From calculus to engineering, Wolfram Language can assist in solving a variety of real-world problems and questions in many different fields. Staying up to date with the latest books written by our users is a great way to see how Wolfram Language can help in your work, studies or hobbies. The authors of the eight new books featured here demonstrate the widespread use of Wolfram Language. We were also able to catch up with three different authors, including Eric Schulz, to speak about their experiences with Mathematica and Wolfram Language and how they use them day to day.

Calculus, Calculus: Early Transcendentals and Precalculus

Eric Schulz, longtime Mathematica user, 2011 Wolfram Innovator Award Winner and mathematics instructor at Walla Walla Community College, first recognized the benefit of using interactive visuals to teach calculus while working with his own students. In 2010, his Wolfram Language skills and passion for helping students resulted in a brand-new textbook and online access package that met the challenge of helping students visualize calculus in a big way.

How did he do it? Schulz leveraged the power of Wolfram Language to not only create his visualizations but also build palettes of tools for use in his publishing process. The result was a textbook that teaches Wolfram Language and is also an interactive learning platform with hundreds of visualizations using the language to teach Calculus 1–3.

In 2021, Schulz and his coauthors—William L. Briggs, Lyle Cochran and Bernard Gillett—introduced the third editions of Calculus and Calculus: Early Transcendentals. Today, the calculus volumes include over 750 interactive visuals built with Wolfram Language that students can access through Pearson’s MyLab Math. Even more recently, Schulz worked with coauthor Julianne Connell Sachs to produce the second edition of another Pearson e-textbook, Precalculus, with another 350 interactive visuals to help students grasp concepts even earlier in their calculus education.

These editions include new concepts and revisions based on student and instructor feedback, as you’d expect with any revised textbook, but there’s more too. The use of Wolfram Language has also evolved over the years. While the dynamic visuals used to be available through CDF downloads that students could run in Wolfram Player, they now run seamlessly in the Wolfram Cloud with no extra software download required. This has opened up the content to be useful to students on all kinds of devices and expanded equitable access to the material.

Let’s see what Schulz had to say about the books and his use of Wolfram Language.

Q: I understand you built all the interactive visualizations yourself using Wolfram Language—more than one thousand overall. What was your process?

A: I coded each and every one of them by hand. An artist creates a painting of what they envision in their mind using oils and brushes. Similarly, I have a picture in my mind of an interactive visual and the story I want it to tell. I know Wolfram Language well and write code to bring the interactive visual alive. My artistic tools are in Wolfram Language. The process is very creative and artistic.

Q: What are a few examples of new content in the third editions of the calculus books?

A: From the second editions to the third editions, there’s new content and new visualizations in there. For example, we had not included a dynamic visualization of Simpson’s rule in the numeric integration section of earlier versions. In the third edition, there is now a beautiful interactive visualization of Simpson’s rule. Once a book is published, you get lots of feedback—“why didn’t you do this, why didn’t you do that” and you didn’t—so you make modifications.

Q: What’s next for these books?

A: On the most recent release, which this fall will be the first time it will be used, the text itself did not change. You could think of it as a 3.5 edition. What we worked on is content in MyLab, where there are thousands of homework questions. Many of these have what Pearson calls “learning aids.” If a student is stumped on a homework question, they can click a button and be guided through a solution. For this last edition, we spent a year and a half rewriting hundreds of existing learning aids as our own. We also created hundreds of new MyLab questions and wrote the associated learning aids in our own voice, using techniques we wrote about in the book.

Q: Will users with access to the third edition be able to use this updated content?

A: Yes, the new digital update is still based on the third edition of the text, so if they have access to the third edition, they have access to the new update as well. The MyLab codes that Pearson sells are not for a course; the codes are good through the life of the edition of the text.

Q: Are there any other textbooks you’d like to recommend in terms of great visualizations?

A: Yes, a linear algebra text published by Pearson, titled Linear Algebra and Its Applications. The authors are David Lay, Steven Lay and Judi McDonald. Judi McDonald teaches at Washington State University, which is a couple hours from me. Washington State University adopted our calculus text, and after seeing the Calculus interactive text, Judi reached out to me because she wanted to create something similar for the linear algebra text. I met with Judi and taught her about interactive figures and using Mathematica to typeset the text material. I worked behind the scenes on the linear algebra text. I took her source files and built the files that are deployed in MyLab using the Wolfram Cloud. The build process is similar to the Precalculus and Calculus titles; however, I’m not an author on the linear algebra text.

Q: I see that book is in its sixth edition. Did it always include Wolfram Language–built visualizations?

