The Earth's dynamical behaviour is a complex and fascinating subject with many practical ramifications. Its description requires the language of mathematics and computation. In this book, we attempt to make the theoretical foundations of the description of Earth's dynamics as complete as possible, and we accompany the theoretical descriptions with computer code and graphics for the implementation of the theory.

Scalar, vector and tensor analysis

We will make extensive use of scalars, vectors and tensors throughout the book. In this section, we will summarise the properties most often used. It is assumed that the reader is familiar with the elementary results of vector analysis summarised in Appendix A.

Scalars

Physical quantities determined by a single number such as mass, temperature and energy are scalars. Scalars are invariants under a change of co-ordinates, they remain the same in all co-ordinate systems. They are sometimes simply referred to as invariants. A scalar field is a function of space and time.

Vectors

Vectors require both magnitude and direction for their specification. They may be described by their components, their projections on the co-ordinate axes. An arbitrary vector then associates a scalar with each direction in space through an expression that is linear and homogeneous in the direction cosines.

In general, a vector can be defined in a space of arbitrary dimensions numbering two or greater. Our applications will be confined to a space of three dimensions and we will adopt this limitation.