# Reference requests: Is there a text that is even more advanced than books on “advanced engineering mathematics”

Advanced engineering mathematics is a subject of its own, building up from simple notions of functions, series, integration techniques and brief review of linear algebra which leads to transform methods (Laplace, Fourier, Z and sometimes Hilbert), with strong emphasis of course on ODE and some elements of PDE and finally ends with complex variables and conformal mapping.

With that I've just described the table of content of probably 99% of books out there on advanced engineering mathematics. Kreyszig, Zill, Stroud, Duffy et. al. falls squarely in this category. This is not to say such an approach is without value, in fact the standardization and consistency of a subject is probably a sign of very mature development of the field which is great for an introduction to engineering mathematics and provides great motivation for applied mathematics in general.

But the lack of variation between these texts leaves me wonder if there is a book that covers material that is even more advanced, more comprehensive than what has been described above. For engineering students, it is often crucial to have motivation when it comes to studying a subject and such motivation usually comes from engineering examples (circuit theory, mechanical system, modeling) so an advanced text that is typically used and cherished by math majors just won't do.

Can someone recommend a text that is most advanced or comprehensive when it comes to engineering mathematics covering topics that goes beyond the standard curriculum that could potentially be engaging for a mathematically mature engineering student who wish to know more about techniques used in mathematics for engineering applications? Please let me know if this is a bit too much to ask.

I am unaware of standard mathematical texts for engineers at a level beyond that of the usual advanced engineering mathematics texts. Perhaps the closest thing in some sense might be Strang's classic Introduction to Applied Mathematics or his more recent Computational Science and Engineering. While the aforementioned books do overlap with typical advanced engineering mathematics, they also introduce new topics and insights, in particular revealing the ubiquity (or at least broad applicability) of linear algebra and its structures.

My experience is that engineers tend to specialize and there is no one text at above-standard-advanced level to cover the main methods used in all domains. Springer does have a fairly recent Mathematical Engineering series that could be of interest, but many of the titles cover fairly narrow areas, fascinating as they are. There's also Chirikjian's Stochastic Models, Information Theory, and Lie Groups in two volumes, which could be interesting to engineers wanting to learn about stochastic processes and differential geometry, and might be a nice complement or lead-in to Arnold's Mathematical Methods of Classical Mechanics mentioned by Joseph O'Rourke in his answer.

Not certain this satisfies your set of criteria, but... This is a challenging but wonderful book. Arnold emphasizes the geometry of manifolds throughout: differential forms, Riemannian geometry, symplectic geometry, Lie groups and Lie algebras, dynamical systems, integrable systems.

Arnold, Vladimir Igorevich. Mathematical Methods of Classical Mechanics. Vol. 60. Springer Science & Business Media, 1989. (Springer link.)

Here is an excerpt from Liviu Nicolaescu's review on Amazon: "If you are looking for an easy read, this is the wrong place. ... This book frees your imagination, and it forces you to ask yourself many questions, ... This book is as eloquent an argument on the depth and beauty of geometry as you could find anywhere."

• AI Arnold is a classic and I have heard a lot of "free your imagination" talk surrounding this book not only from you but also several professors one working in differential geometry. Can you be more specific as to what Liviu Nicolaescu mean by "frees your imagination"? – Carlos - the Mongoose - Danger Jul 18 '15 at 7:08
• @IllegalImmigrant: Arnold emphasizes intuition and insight, and writes with great clarity. – Joseph O'Rourke Jul 18 '15 at 12:50

Geometric Methods and Applications for Computer Science and Engineering, by J. Gallier, does the job for what concerns classical differential geometry.

Elasticity and Geometry by Audoly-Pomeau addresses a classical problem with many important applications with quite advanced math techniques.

As for Linear Algebra, I'd like to mention Applied linear algebra by Olver-Shakiban. In this direction, however, it is quite different to draw a line between good mathematical texts with solid reference to applications and introductory books to numerical linear algebra.

Surprisingly not yet mentioned is

Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A. Orszag

I had a course out of this book 23 years ago. If someone or some company had approached me (or I had known who to approach) to go into applied mathematics and then do the kinds of things in this book for them as a career, I might very well have followed a different career path than I did. I loved the stuff in this book (we only did about half of it in the one semester graduate course I took) and I was constantly amazed at how certain techniques were able to squeeze out so much information in situations you'd think were just hopeless. A lot of the exercises are old Putnam Exam problems also, not that it helped me any -- my Putnam days were over by then, such as it was.

