It is said that you should accomplish three things in life:

Plant a tree. Have a child. Write a book.

Out of the three, I have completed at least one. At the time of writing this Insight, my textbook “Mathematical Methods for Physics and Engineering” just hit the virtual online shelves. This Insight will describe the process leading to its creation, from the first seeds of an idea to where we are today. Although each textbook has its own story, I hope this one can satisfy the curiosity of anyone who wonders how some textbooks come to be.

The search for a textbook

As many of those who frequent the Physics Forums homework sections are aware of, I am a university lecturer working in a theoretical physics department. When I started my tenure track position five years ago at my alma mater, I took over one of the courses I appreciated the most as a student and faced a conundrum that many educators will face when giving a course for the first time: Which textbook should I use? In this context, it should be mentioned that (unfortunately) students in the program I was teaching generally expect the course to be defined by the textbook and rarely acquire or consult textbook sources apart from the main course book. The choice of textbook was therefore crucial.

The course covered four main topics: modelling physical systems using partial differential equations (PDEs), solving PDEs using orthogonal sets of functions, Green’s functions, and variational calculus. The course had been using a textbook in Swedish for some time, but it did not really cover variational calculus in a satisfactory manner and so I set out to search for an alternative. Among the books I looked at were classics such as the books by Arfken et al, Boas, and Riley et al, but none of the books I found really satisfied the needs of the course. While some covered the necessary material, they often covered more material than necessary, did not cover some of the required topics in sufficient depth, or presented a majority of its examples using physics that my students were not yet familiar with (mainly quantum mechanics).

Due to its extensive coverage of topics, I ended up trying out Arfken as the main textbook the second time I gave the course. However, I found it difficult to select problems that the students would have a sufficient physics background to solve and find meaning in and the reaction from the students was very mixed. Apart from the physics background necessary for the problems and examples, students found it annoying to buy a book that contained much more material than what was covered in the course. Several chapters in the beginning of Arfken discuss topics with which the students were already familiar from dedicated courses in mathematics and many chapters towards the end included material that was not relevant to the course.

Lecture notes

Without a really good alternative, I decided to return to the Swedish textbook and write a set of lecture notes on variational calculus to supplement it with the missing material. Hoping that the lecture notes might help a larger set of students than those I was teaching at the time, I decided to write in English rather than Swedish. This turned out to be the best fit for the course that I had tried so far.

At this point I had already been using Physics Forums for some time. I had spent quite some time in the homework forums and people appeared to appreciate my written replies, which was also soon underlined by the homework helper and science advisor badges. Furthermore, I was on the lookout for a hobby project. Where some people learn how to play a musical instrument or develop an affinity for stamp collecting, I decided to dedicate my time to extending my lecture notes to cover more parts of the course and eventually the idea of writing a full text specialised to my students’ needs took shape.

Deciding on topics

The first thing that was necessary was to define an outline of exactly what I wanted to write. One possible option would have been to just cover the topics of the course itself, but there was also other material that I wanted to include. My department was also giving an introductory course on vector and tensor analysis earlier in the academic year and I was also teaching an elective follow-up course covering Lagrange’s and Hamilton’s formulations of classical mechanics. Furthermore, I was also giving a course in special relativity at master level, where I found that one major hurdle was that students had forgotten (or never really learned) most of their tensor analysis. In order to create a text that would cover most of my students’ needs, I decided to include the following topics:

Vector analysis. Tensor analysis. Modelling physical systems using PDEs. Function spaces. Series and transform solutions. Green’s functions. Variational calculus. Calculus on manifolds. Classical mechanics. Electrodynamics. Special and general relativity.

I had intended the last three topics to introduce new physics subjects to the students, using the mathematics introduced earlier in the text. However, as time went by, I realised that the two latter topics would make the text very extensive and would essentially result in books of their own if I covered them to the extent that I wanted to. If you check out the table of contents for my book, you will find that the structure above is essentially the same, apart from the electrodynamics and relativity parts. A chapter on symmetry was later added as I found it to be a topic in which many students in the master program were lacking knowledge. It also provided me with an opportunity to discuss the use of symmetries in modelling and allowed the addition of some nice insights later in the text.

Once the scope of the text had been defined, I needed more structure and an idea of what would be included in each chapter. Before writing anything, I sat down with a notebook and dedicated one page each for writing down the topics I wanted to cover in every chapter and their ordering. I must admit that I at this time did not have a clear understanding of how long it would take to complete the text, but I did realise that it would be a massive undertaking.

