The role of Mathematics in Engineering

“Scientists dream of doing great things while engineers do them.”

Before you read any further, it is imperative that one understands what mathematics is. Mathematics is the only science that can exist independently. There exists no other discipline in the field of science that can exist independently. It is someway or the other tied up to mathematics. The part to which each of these applications are tied is called Applied Mathematics while the other section is called Abstract Mathematics which deals mainly with postulates, lemma etc. Abstract mathematics mainly helps one to enjoy the mystic beauty of the subject independently. While applied mathematics helps to apply the same to practical problems, these problems are substantial with regard to engineering. Nevertheless, it would be wrong on my part to write that the problems are substantial with regard to engineering alone. Applied mathematics finds purpose even in medicine, data collection and analysis and so on. One can go on and on praising this and arguing against. Now the popular question comes- How do I use it in engineering? A popular question among my school mates and college buddies was this: “I really don’t understand why we read and study all this. I’m never going to use this.” Allow me to write further where each of the things come in practice and how good engineers use mathematics on a daily basis. To begin with, let me start with Engineering Mathematics- the subject that haunted several students from first semester to sixth semester during the course of undergraduate studies.

The first topic we studied probably is Matrix and Determinants. Upon first sight, we view them as collection of numbers in rows and columns. While that’s what it is, it is one way of looking at it. Let me go a step deep into what eigen value and eigen vector is. I write this because I along with my friends always thought – “Why do I subtract a square matrix from an identity matrix multiplied by an unknown scalar and find out the unknown by taking the determinant and equating it to zero(in case you didn’t understand what I wrote, I was writing about the step to find eigen value).” At first, it didn’t seem to make sense. It does now. Consider a photograph. Let’s say a painting. Each point can be considered as a vector with origin being the center of the photograph (origin is assumed here). Let us to an operation called shear mapping (writing about this would make the article long. You may google it). What happens is that all the vectors above the x-axis is displaced and the ones below are displaced in the opposite direction proportionally. Nevertheless, the vectors having no vertical component stay as it is. With regard to this application, eigen vectors are those vectors that stay as it is. This is because these vectors don’t have the effect of shear mapping on them. This is one of the fundamental concepts in image processing- a subject of special interest for Computer Science engineers. Let’s look at a mechanical engineering point of view. Consider a 3-D body subjected to a state of stress. One can always write the state of stress on a body in a matrix form. The eigen value would give the principal stresses while the eigen vectors would give the direction of those principal stresses. Again, principal directions refer to the plane characterized by the direction where no shear stress acts.

Let’s look at the next most important stuff we learned. This is probably the one that made people wonder the purpose of studying math. It’s called Probability and Statistics. Our understanding of probability is limited. We look at it as a subject where we are asked to find out the probability if rain would come or not (haha.. kidding). And statistics as simply mean, median and mode. Let’s look at distribution and correlation-regression for instance. Assume I have a set of values from a series of observations. I need to make sense out of these observations. Well, let me put it this way. If I can get one dependent value for every independent value and there are several such sets, there should be a relation between this that at the least closely satisfies. This concept is called correlation and regression. As far as engineers are concerned, we deal with a lot of data. And it’s our job to make meaning out of this data so as to predict and take necessary action. This is just regression for one variable. It can be extended to multiple variables. Let’s look at an application of probability. To make it interesting, let’s consider the movie “The Matrix”- In one scene, the rotor of the helicopter is seen to be striking a glass panel of window and the cracks are shown propagating. In an article, I read that engineers spent hell lot of time studying the mechanics behind the fracture in glass. This enabled them to show the crack propagation in a realistic way. Considering glass to have uniform properties throughout, cracks are supposed to propagate equally in all directions (at least in the first sight). But cracks don’t propagate in that fashion most of the time. Some cracks (in some directions) propagate faster while some propagate slow. As fracture is associated with failure of a structure, it is imperative for the structural engineer to find out the direction(instantaneous) along which the crack shall propagate at the fastest rate. This is done through probabilistic analysis. This enables him to make the design such that failure is minimized.

Last but one example: Differential equations and transforms. This is another subject that I scored great marks in but didn’t understand much of its practical importance. Lot of phenomenon that takes place in the universe can be characterized mathematically. Now, the variation of variables in a phenomenon cannot be quantified or characterized in general, they can be characterized only for infinitesimally small changes. Such equations, that characterize changes taking place in infinitesimally small rate is called a differential equation. This is the representation of the phenomenon. Its solution represents the equation in a given domain. Solution to differential equation helps an engineer find out the value at any desired point within the domain. This helps the engineer take necessary actions or steps so that the desired purpose is accomplished. Talking about transforms, not all differential equations are easy to solve. There exist some transforms that transforms differential equations to simple algebraic equation and thereby help to solve it at ease. An example of such transform is Laplace transform. Transforms are typically methods that help one engineer to make the process of solving some equation easier. Nothing more to it. At least, that’s how I view it.

Last and most powerful topic: Numerical methods. Not all differential equations (or any phenomenon) have analytical solutions. That’s why I always used the word “close solution”. This means the solution is close but not exact. Also, with the advent of super-fast computers, it is essential that some methods are found out such that a computer understands them. Computers understand only binary. That is, for any given phenomenon, computers typically solve it and deliver the solution to a particular point in the domain unlike, analytical solution, that gives a general solution and the value at a point in the domain can be found out at a desired point by simply substituting them. Some examples of numerical methods are Finite Difference methods, Finite Volume Methods, Finite Element Methods. Other examples are LU decomposition (solving system of linear algebraic equations), Newton-Raphson method (solving higher order degree equations) and Trapezoidal rule (numerical integration). These methods when implemented wisely give close solution such that the error can be neglected. As engineers, we deal with errors and our job is to optimize the solution such that the error can be neglected.

Now that I have covered the application of major topics of applied mathematics in engineering, let me give you the sole reasons why good engineers use it almost daily.

PS: No engineer builds stuff from scratch. Hobbyists do that. We mix and match components and follow standards. As engineers our job is to build products that move intelligently (or be a strong contribution towards the same).

The only reason why good engineers use math almost daily is this:

“Numbers don’t lie”

In a popular book titled “The Mckinsey Way” by Ethan Rasiel, he writes – facts can cover up for the failure of gut instincts (or intuition). This isn’t the exact line but it almost means the same. As engineers, it is the numbers that serve as facts. And numbers don’t lie.

“Math is the only place where truth and beauty mean the same thing”