# How much Math do you REALLY do in your job?

I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I get the "most amount of math" you can get while not studying math.

In my courses I have done deep theoretical analysis for numerical concepts and application in CFD. But currently I am starting to wonder, how much the e.g. Calculus of Variation course really helps me in my future career. The theory you learn at university seems to get only a little application in the real word.

Example: In my numerics for PDE class I have spent (wasted?) so many hours on trying to figure out the CFL number of certain schemes, but what I am doing right now has nothing to do with that. Oh your simulation does diverge? Well let's take 2 instead of 4 as our CFL number. Furthermore, I am not really programming stuff as I hoped I could, but I am rather scripting. Fact is, 99 out of 100 people are not going to program a CFD solver. You rather use the code and apply it to your needs.

I am aware that university always follows a way more theoretical path than industry, but I am actually disappointed how little math I am really doing. Okay you might, say that's due to the fact that I am an intern and of course you are right. But I am in the lucky situation, that my team comes really close to research. Most of the members hold a PhD and studied engineering or math, and the focus is definetely on research ( in this departure of the company). But if the amount of math is that small in such an environment, where are you really able to make use of what you have learned at university.

So here comes my question

How much math are you actually doing at your job? And I don't mean, how much math is helping you to understand things, but how often does it happen, that you sit down and really do math in your non-academic job?

Personally I get the impression that I could do the exact work without having heard most of my courses. Don't get me wrong, I really enjoy the theory, but currently I am rather frustrated.

Note: As this is my first Question, I hope I did not screw up completely. I did not found similar questions on this side. And feel free to edit or ask questions if thinks are not clear.

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Don't buy the 'oh that gets used in lots of jobs' standard line. Not even in the bargain bin for 1$. Many pure mathematicians don't even try saying stuff like that which might be a bit more honest. If CFD doesn't do it for you, drop it and do something else. However, solving PDE's numerically does have a few other applications: other physics, computer vision - horn schunk method for optical flow or poisson image editing, graphics - radiosity global illumation and poisson surface reconstruction... – Peter Sheldrick Jan 29 '13 at 14:53 The theory in university courses is about understanding how things work (instead of teaching magic-black-boxes usage), so that whatever job you will have, you could figure it out. There is such a variety of tools, that no college would be able to prepare you just for that job you will have. Later usually the simplified approaches are used, because it is easier to apply, modify and extend. However, knowing all this sophisticated stuff you should be able to reason why it is alright to put into your formula$2$rather than$4\$ and have some intuition what the effect will be. –  dtldarek Jan 29 '13 at 15:02
@dtldarek yes you are right, university teaches you to understand the underlaying basics. But it would still be nice to hear from someone who applies the theorie learned at university for his/her job. –  sonystarmap Jan 29 '13 at 17:08
@PeterSheldrick SF means? –  sonystarmap Jan 29 '13 at 18:00
If you want to work in CFD, don't work for the companies using CFD. Work for the companies writing CFD software. –  Arkamis Feb 6 '13 at 19:40

I've spent the past twelve years as a professor. However, for five months last spring I spent part of a sabbatical working in the long-term forecasting group of an investment firm. I used a lot of math in those five months. (Admittedly, it was mostly a research-type position, and I gravitated toward the math-heavy problems.) Here are some problems I tackled on this job that required me to use math.

1. We have huge gaps in our set of stock prices because countries change currencies or come into existence or cease to exist or stocks move in and out of major indexes or name your reason. We need to know how these stocks rise and fall (or don't) with each other. How do you calculate a covariance matrix in the presence of missing data? The best solution often results in the matrix becoming singular. This is a big problem because your model requires you to invert it. Do you try one of the other solutions to avoid the singular matrix problem and accept the resulting drawbacks, or do you try to "fix" your matrix somehow? If the latter, what are the best ways to do that? Answering this question required a great deal of understanding of (well, to be honest, learning about) numerical issues in linear algebra.

