Описание
xiiâ•…â•…Preface
reasonable standard rather than to prove them absolutely. To achieve simplicity, there
have been a few departures from convention and some changes of emphasis:
• The use of indices is kept to a minimum, for example, basis elements are
written as x xy , rather than e e 1 12 ,
• The basic intuitive ideas of parallel and perpendicular are exploited wherever
this may be advantageous.
• The term “translation” is introduced to describe a mapping process between
spacetime and 3D as distinct to the spacetime split.
• A notation is introduced whereby a vector underscored with a tilde, for
example, u R, , is to be identified as a purely spatial vector. Since such vectors
are orthogonal to a given time vector, this contributes to the aim of exploiting
parallel and perpendicular.
• To maximize the readability of equations, a system is introduced whereby
SI units are retained but equations are simplified in a way similar to the
mathematical physicist’s convention taking the speed of light to be 1.
A geometric algebra is a vector space in which multiplication and addition
applies to all members of the algebra. In particular, multiplication between vectors
generates new elements called multivectors. And why not? Indeed, it will be seen
that this creates valuable possibilities that are absent in the theory of ordinary linear
vector spaces. For example, multivectors can be split up into different classes called
grades. Grade 0 is a scalar, grade 1 is a vector (directed line), grade 2 is a directed
area, grade 3 is a directed volume, and so on. Eventually, at the maximum grade,
an object that replaces the need for complex arithmetic is reached.
We begin with a gentle introduction that aims to give a feel for the subject by
conveying its basic ideas. In Chapters 2–3, the general idea of a geometric algebra
is then worked up from basic principles without assuming any specialist mathematical knowledge. The things that the reader should be familiar with, however, are
vectors in 3D, including the basic ideas of vector spaces, dot and cross products, the
metric and linear transformations. We then look at some of the interesting possibilities that follow and show how we can apply geometric algebra to basic concepts,
for example, time t and position r may be treated as a single multivector entity t + r
that gives rise to the idea of a (3+1)D space, and by combining the electric and
magnetic fields E and B into a multivector F, they can be dealt with as a single
entity rather than two separate things. By this time, the interest of the reader should
be fully engaged by these stimulating ideas.
In Chapter 4, we formalize the basic ideas and develop the essential toolset that
will allow us to apply geometric algebra more generally, for example, how the
product of two objects can be written as the sum of inner and outer products. These
two products turn out to be keystone operations that represent a step-down and stepup of grades, respectively. For example, the inner product of two vectors yields a
scalar result akin to the dot product. On the other hand, the outer product will create
a new object of grade 2. Called a bivector, it is a 2D object that can represent an
Детали
- Год издания
- 2011
- Format
