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October 16th, 2018, 05:13 AM   #11
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The best older books tend to get revised to correct any mistakes, to include new methods and results and to simplify complicated proofs.
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October 16th, 2018, 05:18 AM   #12
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Originally Posted by Micrm@ss View Post

Or keep with differential geometry. Earlier books, like Kreyszig, often presented differential forms or tensors in a very unintuitive way. I much prefer the more modern point of view here. Sure, it's personal.
Side stepping for a moment.. what is the ‘intuitive way’ for forms in your opinion? I’ve been wrestling with these things for a while and I can’t seem to grasp the intuition let alone do something practical.

There’s a very informal ‘geometric approach’ by David Bachman which has helped me a bit. I’ve even been recommended Arnold’s take on them in one of his books on classical mechanics, it was quite brief. Happy for further suggestions.
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October 16th, 2018, 05:32 AM   #13
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Side stepping for a moment.. what is the ‘intuitive way’ for forms in your opinion? I’ve been wrestling with these things for a while and I can’t seem to grasp the intuition let alone do something practical.

There’s a very informal ‘geometric approach’ by David Bachman which has helped me a bit. I’ve even been recommended Arnold’s take on them in one of his books on classical mechanics, it was quite brief. Happy for further suggestions.
The best book for an intuition behind forms is Hubbard: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
I think it really motivates them well. Don't get the first or second edition, the later editions are much much better.

Although to be honest, while I think Bachman and Hubbard has nice intuition of forms, I've found that the best thing to do was just to stop looking for an intuitive understanding of them at every step and just see them as the most natural "thing" to integrate.

The smooth manifolds book by Lee deals with forms very nicely. He doesn't explain them at all intuitively, but the main stuff is there and if you reflect a bit, you will start to see why they are useful to us: they can be integrated but they're coordinate invariant, unlike functions which if you specify them you also need to specify the coordinates you work with (polar, spherical, ...). With forms, it takes all of this into account.

If you want to discover the sense behind forms, I think it would be a very very good thread where I can perhaps also learn things. It would be a nice change from all the disproofs of cantor
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October 16th, 2018, 06:38 AM   #14
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There is a readable basic introduction to forms in Marsden and Tromba p537 ff in my 4th edition.

This is set in mdodern mapping terminology.


Somewhat older is the more complete

Differential Forms with applications to the Physical Sciences

by

Harley Flanders


This starts off with a comparison between forms and tensors (they overlap).

Much of the book is set in conventional calculus and vector calculus notation so that crossover to more modern notation can easily be achieved.
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October 16th, 2018, 11:09 AM   #15
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Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
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The best older books tend to get revised to correct any mistakes, to include new methods and results and to simplify complicated proofs.
Of course, it could be my lack of understanding, but I thought this page (with the notation) was confusing. The way he writes the union of functions, and the supremum of a sequence of real numbers.

I've not heard of taking a union or intersection of two real valued functions. So, maybe I'm just missing something. I've heard of lim sup_n, but the way it's written here I find unusual.
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October 16th, 2018, 11:29 AM   #16
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Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!
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I don't know the book but it was pretty standard back in the day. I can't imagine measure theory changing all that much, at least at the introductory level. I could be wrong. But Halmos is known as a clear writer. His Naive Set Theory is a classic and well worth reading today.

So I would say that without actually knowing the book in question, I would recommend that book. At least start working with it and decide for yourself.

Can you say how his notation was different? A sigma algebra is a sigma algebra I'd think.

ps -- Your first sentence said, "I'd like to read the book!" That's the best possible reason to read it. If all you want is a simple introduction to basic measure and integration theory, I'd suggest Royden (the classic) or Folland (the modern classic).

But if what you want is to read Halmos's book ... then you should read it! I'm sure such an enterprise would be richly rewarding.
I ordered it because of Paul Halmos's biography and the awards that he won because of his mathematical writing. He also had a background in math that I could identify with, and I became interested in him as a person and mathematician.

I will definitely check those out. I have Folland, but it's intimidating to me, of course I haven't looked at it in a while.
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