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I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in university they seem to lack intuition, or simply they are taught to smother their own intuition with formalities they don't really understand.

I can occasionally come up with intuitive ideas, examples, pictures. Sometimes they come up with their own ideas, and ask me to check "if they got right what is behind". But this does not happen often, because they (and me) don't have much time to waste (or invest) in such "games".

A full book which focuses on the intuitive aspects, in addition to their own official text, sometimes is exactly what we need. I am particularly fond of the book "Visual Complex Analysis" by T. Needham, for example.

Are there any other books you know which focus on intuition, visualization, and understanding, rather than rigor and formalism?

Topics that would "call" for such a treatment are, in me and my students' opinion:

  • Differential forms and de Rham cohomology
  • Linear Algebra
  • Differential Geometry of Curves and Surfaces
  • Riemannian Geometry
  • Lie groups and Lie algebras (maybe with a focus on their applications to Mechanics, for physicists and engineers)
  • Relativity (special and general)
  • Probability and random processes.

Other topics are very welcome, too! (Also more advanced, if they exist.)

We could rephrase the question as: What are the introductory books you wish you had known before? Thanks.

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A Primer of Infinitesimal Analysis, J L Bell –  mistermarko Aug 10 at 3:35

3 Answers 3

For Differential Geometry a combination of Elements of Differential Geometry by Millmann and Parker and Elementary Differential Geometry by Andrew Pressley is very good for developing geometric intuition. Similarly for Riemannian Geometry DoCarmo's book on Riemannian Geometry is very good (one need to do a lot of exercises to extract concepts). For Abstract Algebra $Topics\ in \ Algebra$ by Herstein is the best (though good for a second reading). For topology, apart from standard Munkres' Topology I liked $Topology$ by Klaus Janich.

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I personally think Doug West's Graph Theory text is a great introduction to the subject. Godsil and Royle's Algebraic Graph Theory is a nice text as well, I think. It's quite an easy read for undergraduates with some linear and abstract algebra, as well as a bit of graph theory. I personally like Dummit and Foote for Abstract Algebra, but it's a bit sophisticated. Durbin is perhaps an easier read for those who are having some trouble.

Regarding linear algebra, I find graph theory and combinatorics to be an excellent precursor to explaining the concepts. Linear independence is analogous to acyclicity in a graph, if you consider Matroids. This makes it easy to visualize bases as spanning trees, which I think are less abstract. When talking about linear transformations, I find combinatorial intuition quite helpful. When seeing isomorphisms, teaching students to "see" the bijection can be helpful. It's also useful to use combinatorial insights for non-bijective transformations, such as $T: \mathcal{P}_{3}(\mathbb{R}) \to \mathcal{P}_{2}(\mathbb{R})$ by $T(v) = \frac{dv}{dx}$. When seeing the difference in dimension, it is easier to visualize combinatorially why such a transformation can be at most onto, but never one-to-one. Sorry if this is a bit off-topic, but I figured I'd share!

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The following might be helpful:

All the Mathematics You Missed: But Need to Know for Graduate School by Thomas A. Garrity and Lori Pedersen

Lectures in Mathematical Physics Vol. 1 and Vol. 2 by Robert Hermann (and other books by him)

However, I can't help but wonder about someone studying things like differential forms and de Rham cohomology who "didn't bother with mathematics in high school". I'd almost say that such people don't exist, except I've encountered a few. But only a very few. Almost always, in my experience, anyone that makes it to mid-level graduate mathematics was either a standout in mathematics throughout high school (unless they attended a very elite and/or special admissisons high school) or they had sufficient interest in mathematics to overcome a lack of top-level ability.

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Thanks for the answer. I have been a little unclear. With "while in high school..." I didn't mean the same students that I help in university. The people I help with differential forms, as you said, were quite good in high school, enough to choose such a major. –  geodude Mar 19 at 15:52
@geodude: I encourage you to look at the Robert Hermann books, both those I cited (which I actually own copies of) and some of his (very) many other books. I personally don't like his style, but the books are intended for those outside of mathematics who want to learn about certain aspects of differential geometry. Most university libraries (in the U.S., at least) have many of his books. As for the Garrity/Pedersen book, that's sufficiently recent and well known that you can easily google up information about it. –  Dave L. Renfro Mar 19 at 16:13

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