic examples of the use of Newton's laws of motion, including projectiles, vibrations, orbits, the motion of a charged particle in crossed electric and magnetic fields, and rotating frames of reference.
We now come to vector field theory which is concerned with vector functions of position in regions of space and the important theorems first proved by Gauss, Green and Stokes. This subject enables us to solve problems in Newtonian gravitation, electricity and magnetism, and fluid dynamics.
We then return to the solution of problems in classical dynamics through the use of Lagrange's equations and Hamilton's equations, as well as Hamilton-Jacobi theory which is fundamental to quantum theory discussed towards the end of the book.
Although the subjects of Fourier series, Fourier and Laplace transforms, and integral equations, are not strictly applied mathematics, they are essential for the study of wave motions, including vibrating strings, sound waves and water waves, and for the study of heat conduction. Thus the boxes labelled wave motion and heat conduction cannot be opened before the box containing Fourier series has been examined in some detail.
Next we turn to tensor analysis, which enables us to express Maxwell's electromagnetic equations in a relativistically invariant form, and to develop the special theory and general theory of relativity.
At this stage of the book we show how to treat problems in elementary quantum theory and finally we look at the solution of problems by using variational principles and variational methods which are extremely important for many aspects of applied mathematics and theoretical physics. This last chapter contains problems associated with many of the preceding chapters and relates them by means of the calculus of variations.
There are many applied mathematics boxes which we have not explored in this book. Several contain more than one box immediately within, some of which we have not opened, such as for the subjects of elasticity and statistical mechanics, for example. ...
To summarize, my aim has been to attempt to bridge the gap between a course of lectures in which concepts are developed and the basic applied mathematics analysed, and the problems which are provided for students to test their understanding of the subject being studied.