calculus is the mathematics of change: it assigns exact meaning to two questions that stumped everyone from zeno to the seventeenth century — how fast is a quantity changing right now? and how much of it has accumulated so far? 𐃏 both questions are answered by the same primitive operation, the limit, applied in two different directions. everything on the pages below is a variation on that one move.
Limits
the calculus of one real variable, assembled in logical order: limits, then continuity and its two workhorse theorems (ivt, evt), then the derivative and the mean value theorem, taylor’s theorem with an honest error bound, and finally the riemann integral and both parts of the fundamental theorem. everything downstream — multivariable calculus, differential equations, every convergence argument in machine learning — leans on the theorems here. proofs are given where they are short and instructive; the deferred foundations live in real analysis (Courant, Richard, 1996).
I am finding Real Analysis to be more difficult than any other mathematics that I have studied before. I can seem to verify the truth of statements because they seem right; but I am having a difficult time producing rigorous and correct proofs.
It seems that High-School children (on the internet) are able to self-study Fomin with success. Bitterly, we remind ourselves:
“Comparison is the thief of Joy”—Theodore Roosevelt (probably)
Backlinks (2)
1. Wiki /wiki/
Knowledge is a paradox. The more one understand, the more one realises the vastness of his ignorance.