*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.*

Preface 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.

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.

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. 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.A case is presented for the importance of focusing on (1) average ability students, (2) substantive mathematical content, (3) real problems, and (4) realistic settings and solution procedures for research in problem solving.It is suggested that effective instructional techniques for teaching applied mathematical problem solving resembles “mathematical laboratory” activities, done in small group problem solving settings.Here we collect 10 essential applied mathematics problems to get you thinking before you embark on your degree, although many of the problems from the physics and engineering areas will also be relevant.They will be very useful to anyone intending to study mathematics, physics or engineering: they will give you a good mathematical grounding in some of the topics likely to arise in your degree course and refine your problem solving skills.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.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.It was fascinating to learn how to solve certain classes of differential equations in one class, and then go to this class, set up some differential equations using conservation laws and common sense, and using what I just learned in the other class to solve them! Yes, from the perspective of the differential equations we set up. At this point, you are easily mixing tools from 4–5 different upper division courses.But there is something funky going on when you analyze the solutions more in depth. This subject is much more fun and fascinating than just learning numerical methods and grinding through monotonous calculations. You are given tools that can be applied to countless topics, and as a consequence you get glimpses into many different disciplines.

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