Books

Spiegel, Schaum's Outline: Mathematical Handbook (McGraw Hill, 1968...). A very useful collection of mathematical results and formulae. Not too long; easy to use; I keep copies both at home and in the office.

Zwillinger, CRC Standard Mathematical Tables and Formulae (CRC Press, 2002). An alternative to the Schaum Handbook that is more comprehensive, but it takes more time to search.

Courant, Robbins, and Stewart, What is Mathematics ? (Oxford University Press, 1996). This book shares gems of fundamental mathematics with the reader; there is no talking down from on high; everything is explained and proved. For example, there is a nice winding-number proof of the fundamental theorem of algebra.

Arthur Koestler, The Sleepwalkers (Peregrine Books, 1986). A relaxed historical description of scientific discovery. Particularly good on the story of mechanics. Tells of Kepler having to rush home to defend his Mother from witchcraft accusations, and of his almost reluctant acceptance that the paths of the planets are conic sections. Then there is Newton.

Edmund Landau, Differential and Integral Calculus (Chelsea, New York, 1980). By a great master, this book, originally published in 1934, has no sentence in it that could safely be removed; it has the clarity of ice crystals on a sunny day in the Canadian Arctic. By chapter 5 one reaches the derivative: the first example is a Weierstrassian function that is everywhere continuous but nowhere differentiable; thus one learns that differentiability is a radically new property. There is much mathematics here, including infinite series, special functions, and Fourier series; but nothing on functions of a complex variable or of several variables. We note that there are different ways to look at things.

D. Bleecker and G. Csordas, Basic Partial Differential Equations (International Press, Cambridge MA, 2003). A thorough, enthusiastic, and readable book with many mathematical ideas and plenty of interesting applications. There are also lots of problems and solutions. Highly recommended for the autodidact.

Julie Levandosky, Stanford Lecture Notes In particular, see the notes for 1st-order linear pde