Monday, January 26, 2015

A Mathematical Vocabulary for Modern Engineering

My adviser and I sat down today for one of our typical meetings. After we had talked about some of my new results and discussed a paper (not mine, thankfully) which he referred to simply as "a mess", the discussion shifted. He started asking me about my background and what sort of math and physics courses I had taken while an undergraduate. We have similar backgrounds in a way--both physicists turned engineers, both interested in the more scientific side of engineering, and both less persnickety about obtaining rigorous numerical results in any given calculation than we are about making sure our concept is right. But one huge difference in our backgrounds is where we were educated. I went to an American public university while he was educated in the Soviet Union. I graduated just over a year ago, having the full blessing (and curse) of powerful computing tools at my side, while most of his computations were done with pencil, paper and a lot of careful thought. I knew the main point in his asking me about my education was to try and estimate the skill level of the students in a course he is currently teaching in plasma physics, and his realization that many American engineering students today are woefully uneducated or undereducated in the physics and math they need to understand and solve modern "high-tech" engineering problems.

He asked me how many semesters of general physics were typically included in our curriculum, I said three. He was clearly disappointed and said "there should be five." He then asked me what sort of math most engineers take. I said "three semester of calculus, one of linear algebra, and one of differential equations."

Him: Any probability or statistics?

Me: Not required.

Him: What about PDEs? How many semesters of ODEs?

Me: PDEs are also optional and only one semester of ODEs.

Him: Complex variables?

Me: A few weeks in some math methods class if at all.

He frowned again. "when I was an undergraduate we had two semesters of both PDEs and ODEs, and another for complex variables." My heart sank with his as I too realized how little most American engineers are taught about math. I chose not to inform him of how little regard most of them even have for math--useful math like series expansions, special functions, and complex integration; not just obscure modern mathematics, which can only rarely be used to gain useful results. He told me some students in the class couldn't even understand how to compute the average of a function, that others had no idea what Bayesian probability meant, that still others had no idea what the divergence or curl of a vector field was. This meant he would have to tailor his discussion of concepts involving these basics to a level far below that of a graduate course, something that myself as a student enrolled in the course found disappointing. It is a sad state of affairs.

On my way home I started thinking about how this could be changed. There's certainly no shortage of stuff to do on this count, but I thought that the first step might be establishing a basic mathematical vocabulary for modern engineering. This would be a list of mathematical topics to serve as an educational guide for people who want to be good engineers (as opposed to marginal, unqualified, or simply bad engineers) and would include actually "advanced" topics rather than the basic topics of calculus, power series, rudimentary linear algebra and ODE theory--which are typically taught in upper-division engineering math courses, but in reality should be assumed knowledge from freshman courses and high school. I've decided to list these topics and post them here. If I get enough time I might try to find good resources for learning what you need to know about the topics I've listed, but for now you're on your own. In order to be complete, I'll go ahead and list the basic topics under "Basic Math" but hopefully everyone with an engineering degree has at least seen those. The highly recommended (but not typically taught) subjects are classified as "Mid-Level Math"; these are math topics which can be learnt once the courses in Basic Math are completed and should be the bulk of the higher level math knowledge of the typical good engineer. Finally comes "High-Level Math" which may not be extremely useful for every engineer but is a good idea to learn to get more of a math background and is essential for some of the more advanced parts of engineering.

Basic Math (currently required at most schools):

--High School Math (basic algebra, geometry, trigonometry, and "precalculus", e.g. analytic geometry and some properties of functions.)

--Calculus (3 semesters worth hopefully with an introduction to vectors and multivariate functions).

--Linear Algebra including the concept of a vector space.

--Ordinary Differential Equations.

--Rudimentary probability and statistics (even this is not always required however)

Mid-Level Math (optional or not taught at most schools but needed frequently):

--Calculus-based probability and statistics.

--Partial differential equations, including eigenfunction expansions and special functions

--Linear operator theory

--Detailed theory of function spaces, Fourier theory, integral transforms, and completeness

--Complex variable theory

--Detailed vector calculus theory

--Basic numerical analysis

--Basic optimization techniques

High-Level Math (typically not offered for engineers):

--Tensors and tensor calculus

--Applied group theory (do not learn this from a pure math book. Instead look for a physics book)

--Differential forms and curvilinear spaces

--Stochastic processes and measure theory (again making sure it is not from a pure math book)

--Classical and modern theory of the Calculus of Variations (could really be "Mid-Level")

--Topology and mathematical analysis (needed mostly in control theory)

--Numerical methods for ODEs and PDEs (CFD or other computational fields)

Is this a lot? Well, yes. Unfortunately it doesn't change the fact that modern engineering problems typically deal with math from at least the "Mid-Level" category, if not the "High-Level" category (I recently was surprised to find the group at Purdue which focuses on tracking Space Debris using hard-core measure theory to construct their models of the debris field.) High school, as you see, barely teaches you the most rudimentary aspects of the most basic mathematics which you will use as an engineer. The math courses you are forced to take in college cover less than a third of what you will probably have to use and less than a quarter of what you might have to know to be a top engineer in your field. To circumvent this failure of the typical engineering curriculum you will have to put in quite a bit of elbow grease and more than a few nights of voluntary studying. But if you are patient, dedicated, and reasonably intelligent you will find the benefits to understanding real engineering mathematics are enormous. With a definite and extensive mathematical vocabulary we will make the first step toward building much more widespread mathematical literacy in engineering, but the hard parts are yet to come.

Good night and good luck, folks.

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