The Math of Life

I apologize for a longer-than-acceptable absence. My academic progress is getting in the way of so much of my life right now, including but not limited to this blog. I will, however, persevere in continuing with it – both the academics as well as the blog. I have also been working on a few projects outside of academia and outside of this blog. This includes a new website which incorporates poetry, chemistry, undergraduate papers, ACS news, and my tutoring in chemistry. For those of you curious, it can be found at http://poetchem.alexandersdesigns.net/.  Over the course of the next few months, I will be transitioning this blog over there.

For this blog entry, I will detail how one of my classes – in particular, Calculus III – relates to the real world. This is one project I have been working on for the past few semesters.  This will also be my final extra credit project for that course.

There is something you have to understand about the realm of mathematics, which seems to be either understated or unstated in all mathematics courses. Mathematics is basically logic. The logic which can be seen in all aspects of life.

The first concept we cover in calculus is what is called a vector. For those of you who are unfamiliar with it, a vector is just a number with a direction. You drive down the street going west towards your job at 35 mph (56 kph, 200 mps), that is a velocity vector, because it describes speed in a particular direction.

When we want to plot the course of an air plane, we have to input into it any potential course changes to maneuver around predicted weather conditions which would cause turbulence. To do this, the change in latitude, longitude, and altitude have to be taken into consideration. Therefore, when those course corrections are input ahead of time, they are input as partial derivatives of the position and velocity vectors with respect to latitude, longitude, and altitude.

When we want to make a better engine with more work per fuel used (in other words, higher fuel efficiency), we use what is called the dot product. The work output of an engine is given by the product of the force vector and the velocity vector at the piston head where it faces the cylinder (and thusly the exploding gas, a chemical process). The higher the velocity, the higher the force, or the higher both are, the grater the work done per cycle of the cylinder. The more the force and velocity vectors are directed towards the perpendicular line of the head of the piston, the greater the work is. For all engineering projects, including engineering new engines and modifications to modern engines, this dot product of two vectors to produce a scalar value of work.

Work done over distance traveled with this engine must be determined by using what is called the multiple integral. This is basically taking the route you choose and splitting it up into zillions of tiny little pieces (and yes, zillions is a technical term), and taking the sum of the work through all of those zillions of little pieces. When engineering a new engine, this long term summation of work over from the double integral (the function of adding together those zillions of pieces) of the cross product of the force and velocity vectors (as stated above) and maximizing that value while minimizing the wear on the parts of the engine.

In biological cells, there is this concept called the proton gradient. It aides greatly in the production of the production of ATP, the final energy currency in the body of you and me and every living thing on the planet. This proton gradient is described by Calculus as the change of proton concentration for the change in position vector relative to height, width, and length of the cell of the body. This change in concentration dictates how much ATP (and therefore, change in energy for you to do stuff) produced at any given spot can be described and predicted by calculus.

In environmental studies, there is this concept which calculus calls vector fields. This is basically the graphical representation of the vector of one aspect of the environment at each point on the planet. The vector field of wind, for example, is a description of how fast in what direction the wind is blowing in a city, in a state/territory, in a nation, or, if we so choose, the entire planet. The same thing can be applied towards oceanic motion.

These is the basic concepts of calculus which have real world applications. My prompt for you today is for you to find real world applications of mathematical concepts you are at least aware of. You will be surprised what you find out about math when you do this.
Until next time, don't forget to be awesome.

Take that as you will.

K. “Alan” Eister Δαβ