Math-for-physics books recommendation plea - with two twists

Question

As in the topic, I'm asking for recommendations for a books (supplanted perhaps with other resources) given my uncommon situation. The explanation is a bit lenghty, please bear with me.

Motivation and general aims:

I'm a freshly minted PhD in biology, I focus on protein structure and evolution, and bioinformatics is my main tool, from which Molecular Dynamics simulations is one of my favourite - so You can see where my interest in physics comes from - simulations of systems of rigid bodies, hydrodynamics simulations, force field and model building for MD (and getting parameters for those from quantum chemistry calculations), and so on. I am also fascinated by superfluids. Alas, as great as my undergraduate courses may have been, they have severly lacked math and physics - we only got half a year algebra, half a year of analysis and our physics course was, really... high school level at best.

Having finished my PhD and moving away from destitute country of my birth, I can now afford to try to grow, and that is what I am going to do.

Big Issue 1#

The issue, however, is that I am dyslectic, and strongly. In particular, the tests I've taken as a kid showed a striking disability for memorizing abstract symbols and associating them with their meanings (fun fact: it took me years to memorize for whom the triangle and circle stand for on toilet's doors). One can easily see how that can be a problem for self-learning math (and japanese too, unfortunately) from an average book - I've been trying Abbot's "Understanding Analysis" recently and rebounded like a ball from a wall, even tough I quite liked the conversational style. The problem is compounded by a fact that I am a postdoc and most of my time to learn anything is when I ride a bus to/from facility and read ebooks on my kindle - so I have to make all the transformations in my head.

Big Issue 2#

Let's be honest, my mathematical skills are rusty as hell. It's been 10 years since I've last had a proper math course, even more since I've really systematically cracked problems and thus I noticed that more often than not even tough I understand the idea behind some proof or derivation, I have incredible difficulty following it because author goes from equation x to equation x+1 without explaining HOW, because, well, it should be obvious. This comes with practice, but to practice one needs examples with explanations and exercises of particular problems with answers. (Abbot's book seems to be targeted for mathematicians and has exercises of proof building).

Wrapping it up: I am a biologist specializing in a field close to physics. So I want to learn/do some physics too, but first I need to build up my math. As a kid, i loved math, I had a great intuition for it and I am not scared. As a first year student, I've toyed with complex numbers, matrices, derivatives and integrals, but that was short and was 10 years ago. I need to rebuild my problem solving skills, relearn the facts I had (so how exactly do You make Gauss elimination again?) and I need to do it despite:

• my brain obnoxiously refusing to remember abstract symbols (so, figures, schemes and lots of text help, as well as operations as explicit as they can)

• having no one to ask for help/advice with every problem i stumble upon (explicit operations again, as well as answers to excersises, examples of solutions, etc.)

I am purposefully not listing the branches of math's I need - please refer to Motivations section where I describe the branches of physics I am interested in and use this as a guide. I think it is clear why I am posting this question here instead of mathematics.se

I will be happy to clarify should a need arise, please ask in the comments!

I would be very grateful for advice!

Answer 1

Not an answer or at least not a full one (especially since I don't address the requirements of your dyslexia, focusing instead on how to ease you into the mathematics), but far too long for a comment.

We can recommend molecular dynamics textbooks, but most assume physics knowledge that in turn is explained in physics textbooks that assume mathematics knowledge. Based on these two molecular dynamics textbooks (i.e. the first two suggested here) I'll bullet-point some scary-sounding jargon that summarises what you'll encounter, with some Wikipedia links you don't need to read in the entirety but can occasionally use as reference material:

• At a minimum you will need differential and integral calculus and differential equations, so you can understand Newton's second law as a second-order differential equation and the Taylor series motivation for approximating the force in Hooke's law as a linear function of displacement. This results in simple harmonic motion.
• You'll also need vectors and matrices for molecules' normal modes.
• A little knowledge of thermodynamics will be needed too, enough to compute a partition function and its mean energy; this applies calculus to (among other things) Gaussian and trigonometric functions, the latter measuring angles in radians instead of degrees.

I think what may help you the most is to have several types of textbook that complement each other, so that if you hit something unfamiliar in one book you can go to another for some background information. For example, Mathematics for Physicists and Engineers is a great resource for looking up any new mathematics you encounter, but just use it to look things up rather than reading it from cover to cover. For the physics you'll need to look up as you go, you'll probably find Wikipedia enough for starters.

I linked to two molecular dynamics textbooks above, but I'm not saying those ones will be the best for you. (It depends on what you want to read about, e.g. few books specialise in superfluids.) What I did notice is they don't seem to require you to do much if any quantum mechanics (which would require a lot more background reading), but I'm sure you'd have to get into that eventually. For now, here's a non-mathematical summary of the important physics:

• Quantum effects complicate molecular dynamics, but can be approximated away "at first order", and we can then just use older physics with a specific choice for how forces vary with distance.
• When a system is disturbed a little from its equilibrium position, we can approximate its behaviour like a lot of vibrating springs, because in terms of the mathematics of forces it comes to the same thing (this makes use of an aforementioned linear approximation).
• For van der Waals forces between atoms there is sadly no one-size-fits-all model, but this is a common model of the potential (which you differentiate to get the force).