Exploring the Benefits of Pursuing a Double Major in Math and Engineering

[Angry Citizen]

Math + Engineering = ?

I'm an aerospace engineering major. My career will be in aerospace engineering. But recently, I've thought about doing a double major in pure math.

My main reasoning is that it would only be an extra twelve classes, and I've always been highly curious about 'proper' math. My other reasoning is that it would give me something to do while catching up on my engineering classes. As a transfer student, I've already completed all but two of my general education courses (which under a normal degree plan are interspersed throughout my college years), and will have completed all my basic, foundational science classes (intro physics, intro chem, and all three basic calculus courses). This leaves my first year entirely blank, save for two small intro engineering courses which must be completed sequentially (one in my first semester, the other in my second) before I can take any upper division engineering courses.

But I think I need a third reason. Is a pure math degree of any use in engineering? The extra classes I'd be taking are as follows:

Foundations of Mathematics: Foundations of mathematics including logic, set theory, combinatorics, and number theory.

Linear Algebra I and II: Linear equations and matrices; real vector spaces, linear transformations, change of bases, determinants, eigenvalues and eigenvectors, diagonalization, inner products (I); Eigenvalues, similarity and canonical forms, applications to differential equations and quadratic forms (II). Note: It is not necessary under my engineering degree plan to take linear algebra, and an 'intro' course is offered in the subject that is presumably less rigorous, but as I understand linear algebra is very useful in engineering, I think I'll be taking this even if I don't go for the math degree.

Advanced Calculus I and II: Axioms of the real number system; point set theory of R1; compactness, completeness and connectedness; continuity and uniform continuity; sequences, series; theory of Riemann integration (I); Differential and integral calculus of functions defined on Rm including inverse and implicit function theorems and change of variable formulas for integration; uniform convergence (II).

Modern Algebra I and II: Groups, rings, fields (I, II).

Theory of ODE's: Existence and uniqueness of solutions to differential equations, linear systems, nonlinear equations, stability analysis, qualitative behavior of solutions, and modeling with differential equations.

Theory of PDE's: Formulation and solution of partial differential equations of mathematical physics; Fourier series and transform methods, complex variable methods, methods of characteristics and first order equations.

Introduction to Topology: Metric spaces; continuity of metric spaces; topological spaces; basic notions; separation axioms; compactness; local compactness; connectedness; basic notions in homotopy theory; quotient spaces, paracompactness and topological manifolds.

Numerical Analysis I: Linear systems, matrix decomposition and eigensystems, numerical integration, interpolation and numerical solution of ordinary differential equations.

Programming in C: Basic concepts, nomenclature and historical perspective of computers and computing; internal representation of data; software design principles and practice; structured and object-oriented programming in C; use of terminals, operation of editors and executions of student-written programs.

Taken from:
http://catalog.tamu.edu/09-10_UG_Catalog/course_descriptions/math.htm
And:
http://catalog.tamu.edu/09-10_UG_Catalog/science/math/math_bs.htm

Will any of these be useful? I understand that aerospace engineering, especially the propulsion sub-discipline, often uses differential equations, so I've tailored many of my electives around a thorough treatment of ODE's and PDE's so that I could at least be completely comfortable using them for engineering. I will, of course, be discussing this with my adviser as well, but I'd like the input of the community here before I proceed. Apologies for the somewhat long post.

 

[╔(σ_σ)╝]

The only courses that would be useful to you in aerospace are basic linear algebra, ODE some PDE, for sure numerical analysis and C programming.

At my school aero's must take the above courses with the exception of PDE's.

Analysis (Advance calculus) would be useless to you i terms of helping you in engineering unless you want a deeper unstanding of the foundations of calculus.

Foundation of math, Topology and abstract algebra would be completely useless to you.

Why don't you take a minor instead of a double major ?

If you want to learn math that would be usefull to you in aero you should take applied math courses not pure math courses.

 

[Jokerhelper]

also, make sure you take a more "applied" ODE course if you can.

 

[Chaostamer]

From what it looks like, the most useful courses for engineering would be the Linear Algebra sequence, ODEs, PDEs, Numerical Analysis, and Programming in C. That's six of twelve classes that will be directly applicable. In that sense, it might benefit you to search for an Applied or Computational Math track instead.

However, I know a girl who's currently pursuing a Ph.D. in Computer Science at my school. She got her undergrad, though, in Pure Math and says that she's glad she went that route as opposed to a CS undergrad. She said that it gave her a lot more perspective and a good bit of insight into various problem-solving techniques that have been serving her well in her classes and her research.

I also know a teacher at my school who works primarily as a mathematician for Lockheed (teaching one class per semester on the side). He works alongside engineers on missile defense and imaging projects and he says that taking "pure" versions of certain classes (Linear Algebra, ODEs, and PDEs, for example) has served him well because understanding the theory behind the math has helped him use it more effectively because all he needs to do is put the math into "engineering language". So there are two accounts from Pure Math majors who diverted from that path to some extent but still found their mathematical foundation helpful.

