Some essential mathematics

This chapter introduces most of the important mathematics that will be used repeatedly throughout this book. Whilst scalars like the electric potential Ф and the vector fields E and B are familiar quantities in electricity and magnetism, it is not always known that they are examples of a mathematical form termed a tensor. More sophisticated tensors are required in some descriptions and theories in electrodynamics. As tensor notation is compact and usually allows for a simple derivation of a result, the chapter begins with a series of questions involving Cartesian tensors. The results obtained here will facilitate the solution of many questions in subsequent chapters. Readers who are not familiar with tensors are advised to consult Appendix A before proceeding. Towards the end of this Appendix a ‘checklist for detecting errors when using tensor notation’ is given, which will hopefully provide some help for the uninitiated.