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.