3 ms·
In North America, traditionally Calculus is where you learn how to use the techniques to solve problems, and analysis is where you actually learn how to prove t
by btilly 2y ago
In North America, traditionally Calculus is where you learn how to use the techniques to solve problems, and analysis is where you actually learn how to prove that it all is true.
But I have to correct you on the history. Essentially none of the material in a modern analysis course is the same as in Cauchy's day. You identify integration as coming after. But Cauchy didn't have a definition of a limit, axioms for the real numbers, a construction of the real numbers, set theory, the modern idea of open and closed sets, and so on and so forth.
- dan-robertson 2y agoAh thanks. My understanding was the whole epsilon-delta stuff one typically learns in a first course in analysis was due to Cauchy.
- btilly 2y agoNo, epsilon-delta in the modern form was invented by Karl Weierstrass in 1861. Cauchy's approach was to define an infinitesmal as a sequence approaching 0, and then to use the old infinitesmal definition of a derivative. This almost works. One of the places where it falls apart is defining the chain rule when the inner derivative is 0, because the sequence may wind up with 0/0 infinitely often. But it is from this definition that we get Cauchy sequences today.