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One of the things I remember from this book is that it's often easier to analyze asymptotic and qualitative behavior by directly working with a differential equation than with an integral or a series expression.

• The Amazon link you provided states that the book is currently in its 1999th edition. :) – Joel Reyes Noche Jul 22 '15 at 0:17

I'm currently looking at Steinmann, Geometrical Foundations of Continuum Mechanics (Springer 'Lecture notes in Mathematics and Mechanics' series), as a way of making differential forms relevant and palatable to senior engineering undergraduates.

Let's take Kreyszig as the reference case, since I'm familiar with it. In terms of next harder books, I would say:

1. Arfken and Weber: unfortunately I HATE this book as do many physicists. (It's best use is as a door stop or ganglion basher.) It shows the signs of being written by someone(s) who was not a good math teacher or writer (not even talking about rigor, just basic math textook writing as for example Thomas Calculus or any regular math book). It is sort of a grabbag for referring to and not that good to actually study from. But it does have some harder topics that Kreyszig eschews. But where there is overlap to K, stick with K. That is a real book that is designed to actually teach math topics, not be a glorified, poorly organized reference. So, while I hate it, would be remiss not the mention it as many people see it as next hardest book. And it is on my shelf. But you have been warned about its flaws (see the rips on Amazon also.)

2. Books on asymptotics and perturbation methods. The one by Bender is best known but there are a few other good ones that are similar content. [This is probably the closest to a real "next book" in spirit.]

3. Full texts on some of the topics in Kreyszig (PDEs, complex analysis, numberical analysis, linear algebra, ODEs) Note, I think K actually does a rather good job of covering these topics. (High bang for the buck.) But there are standalone books that are more encyclopedic in terms of methods. This is even without getting into books that are super rigorous or that go deeply theoretical. Just more computational content. Some are quite good, with lots of problems and answers. In addition to PDEs and complex analysis, there are whole long good books on numeric analysis (including by Hamming). Some of these even have programs or discs.

4. A good primer on Calculus of Variations: there are a few that come up as most popular on Amazon. Stick to the easier ones. K doesn't cover it. Arfken (as usual) does a bad job of covering it.

5. A primer on integral equations. (Some of the advanced math methods books similar to Bender already have this.)

6. Handbooks on special functions or on differential equations or integrals. Some of these are really just like CRC tables. Others have some good discussion of usage or derivation or manipulations. Don't know any that have drill problems. Some names: Abromowitz and Stegun, Bates manuscript project, NIST website, Zwillinger. There are others. [As with all of these texts, check out the description on Amazon along with a few reviews and read inside to the preface description. Once on Amazon, you will usually get a list of competing similar books.]

7. Whitacker and Watson second half has a decent set of chapters on special functions. Some content above what you get in K. Also has a section on Mathieu functions.

8. Monograph by Watson on Bessel functions is whole book just on that. Also, can look at Byerly treatise on Fourier series and spherical harmonics. Both of these lack homework problems though. But they are pretty straight forward and have insights used in Jackson E&M problems or the like. There is one well known text on Mathieu finctions by McLachlan (not sure if it has problems.)

9. Some specialized texts on diffy Qs: Ince ODE and Forsyth Treatise of DE and Theory of DE (6 volumes). These have some theoretical aspects but are really mostly in the realm of engineers who want to solve equations. Not Fraktur letter craziness and Sobolov spaces.

10. Joseph Edwards 1000 page (!) text on Integral Calculus. Full of Cambridge math problems and tricky integrals. Sort of like having a handbook with problems. More tricks than in a normal course like Thomas or Granville or the like.

I would also say that there are some topics that are not really in the main line of Adv Eng Math (so not logically "next) but worth considering as forks for an engineer or scientist who just wants more math tools:

-Probability and statistics is a whole wide field from intro courses to all kinds of applied statistics and design of experiments. The book by Box, Hunter, Hunter is a great one for DOE.

-Operations research (search problems, queuing, etc.) Emphasize a text that is practical and from an industrial engineer perspective or military operations. Not super theoretical.

-Group theory. Some math guy will say entire topic of abstract algebra but that is way overkill for a technical guy and maybe even unpleasant. stick to a course or text on group theory for physics or chemistry.