If anyone gets to the 495th or so comment, as a former mathematician, I recommend a pen and paper. If it helps you to have it in ASCII text (for searching, etc...) then take a few minutes at the end of the day to transcribe it.

My rationale -
A) Basic mathematical notation has evolved over a few hundred years to be concise and easily manipulated. It once was that equations were all described in text, so that even the description of the square root of a number was cumbersome ('a quantity, which when multiplied by itself yields the original...'). Expressing mathematical entities in textual psuedo notation, while quite a bit better than in prose, is a step backwards.

B) As you get more advanced in mathematics, you will most likely find the need for diagrams anyway (depending on the field). For instance, commutative diagrams in algebra, or all the bizarre pathologies in analysis (such as the 'walking ghosts'). There may be packages to do specific kinds of diagrams, but they will likely be unwieldy.

So get a nice pencil and paper. Or, like has been mentioned many times in this thread, get a Pulse pen. It's no worse than a pen ('cept for price), and you might find it useful.

(mini-review)
  I recently got a Pulse pen, and while I'm no longer taking class notes, it's quite handy. It's a tiny bit large for my hand, but fairly comfortable still. You can search for the textual parts of your equations, and it mostly finds them. I've downloaded the MyScript OCR free trial (for Mac), but I don't think I'll buy it, since it does a poor job with my poor hand.

Your comments are valid, but if one is concerned about security, there are a number of steps to take. First, in regard to the STL (well, to be accurate, that part of the library which used to be called the STL), you can test and debug with a 'checked STL' implementation. Such an STL will detect all out of bounds accesses (not just when using 'at'). They can usually check a number of common iterator problems such as using invalidated iterators, an invalid iterator range (such as foreach(a.end(),a.begin()) ), mismatched iterators ( foreach(a.begin(),b.end()), and so on. STLPort provides one such implementation.

As for problems with uninitialized variables and order-of-initializer problems, one should compile with full warnings on. These types of problems are easy for the compiler to catch, and once warned, usually easy for you to fix.