Book with novel approaches to analysis

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.

Solution 1:

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!

Solution 2:

There is something to be said for Calculus in Context by Callahan et al.

It emphasizes applications in the sciences. Students who will not use calculus in science or engineering courses that they take later should learn why calculus is important and considered a great achievement, rather than just learning to chant "n x to the n minus one", as in the conventional course. This book does something toward that purpose.

Solution 3:

I may recommand the series books of Analysis written by E.Stein. It consists of four books, Fourier analysis, complex analysis, real analysis and functional analysis. Comparing with the books likes principles or real and complex analysis of Rudin, you will find these 4 books follow a really different approach. It never lists many theorems of great generality, rather, it gives you an intuition of why we should learn such things, how can we apply these ideas to deal with another similar circumstance. More precisely, when you read these books you will encounter many concrete examples in advance which leads you to think by yourself about how to do if you were the first one who found it. Such concrete problems come from many different brunches within mathematics, including Fourier analysis, PDEs, geometry, or even number theory. You may not know many general theorems after reading it, but you'll have a fairly large probability to solve the general case by self when needed.

Solution 4:

From my side Thomas calculus is best book for calculus which will give rigorous explanations of differentiation using geometry. It starts from the basics of differentiation and goes on to advanced level like vector calculus.

Not only geometrical approach but it also provides writing exercises. A hallmark of this book has been the application of calculus to science and engineering.

Moreover it also provides computer explorations for math problems.

Solution 5:

You can find books on calculus from section Analysis/Applied Mathematics from the link below:

http://digital.ipcprintservices.com/publication/?i=140026&p=24