### Advanced Engineering Mathematics (Part III) - Michael D. Greenberg

I bought this book way back as a first-year undergrad ('97-'98) when I was having difficulty with the applied part of maths - when we switched from the theory of analysis to applied multi-dimensional calculus, and the formality turned into hand-waving.

I never read this book at the time, and maybe I should have. It's definitely "engineering mathematics", where formality is dropped, and hand-waving whole-heartedly embraced. The other side of this trade is that it goes further than the formalised approach allows you to do in reasonable time, and it concentrates on intuition and real-world examples, both of which tend to get squeezed out in the more formal approaches.

Or maybe I should have got Kreyszig's Advanced Engineering Mathematics, which I understand to cover the same material, but is better regarded (and looking through the table of contents, covers a lot more). Oh well. This is what I have.

It's a real doorstop of a book, at 1324 pages. In my current state of health, I can hardly lift it while reading! I'm only reviewing Part III: Scalar and Vector Field Theory (of five parts). This is stuff I should know like the back of my hand, but, well, it's a little messy and nice to review from time to time.

The reason I'm interested in this topic, now, is that I've been working on understanding curved space, and really want to work through Darling's Differential Forms and Connections. The latter is a generalisation of 3D vector calculus, and you could argue I should jump straight to that, but it's nice to have something that I have a stronger intuition for that I can compare it with.

Overall, I would say that the presentation is good. It starts with differentiating functions of multiple variables, moves on to dot and cross product in 3D, integration over curves, surfaces and volumes, and then finishes off with "scalar and vector field theory", covering div, curl, grad, Laplacians, the divergence theorem, Stoke's theorem and irrotational fields. Everything is mode nice and clear with intuitive explanations, diagrams, examples and exercises.

As an engineering textbook, it loves reformulating everything in terms of Cartesian coordinates, cylindrical coordinates and spherical coordinates. While not useful for my interests, these provide good examples of how to manipulate the symbols, and is much trickier than the usual change of basis discussed in books for mathematicians.

It does accelerate somewhat, taking whole chapters to cover the basics, and then cramming an awful lot into that last chapter. Irrotational fields are a nice place to end up, as they're elegant and useful, although you might know them as conservative fields, with path independent integrals, being the grad of a scalar potential field.

As a mathematician, it's not completely satisfying, though. You get the feeling that an arbitrary vector fields should be able to broken into a not-curly (irrotational) part and an only-curly part (zero divergence, maybe?). The book says nothing, presumably because there's no engineering application, but a bit of Googling reveals the Helmholtz decomposition, AKA the fundamental theorem of vector calculus!

In fulfilling the goals of this part, I can't fault the book. Assuming the rest of the book meets the same quality bar, and ignoring alternative texts, I think I could recommend it.

Fundamentally, though, I still have a problem, mostly with the content and notation. Partial derivatives are a mess, with "How do I chain these?" depending on thinking hard, rather than standard, clear rules. Many variables are ambiguous, being used in multiple ways (although the text tries hard to catch and explain these). 3d calculus is treated as a special case, making generalisation much harder. The cross product and curl return vectors, when they should be returning bivectors (oriented areas), but instead we rely on vectors and bivectors being duals in 3d space. Really, it should be using some basic exterior calculus.

Notational messes crop up in the 3d calculus, where there are a huge number of "n is the normal of dA"-style notes. Del is an operator that is treated like a vector in order to create div, curl and grad, but what this means is never really dwelt upon. Proofs take a tour through various different notational approaches in order to find what is needed.

None of this is the fault of the book. Indeed, this is the clearest exposition I've seen of this notational minefield. I think back to e.g. predicate calculus, or various programming language formalisms, and there is just a world of difference. Fundamentally, this is bad notation that has accreted, and become both something people have become used to, and something expected if you are to communicate with other mathematicians.

I am vaguely hoping that differential forms (and to a lesser degree external algebra) will reduce the ambiguity, and hope to tackle Differential Forms and Connections next, but I suspect basic integral notation messes will remain. Oh well.

Posted 2024-07-03.