Selecting the tools and starting to write

Being used to writing scientific articles and having some experience of larger projects from my PhD thesis, I knew that there was really only one option for typesetting and organising the text, ##\LaTeX##. As it would later turn out, the publisher fortunately agreed with this particular choice. For the figures, I decided that the best course of action was to create everything from scratch in order to avoid possible permission issues further on. At this stage I had not yet decided exactly what I would do with the final text and one option that was on my mind was to make the final PDF available online. I will discuss the final decision later on. Most of the figures in the book were created using Xfig, but there are some notable exceptions, where I took photos that were later edited in Gimp.

With the process of writing my PhD thesis in (not so fresh) memory, I also decided to write every chapter in a separate ##\LaTeX##-file and include them all using a master file. The first order of business was to set up this master file and a structure that I could work with. I chose to use a generic ##\LaTeX## book template and created one file for each chapter, in which I introduced all of the structure I had earlier written down in my notebook. I now had a skeleton of a textbook with the title “Introductory Mathematical Methods for Physicists.” Although I would review the structure of each chapter before writing it and made some changes along the way as I realised that some additional material was needed or that things were better presented in a slightly different order, this skeleton was very reminiscent to how the book is structured today.

Once the structure was in place, I started writing from the beginning and slowly worked my way towards the end. I vividly remember starting to write the first chapter on a weekend when I had nothing else to do. Before long I also needed to start constructing different figures, which I initially did when necessary. Although Xfig is a great tool for making vector based graphics, I soon realised that I also wanted something more. In particular, I needed a figure of a hand in order to demonstrate the right-hand rule in what became Figure 1.3. Being a lousy sketcher, I decided to base the figure on a photograph instead. Applying several Gimp contour filters, importing the result into and adding the arrows in Xfig gave the final result. Of course, before taking the photo, I needed to find an appropriate hand to act as model. Luckily, I have two of my own and the hands you will find in the book is mine. I say “is” since I needed the other hand to hold the camera. The final figure also has a mirror image of the original image in order to also demonstrate a left-handed set of vectors. I will leave it up to the reader to figure out whether the hand depicted in the figure is actually my left or right one.

Later I decided that it was more efficient to leave the figures empty with a comment describing what I wanted to show and construct all the figures of each chapter at the same time.

Once I had written about 20 pages in the format I was using, which contained slightly less text per page than the format you will find in the printed book, I decided to print what I had so far to get a feeling for how it might look in the end. I was rather happy with the resulting look and found some encouragement in that. Had this not been the case I am not sure that the book would ever have been completed. With this, the long process of writing had begun.

The long process

The actual process of writing a book of the size in question is a rather outdrawn one where not much of particular interest occurs. Rather than going through it chapter by chapter, I will just summarise how I approached the task.

As already mentioned, the book was my hobby project. At several occasions, my wife has told me she was astounded by the fact that I could write at the same time as I was watching TV, but whether that is a result of efficient multi-tasking or a testament to the average information conveyed by your average movie or TV-series I will leave unsaid. Apart from evenings and weekends, my daily commute was of the order of one hour one-way. Being otherwise wasted time (posting on Physics Forum using my phone aside), I started filling it with tasks related to my text. At first, the tasks I would do during the commute were mainly to bring out a notebook to go through the maths behind the next section, in particular when longer arguments were needed and I wanted to get the idea right before putting it in print. However, as time went on, this evolved into also taking out my laptop and writing entire sections. In the end, a non-negligible part of the book was actually written in a subway car.

For the figures, I would generally complete the text of a chapter before constructing its figures. Once it was time to do so, I would go through the entire chapter and make small rudimentary sketches of what I wanted to show in a notebook. Often this would result in several pages of tiny pictures, which were then realised in Xfig or by other means. With the order of 30 figures in a chapter, this would generally keep me occupied for an entire weekend per chapter.

The long process naturally also involved proofreading the text. While being a tedious task, it did give me something different to do and provided some variation, although I must admit I was lagging behind by a few chapters towards the end.

Completing the text

After roughly one and a half year of spending a large portion of my free time on writing about physics (do not feel bad for me, I enjoyed it immensely), I was approaching the end. The page count was rapidly approaching 900 and I had decided to leave electromagnetism and relativity out of the text for several reasons. Apart from making the text immense and likely too broad in scope, I felt that it would be sufficient to include the classical mechanics chapter as illustration of how the methods in the book can be applied to new settings. I would also be lying if I said that there was not a large part of me also wanted a completed project at that time. Once all the text was in place and I had been through an entire round of proofreading, the time to decide what to do with the text was approaching fast.