Plus, if you're truly interested in it, at least give it a try. Like I just said in another thread, studying pure math will always benefit you in some way if you do anything remotely "technical". And if you try it and don't like it, you can always switch to another math track or even a minor. But I'd say that if you're curious, go for it.

 

[AlephZero]

Foundation of math, Topology and abstract algebra would be completely useless to you.

I disagree. Yes they will be "completely useless" for doing the handle-turning stuff you learn in a typical engineering course. On the other hand they are far from useless if you ever progress to doing some ceative thinking about hard engineering problems that need INSIGHT to make sense of what is going on, not just the ability to make pretty pictures using commercially available computer software.

YMMV, but I'm speaking from my own personal experience here.

 

[Angry Citizen]

At my school aero's must take the above courses with the exception of PDE's.

Well, I suppose I should mention that I'm already going to be taking an ODE course and a course with PDE's in them, presumably the applied versions of the courses I listed above.

Why don't you take a minor instead of a double major ?

Because it'd only be one more course (the intro-to-linear-algebra course I mentioned in my OP). Reason #2 (giving me something to do) is a pretty big deal to me. Besides, I've heard minors are essentially useless unless they're a foreign language minor. Presumably a math major would make me at least slightly more marketable to an employer.

 

[Angry Citizen]

In that sense, it might benefit you to search for an Applied or Computational Math track instead.

My applied PDE course is sort of a mish-mash of stuff, including computational mathematics.

The anecdotes were very helpful. Thank you for sharing them.

 

[Angry Citizen]

I disagree. Yes they will be "completely useless" for doing the handle-turning stuff you learn in a typical engineering course. On the other hand they are far from useless if you ever progress to doing some ceative thinking about hard engineering problems that need INSIGHT to make sense of what is going on, not just the ability to make pretty pictures using commercially available computer software.

YMMV, but I'm speaking from my own personal experience here.

Could you explain what you mean by 'insight'? You mean understanding, for example, the theory behind diff EQ's that make them applicable to a certain problem, so that one could find an unconventional use for them?

 

[╔(σ_σ)╝]

I disagree. Yes they will be "completely useless" for doing the handle-turning stuff you learn in a typical engineering course. On the other hand they are far from useless if you ever progress to doing some ceative thinking about hard engineering problems that need INSIGHT to make sense of what is going on, not just the ability to make pretty pictures using commercially available computer software.

YMMV, but I'm speaking from my own personal experience here.

I understand what you meant but I meant useless for the handle-turning stuff and the kind of things aero's do. Aero's have no use for that kind of math.


For example general topology is the language which functional analysis is writing, such a course would be useful to student studing advanced signals and control theory in the Electrical Engineering department.

Because it'd only be one more course (the intro-to-linear-algebra course I mentioned in my OP). Reason #2 (giving me something to do) is a pretty big deal to me. Besides, I've heard minors are essentially useless unless they're a foreign language minor. Presumably a math major would make me at least slightly more marketable to an employer.

You need to be careful with the "giving me something to do" idea considering you can't take all these classes in one year.

Engineering is not the type of program that gives you tons of time to do anything but school work. You may have a hard time jugguling thoes courses once you get into second year.

I myself mildly screwed up my previous semester because I was having too much fun with Analysis ( Advanced Cal). I neglected my engineering courses.

 

[Angry Citizen]

You need to be careful with the "giving me something to do" idea considering you can't take all these classes in one year.

I know. I figure I can take at least half of the ones I need in the first year. Plus, I don't know how summer classes work in universities, but that's always a possibility too. And like I said, the engineering degree plan already had general education classes interspersed throughout all four years. I realize gen ed classes aren't senior-level advanced math courses, but I think it's doable -- especially considering that studying so many ODE's and PDE's will keep my math skills sharp as a razor.

 

[╔(σ_σ)╝]

I know. I figure I can take at least half of the ones I need in the first year. Plus, I don't know how summer classes work in universities, but that's always a possibility too. And like I said, the engineering degree plan already had general education classes interspersed throughout all four years. I realize gen ed classes aren't senior-level advanced math courses, but I think it's doable -- especially considering that studying so many ODE's and PDE's will keep my math skills sharp as a razor.

If so then you are in good shape.

I have to warn you some of the classes would be difficult one. I can attest to for sure, Advanced Calculus. You have to really love math in order not to quit and finish what you started.

Also if you are doing this for interest then go on, however, if you are doing this hoping to apply the courses to engineering think again. If the later is true then I suggest you take more applied math courses.

 

[radou]

If you enjoy learning math, learn as much of it as possible. It doesn't need to be applied, after all. At least, that's my philosophy.

If you'll engage in some serious theoretical research one day, I think any piece of math you learn can be useful.

I have MS in structural engineering and am undergoing a thorough study in general topology in my spare time for the last 4 or 5 months. And I don't care if I'll apply it some day or not (at least not right now). I just enjoy it, that's all.

 

[Angry Citizen]

Well, the applied math degree requires far more time investment, because it has a lot of economics courses and such. I'm open to suggestions as far as other classes to take. I might, for instance, be able to take grad school courses such as Methods of Applied Mathematics I and II rather than some of the more theoretical ones like topology and modern algebra II.

