After 7 years in the making, I have today submitted the final (I hope) manuscript for the book I have been working on with my two colleagues Stig Larsson and Axel Målqvist. The book is the first part in the series Analys & linjär algebra. The book series will be published by Studentlitteratur and if all goes well, the two first books in the series will be in book stores already this fall.
The books cover first-year calculus and linear algebra and set out with the following ambitious goals:
- To go deeper into the mathematical analysis than what is done in most calculus textbooks. This means no short-cuts, dealing properly with convergence and properly defining the real numbers.
- To integrate algorithms, programming, numerical analysis, and applications into the regular mathematics curriculum.
- To create a book that is compact and easy to read.
This is quite a challenge, but quite possible. In fact, (3) is made possible by (1) and (2); by spending time on defining and analyzing convergence and the completeness of the real number system (via Cauchy sequences), we can prove convergence of the algorithms that are presented in the book (bisection, fixed-point iteration, and Newton’s method) and, conversely, by spending time on programming and application of the algorithms, the understanding of the mathematical analysis is greatly improved.
For example, the proof of the Banach fixed-point theorem is carried out by executing the fixed-point algorithm and observing that a Cauchy sequence is formed. And since the students work actively with generating Cauchy sequences (by programming), the concept of Cauchy sequence becomes both natural, practical, and understandable. In this way, proofs and algorithms blend into one.
It’s a big undertaking to write a book (let alone four books) and it’s been a massive effort by my co-authors and myself. Here’s a glimpse of how the work has progressed since the project was started in 2013.
Now we look forward to completing the remaining three books in the series. The titles of the four books in the series will be:
- Differentialkalkyl och skalära ekvationer
- Integralkalkyl och ordinära differentialekvationer
- Linjär algebra och linjära ekvationer
- Flervariabelanalys och partiella differentialekvationer