A tough decision

The decision on what to do with the text was not an easy one. When I took the decision to write in English rather than Swedish, my idea had been to try to reach as many interested students as possible, but how would I accomplish this? There were three options that crossed my mind:

I could put the final PDF on my homepage and advertise it sporadically by linking to it when called for in Physics Forums threads.

I could try to self-publish in the same style as former Physics Forums member Benjamin Crowell has done with his set of books.

I could contact a publisher to try to get it published as a textbook.

All of these options came with different pros and cons. If I had put the final PDF on my homepage, it would have been freely available for anyone, but how would they find it and read it? After all, it was a 900-page document and you can hardly expect anyone to print all of that and most people will not want to read that on their screens. In fact, already when on the lecture note stage, which only covered parts of the final document, students asked me for a printed version. Add to that the complication of people finding it and relying on awareness of the text’s existence being spread mouth-to-ear. I did consider self-publishing for some time, but in the end, I decided that the best method to spread the text was probably still to rely on a professional publisher with an economic incentive for promoting it and having a good quality print.

Enter CRC Press

A few years prior, I had been in contact with an acquiring editor at CRC Press asking me if I had any ideas for textbooks and encouraging me to submit a textbook proposal. Therefore, after making the decision to try to get the text published as a book, I decided to explore this path further and sent an e-mail to the editor asking whether they would be interested in the text or not. The e-mail immediately bounced back with an automated reply that the editor was on parental leave. A few days later I got a reply from the editorial assistant telling me the editor would return in a few weeks and that they would be happy to discuss the issue once she was settled in again.

In the end of September 2016, the editor replied and after some short discussions I was encouraged to submit a textbook proposal. Such a proposal involves providing quite an extensive amount of information about the textbook, such as the number of pages, the number of figures, and the number of equations as well as making summaries of the target audience, the typical expected price, what classes would make use of the textbook, etc. Furthermore, a survey of the possible competing texts already available on the market should be provided.

Although I essentially had a completed manuscript at that time, it was my understanding that it is generally more common to make the proposal before the bulk of the text is actually written. However, having written most of the text already, many of the quantitative questions were rather easy to answer. Additionally, the proposal should include summaries of the intended contents of each chapter and if possible any samples of text that could aid in the evaluation. Since I already had the text, I provided two of the chapters (Chapters 3 and 9) as sample material. After submitting the proposal in the end of October, a period of about five weeks when I did not spend much time with the manuscript and mainly waited for the response started.

A mountain of reviews

In the afternoon of December 6, 2016, an email from the editor appeared in my inbox with the subject “proposal reviews received!” Like most people, I am not a big fan of being evaluated, but the exclamation mark in the subject already took some of the nervousness away. Opening the email, it turned out to be quite long with seven(!) reviews attached and an extensive analysis of what they said. Being used to having one reviewer when publishing scientific papers, having seven took me a bit by surprise, but after thinking some time about it, it does makes perfect economic sense from the publisher’s side. While the revenue from publishing scientific journals is based on subscriptions and not the overall quality of a single paper, a publisher needs to ensure that there is a high quality and a reasonable demand for a textbook before spending resources on the publishing procedure.

Overall, all of the reviews were quite positive and provided a large amount of useful feedback, both for the publisher and for myself in terms of included material. Just like the textbook proposal involved answering a large number of questions from the publisher, so apparently did making a proposal review. The questions put to the reviewers included everything from the quality of the material itself to questions about the market, the most direct of them being “Do you recommend that we publish this book?” to which none of the reviewers had answered “no” and most were quite emphatically positive to. However, my personal favourite reply to the question was the honest “I am a physicist, not a book publisher.”

Apart from being generally positive, there was one thing that all of the reviewers agreed on. The word “introductory” should be removed from the title and the target audience of the book should be graduate or advanced undergraduate students. Since I did not have any particular problems with this, I was more than happy to adjust to this assessment.

I spent the next few evenings going through the reviews in detail and considering what they had to say about the text in particular. After discussing my thoughts on the reviews with the editor, she decided to go ahead and propose to the editorial director that the book should be published and that I should be made a contract offer.

Signing a contract

I received the first contract proposal a week or so before Christmas. At this stage, several things had to be settled, since the contract is what binds you to the publisher and vice versa. Many details about the final manuscript, such as the intended number of pages, were laid out in the contract. After discussions regarding several of the paragraphs, including the deadline, how the rights for the manuscript would be handled, how possible future editions would be written, and royalties, we ended up with a contract that was acceptable for both parties. Regarding the deadline, it was initially suggested that it should be put in mid-2017 since the manuscript was essentially completed already. However, in order to make sure to be able to deliver, I pushed to have it at the end of 2017 as I felt that there were still several things that needed to be done based on how the manuscript needed to be updated and proofread before submission. Finally, the contract was signed electronically by myself on Christmas eve and by the editorial director five days later. As fate would have it, I think I am about to receive my author copies almost exactly one year later.