 

[Angry Citizen]

If you'll engage in some serious theoretical research one day, I think any piece of math you learn can be useful.

I'm considering the research track for developing propulsion methods like VASIMR and ionic drives. I'd say that is serious theoretical research.

Honestly, I'm not sure if I enjoy it or not. I've never taken a pure math course with proofs. I've never even done elementary proofs (didn't take geometry). I do know I'm interested in it.

 

[radou]

I'm considering the research track for developing propulsion methods like VASIMR and ionic drives. I'd say that is serious theoretical research.

Honestly, I'm not sure if I enjoy it or not. I've never taken a pure math course with proofs. I've never even done elementary proofs (didn't take geometry). I do know I'm interested in it.

You have to see for yourself. I believe doing mathematics without proofs is senseless - you gain a specific set of tools which you can use in specific situations, but without real understanding, the tools are more or less useless.

Again, that's only my oppinion. Since I basically don't know anything about the nature of the material you're studying, perhaps I'm wrong.

 

[nlsherrill]

I'm considering the research track for developing propulsion methods like VASIMR and ionic drives. I'd say that is serious theoretical research.

Honestly, I'm not sure if I enjoy it or not. I've never taken a pure math course with proofs. I've never even done elementary proofs (didn't take geometry). I do know I'm interested in it.

You should take the course Foundations of Mathematics that you listed to see if you want to go the pure math route. I take it this is the equivalent of an introductory course on proofs, and if that is true then this course will let you know whether or not you will want to continue or not. I have spoken with a mathematics professor and he told me that if you couldn't get a B fairly easily in the introductory proof course, then mathematics is not for you. I plan on taking this course next year too see if I want to double major in math and physics...

Apparently you haven't come across this but if you look in some of your engineering, or most likely physics textbooks, there should be a problem or so at the end of every chapter that says something along the lines of "proof that this relation holds true under these conditions" or "show that the divergence of the curl of a function f is = 0." Try these problems and this may give you a taste of what kind of problem solving you will run across with in mathematics.


Furthermore, more math is never a bad idea in my opinion...unless it is conflicting with your engineering passion that is. While some posters in this thread do have good points on how relevant topology would be to an aerospace engineer, it is still to the advantage of the student to challenge themselves with tough problems/subjects to broaden their "arsenal", if you will, of mathematical tricks/techniques to solving hard problems.

Apparently their is even a field called Mathematical Engineering. Sounds cool.

http://www.postgraduate-courses.net/articles/mathematical_engineering.htm

 

[Chaostamer]

You should take the course Foundations of Mathematics that you listed to see if you want to go the pure math route. I take it this is the equivalent of an introductory course on proofs, and if that is true then this course will let you know whether or not you will want to continue or not. I have spoken with a mathematics professor and he told me that if you couldn't get a B fairly easily in the introductory proof course, then mathematics is not for you. I plan on taking this course next year too see if I want to double major in math and physics...

I agree with a bit of a caveat. I wouldn't let Foundations of Mathematics alone determine whether you're interested in or capable of handling Pure Math. If your class is anything like my Logic and Proof in Mathematics class, it will probably cover set theory and the abstract notions of functions and relations at the very least--along with a foundation in mathematical logic and an introduction to various types of proof (direct, contradiction, contrapositive, induction, etc.). That said, a lot of students I know tend to find the class a bit too abstract and are very challenged by it. The material taught in such courses is the sort of material on which all the rigorous pure math courses are developed, so you'll be seeing it constantly as you progress through the program and it will begin to make more and more sense as you put it in context.

Basically, try your best in the class and try to have fun with it, but don't be scared away from the entire program if you struggle or even if you dislike some of the material. I got an 'A' in my class, but the material didn't really click with me until I started learning Linear Algebra and Combinatorics. It could very well be the same for you.

 

[Angry Citizen]

I appreciate the advice. Even if I don't decide to take the double major route, I'm definitely going to take the theoretical linear algebra sequence, which is almost certainly going to involve proofs, proofs and even more proofs. I'll probably know then whether or not I'm cut out for topics like advanced calculus. My first semester won't involve any of those ridiculously difficult courses, and is going to consist of an ODE course (presumably applied), the foundations of math course, and I'm going to try begging the department into allowing me to take the first linear algebra course as well. Hopefully I can convince them, since I'll have read at least one book of proofs this summer, and will probably be pretty well versed in proofs by the time I start.

 

[nlsherrill]

I appreciate the advice. Even if I don't decide to take the double major route, I'm definitely going to take the theoretical linear algebra sequence, which is almost certainly going to involve proofs, proofs and even more proofs. I'll probably know then whether or not I'm cut out for topics like advanced calculus. My first semester won't involve any of those ridiculously difficult courses, and is going to consist of an ODE course (presumably applied), the foundations of math course, and I'm going to try begging the department into allowing me to take the first linear algebra course as well. Hopefully I can convince them, since I'll have read at least one book of proofs this summer, and will probably be pretty well versed in proofs by the time I start.

Sounds like you have a good plan. Good luck with your decisions.