# University-level books focusing on intuition?

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

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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