Engineer Studying Pure Math: Benefits and Application

[zeralda21]

I have recently completed my first year of Eng. Physics taking the standard math courses: Calculus, Linear Algebra 1 and 2, Multivariable Calculus and Numerical Analysis.

Recently though I have been self studying Rudin's "Principles of Mathematical Analysis"/Abbott on my own and I enjoy this "kind" of proof based mathematics a lot more than traditional calculations. Does it ever makes sense for an engineer to actually study pure mathematics of this kind or is it,"application-wise" , a waste of time?

 

[Useful nucleus]

The following thread touches the same point:

https://www.physicsforums.com/showthread.php?t=459575

In my opinion, this interesting blend of Engineering and Pure Mathematics if can be present in the brain of a single human being, it can help in finding new applications and can also lead to the creation of new interesting theoretical questions.

 

[MathematicalPhysicist]

I have recently completed my first year of Eng. Physics taking the standard math courses: Calculus, Linear Algebra 1 and 2, Multivariable Calculus and Numerical Analysis.

Recently though I have been self studying Rudin's "Principles of Mathematical Analysis"/Abbott on my own and I enjoy this "kind" of proof based mathematics a lot more than traditional calculations. Does it ever makes sense for an engineer to actually study pure mathematics of this kind or is it,"application-wise" , a waste of time?

It's not a waste of time if you enjoy studying pure maths.

 

[lurflurf]

It is not clear what you mean by "application-wise" and "Pure math". Rudin's "Principles of Mathematical Analysis" is a (pompous and not particularly good in my opinion) calculus book. Calculus is quite useful for engineers so I would not say it is pure or a waste of time (though another book might be a better use of time). Pure math becomes applied math when a use for it is found.Some subjects like graph theory and geometry are quite useful at times in engineering. That said time is limited so it is wise to focus on basic math and science that are known to be most useful rather that obscure subjects of unknown value.

 

[MathematicalPhysicist]

It is not clear what you mean by "application-wise" and "Pure math". Rudin's "Principles of Mathematical Analysis" is a (pompous and not particularly good in my opinion) calculus book. Calculus is quite useful for engineers so I would not say it is pure or a waste of time (though another book might be a better use of time). Pure math becomes applied math when a use for it is found.Some subjects like graph theory and geometry are quite useful at times in engineering. That said time is limited so it is wise to focus on basic math and science that are known to be most useful rather that obscure subjects of unknown value.

Innovation doesn't come when you're stuck with the same known methods!

It's all about taking risks, most people prefer to take it safe and the sure bet. But humanity didn't come this far by just doing the sure thing.

Anyway also abstract algebra has applications in CS and Engineering, even if a UG Engineering might not see the use in it.

 

[zeralda21]

It's not a waste of time if you enjoy studying pure maths.

Yes, certainly. My point was that I would rather(especially since I am in an engineering program) study mathematics/physics/whatever that will be of benefit in my future jobs rather than "I know analysis really well but I can't do anything with it". I didn't mean it is a waste of time, rather that it is a lot of time and struggle spent giving me nothing else than enhanced problem solving skills (and of course amusement). If that is the case, please tell me. Is analysis and "pure" math(think of the structure of Rudin) only should be red by future mathematicians?

 

[MathematicalPhysicist]

Well, if you want to understand the theorems, and by that I mean understand their proof, you need to understand books like of Rudin's.

But there are better books than of Rudin's, Spivak's and Courant and John's are better as I see it, and they are less terse than baby Rudin's.

 

[jasonRF]

Yes, it is useful. Most electrical engineering PhDs I know that specialized in signal processing / communications / controls took at least one intro real analysis course on that level.

Sure, some of the results are useful, but I think that the primary benefits are: 1) ability to comprehend the more theoretical literature and apply those techniques appropriately, and 2) having the ability to really do proofs allows the engineer to obtain deeper understanding of techniques and sometimes prove bounds that can more efficiently drive the design process than simply doing a ton of simulations.

I have self studied a little pure math on my own since being in industry, primarily for reason 1) above, but have found that it helps with 2) as well.

 

[lurflurf]

Innovation doesn't come when you're stuck with the same known methods!

It's all about taking risks, most people prefer to take it safe and the sure bet. But humanity didn't come this far by just doing the sure thing.

