# Interning at Wolfram: My Regeneration as a Theoretical Scientist

How does it feel to be an intern at Wolfram?

Most undergraduate college students chase opportunities for internships in New York, Miami, Seattle and particularly San Francisco at very young but large high-tech companies like Uber, Pinterest, Quora, Expedia and similar internet companies. These companies offer the best salaries, perks, bosses, coworkers, catered lunches and other luxurious amenities available in such large cities. You would seldom hear about any of these people pursuing opportunities in small, lesser-known towns like Ames, Iowa, or Laramie, Wyoming—and Champaign, Illinois, where Wolfram Research is based, is one of those smaller towns.

Many students want to go into computer science, as it’s such a rapidly developing field. They especially want to work in those companies on the West Coast. If you’re in a different field, like natural science, you might think there’s nothing beyond on-campus research for work experience. At Wolfram Research, though, there is.

Wolfram Research is a tight-knit company with a relatively small office where everyone can easily get to know each other. Fortunately, I have been in good company for the nearly two years I have worked here. Most of my full-time colleagues are highly qualified, with prestigious master’s or doctoral degrees in science or engineering. I have picked up a lot from their diverse knowledge related to the subjects I intend to pursue. Like most of them, I am a highly theoretical physicist, an applied mathematician and somewhat of a computer scientist myself, and this is what has compelled me to keep interning here instead of at other companies such as Intel or Boeing. Access to our company library, with its vast collection of books on modern computer science, applied science, mathematics, statistics, intelligent systems and various other subjects, allows everyone to learn about activities happening at much larger companies. This gives an incentive to keep picking up new skills and diversifying one’s industrial skill set for more outside projects. It’s thanks to the good influence of the library and the coworkers on me that I have stayed at Wolfram much longer than most other interns here in order to develop my skill set.

## A Little about Myself

My name is Parik Kapadia and I have been an intern in the Algorithms R&D department at Wolfram Research for nearly two years now. I am also a student at the University of Illinois at Urbana-Champaign, majoring in electrical and computer engineering with a minor in statistics. I’m also about to complete a Certificate in Data Science offered by the university.

## My Internship Projects

Throughout the time I’ve been working at Wolfram, my projects have allowed me to return to subjects I hadn’t studied for two years, since they related to high-school or first-year-college courses. I’ve also worked on projects far beyond what an entry-level employee would take on. Working on such projects as an intern has given me more experience than I would receive at other companies.

## Benchmark for Calculus

My initial training consisted of solving around 1,200 examples for intermediate and advanced calculus from a well-known textbook, *Calculus* by James Stewart, used by millions of students all over the world. It had already been nearly two years since I completed the three-course sequence of calculus required for engineering majors like myself, and this put me in a prime position to carry out this project. It turned out that the chapters covering Calculus I and II (or AP Calculus AB and BC) had already been completed by previous interns. This meant that I only had to complete the last six chapters of the course, which consisted of Calculus III. This final segment covers topics in multivariable calculus, such as vectors themselves, vector integration and differentiation, partial derivatives, multiple integrals, and vector calculus itself. In turn, these topics are used in physics and engineering, most particularly electromagnetism—one of my favorite topics in theoretical electrical engineering.

The result of this benchmarking project was a huge collection of around 6,000 problems completely solved using the Wolfram Language. The success of this project eventually led to a course in calculus that was released in September.

Since then, I have been coding an endless string of problems in advanced and applied mathematics from a collection of mathematics textbooks. Although I am an engineering major, I have seldom had the opportunity to study these textbooks for any major projects, although research at the graduate degree level for engineering may require the application of these as research becomes more interdisciplinary. Having coded more than 10,000 of these examples since the completion of the calculus book project, I have become a born-again applied mathematician and theoretical physicist, and now feel the need to return to those roots that I previously nourished when I was a high-school student. I am now hoping to potentially work full time at Wolfram Research in the near future.

## Mathematics Stack Exchange

My first formal, full-fledged project after the end of the calculus project was to collect a large number of examples regarding the use of the `RSolveValue` function to determine the limiting behavior of recursive sequences. In order to do this, I looked at all the relevant examples on Mathematics Stack Exchange and compiled a notebook showing how well they worked with `RSolveValue`. This was a very satisfying project, and you can imagine my thrill when Stephen Wolfram used a similar example to one I collected in his blog post announcing Version 11.2.

## Multivariate Limits

Another interesting project assigned to me was to check more than 1,200 examples for multivariate limits, which was a new feature in Version 11.2 of Mathematica. This required carefully going through all of the examples manually and doing sanity checks to make sure that the results and plots agreed, and to tweak the plots so that they looked elegant. Here, my knowledge of multivariable calculus came to the rescue, and I helped to select 1,000 examples that were used for the blog post “Limits without Limits in Version 11.2.” As you can see, people like myself work in the background at Wolfram to make sure that all publications and products are of the highest quality, and we take pride in maintaining the highest standards.

## Asymptotics

As a final project, I would like to mention my role in setting up large benchmarks for the asymptotics features in Version 11.3. I did this by collecting examples of differential equations and integrals from around 15 books, starting from undergraduate mathematics and engineering texts to advanced graduate-level discussions of asymptotic expansions. The challenge here was to make sure that the results from the new asymptotics functions agreed with the intuition and some numerical or symbolic comparisons with built-in functions such as `DSolve`, `Integrate`, `NIntegrate` and `Series`. The complete benchmark ran into around 4,000 examples and boosted the developers’ confidence in this exciting new set of functions, and some of the examples were used in a blog post after Version 11.3 was released.

## Future Plans

A total of 21 months have passed since I began my internship at Wolfram Research, and I am now looking forward to future plans. This internship has facilitated the process of my rebirth into a budding theoretical physicist, applied mathematician and computer scientist, and I shall build my career out of my growth into this level of achievement and how I have established myself while interning at Wolfram Research.

There are endless possibilities for me to hone everything I did here, of course—I gained experience in research and development as well. Government jobs, software and internet, data analytics, health care, bioinformatics, computer hardware and a diverse set of industries will need the expertise I have gained. The projects I have done as part of my internship have given me opportunities for moving on to quality assurance tasks, including debugging and testing Mathematica. Interning at Wolfram has given me the opportunity to make use of my skills and education, learn more about the directions I’d like my career to move in and build mutually beneficial relationships. If you’re a college student looking for a similar experience to my own, apply for an internship at Wolfram.

Find an internship in your field at Wolfram. Available opportunities are continually updated.

Great post. Straight from the heart.

” I am a highly theoretical physicist”. what is a HIGHLY theoretical physicist? are there theoretical physicists who are less than highly theoretical?

Really enjoyed this very interesting & well written post! The Wolfram Calculus Course Was Absolutely GREAT!! So glad to be able to actually THANK the guy who developed it! Thank You So Much!