October 16th, 2018, 04:13 AM  #11 
Global Moderator Joined: Dec 2006 Posts: 20,810 Thanks: 2153 
The best older books tend to get revised to correct any mistakes, to include new methods and results and to simplify complicated proofs.

October 16th, 2018, 04:18 AM  #12  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,829 Thanks: 648 Math Focus: Yet to find out.  Quote:
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.  
October 16th, 2018, 04:32 AM  #13  
Senior Member Joined: Oct 2009 Posts: 839 Thanks: 323  Quote:
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  
October 16th, 2018, 05:38 AM  #14 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
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. 
October 16th, 2018, 10:09 AM  #15  
Senior Member Joined: Oct 2016 From: Arizona Posts: 207 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  Quote:
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.  
October 16th, 2018, 10:29 AM  #16  
Senior Member Joined: Oct 2016 From: Arizona Posts: 207 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  Quote:
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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