How To Solve Applied Mathematics Problems
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.
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.
Remember, these problems are designed to make you think and there is not necessarily a 'right' answer. Take them to a level that feels comfortable for you.
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.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.One fun topic that came to mind when I read this question is traffic flow.I spent a third of a semester in one class focusing solely on setting up, interpreting, and solving a model for freeway traffic flow. Whatever it is, you likely have the tools to solve the complicated optimization problem that emerged either exactly, or with some numerical method you learned.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.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.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.