A New Perspective on Mathematical Analysis

[franznietzsche]

Looking for a Math book...

As the title suggests, I'm looking for a math book (or set of books). I'm not looking for a book on just one subject though. What I'm trying to find is a book that focuses on interrelationships between different ideas, particularly in analysis, but also in linear algebra.

As an example, the traditional approach to teaching vector calculus involves doing green's theorem in various form in two dimensions, then Stokes's theorem, then Gauss's theorem, and then maybe in the last five pages of the book talking about differential forms, and how they're all just one theorem. What I'm looking for is something that goes in the other direction, say starting with the idea of differential forms, and showing the special case consequences of the unifying idea.

Any suggestions for books that take such an approach or at least focus heavily on interconnects between mathematical ideas, especially in Analysis and Linear Algebra? Not necessarily intro books either, I'm looking for texts that offer deeper insight into the subject.

 

[mathwonk]

try spivak, calculus on manifolds.

 

[George Jones]

You might want to look at https://www.physicsforums.com/showpost.php?p=993751&postcount=4", although I don't think it contains what you're looking for.

Mathwonk: I'd be quite interested in hearing your opinion on Geroch in the above post, so if you find yourself at the library looking for a book to check out... It's there - I checked.

 

[franznietzsche]

You might want to look at https://www.physicsforums.com/showpost.php?p=993751&postcount=4", although I don't think it contains what you're looking for.

Mathwonk: I'd be quite interested in hearing your opinion on Geroch in the above post, so if you find yourself at the library looking for a book to check out... It's there - I checked.

I'm less interested in techniques, than in the ideas behind the techniques. To be honest, I'm absolutely sick of math texts and classes that focus on techniques. Memorize this technique. Memorize that technique. What I want is something that focuses on the underlying ideas behind the techniques, and how they interrelate; i.e., I want to understand what I'm doing, and see how it relates to the 'big picture'.

 

[George Jones]

I'm less interested in techniques, than in the ideas behind the techniques. To be honest, I'm absolutely sick of math texts and classes that focus on techniques. Memorize this technique. Memorize that technique. What I want is something that focuses on the underlying ideas behind the techniques, and how they interrelate; i.e., I want to understand what I'm doing, and see how it relates to the 'big picture'.

Then I really think you shoud get a hold of a copy of Geroch and at take a close look at it. Maybe also Isham's book on differential geometry. These are very nice books (by math standards) and are certainly not of the "memorize this; memorize that" type.