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This book was bom out of my fascination with applied mathematics
as a place where the physical world meets the mathematical structures
and techniques that are the cornerstones of most applied mathematics
courses. I am interested largely in human-sized theatres of interaction,
leaving cosmology and particle physics to others. Much of my research
has been motivated by interactions with industry or by contact with
scientists in other disciplines. One immediate lesson from these contacts
is that it is a great asset to an interactive applied mathematician to be
open to ideas from any direction at all. Almost any physical situation has
some mathematical interest, but the kind of mathematics may vary from
case to case. We need a strong generalist streak to go with our areas of
technical expertise.
    Another thing we need is some expertise in numerical methods. To
be honest, this is not my strong point. That is one reason why the book
does not contain much about these methods. (Another is that if it had
then it would have been half as long again and would have taken five
more years to write.) In the modem world, with its fast computers and
plethora of easy-to-use packages, any applied mathematician has to be
able to switch into numerical mode as required. At the very least, you
should learn to use packages such as Maple and Matlab for their data
display and plotting capabilities and for the built-in software routines for
solving standard problems such as ordinary differential equations. With
more confidence, you can write your own programs. In many cases, a
quick and dirty first try can provide valuable information, even if this
is not the finished product. Explicit finite differences (remember to use
upwind differencing for first derivatives) and tiny time steps will get you
a long way.
Who should read this book? Many people, I hope, but there are some
prerequisites. I assume that readers have a good background in calculus
up to vector calculus (grad, div, curl) and the elementary mechanics of
particles. ! also assume that they have done an introductory (inviscid)
fluid mechanics course and a first course in partial differential equations,
enough to know the basics of the heat, wave and Laplace equations
where they come from, and how to solve them in simple geometries).
Linear algebra, complex analysis and probability put in an occasional
appearance. High-school physics is an advantage. But the most important
prerequisite is an attitude: to go out and apply your mathematics, to see
it in action in the world around you, and not to worry too much about
the technical aspects, focusing instead on the big picture.
    Another way to assess the technical level of the book is to position
it relative to the competition. From that point of view it can be thought
of as a precursor to the books by .Tayler [58] and Fowler [19], while
being more difficult than, say, Fowkes & Mahoney [18] or Fulford &
Broadbridge [22]. The edited collections [9, 38] are at the same general
level, but they are organised along different lines. The books [40, 56]
cover simil