## Math 341, Advanced Calculus, Spring 2014

Section: 17903

Official departmental syllabus

These are the notes I make for class, they are probably not of much use to anyone else.

Found on the internet:

One of the motivations for the group homeworks was Treisman's paper. Here's a typical extract:

[Successful students] studied calculus for about 14 hours a week. They would put in 8 to 10 hours working alone. In the evenings, they would get together. They might make a meal together and then sit and eat or go over the homework assignment. They would check each others' answers and each others' English. If one student got an answer of "pi" and all the others got an answer of "82," the first student knew that he or she was probably wrong but could pick it up quickly from the others. If there was a wide variation among the answers, or if no one could do the problem, they knew it was one of the instructor's "killers."

It was interesting to see how the Chinese students learned from each other. They would edit one another's solutions. A cousin or an older brother would come in and test them. They would regularly work problems from old exams, which are kept in a public file in the library.

Here's some motivation for why we have definitions and rigorous arguments:

To a first approximation the method of science is “find an explanation and test it thoroughly”, while modern core mathematics is “find an explanation without rule violations”. The criteria for validity are radically different: science depends on comparison with external reality, while mathematics is internal.

The conventional wisdom is that mathematics has always depended on error-free logical argument, but this is not completely true. It is quite easy to make mistakes with infinitesimals, infinite series, continuity, differentiability, and so forth, and even possible to get erroneous conclusions about triangles in Euclidean geometry. When intuitive formulations are used, there are no reliable rule-based ways to see these are wrong, so in practice ambiguity and mistakes used to be resolved with external criteria, including testing against accepted conclusions, feedback from authorities, and comparison with physical reality. In other words, before the transition mathematics was to some degree scientific.

This is an example of the incomprehensible nonsense that precise definitions replaced:

When you need to think very hard about something, try:

Lying down

Mathematicians usually have a hard time explaining to their partner that the times when they work with most intensity is when they are lying down in the dark on a sofa.

or maybe:

Walks

One very sane exercise, when fighting with a very complicated problem (often involving computations), is to go for a long walk (no paper or pencil) and do the computation in one’s head (irrespective of the first feeling “it is too complicated to be done like that”). Even if one does not succeed it trains the “live memory” and sharpens the skills.