Now I'm studying Rudin's Principles of mathematical analysis, but I'm searching for a book that offers geometric, physical or otherwise non-standard approaches to topics in analysis. Also, I'm looking for some book (like Bell's), that describe calculus techniques from a novel perspective (possibly emphasizing their applications).
Note: I'm not searching only for books that emphasize the application of analysisto physics, but also the other way round: a book that emphasizes the applications of physical arguments or geometry or any other non-standard approach to solve problems that should require a "standard" technique. For example, something like New Horizons in Geometry.
Best Answer
Zeldovich-Yaglom's Higher Math for Beginners (Mostly Physicists and Engineers) is an extraordinary book that should suit your purpose.
Although it starts at a very elementary level it discusses more advanced topics like Dirac's "function".
The emphasis is on Physics and the book is shock full of mathematical development of topics like jet propulsion, radiation in thermodynamic equilibrium, brownian motion, lasers, and of course all the standard stuff on center of gravity, moment of inertia, etc.
The book is written by the eminent mathematician Yaglom and the physicist Zeldovich.
And did this Zeldovich guy know what he was talking about?
Well, he got 4 Stalin prizes (better than 20 years in a Gulag camp or a bullet in the neck at the Lubianka) and 1 Lenin prize.
And not coincidentally he was one of the creators of the Soviet atomic and hydrogen bomb, along with Kurchatov and Sakharov.
Eerie, eh, where Calculus leads!