## What if?

Sara, Spencer and Cam (left to right) discussing their projects in office hour

What if?” is a truly powerful question. It is the question that separates the child from the adult; the student from the professional. Average students will go through the motions of a course and ask themselves many times: “What is the point of all this?” On the other hand, the notable students on their way to excellence will ask themselves: “How can I profit from this?” It is these inquiring minds who make it at the end: they are a pleasure to work with, they have the drive and the passion to get the job done, enjoy the process, and they are more likely to give their future employers more of their time in pursue of solutions—not because there is monetary or status gain alone, but because their commitment is only matched with their skill and curiosity.

It is a thrill to witness your own students pose that “What if?” question to themselves, and take steps to accomplish that little dream with the knowledge obtained in class. This semester, I had been blessed with a group of extremely talented people in all my different teaching assignments—especially those in my course on elementary differential equations.

Take Christina, for example: a sophomore chemical engineering student (originally from Greece!) who is doing a mini-grant-award-winning research on catalysis for her department. At some point she realized that the acquired knowledge on differential equations gave her a much better understanding of the way her catalysts work. She then applied that knowledge to further her comprehension on this topic, and did a side-project with us where she explores the mechanisms of diffusion and reaction on catalysts with different geometries. Some of her results might even be surprising to the scientific community! The fact that she obtained them all by herself speaks volumes of her skill and determination.

Take also the case of Spencer: A Texan whose family is in the energy business (oil, or course). He once explained to us how they set up these drills called “oil christmas trees.” He commented on the rare security hazards associated with such complicated structures, and for quite some time he travelled to different sites all over the country to test them. His goal was to find where weaknesses could lead to disaster. He developed a working theory that listed the main weaknesses of the drills, but his knowledge at the time was not strong enough to actually prove it. Shortly after revisiting the theory of mechanical vibrations in our class (this time through the scope of differential equations, rather than simply applications of the formulas), he found the missing link between his field work and the theory. He realized that a better model was possible, and that he had the skills to explain the impact of vibrations on each section of the drills independently (especially on nuts and bolts, which seemed to be the obvious weakest link). He then embarked into a search for reading material that could support his claims, and finally established communication with an Australian engineer—Dr.Saman Fernando—who developed a model with which Spencer feels confident a solution is approachable. The result of this collaboration could lead to a very simple solution to finding more secure structures in the future. I can’t wait to see how far Spencer will go with this project.

Tim is also from Texas, and also very passionate about oil but in a different way: It didn’t take him long to realize that one could use all the power of differential equations to explain what factors affect the size and shape of the oil spill in the Gulf, as well as its ecological impact. He managed to bring together six different research articles that explain completely unrelated properties of the damned spill, and create a unifying differential equation that accurately models its behavior… Yes, that good: The more we talk about it, the more accurate this equation becomes. I am looking forward to working with him on a numerical solution of this problem, and through the manipulation of the different parameters existing on his model, gain some insight on the way to control this huge ecological disaster.

Sara is another ChemE student, that once realized that it is possible to turn chemical equations into differential equations. One could then model complex networks of reactions (like those in the photosynthesis), create huge systems of differential equations that express how the different chemicals are created or disappear, and control not only the time that it takes for these networks to operate, but also to keep track of the production depending on the initial concentration of each of the necessary chemicals. In the process, she learned of the different approaches to this problem from other scientists, and why the field of differential equations is most certainly the best way to attack it.

Sam’s case is certainly something to write home about: she is extremely interested in economics and political science—something a priori far from the field of differential equations. At this level it is really hard to comprehend the intricacies of the subtle mathematics that govern the world of finances. Yet, her knowledge of differential equations sparked enough interest to try to understand and develop difficult concepts like Ito’s Lemma or the Black-Scholes equation. Note that, working by herself in this challenging area, she basically discovered partial differential equations and stochastic analysis on her own!

Ronen committed to a very thorough study of the catenary and some novel engineering applications. Unlike Galileo (who thought that the catenary was just a parabola), Ronen has a solid understanding of Physics. By revisiting all his knowledge from the point of view of differential equations, not only he learned how the different formulas were derived… now he can also tell when a “bad approximation” to a formula will not work in certain cases. That was precisely one of the points of taking this course, and he accomplished that beautifully.

Cam’s case is similar: He wanted to develop techniques to test the strength and durability of tree branches, to help households prevent disasters related to trees falling on roofs, cars, etc. This is usually accomplished by modeling the deformation of the branches as if they were beams attached to a wall. He learned in Physics that beam deformation can be expressed as a very simple differential equation of second order: $EI v'' = M(x).$ He was shocked to learn that this “simple” equation is actually not useful at all in his case, since its derivation assumes that $v'$ must be negligible. When dealing with life-death situations, the real differential equation is preferred, of course. He could not find it anywhere in the literature, so he decided to derive it himself and, with the help of the numerical methods learned in class, obtain more robust solutions. Another solid and interesting project, that I can easily see turned into a money-making machine.

I believe there is also a group of them trying to manufacture a clepsydra (a water clock) based on that lecture we had on applications of Torricelli’s Law. If they accomplish this task, I expect to see a video with the details!