A: No, David Lay’s Linear Algebra has been a standard going back to maybe the 90s. It happened for, I think, the fifth edition—after I’d already finished the Calculus and Precalculus titles.

Q: What’s next for you?

A: I seem to always be pushing the leading edge. We authors are thoroughly engaged and not maintaining a status quo. When new things come out, I think about how we can improve our materials. I have considered for many years the challenge that visually impaired students have in understanding concepts that are explained visually. If a student is not able to see an image on the screen of a device, then an interactive visualization displayed on a screen is of no value. Now, with the prevalence of 3D printers, we can address this issue.

This year I have been building a library of 3D-printable objects based on the calculus concepts featured in many of the interactive figures that are in the e-text. For some of the interactive figures, there are multiple 3D objects that together tell the story of how change leads to a calculus result. By feeling these 3D-printed objects myself, I have come to realize the physical 3D models have the potential to benefit students who are not visually impaired too.

A good example would be when you take a curve and spin it around an axis to create a 3D solid whose volume is to be computed. A 3D-printed solid with a smooth surface can be felt so the shape of the solid is understood. A technique in calculus to find the volume of a solid of revolution begins by approximating the volume using some number of cylindrical shells, say five shells. A 3D-printed solid composed of the five cylindrical shells can be felt and understood to be significantly different from the smooth-surfaced solid. Finding the collective volume of the five shells requires geometry and algebra, no calculus.

Increase the number of cylindrical shells from five to 10 and print a second 3D solid. The second solid better approximates the smooth-surfaced solid and its volume can still be computed without calculus; however, even more cylindrical shells would be better. How many shells are needed to find the exact volume? Well, infinitely many. By taking the limit as the number of shells approaches infinity, the exact volume of the solid can be computed; however, this step requires calculus. A set of 3D-printed objects to accompany the computations helps to build a concrete understanding of the process and technique.

Schulz provided us with some photos of his 3D solids. The first image shows the solid created by revolving f x = sin x around the x axis for 0 ≤ x ≤ π. The next three images show volume approximations to the solid with n = 5, n = 11 and n = 101. The exact volume is found by letting n go to infinity:

The textbooks Schulz has been involved with for Pearson continue to be adopted by additional institutions. We look forward to seeing how he and his team continue using Wolfram Language to push boundaries in math education.

What Is Your Model? A Bayesian Tutorial

Dr. Romke Bontekoe, author of What Is Your Model? A Bayesian Tutorial, earned a doctorate in astronomy from the University of Groningen in the 1980s. In the 1990s, he worked for the European Space Agency, then briefly worked in risk management at a Big 5 company, and ultimately became engaged in the life sciences in the 2000s. Now that he is over 60 years old, his goal is to pass on his years of experience to the next generation.

What Is Your Model? A Bayesian Tutorial teaches readers how to perform data science through examples and figures. The book illustrates the inner workings of linear regression, nonlinear models and Monte Carlo methods as well as demonstrates the “art and craft” of data science.

“The approach is a Bayesian one,” Bontekoe said. “So, while every scientist agrees on the value of good data, most shy away when discussing the motivation of their priors. But suitable priors are equally important as good data. This tutorial guides the reader in constructing useful priors.”

The book teaches readers how to choose among different models in cases where many look credible, and the author helps readers along by explaining difficulties and mistakes he’s encountered and overcome.

We were fortunate enough to catch up with Bontekoe and ask him a few questions about his book, how his work ties into Wolfram Language and what motivates him.

Q: How were you first introduced to Wolfram Language? How has using Wolfram Language changed your work life?

A: My first encounter was around 1992 when Stephen Wolfram demonstrated Version 2.0 in the Netherlands. However, I could not get it working for my problems then. I really started in 2009 with Version 7.0. In hindsight, I regret that I did not start using it earlier. The strictness of Wolfram Language is a hidden gem. It has saved me from many errors, thereby saving lots of time.

Q: How do you tie Bayesian inference into Wolfram Language?

A: In principle, one can do Bayesian inference in any computer language. But there are two features unique to Wolfram Language that stand out. First is its arbitrary-precision calculation capabilities that are sometimes needed in the numerical computations of probabilities. Second, its symbolic manipulation capabilities that are essential in dealing with the probability expressions from Bayes’s theorem.

Q. How does What Is Your Model? A Bayesian Tutorial differ from other texts regarding Bayesian modeling?

A: Thirty years ago, I was asked, “What is a prior?” and I had no good answer. In this tutorial, I have tried to explain why prior probabilities are essential, how to construct them and what their deeper meaning is. This is highlighted in chapter 10, entitled “Real-World Priors.”

