How useful is mathematical proof as a mechanical engineer?

[theBEAST]

Specifically I plan on specializing as a Mechatronics engineer. I recently bought the book "Mathematical Proofs: A Transition to Advanced Mathematics" and I plan to study it on my own due to curiosity and interest towards mathematics. I would like to listen on one's opinion on how useful it would be as an engineer to understand the theoretical proofs behind the math.

 

[MarneMath]

I'm not an engineer, but I know many of them. I can say with a good level of confidence that knowing proof behind mathematical concepts helps very little. Knowing how to prove the square root of 2 is irrational or the rising sun lemma does nothing for you when it comes to your every day work.

 

[Hercuflea]

I got this same book and I am planning on previewing the group theory sections over the winter break as a preview for Abstract Algebra. Perhaps we could PM to talk about some of the problems later?

 

[theBEAST]

I got this same book and I am planning on previewing the group theory sections over the winter break as a preview for Abstract Algebra. Perhaps we could PM to talk about some of the problems later?

Yeah sure, but I don't think I have the prerequisites for group theory? I plan to start from Chapter 1 :P

 

[ahsanxr]

It will most probably not be of any direct use to you. However, knowing and understanding how proofs work in math will lead you to a better understanding of the tools you are using in your work. Secondly, it also makes you get used to thinking clearly and also gets you in the habit of stating everything you do very precisely. As a math and physics major, and spending quite some time on pure math-type stuff, physics textbooks can definitely be a bit frustrating sometimes in that they don't state things precisely enough.

 

[chill_factor]

writing things in a pure math notation makes it harder for scientists and engineers to understand, not easier. sure its more "precise" but it does not offer the all-important physical intuition.

Indeed from my own experience, I believe that learning mathematical proofs will be of very little use for physical scientists and engineers.

 

[theBEAST]

What math courses do you guys recommend then. My third year mechanical engineering timetable has no more math courses which is somewhat depressing. What is a fun math topic that would be useful. I was thinking applied linear algebra or complex variable related math which could be useful for mechatronics.

Oh yeah I'm currently in second year mechanical engineering, so by the end of this year I would have finished ordinary differential equations and Calc 3/4.

 

[Hercuflea]

My university only teaches Calculus in 3 sections, so I am assuming Calc 3 and 4 are multivariable and vector calculus at your univ. I just finished a second semester in linear algebra and a first semester in complex variables (<--final in two days). Complex variables was the hardest class I have ever taken in my life, much more so than linear algebra or differential equations. However, it is very fascinating and it would have been nicer if the professor would have taught the applications in the class, but you can always study applications on your own.

Not sure about mechanical engineering but I am pretty sure you can't go wrong with linear algebra and differential equations in engineering. I know complex analysis is applicable in some fields but I think the only thing that you will take from it is being able to solve difficult integrals with residues/contours. I would take numerical analysis if I were you.

 

[chill_factor]

complex variables is also useful for learning conformal mapping which transforms extremely brutal and crazy geometries into not so brutal and crazy ones. that said, computers do that much better than humans do since often you must "guess" the best transform for the job.

 

[espen180]

I think analysis on manifolds can be useful for engineering. If you have a system moving under certain constraints (a robotic arm is a classic example) then the configuration space of the system can be described by a manifold ("generalized euclidean space"), and you can do calculus directly on the manifold with no constraints rather than working in the original constrained system. I recommend checking it out. It is also useful for describing "singular" states" like gimbal lock. http://en.wikipedia.org/wiki/Gimbal_lock

 

[theBEAST]

I think analysis on manifolds can be useful for engineering. If you have a system moving under certain constraints (a robotic arm is a classic example) then the configuration space of the system can be described by a manifold ("generalized euclidean space"), and you can do calculus directly on the manifold with no constraints rather than working in the original constrained system. I recommend checking it out. It is also useful for describing "singular" states" like gimbal lock. http://en.wikipedia.org/wiki/Gimbal_lock

So would this be "Euclidean Geometry"? I have a course at my university called that; it is offered for third years and requires "Mathematical Proof" course as a prerequisite.

 

[espen180]

No, that's another course. This would be differential or riemannian geometry. A good starting point is to learn about differential forms. http://en.wikipedia.org/wiki/Differential_form

 

[Dembadon]

Regarding knowing how to prove theorems: It will certainly strengthen your understanding of the tools you use as an engineer, however I don't think you will find yourself proving theorems on a day-to-day basis. If you plan on doing research in the field of engineering, I can see it being more important, though. You should speak with some of the faculty in the mechanical engineering department and see what they say.

What math courses do you guys recommend then. My third year mechanical engineering timetable has no more math courses which is somewhat depressing. What is a fun math topic that would be useful. I was thinking applied linear algebra or complex variable related math which could be useful for mechatronics.

Oh yeah I'm currently in second year mechanical engineering, so by the end of this year I would have finished ordinary differential equations and Calc 3/4.

