In physics and mathematics, coordinate transformations play an important role because many problems are much simpler when a suitable coordinate system is used. Furthermore, the requirement that physical laws do not change under certain transformations imposes constraints on the physical laws. An example of this is presented in Section 22.11 where it is shown that the fact that the pressure in a fluid is isotropic follows from the requirement that some physical laws may not change under a rotation of the coordinate system. In this chapter it is shown how the change of vectors and matrices under coordinate transformations is derived. The derived transformation properties can be generalized to other mathematical objects which are called tensors. In this chapter, only transformations of rectangular coordinate systems are considered. Since these coordinate systems are called Cartesian coordinate systems, the associated tensors are called Cartesian tensors. The transformation properties of tensors in Cartesian and curvilinear coordinate systems are described in detail by Butkov [24] and Riley et al. [87].

Coordinate transforms

In this section we consider the transformation of a coordinate system in two dimensions. In Figure 22.1 an old coordinate system with coordinates xold and yold is shown. The unit vectors along the old coordinate axis are denoted by êx,old and êy,old. In a coordinate transformation, these old unit vectors are transformed to new unit vectors êx,new and êy,new, respectively.

In physics one frequently handles the change of a property with time by considering properties that do not change with time. For example, when two particles collide elastically, the momentum and the energy of each particle may change. However, this change can be found from the consideration that the total momentum and energy of the system are conserved. Often in physics, such conservation laws are the main ingredients for describing a system. In this chapter we deal with conservation laws for continuous systems. These are systems in which the physical properties are a continuous function of the space coordinates. Examples are the motion in a fluid or solid, and the temperature distribution in a body. The introduced conservation laws are not only of great importance in physics, they also provide worthwhile exercises in the use of vector calculus introduced in the previous chapters.

General form of conservation laws

In this section a general derivation of conservation laws is given. Suppose we consider a physical quantity Q. This quantity could denote the mass density of a fluid, the heat content within a solid or any other type of physical variable. In fact, there is no reason why Q should be a scalar, it could also be a vector (such as the momentum density) or a higher order tensor. Let us consider a volume V in space that does not change with time. This volume is bounded by a surface ∂V. The total amount of Q within this volume is given by the integral ∫VQdV.

From this book and most other books on mathematical physics you may have obtained the impression that most equations in the physical sciences can be solved. This is actually not true; most textbooks (including this book) give an unrepresentative state of affairs by only showing the problems that can be solved in closed form. It is an interesting paradox that as our theories of the physical world become more accurate, the resulting equations become more difficult to solve. In classical mechanics the problem of two particles that interact with a central force can be solved in closed form, but the three-body problem in which three particles interact has no analytical solution. In quantum mechanics, the one-body problem of a particle that moves in a potential can be solved for a limited number of situations only: for the free particle, the particle in a box, the harmonic oscillator, and the hydrogen atom. In this sense the one-body problem in quantum mechanics has no general solution. This shows that as a theory becomes more accurate, the resulting complexity of the equations makes it often more difficult to actually find solutions.

One way to proceed is to compute numerical solutions of the equations. Computers are a powerful tool and can be extremely useful in solving physical problems. Another approach is to find approximate solutions to the equations. In Chapter 12, scale analysis was used to drop from the equations terms that appear to be irrelevant.

I have written this book mainly for students who will need to apply maths in science or engineering courses. It is particularly designed to help the foundation or first year of such a course to run smoothly but it could also be useful to specialist maths students whose particular choice of A-level or pre-university course has meant that there are some gaps in the knowledge required as a basis for their University course. Because it starts by laying the basic groundwork of algebra it will also provide a bridge for students who have not studied maths for some time.

The book is written in such a way that students can use it to sort out any individual difficulties for themselves without needing help from their lecturers.

A message to students

I have made this book as much as possible as though I were talking directly to you about the topics which are in it, sorting out possible difficulties and encouraging your thoughts in return. I want to build up your knowledge and your courage at the same time so that you are able to go forward with confidence in your own ability to handle the techniques which you will need. For this reason, I don't just tell you things, but ask you questions as we go along to give you a chance to think for yourself how the next stage should go.

I have thoroughly revised all the ten chapters in the original edition, both making some changes due to comments from my readers and also checking for errors. I've also added a chapter on vectors which continues naturally from the present chapter on complex numbers.

I wrote the first version of this new chapter as an extension to the book's website (which is now at http://www.mathssurvivalguide.com) building up the pages there gradually. Their content was influenced by emails from visitors, often with particular problems with which they hoped for help. I've now extensively rewritten and rearranged this material. Writing in book form, it was possible to structure the content much more closely than on the Web so that it's easy to see the connections between the different areas and how results can be applied to later problems. The new chapter also has, of course, many practice exercises with complete solutions just as the earlier chapters have.

I'm once again very grateful to Rodie and Tony Sudbery and to David Olive for their helpful suggestions and comments. I must also thank all the people who emailed me, both with comments on the original ten chapters, and also with particular needs in using vectors which I've tried to fulfil here.

I hope that this two-way communication will continue. You can email me from the book's website if you would like to.