The list author says: "Applied math is sort of in a funny place. Since it is applied, it is usually tied to a non-mathematical field (e.g., physics, elastics, fluid mechanics) which eases you into the math, based on your knowledge of that field.
Once you get the math, you start reading applied math books from fields that you don't know, and suddenly you start learning those fields too, because now it is the math that is familiar, rather than the application. Applied math is the language that all engineers speak, no matter what field they come from. That being said, I come from a geology/hydrology background, so the fluids and hydrology books are most familiar for me."
"An incredibly well-written book that is full of gems. It has both wonderful explanations in the text, and really insightful and worked-out difficult problems. The problems make excellent self-study material. Most engineers don't realize there are very elegant and powerful methods to approximate differential equations besides techniques like finite difference and finite element."
"An excellent, concise book on advanced calculus and analysis. Most of the tools you need in applied math are covered here. This book isn't always easy, since it is quite mathematical and sometimes abstract, but always methodical."
"This is a pretty good self-contained introduction to Applied Mathematics for engineers. It is a bit on the big side, but it has good problem sets that often serve to extend the material presented in the text. The two notable things missing are Green's functions and linear algebra."
"This is an excellent non-standard intro to applied math. Strang is conversational in his tone (which can deceive you into thinking the material is easy) and makes interesting connections between various parts of applied math and he covers topics usually not covered in intro books (e.g., Kalman filters and chaos). He jumps around, which makes it frustrating sometimes to look one thing up."
"An incredible book. Just as an example, Lanczos was making up numerical inverse Laplace transform algorithms in the 1950s (most published numerical algorithms are from the 70s). Great description of the eigenvector/value problem and his famous "tau" method (solve an approximate form of the problem exactly)."
"Linear algebra and differential equations are really the same thing. Greens functions, too. This book is both well written and full of incredible insight. Like an older version of one of Strang's books."
"A good modern introductory text on PDEs that does a fairly good job introducing the more difficult theoretical aspects, as well as covering the basics. Good coverage of Green's functions and generalized Fourier series."
"A classic. From the 1960s, but it will give you a real understanding of the foundational material in the subject; much more detail than Golub & van Loan. Wilkinson invented much of the current foundation of numerical analysis of matrices; the guy knows his stuff."
"This is better than most books on special functions. It gives actual examples of them being used, and has a great deal on hypergeometric functions, which are typically skimmed in most books. The coverage on Bessel functions is quite good too."
"Although most of the problems in this book are not worked out in detail, the problems themselves and the hints that are given are valuable. Good coverage of non-standard coordinate systems and integral transforms."
"This is a well-organized and well-written book that systematically explores all the traditional avenues for solving the diffusion equation. While not as famous as Carslaw & Jaeger, it is much better written and does not give results without explanation or derivation."
"A clear but dense treatise on Fluid mechanics, but covering several topics not included in introductory books. It was written for mathematicians trying to learn fluids, so many consider it to be too hard, but it is some good stuff."
"This too is a fluids book, but it is about turbulence. The text is written very well, and it describes what turbulence means and the implications at a somewhat introductory level, although the mathematics in this book is not introductory. Written for mathematicians who want to learn turbulence."
"This is an geophysical fluid dynamics book, and is therefore quite mathematical. He does a good job explaining the mathematics in clear English and giving some physical meaning to things. Hard to find, but worth it."
"This book is also a fluids book, but is more a math text. This is good if you learn math by example and application. Mostly about complex variables in 2D classical fluid dynamics. Very dense and somewhat terse, but excellent."
"Feynman was a brilliant physicist and an excellent teacher. Volumes 1 and 2 of the 3 volume set cover most of physics that engineers would need. Even if you know the material, reading this will give you a deeper insight (e.g., gravitation and wave superposition)"
"Though this book is about groundwater, mathematically it is very rigorous and carefully uses the representative elementary volume and volume averaging approaches to derive most of the aquifer flow equations from basic principles."
"A giant reference on most of the integral transforms in use, slanted towards the Fourier and related transforms. Loads of worked-out examples. Probably the most accessible introduction to the more esoteric transforms (those other than Laplace or Fourier) for engineers."
"a classic unifying textbook about transport theory (momentum, energy, and mass). It does a great job drawing analogies between topics, and is fairly methodical. It has good coverage of tensors (albeit in a non-standard way) and cylindrical and spherical coordinate systems. I have the brick-red first edition."
"a very good introductions to both variational calculus and tensors; things often only skimmed in other books (or they assume you already know it). This assumes you already know quite a bit of physics, but it is pretty gentle with the math it introduces : complementary to books like those by Shadowitz an Graff in this list."
"These are incredibly dense with information and insight. The theme of Green's functions, eigenfunction expansion and integral transforms show up throughout the two volumes. One of the best references available on the use of non-standard coordinate systems and the eigenfunction that arise in them."
"This is a non-conventional book. It covers a very large amount of ground quickly. Most of the sections are comprised of a concept, followed by problems and hints, used to introduce more advanced concepts."
"an excellent self-contained introduction to Electromagnetic fields, including special relativity. You need to know some math before this book will be useful, but it is expertly written to minimize logical jumps and nearly everything is explained succinctly yet fully."
"clear, with loads of examples. This book and its appendices introduce most of the mathematics required for elastic wave propagation (e.g., tensors, integral transforms), albeit in a rather terse way. In some ways this is a more condensed and simpler form of Morse & Feshbach."
"This is a good introductory book to quantum mechanics, if you are comfortable with intermediate applied math. It also is full of fairly detailed examples of using applied math to solve sometimes rather complex problems."
"While this book is hard to find outside university libraries, it is an excellent book on the somewhat esoteric topic of periodic differential equations, including Mathieu, Lame, and other high-order special functions. Well-written and insightful."
"This is the long-awaited revision of Abramowitz & Stegun's epic book on special functions. There is a website that goes with it (http://dlmf.nist.gov/). The book nicely summarizes most of what is known and in general use regarding special functions and applied math methods."
"Everybody needs a "go-to" book for looking up equations or relationships that you know of, but can't remember the details. It can serve as a dictionary of mathematics too, since it has a good index and covers basically everything. More useful than Abramowitz & Stegun for most engineers (unless you are heavy in special functions)."