Q. What are some things that have motivated you in your writing career? What inspired this book?

A: I don’t regard myself as a good writer, and I got lots of help from others. My inspiration for this book, in part, came out of frustration. Namely, “Why is data analysis so tricky?” Again and again, I expected to fit a dataset like this:

And I got a terrible mess like this:

In this tutorial, I have tried to explain data modeling to the bare essentials. The many pitfalls are demonstrated and resolved. I hope that this little book saves others their valuable time by a better understanding and by making fewer mistakes.

Bontekoe is another example of a scientist who has found Wolfram Language to benefit his professional life. Seeing how he has used these tools to improve his work life offers insight into the many ways to put Wolfram Language to work.

Mathematica Beyond Mathematics: The Wolfram Language in the Real World

José Guillermo Sánchez León is an engineer, physicist and mathematics PhD holder. He is currently a mathematics and statistical consultant and teaches mathematical modeling at the University of Salamanca in Spain. He’s contributed to research in many fields, including modeling, optimization, medical physics, astronomy and finance, and he’s worked in the energy industry. In 1999, his project on statistical applications of Mathematica won him a grant in a Wolfram Research–sponsored competition. Since then, he’s helped Wolfram as an alpha and beta software tester. He is also a Wolfram certified instructor and has gained years of experience in both teaching and developing programs for Mathematica and webMathematica.

This thoroughly revised second edition of Mathematica Beyond Mathematics: The Wolfram Language in the Real World covers the new functionality added to Wolfram Language through Version 13.0. The latest edition introduces new features by using real-world examples based on experience the author has gained in his extensive work with Wolfram Language and as an instructor. The examples included in the text demonstrate a balance between the relevance and simplicity of Wolfram Language syntax, allowing readers to build their Wolfram Language skills in smaller increments as they navigate through the book. This edition also covers new topics, including machine learning, big data, finance economics and physics.

Guillermo Sánchez shows how Mathematica’s latest features can be used to answer questions from a range of fields: What sources of energy does the world really use? Are our cities getting warmer? How can we model epidemics, earthquakes and other natural phenomena? What is the best way to compare organisms genetically?

The book also points readers toward online supplementary materials, including code snippets and additional examples.

Guillermo Sánchez was happy to talk about his experiences with Wolfram Language and gave us more insights on his book.

Q: The second edition of your book, Mathematica Beyond Mathematics: The Wolfram Language in the Real World, covers a lot of the new functionality that has been added to more recent versions of Mathematica. What have been some of the more helpful additions that you have found?

A: Not only have new functionalities been added, but also the classic functions and some other fields have expanded considerably, such as Manipulate, curated data, machine learning, natural language and many more.

Q: You are very involved with Wolfram Language as a Wolfram certified instructor. What have been some of the biggest highlights of being associated with Wolfram Language both in your career and personal life?

A: Wolfram Language was an important help in preparing my PhD thesis. I have worked many years in the industry in different fields, such as nuclear energy, economics, environmental modeling, etc., and Wolfram Language is my language. I also teach in the university, and I usually solve problems using Wolfram Language. I took a big step in 1999 when I was in Champaign, Illinois, as a visiting scholar.

Guillermo Sánchez’s insights on how Wolfram Language is used in the “real world” definitely helps demonstrate just how much the latest version of Mathematica can accomplish.

Monte Carlo Methods Utilizing Mathematica

Sujaul Chowdhury, professor of physics at Shahjalal University of Science and Technology in Bangladesh, published this new book in February with Springer. Monte Carlo Methods Utilizing Mathematica shows how Mathematica can be used to solve definite integrals with Monte Carlo methods. According to Springer, “The author covers the acceptance-rejection method using uniform, linear, Gaussian and exponential probability distribution functions.” Readers can also expect a chapter on applying the variational quantum Monte Carlo method to a simple harmonic oscillator. Writing for an audience of mathematics and physics students, Chowdhury pulls together the most essential topics and provides a thorough background and examples to help readers not only grasp concepts but also apply the information to actual problems.

Applied Linear Analysis for Chemical Engineers

Complex Variables for Engineers with Mathematica

When it comes to textbooks on complex variables, publisher Springer tells readers, “Naturally, mathematicians tend to emphasize rigorousness and consistency while less emphasizing applications. On the other hand, books written by engineers often jump directly to the specific topics, assuming that the readers already have sufficient background of complex variables, and the pathway from theory to the application is not clearly elucidated.” Dr. Seiichi Nomura created Complex Variables for Engineers with Mathematica specifically to bridge the gap between books written by mathematicians versus engineers and successfully provides a seamless transition between the two. The text also focuses on showing readers how they can take advantage of Wolfram Language to calculate algebraic equations and visualize complex functions to avoid spending hours doing algebra by hand.