Today in the linear algebra lab I TA, we solved a system of differential equations by using eigenvalues and eigenvectors to transform the system into a series of linear, separable and first order ODEs. Judging by the looks of astonishment and general silence that followed the resolution of the system of equations, the students could not believe how we turned a complex problem into a very simple one by using a branch of mathematics that may have seemed completely unrelated.
We followed a series of simple steps, that each student was able to complete on their own, to find a solution. We used knowledge gained in a first year linear algebra course to simplify a problem into something simple that we could solve.
I think that this is testament to the beauty and versatility of mathematics. Taking a problem that is seemingly unresolvable and transforming it into something that is simple and easy by using a simple series of steps.The way that mathematics are intertwined and how, seemingly, different branches of mathematics can come together to solve a problem absolutely blows me away. I think that is one of the most incredible things about math; the way the different branches work together to create simple and elegant solutions to difficult or unapproachable problems. The more I learn, the more impressed and awed I am by this fact.
There is definitely a certain beauty to solving problems for the solution, using math as a tool to solve problems and describe natural phenomena. Solutions are interesting, and more often than not, useful. It is helpful to be able to predict or model using mathematics. I have definitely used the outcomes from math that I cannot understand to solve problems in physics and I have been very thankful for the fact that I can simply use a solution that someone smarter than me has found.
That being said, I find the true beauty of math to be in the details and the procedures that give birth to the solution. The interdependence of mathematics that is present in so many disciplines is an amazing thing. I feel that I will never be able to understand it, but I hope that I will never stop reveling in the beauty that accompanies a proof or theorem.
My only regret today is that I was not able to help the students find their own way to the series of steps that helped us solve the problem. I was forced to present it. I do not know the leading questions that I need to ask, or where I need to limit my input so that the students can find the result by themselves, albeit with a little help.
That is something I must work on; let the students find their own definition of beauty in math, instead of presenting mine.