2. We have a model that we're happy with that makes short-term predictions, and we have a model that we're happy with that makes long-term predictions. How about the medium term? How do we smooth our short-term predictions into our long-term predictions? To answer this question for us I had to, among other things, solve a couple of differential equations that resulted from trying variations on the logistic curve as models.

3. Our model is giving weird, erratic results. Why? Does it have some fundamental economic flaw? It takes me a couple of days to determine that the answer to this one has to do with the eigenvalues of the covariance matrix at the core of the model. Linear algebra once again.
4. I'm analyzing a set of economic indicators to determine their predictive value. Are they random walks of some sort, or do recent values have something to do with slightly less recent values? This requires time series analysis.
5. The woman sitting next to me is having trouble with her linear regression. I can't remember now exactly what the difficulty was, but I needed some statistics to solve the problem for her.

The job wasn't all math; I spent more time coding than anything else. But I did use linear algebra, numerical methods, optimization, statistics, differential equations, and even some calculus on this job.

So don't give up hope yet. Maybe they are in the research arm of a company, or maybe they require an advanced degree, but there are some jobs out there that require a good deal of math.

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Thank you very much for sharing your experience. This comes really close to what I am looking for. Thanks! –  sonystarmap Feb 1 '13 at 7:05

I've tried to put it into words, but I guess the following picture is much better (of course, please alter the text in your mind so that it fits the current context).

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In my job as a (thus far) applied computer science researcher, I can sympathize with you in that I am not utilizing as much math directly from my courses into my job. Sure, I have to know some basic probability, but much of my job consists of reading research papers where most math consists of the standard operations along with some tricky, but not difficult, summations and integrals. I am a math major, but I could easily perform my research if I had never taken abstract algebra or real analysis. If I ever did need knowledge from abstract algebra, the fact that I took the class makes it much easier for me to "re-understand" concepts again.

But one thing that may be misleading is that, no matter what you major in, you're unlikely to use a significant fraction of whatever you study in your everyday job. In my opinion, it's far more likely that you've encountered a small topic in one course that you end up working on in depth, as deep knowledge of one subject trumps general breadth.

Regardless, I still take math courses as they are among my favorites. And the point of university isn't solely to provide preparation for specific jobs; that's why people do internships. Experience can also trump coursework.

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Yes this really comes close to what I mean. Of course you need to be able to read math, but read isn't write. Sure some abstract classes are only for the purpose of having seen them, but if your major courses, the heart of your studies gets little application, doesn't that frustrate you? –  sonystarmap Jan 29 '13 at 14:54
It doesn't frustrate me too much, as I enjoy the required core courses for my major. And I think dtldarek in the comments above captured the idea nicely; they're not designed to specifically train you for a certain job. Even if one becomes a professor, the classes he/she teaches will differ slightly from the ones he/she took as an undergraduate. And if you ever DO need a certain topic from your courses in research, etc., then having the past experience of taking that subject makes it easier to "re-learn" it for the purpose of a job. –  TakeS Jan 29 '13 at 15:10

A bit of background first: my area is Computer Science, and I've been a student as well as a professional software engineer. As such, my answer is tailored to this profession. As a CS student I'd learnt a fair amount of mathematics : discrete maths, logic, plus the standard engineering stuff: ODE, PDE, Transforms, Operations Research. I've also picked up quite a bit of undergraduate pure maths by myself.

As a software engineer, the most mathematics I had to do was in the form of sanity checks and unit tests for my software. Arguably, the calculations hardly went beyond basic arithmetic, and sometimes involved a certain amount of algebra.No calculus, ODEs, Transforms, etc. Hardly have I "sat down to do maths"!

In the software industry, the core product development does take a substantial knowledge of relevant mathematics concepts, but the work of a typical engineer may not involve any of this. For example, one of my friends at a database firm told me that their core database team knew a great deal of logic and set theory , which is the maths that gets heavily used in databases. But the typical engineer was involved in the peripheral parts, e.g. designing the GUI, and this involved relatively little mathematics. A company is not necessarily interested in what amount of maths is used, but rather what brings in the maximum revenue.