Both of the courses you mentioned would be helpful. Here are a couple more courses that might be useful to you:

- Numerical Methods: error analysis, solutions of linear systems, solutions of nonlinear equations, spline functions and polynomial interpolation, numerical integration and differentiation, and the numerical solution of ordinary differential equations

- Partial Differential Equations: classification of partial differential equations, interpretations of the heat equation, the wave equation and Laplace's equation, solutions by various analytical methods (separation of variables, eigenfunction expansion, the sine and cosine transforms, the Fourier transform, the Laplace transform, method of characteristics, change of coordinates) and approximate methods (explicit and implicit finite difference methods)

 

[homeomorphic]

Depends on what kind of engineering you are doing. There are people who do robotics who try to apply very heavy math, for example. On the other end of the spectrum, there are engineers who don't really use any math at all to speak of.

writing things in a pure math notation makes it harder for scientists and engineers to understand, not easier. sure its more "precise" but it does not offer the all-important physical intuition.

That's too much of a generalization. In some cases, if it makes them harder to "understand", that is only because they are content with plugging and chugging and NOT understanding. In other cases, the extra notation is superfluous, except if you want more generality and so on. For example, if you are working in three dimensions, then the old vector notation does a pretty good job. However, if you are working in high dimensions, differential forms really help. The idea that it doesn't offer the all-important physical intuition is not the fault of notation. That is the fault of mathematicians who refuse to explain the ideas behind things and are only concerned with stating things formally and logically.

Indeed from my own experience, I believe that learning mathematical proofs will be of very little use for physical scientists and engineers.

That's probably because you don't really see the intuition behind the proofs, based on your above statement. I can't blame you. Probably most mathematicians fall into the trap of being too formal and obscuring the intuition, but in my experience, the engineers and physicists were often even worse in that respect, though not always.

 

[Bipolarity]

One thing I noticed is that many ideas in engineering are essentially mathematical ideas, but whereas the mathematician might try to prove the idea using more formal methods, the engineer may skip through the precise details of the proof and yet arrive at the result using "intuition". By intuition, I refer to a process where the engineer can process the main idea of the mathematical proof in his/her mind and arrive at the result without using the actual terminology of the proof that is entailed should one need to actually write down the formal proof. (Sometimes however this results in mistakes that can have big consequences; to prevent this an engineer might use experiments to verify his result holds, which still often saves more time than a rigorous proof).

Here's a trivial example from EE:
Each push-button/toggle switch can control at most 2 possible states. *Then k switches can control $2^{k}$ possible states.* Thus if you want to control m possible states, you need enough switches to satisfy the inequality

$k > log_{2}(m)$.

*Of course, the engineer, or rather, any sane person who is pressed on time would quickly use intuition to arrive at this result, whereas the "rigorous way" of solving this problem would be to use induction, i.e. by showing that the addition of a switch multiplies the state space by 2, and then using that as the inductive case in conjunction with the base case that having a single switch allows you to control 2 states. Using induction to solve the problem makes the proof more formal, yet it takes longer. For the engineer, the important resource is time, not the added bit of certainty that comes from proving the result using more formal means. Of course this varies depending on the problem.

Even the work that most mathematicians do however is not perfectly rigorous. The perfectly rigorous statements are not even expressed in natural languages. They are usually expressed in the form of mathematical logic, for instance the formal definition of a basis of a vector space:

http://img201.imageshack.us/img201/5149/capture1jq.png

Even up to this level, the formalism is done only "for the records". Sane people usually default to intuition depending on the nature of the task. For engineering, you will be defaulting to intuition many times to the point that you do not notice your mind is proving results (such as the EE example above) without even knowing they are coming from the perspective of proof-based math.

I have seen this observation in a ton of places, from computing and economics to physical chemistry, where many logical but not totally obvious steps are skipped for the sake of time/convenience. It's all a matter of intuition vs. formalism, just like the debates between Hilbert and Poincare, and the scientist/engineer must decide prudently which is to exercise in a given situation.

BiP

 

[homeomorphic]

It's all a matter of intuition vs. formalism, just like the debates between Hilbert and Poincare, and the scientist/engineer must decide prudently which is to exercise in a given situation.

I don't think it's "all" a matter of that.

First off, as you said, mathematicians themselves aren't always completely rigorous, even by your stereotypical standards. Secondly, there's the question of subject matter. A lot of the subject matter won't be relevant to engineering in most cases. The question then, if you want to use the math is what areas of engineering use the math? Thirdly, there's the question of taking stuff on faith to save time, which I like to minimize and engineers who like to minimize that would find more math to be useful. Fourthly, there's the idea that engineering may often rely more on practical experience than on theory. There's a lot of stuff that you might not calculate. You just get a feel for it through experience.

I think in cases where not much math is used, you can always ask if more math would help. Sometimes, it will, sometimes it won't. The question doesn't always have a straight-forward answer. I'm sure there are a million cases out there in science and engineering where people are being too mathematical and equally many where they are not being mathematical enough. That's my suspicion. It all depends on the specifics of what you are trying to accomplish. There's no ready-made, one-size-fits-all answer to this question.