[1] Kundu, P. K., (1990) Fluid Mechanics, Academic Press, San Diego.
No science related book has had a larger influence on me than the first edition of this book. After P.K. Kundu's untimely passing, the book has undergone many changes through multiple editions that have largely left it unrecognizable, but the first edition reflects the author's passion for fluid mechanics seen through the prism of the natural environment and I don't think I go a week without leafing through its pages. The end of Chapter notes are often a treasure trove of older papers well worth reading.
[2] Johnson, R. S. (1997) A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Press, Cambridge, UK.
A tour de force of theory, with the poetic version of the Legend of Arthur heading each chapter, this book is very much something I could not write. But with its many exercises it did allow me to develop as a student of wave theory, and forge my own identity as a numerical experimentalist. The Bibliography is impressive in coverage (e.g. Russel's original report on the observation of a solitary wave, the original Korteweg and de Vries paper, and the original Fermi, Pasta, Ulam report), and the historical notes cover many of the players in classical wave theory.
[3] Trefethen, L. N. (2000) Spectral Methods in MATLAB, SIAM Press, Philadelphia.
A close second, in terms of personal influence is this tidy little gem by Nick Trefethen. The idea that real science and real mathematics can fit on a (small) single page of code, was so provocative when I first found the book I used to read it `in hiding' as if the professors that taught me functional programming would find me and cast a disapproving eye! The references, while not as tidy in terms of presentation are excellent (e.g. the Fornberg and Whitham paper mentioned above).
[4] Boas, M. L. (2006) Mathematical Methods in the Physical Sciences, John Wiley.
This is a book of mathematical methods, and comes recommended from a colleague who has taught such material out of a number of sources. It has the advantage of being quite comprehensive in terms of material.
[5] Whitham, G. B. (1974) Linear and Nonlinear Waves, John Wiley.
The standard reference on the theory of dispersive waves and how they contrast with hyperbolic waves. Still a good read.
[6] J. P. Boyd. (2001) Cheyshev and Fourier spectral methods. Dover publications.
Boyd's book is the Yang to Trefethen's [3] Ying. It is broad in coverage, and agnostic as to implementation language. As such, it makes for interesting re-reading many years after you first acquire a copy.
[7] Lovrić, M. (2007) Vector Calculus, John Wiley.
Vector Calculus is taught from a variety of perspectives, depending on the eventual primary application of the material. This is by far the best introduction to vector calculus for our purposes.
[8] Taylor, J. R. (2005) Classical Mechanics, University Science Books.
An excellent introduction to the subject, and one of the best written textbooks of all time. Not really our area, but a student who wishes to be well rounded should spend some time on it.
[9] Trefethen, L. N. The (Unfinished) PDE Coffee Table Book
We build our intuition from concrete examples. Every section of this book is an exploration of a different partial differential equation. A good resource to become familiar with some of the core content of the subject.
[10] Lin, C. C. and Segel, L. A. (1988) Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM
A classic overview of the use of applied mathematics methods. Wide ranging and with enough depth to see the value of these approaches in a variety of contexts.
[11] Iserles, A. (1996) A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press
As well as being a first course, it also makes a great general reference. It builds from ODEs to PDEs, deriving methods from first principles. Highly recommended.
[12] Leveque, R. J. (2002) Finite Volume Methods for Hyperbolic Problems, Cambridge University Press
The focus of this book is hyperbolic PDEs, which describe an enormous range of transport and wave-propagation phenomena. While Iserles [11] gives primarily mathematical examples, Leveque gives examples derived more directly from physical contexts including fluid dynamics.
[13] Strauss, W. A. (2007) Partial Differential Equations: An Introduction, John Wiley
A fantastic introduction to a subject which can be approached in an enormous number of ways. The clear exposition, simple figures, and well-chosen examples make this a joy to read. The focus is on understanding how PDEs are used to describe phenomena, rather than the thorny issues arising from the demands of total mathematical rigor.
[14] Feynman, R. P., Leighton, R. B., and Sands, M. (2010) The Feynman Lectures on Physics, Addison-Wesley
These are undergraduate lectures which Richard Feynman gave at Caltech from 1961 to 1963. Anyone interested in a basic understanding of phyics should go through this at least once. There is a rather expensive box set of three volumes, but it is now also available online at the provided link.
[15] Bender, C. M. and Orszag, S. A. (1999) Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer
Many differential equations cannot be solved exactly. This book explains how to obtain approximate analytical solutions in such cases. A very satisfying subject in itself, but also a very fruitful way of thinking conceptually. It also serves as an implicit introduction to experimental mathematics, as the goal is obtaining verifiable results rather than proceeding through more formal theorem and proof methods.
[16] Trefethen, L. N. and Bau, D. (1997) Numerical Linear Algebra, SIAM
An introduction to making a computer do linear algebra for you. There are still lots of theorems and proofs, but the focus here is on understanding everything constructively, algorithmically. You may be noticing a pattern in this list: Trefethen shows up a lot. Read the books and you'll find out why.
[17] Kutz, N. J. (2013) Data-Driven Modeling & Scientific Computation: Methods for Complex Systems and Big Data, Oxford University Press
In this case the title is an excellent description of the content. Many applications are covered, each including enough theory to get the point, along with algorithms and MATLAB code throughout. Very useful as an introduction to the flavour of work we often do in the lab.
[18] Ipsen, I. C. F. (2009) Numerical Matrix Analysis, SIAM
A short, powerful book dense with content. The author is careful and systematic in their development of the material. Includes facts difficult to find elsewhere.