Anyway also abstract algebra has applications in CS and Engineering, even if a UG Engineering might not see the use in it.

Yes I think we mostly agree. There is a lot more math that anyone can learn. Deciding what to learn is a problem for mathematicians and even more so for engineers who generally learn less math because they learn more engineering and science. Algebra, geometry, and even arithmetic have known uses and (depending on emphasis) can be considered applied math. I think you will have to admit that on average an engineer will benefit more from reading a book on partial differential equations than one on K-theory. Maybe some iconoclast will revolutionize engineering making K-theory of central importance. Something I find funny is that in the nineteenth century engineers seemed interested in topology before mathematicians had done much with it. Now that mathematicians in the last hundred years have developed so much topology that sits ready for engineering use the engineers seem to have lost interest.

 

[Bipolarity]

It depends on the area of engineering. Signal processing makes heavy use of math that many would consider "pure" mathematics, in the sense that it is usually studied by mathematicians/physicists, and most non-PhD engineers wouldn't have a sufficient knowledge of mathematics to explore signal processing at this level.

If you are a lover of math and engineering, I will recommend signal processing to you.

BiP

 

[MathematicalPhysicist]

Yes I think we mostly agree. There is a lot more math that anyone can learn. Deciding what to learn is a problem for mathematicians and even more so for engineers who generally learn less math because they learn more engineering and science. Algebra, geometry, and even arithmetic have known uses and (depending on emphasis) can be considered applied math. I think you will have to admit that on average an engineer will benefit more from reading a book on partial differential equations than one on K-theory. Maybe some iconoclast will revolutionize engineering making K-theory of central importance. Something I find funny is that in the nineteenth century engineers seemed interested in topology before mathematicians had done much with it. Now that mathematicians in the last hundred years have developed so much topology that sits ready for engineering use the engineers seem to have lost interest.

Do you refer to Kirchoff rule for electrical circuits as an example of topology use in engineering?

 

[MathematicalPhysicist]

It depends on the area of engineering. Signal processing makes heavy use of math that many would consider "pure" mathematics, in the sense that it is usually studied by mathematicians/physicists, and most non-PhD engineers wouldn't have a sufficient knowledge of mathematics to explore signal processing at this level.

If you are a lover of math and engineering, I will recommend signal processing to you.

BiP

Mathematics is useful in any branch of engineering and physics.

 

[Bipolarity]

Mathematics is useful in any branch of engineering and physics.

Yes my point merely was that certain branches may more use than others. In this regard signal processing is known to make the most use of abstract mathematics. Concepts from funcitonal analysis and measure theory show up very often in signals. Not so much in other fields of engineering.

BiP

 

[JakeBrodskyPE]

How about a been there, doing that point of view?

When I first started engineering professionally, my mentors told me that if I needed anything more than a plain scientific calculator to figure out the primary designs, then I was probably doing something very wrong or very unique. Incidentally, he has a strong background in mathematics.

It is usually best to stay within the well known regions of design. If you're writing out equations for those regions instead of using rules of thumb, then you're far more likely to be wasting time and making mistakes. If you're heading off the beaten path into regions of design where few have been before, then yes, you'll need your understanding of mathematics, but you need to be absolutely certain of what you're doing and why you're forging ahead on your own.

In 27 years of practice in the field, I have had a few of those unique situations. But the actual math that you cite is not common in practical, hard-nosed, engineering. Either these functions are canned in a piece of software, or there are rules of thumb that you probably didn't know. Those rules of thumb typically also take tolerances and wear into account. If you're not leaving some margin for error in your designs, then you're doing what we call "Engineering with a VERY sharp pencil."

That said, without the mathematics foundation that you are learning, you'll have no clue where those rules of thumb came from; and thus you won't know where to expect those rules of thumb to break down. Eventually you will reach a point where they do break down. If you're a very responsible and practical engineer, you'll usually stay away from those regions of design. But sometimes there is a compelling reason to forge new designs into new territory. I've been there in a few situations over the last 27 years of my career.

To summarize: All those mathematical tools are VERY useful for understanding what you're doing. However, do not expect to be drawing out equations at every turn. Real engineering is done either with software, a plain old scientific calculator, or even a slide rule if you just want to make a point.

Enjoy the math. But don't forget the practical applications in physics. This is WHY you're studying that math in the first place.

Good Luck!