Working to complete the manuscript

Already before signing the contract, I was aware that a solutions manual would probably be required. It had also been a request from several of the reviewers (although some did not like the prospect of students possibly getting access to solutions – there are always several different opinions on this among teachers) and so in parallel to the process of signing the contract, I started writing up solutions to all of the problems. In particular, I spent a lot of time doing this over the Christmas holidays, completing of the order of 250 solutions over the course of two weeks. We all have our hobbies.

Although the solutions manual had a later deadline than the manuscript itself, I did discover some issues with the problem statements while writing the solutions and so I made it a priority to work through all the problems to make sure that they were solvable with the given information and at a more or less appropriate difficulty level.

In connection to the contract discussions, I also got access to the ##\LaTeX## template to be used for the book. Although it decreased the page count from a mighty 900 to around 650, I immediately took a liking to the visual impact. It was probably at this point that I finally let myself change the reference to the text in my head from “lecture notes” to “book”.

After completing the solutions, I identified a number of issues that had to be dealt with and ordered them in terms of what would be the natural work flow, expecting each step to take between one and two months based on how much time I would have during evenings and weekends:

Converting all figures to grayscale. While just keeping the manuscript as lecture notes, I had initially constructed all figures using colours. However, to save printing costs and thereby making the book more competitive, it was necessary to convert all the figures to grayscale. At this point I also took some extra time to improve upon several of the figures, essentially ending up with the final versions that you will find in the book.

Adding new material. Based on the comments of the reviewers, several new passages had to be written and incorporated in the text while maintaining a natural flow. At the same time, the new material could not be too extensive in order to respect the set page limit.

Proofreading. I wanted to go through all of the text at least one more time and make the changes I found necessary. One problem I did run into at this point was to find a reasonable way to carry the text around. In particular, I wanted to be able to carry all the text with me on public transport as I would largely spend my commute reading. Carrying 700 pages that are not bound together around was clearly unfeasible and to be able to do this I decided to take up the art of bookbinding. After reading about some different techniques on the internet, I made an attempt and was actually pretty satisfied with the result (see figure). However, I am very sure that CRC Press will be better at checking that all pages are in the correct order and with the correct orientation than I was – a major part of chapter 9 turned out backwards. Still, there was something special about holding what essentially amounted to a book that I had both written and bound myself.

Time to submit

At long last, after revising many details of the book again and going through all of the figures to check for consistency in style and presentation, the time had come to let go of the manuscript. It is not an easy thing to do when you have worked on a project for such a long time to tell yourself that it is completed and I can assure you that this case was no different. Although I knew that I would have opportunities to make minor revisions at steps along the way, there is always the feeling of insecurity mixed with dread of losing control. I probably hovered over the submit button in the CRC Press file uploading system for several minutes and then had my wife do it for me. She seemed to have no problem whatsoever in sending away something that had been competing for my free time.

In the end, I did submit the completed manuscript only a few weeks after the originally proposed deadline and probably could have forced myself to complete it sooner. However, it never hurts to have some extra time in order to avoid a feeling of stress in the last moments of preparation.

The production process

The manuscript was now off my hands and to be fair, I cannot tell you much about the production process other than the fact that I have not been very involved in it. The major things I have had to do since submission has been to accept the title proposed by the publisher (actually, they do not really need me to accept it, it is one of the few things you really do not control as an author – the contract only requires the publisher to consult with the author), giving my input and thoughts on the cover art, and make some alterations to the manuscript based on the result of the publisher’s proofreading. In the end, the cover art is a photograph of a catenary somewhere in Stockholm with a stream running behind it – along with my own scribbles of some mathematics that can be used to describe them. Luckily, my own proofreading seemed to have been relatively thorough as there were actually not many things to take care of apart from some minor comments every three or four pages on average and some restructuring of the book’s front matter.

Here we are today

So finally, the time has come. The book is now in print and most likely available in an online bookstore near you. Just for reading to the end of this Insight, CRC Press has been kind enough to provide the readers of Physics Forums with a discounted promotional price if you want to buy the book:

Save 20% off the purchase of ‘Mathematical Methods for Physics and Engineering’ when you order online at https://www.crcpress.com/9781138056886 and enter Promo Code MPE18. Free standard shipping when you order online.