I write to you now from my office at the University of Notre Dame. It has been a long five years and as I sit here at my desk, I reflect back on teaching mathematics to the students of this incredible university. One of these reflections centers around maintaining the mathematical scrutiny of my class and their drive to discover.
Why are they here in the first place, you might wonder. Well, when your class consists primarily of engineers, about the only thing you can see in their eyes are dollar signs. Their thought process: Memorize the formula, excel in your technological prowess and you will be well on your way to a very comfortable lifestyle. And while this may to some extent be true, they are not sitting in the classroom for the right reason. They are not asking the right question: why? Why do these equations and these mathematical techniques exist? Where do they come from?
I say these things because I once thought this way and that's certainly not the right route to take when learning about science and mathematics. Let me take you back in time about two years ago when I was down for the holidays visiting with my family in the low country of Georgia. I was spending time with a good friend of mine in the area who works at the Federal Law Enforcement Training Center in Brunswick. We were sitting around sipping on some of his fine scotch when he started asking me questions about my research.
That's when I gave him the following set of equations \[ \begin{cases} \partial_tu+u\partial_xu+(1-\partial_x^2)^{-1}\partial_x\left[u^2+\frac{1}2(\partial_xu)^2+\frac{\sigma}2\rho^2\right]=0 \\ \partial_t\rho+\partial_x(u\rho)=0. \end{cases} \] I told him that here $u(x,t)$ represents fluid velocity and that $\rho(x,t)$ represents the fluid surfaces' deviation from equilibrium. From there, I began to explain everything that I've proven about this system of partial differential equations, but before I could finish he asked the right question: why? Why am I looking at this particular set of equations and where does it originate? To be honest, I didn't have an answer for him.
So, I hopped on a plane back to South Bend and my journey in search of the origins of these equations began. It didn't take long to discover that they stem directly from Newton's second law: Force equals the change of momentum or \[ \vec{F}=\frac{d\vec{p}}{dt}=\frac{d(m\vec{v})}{dt}=m\vec{a}, \] where $\vec{p}=m\vec{v}$ is momentum, $\vec{a}$ is acceleration and mass is understood to be constant. It was in 1687, that Sir Isaac Newton wrote down his three fundamental laws of motion and has been the basis for all classical mechanics ever since.
Then in 1757, Leonard Euler took it a step further and wrote down the governing equations for the motion of an ideal, inviscous fluid \[ \begin{cases} \partial_t\vec{u}+\vec{u}\cdot\nabla\vec{u}=-\vec{\nabla p} \\ \nabla \cdot \vec{u}=0, \end{cases} \] where $\vec{u}(\vec{x},t)$ is the velocity vector field and $p(\vec{x},t)$ is the scalar pressure acting orthogonal to the fluid region.
Unfortunately for Euler, very few, if any, fluids are inviscous and it took until 1821 for a gentleman by the name of Claude Navier to jump in and add viscosity to the set of equations and another 24 years till a rigorous derivation of the following set of equations that Navier produced \[ \begin{cases} \partial_t\vec{u}+\vec{u}\cdot\nabla\vec{u}=-\vec{\nabla p}+\nu\Delta\vec{u} \\ \nabla \cdot \vec{u}=0. \end{cases} \] was given by Sir George Gabriel Stokes. Here, we have $\nu$ is our viscosity coefficient.
It was at this moment that I saw where my set of equations came from. By taking the divergence ($\text{div}=\nabla \cdot$) of the Euler equations and solving a Poisson problem, one can see the similarity between the set of equations I posed to my friend and those that I just mentioned. It's all about the physical context that you are viewing the equations. A particular scientific lense that one looks through to view the birth of the set of equations that I work with from Euler, its father.
Returning from this short digression to the matter at hand, we must always ask our students the same questions my friend was asking me. Have them travel further on down the rabbit hole in search of the truth. To mathematically scrutinize all the models, methods and scientific madness that fills their courses. For it's not just a memory game of who can store all the equations in their head. It's the one who thinks about the situation, gets enveloped in the physical context and learns something.
We all have different opinions as to what fills our life with great purpose. For me, it's about learning as much as I can of the world that surrounds me and beyond. What better way to do so than through the looking glass of science. My only hope is that through the time I devote lecturing my students about this particular philosophy, some will take it to heart, break free from the temptations to just memorize formulas and actually participate in the field of mathematics.
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