Also, I've often encountered "maths heavy" (or otherwise arcane) software code wrapped up in easy-to-use programming libraries, e.g. image processing. In such a case, I'm the user , not the designer, and have to know relatively little maths. Such "packaging" saves the programmer from thorny implementation details and having to reinvent the wheel.

I totally agree with Peter Sheldrick's comment on your question. But it's important to point out that some of the benefits you gain by taking maths-heavy courses may be quite abstract. For example, almost every aspect of Software Engineering involves substantial abstraction( I dare say beyond algebraic geometry!) ; debugging is sometimes like finding an error in a proof. Mathematics courses encourage you to do both.

I also strongly recommend Keith Devlin's article on whether a software engineer needs maths, and suggest you ponder over this. HTH, and sorry for the long post!

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I liked the following quote from your link " The main benefit they got from the mathematics they learned in school and at university was the experience of rigorous reasoning with purely abstract objects and structures. " –  sonystarmap Feb 1 '13 at 6:56

Math has been useful to me in software business for

• the ability to organize complex systems analysis into parts (aka modules or lemmas)
• to be able to keep working on a very complex problem when the end is not in sight
• to have intuition when results don't look right
• to be very careful to get every single character in a program right. (Neither computers or proofs like typos)
• to layout code (or proofs) so that they are easy for me or others to read and check
• to learn how to learn by myself (programmers and mathematicians are always learning new ideas)
• how to debug or find errors in proofs
• to quickly pick up new syntax or notation, and to be able to create my own notation that is both expressive and powerful
• to be able to focus single minded on a problem for long periods of time
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Before I was in academics, I was a software engineer for a signal processing company. I also didn't "do" a lot of math. However, I had to know how to incorporate pre-written mathematical algorithms into my own code, and I had to use my strong logic to write and debug good code.

Don't discount the fact that you know math and use it to understand things. Knowing when an automated result from your software package seems fishy is way more important than being able to sit down and do integration by parts again. People without strong math and logic skills will see "p<.5" from stata and count their result statistically significant without thinking twice.

Just like D Seita said, it most jobs you are unlikely to really use exactly what you learned in school. But school should have taught you how to think critically and how to learn new things that come your way. If some academic somewhere somehow figures out how to apply reproducing kernel hilbert spaces to fluid dynamics, you'll be right on top of that!

Now, if what you really want is to do integration by parts, then come back to academics :)

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I have specialzed in High Voltage Engg.I have worked for about 8 years in Industry and more than 30 years in Academics. I used One percent of whatever Maths I have learnt in Industry and about Ten percent in Academics.

I cannot blame the designers of University Courses ,or the Industry or the Govt. for this Mismatch between Theory and Practice . It seems Education is in a process of Evolution --it is trying to meet the needs of Practical World . Similarly , the Industrial world is trying to get the Maximum out of the Educational System -they wil pay the freshly recruited students the maximum,if they can get an equivalent return from them.

I have worked for about ten years in USa and Europe , and the rest in India. The situation seems to be the same Everywhere.If we apply Darwin's theory of Evolution to this problem , we will come to the conclusion that Optimal use of Theory and Practice -is a continuous search or Evolutionary proess. No one knows the future -- every body is learning to face the uncertaintis of future , by a trial and error process.This is what I believe in .

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There's one (excellent!) real-life application of derivatives, the one that should be explained to people as early as in high school, I guess: neural classifiers aka ANNs.

My doctoral degree was, like, 80% mathematical and 20% computational, but my postdoc jobs are the exact opposite (probably even 90% computational). Nevertheless, I work with some real-life software (automatic machine translation) that is comparable in its performance to Google Translate that uses plenty of computational resources.

The main optimizer is backpropagation that uses derivatives ofthe activation function to find the delta-value used to update the weight of the connection between neurons. One such activation function is hyperbolic tangent:

$$h(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}$$

Without knowledge of how to take the derivative of this (and other) function you won't be able to develop even a simple neural network, let alone a complicated one! And the area of applications of them is very broad: from banking (classify the applicant as genuine vs fraudulent) to language translation to signal processing.

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