on studying, remembering, and what to remember.

After a studying session for my prelim, I've observed a few things:

1. I'm making slow but steady progress. This might be illusion or delusion, but it honestly feels like I'm gaining some degree of intuition from the arguments that I've been reading. I can only hope that my memory serves and I don't forget it all, but perhaps that can be corrected by how I've been processing this information.
   
   
2. I've given up on remembering the technical lemmas which are used to prove big theorems. Instead, I've been trying to remember the big theorems ..
   
   
   (.. which aren't too hard; they are big, after all.)
   
   and what is the "right way" to prove them; this also allows me to build intuition on how to work with certain concepts and properties. If I remember the "right way," then I'll have to collide into what smaller claims I'll need to prove it, and those are the very technical lemmas that I would fumbled to remember. It's the difference between a priori and a posteriori knowledge, I suppose.
   
   Perhaps I am slow-witted and should have thought of this long ago. I've learned it now, though, and it suggests that I might not be a wholly hopeless cause .. (:
   

In particular, I never realised how cool the Area Formula for Lipschitz functions is. From the approach in the Evans-Gariepy text, the idea is to dissect the domain into Borel subsets, where the given Lipschitz function, when restricted to one of these sets, is "essentially" bi-Lipschitz with controlled constant.

One proceeds to argue that the Lipschitz images are comparable in measure to images from linear transformations (relative to these Borel subsets), and then we apply this linearity with impunity and without apology.

Fine ideas, if I've quoted them correctly (I've been wrong before), and it seems reasonably general (though the use of linearity worries me).

What strikes me most is the "optimism" or "courage" behind this type of argument: subdividing the domain as necessary, Lipschitz functions behave like bi-Lipschitz functions or linear transformations in the sense that we want them to. How does one learn such bravado, and learn to get away with it?

So let me say that I'm getting better at remembering, if only because I'm learning to reduce to the essentials. This could mean that I'm becoming a better student, but that doesn't mean that I'm becoming a better mathematician.

Oh well. It's something, at least.