Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box

Appendix E

Trade secrets: Tools for dealing with vectors and vector identities

If anything has magnitude and direction, its magnitude and direction taken together constitute what is called a vector.

—Josiah Willard Gibbs [230]

Del treats them all the same

—Steve Clark [106].

Before the last decades of the nineteenth century, vectors were not used in physics. Back then, physicists would write clumsy expressions, all in terms of components. That is how Maxwell wrote most of the equations in his famous treatise (the now popular vector form of Maxwell's equations is due to Oliver Heaviside). Then Hamilton invented quaternions (see Appendix C), which offered a very compact way of writing four-dimensional quantities. The vector analysis that we use in physics today was introduced by Gibbs and Heaviside, who based it in Hamilton's quaternions and in Clifford's and Grassmann's work.

In this appendix we review briefly some useful techniques for working with vectorial expressions and for deriving vector identities. If you are familiar with the vector identities used throughout this book, you should probably skip this appendix. However, you might still want to give it a quick scan first, for there is some material here that you would not normally find collected together in other books and which most lecture courses do not usually cover.

E.1 RELATION BETWEEN VECTOR PRODUCTS AND DETERMINANTS

A useful tool for working out vector products is to write them down as determinants. Here is how the ...