# Are there any textbooks on multivariable calculus that introduce all non-trivial definitions and all non-trivial proofs with a “first draft”?

The idea is that a student who is reading from beginning to end will always be provided with a train of thought. The student will have no motivation to stop, and memorize an unexplained product of thought.

In many cases, it may be excessively ambitious to hope for students of multivariable calculus to learn to avoid making mistakes that were made by the original creators/discoverers, so it would make sense -- if some parts of the actual historical path of discovery/invention are used -- to include consideration of ideas that don't work, although it may not be immediately obvious that they don't work. Trains of thought don't proceed by the shortest path from beginning to end, but will naturally involve some dead-ends and changes of direction, but the changes of direction occur only after somebody detects a potential error, and confirms that there is an error.

It's true that students who have learned to succeed by memorizing may fail to read carefully, and may memorize some errors, but perhaps there is more to be learned from errors than from unexplained products of thought that were created by means of mysterious, undisclosed trains of thought.

• It might help to provide some examples of mistakes students make that you think are not addressed in standard texts. The only things I can think of involve nuances that are not appropriate for the intended reader. Also, there is a difference between actual incorrect statements and statements whose hypotheses can be significantly weakened. For example, the book may state a theorem with the assumption of continuity at a point, but maybe it's known that all we need is semi-continuity relative to each of co-meagerly many lines through the point. – Dave L Renfro Oct 29 '19 at 11:31
• Also, do you have any examples of single-variable calculus books that do what you are suggesting? That would help a lot in comparing - I'm not quite sure what you mean here. There are a number of interesting single-variable books that claim to take a "genetic" approach but I'm not sure if that's what you're referring to. – kcrisman Oct 29 '19 at 11:59
• Personally, I prefer to develop the material the way it "ought to have happened" historically and to avoid the errors and the dead-ends that actually happened, for the most part. – Peter Saveliev Nov 5 '19 at 17:00

## 1 Answer

One well-known work that might interest you is Schey's Div Grad Curl and All That. It discusses elementary facets of electrostatic fields as a motivation for developing key ideas in vector calculus. There is many points in the "narrative" where they describe why different strategies for solving a problem wouldn't work, which gives some sense of false paths that theorists go through. And it has helped generations of students to get an intuitive knowledge of how the formulas are used in applied engineering.