Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 1

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Introduction and theoretical background
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 topic, 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.
1.1 Scalar, vector and tensor analysis
We will make extensive use of scalars, vectors and tensors throughout the topic. 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.
1.1.1 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.
1.1.2 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 arbit-
rary vector then associates a scalar with each direction in space through an expres-
sion that is linear and homogeneous in the direction cosines.