New book can teach anyone how to derive a formula

(Hayley Dunning for Imperial College London)

Two Imperial academics have written a book to help students gain a deeper understanding of mathematics through the art of deriving formulae.

In the age of machine learning, artificial intelligence, ‘internet-of-things’, and massive computer simulations, you could ask whether we still need formulae to describe physical laws.

Professor Alexei Kornyshev and Dominic O'Lee from the Department of Chemistry passionately believe formulae still have their place, and hope to inspire students to explore the art of formulae and appreciate how they can help check scientific simulations and experiments.

We spoke to Professor Kornyshev about the motivation for the book, How to Derive a Formula, and how it differs from most mathematical texts.

Why did you want to write a book about scientific formulae?

Great formulae, resulting as solutions of great equations, have totally changed our world. For example, the solution of Maxwell equations means the world is now connected via the Internet; the aerodynamic formulae that explained the origin of lift force and what shape wings should be to amplify it connected the world physically through flight; the formula for the rectification of the current at a transistor’s p-n junction opened the way to computing; and the formula that described X-ray scattering patterns from DNA helped to decipher the structure of DNA, giving rise to the birth of molecular biology – to name a few!

Deriving formulae is a fine art. Is this art only the preserve of intellectual giants? Actually, its basics can be learned, with diligence, logical discipline, and analytical skills. How important and useful your formulae will be is, of course, a different matter; they may not bring you the Nobel Prize, but they must at least be correct! This two-volume project is an attempt to teach this art from almost high school/A-level to undergraduate and postgraduate level. Volume 1 is about the necessary basics. Volume 2, to be published soon (work in progress) proceeds to more advanced and challenging methods for deriving formulae.

Why is deriving formulae an important skill?

Over the years of teaching, we understood that many, mainly non-math students, even those diligently studying mathematics, often cannot really understand how it works. They do not have a feeling for how the things are related, and often check their results by inspection of the derivation rather than by analysis of the predictions, order of magnitude of results, or their limiting behavior in the known cases.

Thus the book is not only about deriving formulae, but about teaching students how to understand what they have obtained; whether the results make sense, how they show themselves in the obvious well-known cases, and what their limitations are – in which domain of parameters they are expected to work and where they are not.

The book could not avoid introduction of a mathematical basis, but this is done with the intention to systematically ‘trivialize’ it, making it obvious and simple, with the main focus on teaching skills, and developing intuition and a common-sense attitude to understanding what you have obtained.

Who is the book aimed at?

Particularly ambitious high school/A-level students majoring in science; undergraduate students of physics, chemistry, and engineering, and even biologists interested in quantitative methods; and PhD students, both theorists and experimentalists.

We also hope that the book would be useful for lecturers teaching mathematical methods to non-mathematicians; mathematicians, except for applied mathematicians, have their own way of thinking and we do not want to interfere and mess around there.

With Volume 1 we are starting practically from scratch, which requires only the knowledge of elementary algebra, geometry and calculus, gradually increasing the complexity of the material. In that respect the book is self-contained.

Would you recommend the book to people who are math-phobic?

Very much so! This was one of the motivations to write it, as, sadly, many young students are phobic to math because they do not understand how its different parts work together, and have difficulty seeing the forest for the trees.

Also, with the opportunity to solve problems by pressing keys on your computer, using ready-made programs, one may be seduced into leaving the job to the computer. However, to understand those results, learning what is described in the book may be very helpful.

For creative scientific programming, which requires a lot of discipline but different skills and thinking, learning the analytical skills covered in this book may bring additional strength for the same reasons, and particularly for debugging of results that do not to reproduce the formulated laws that are expected to work.

What makes your book different?

Starting this project, we were motivated by typical student errors, their inability to match different concepts, and their lack of training in understanding their results. We furthermore kept in mind the approach taken in the great Feynman lecturers in physics. We wanted to do something similar for applied mathematics.

We hope that professional mathematicians will not blame us for this. The book was written by users of mathematics, who in the past experienced the same difficulties as our current students, and thanks to their great teachers managed to get through.

Wherever possible, we wanted to make the book amusing and fun. We wanted to show the beauty of some constructions and logical procedures, in the hope that our readers will become fascinated with them. There are historical diversions and humor, supported by entertaining cartoons and allegories throughout the book (judge us not!), to make it less dry and engaging for the young, and to share our love of this machinery.

Last but not least, it teaches using solved examples – purely mathematical (what we call “playing scales”), and physical applications, split by exercises for obvious intermediate derivations or side cases. Systematically, approximate analytical formulae are compared with exact numerical solutions. The physical examples are usually quite natural, but some, intended to demonstrate a particular mathematical trick, have ended up looking a bit contrived (forgive us for that, though Feynman perhaps would not!). Half of this 700-page book are those examples.

Whether we succeeded in making this book really different, not in any way competing with rigorous mathematical books, is for readers to judge. We did what we felt is drastically needed nowadays and, as best as we could in the limited time that we had.