Of Engineers and Math

Just like mechanics need to know how to use their wrenches and hammers, engineers need to know how to use their mathematics. Engineers need an intuitive (as opposed to abstract) understanding of mathematics - even more in this era of specialization. Almost all empirical work has been done preservers like Edison more than 100 years ago. Edison's mathematical illiteracy could not let him see the Benifits of Nicola Tesla and Alternating Current. His approach to solving problems was not good enough even during his own life!

Today, almost every simple economically viable thing has been invented. All further "empirical" work is best left to managers, not engineers. Engineers (the good ones at least) emphasize more on understanding and exploiting complex phenomena. And the understanding of complex phenomena often implies a comprehensive and intuitive understanding of Mathematics - as opposed to the abstract understanding (or lack thereof) that mathematicians restrict themselves to.

It is this sentiment that has suddenly dawned upon yours truly. I have realized that I have wasted a significant portion of my life developing only a cursory understanding - certainly not comprehensive and intuitive - of mathematics. Am I letting myself atrophy into one of those mediocre, uninspiring researchers whose papers I see so often in journals?

It is here that I can start to count my blessings. I am in one of the World's largest universities which offers a plethora of courses in every conceivable field. Mathematics. Physics. You name it. I am now going to build a plan for my doctorate. Because nothing inspires more than old fashioned classes.

I need to cover the following more topics in Physics, Math and Engineering Before I pack my bags from here.

1. Hydrodynamic Stability Theories; Transition to Turbulence
2. Multiphase Flow
3. Statistical Thermodynamics
4. Microfluidics
5. Radiative Heat Transfer in detail
6. Convective Heat Transfer in detail

Math:

1. Tensor Analysis (I've done enough to "develop a feel" for it in Turbulence and CFD).
2. Complex Analysis
3. Perturbation Methods
4. Dynamical Systems and Chaos

It might look a little ambitious - but that's just 10 more courses. With a more comprehensive understanding of math, I will no more bow my head in shame when someone talks of Banach Spaces, Cayley-Hamilton Theorems, Christoffel Symbols (which I have been seeing a lot of late in the CFD class).

To the engineer, a formula is of little use unless it comes with an intuitive meaning. Take the Fourier transforms, for instance. Or let's just stick to Fourier series. If you ask a mathematician what the integral over the period of the function actually means, he/she will wince uncomfortably. But a competent engineer will come up with his but of "intuitive" bit of understanding which could go something like:

Consider a periodic funtion with a zero mean value. The integral over the period of the periodic function with sin (or cos) tells you how "similar" the function is to the sine or the cosine. The largest possible absolute value is when the function is an amplitude-scaled replica of the sinusoidal function, the lowest when there is no similarity at all - zero. The integral actually gives you the "contribution" of the sine signal of that particular frequency to the total signal. This can be repeated for other frequencies. And a Fourier series can thus be constitutive. That these amplitudes are additive stems from the orthogonality of the sine functions at different frequencies.

It would be great if I could put meanings to equations more comfortably. And you know what that is called: PHYSICS. After all, an engineer is a physicist with a more comprehensive understanding of economics than math.