Best upper level math for engineering?

[rhino1000]

So, I can take 4 technical electives in my chemical engineering program, but one of them is pretty much required (so much for being an elective). I am interested in using at least one of the remaining on upper level math classes.

The interesting options are:
Partial differential equations (leaning towards this one, because of its supposed utility)
Emphasizes partial differential equations w/physical applications, including heat, wave, Laplace's equations. Interpretations of boundary conditions. Characteristics, Fourier series, transforms, Green's functions, images, computational methods. Applications include wave propagation, diffusions, electrostatics, shocks


Introduction to analysis ii (I think i is Real analysis, which I guess is also an option, but it's for some reason not on my school's curriculum guide for elective options). Which is better if I had the option?
Riemann-Stieltjes integration. Sequences/series of functions, uniform convergence, equicontinuous
families, Stone-Weierstrass Theorem, power series. Rigorous treatment of differentiation/integration of multivariable functions, Implicit Function Theorem, Stokes' Theorem

Numerical methods
Solution of nonlinear equations in one variable. Interpolation, polynomial approximation, numerical integration/differentiation, numerical solution of initial-value problems.Probability and Statistics (theory)
Logical development of probability, basic issues in statistics. Probability spaces, random variables, their distributions/expected values. Law of large numbers, central limit theorem, generating functions, sampling, sufficiency, estimation.

Applied linear algebra
Systems of linear equations, vector spaces, subspaces, bases, linear transformations, matrices, determinants, eigenvalues, canonical forms, quadratic forms, applications. Which ones do you guys think I ought to take/not waste my time with? The two classes that most intrigue me are the first two. How crucial is it for a guy to know his probability and statistics?

I give you the open-ended question: which of the above upper level math classes ought I to take as a chemical engineering major? (Keep in mind that I plan to use this degree to enter into a health-related field, which may or not influence your opinion on how rigorously I should strive to stay within the bounds of purely chemical engineering related courses). Would it be better for me to take more strictly chemical engineering courses if I were to enter into industry?

 

[SteamKing]

Actually, as you are a ChemE undergrad, I'm surprised you're not required to take Numerical Methods, ProbStats, and Linear Algebra at some point before getting your Bachelor's.

Most PDE courses are encountered at the graduate level, although you can brush up against them in undergrad studies, particularly for heat transfer or structural dynamics. I wouldn't recommend taking a PDE course unless you have done the Numerical Methods and the Linear Algebra first. ProbStats will be useful especially if you desire to go into any manufacturing or process control fields, which include health-related areas.

 

[rhino1000]

Actually, as you are a ChemE undergrad, I'm surprised you're not required to take Numerical Methods, ProbStats, and Linear Algebra at some point before getting your Bachelor's.

Most PDE courses are encountered at the graduate level, although you can brush up against them in undergrad studies, particularly for heat transfer or structural dynamics. I wouldn't recommend taking a PDE course unless you have done the Numerical Methods and the Linear Algebra first. ProbStats will be useful especially if you desire to go into any manufacturing or process control fields, which include health-related areas.

Ah yes, sorry. I have indeed taken a differential equations/linear algebra combined course. The linear algebra course as listed above I think is an upper-level class.

 

[rhino1000]

Actually, as you are a ChemE undergrad, I'm surprised you're not required to take Numerical Methods, ProbStats, and Linear Algebra at some point before getting your Bachelor's.

Most PDE courses are encountered at the graduate level, although you can brush up against them in undergrad studies, particularly for heat transfer or structural dynamics. I wouldn't recommend taking a PDE course unless you have done the Numerical Methods and the Linear Algebra first. ProbStats will be useful especially if you desire to go into any manufacturing or process control fields, which include health-related areas.

Also, by the way you are right to be surprised. Apparently there is a numerical methods class that has specificity to chemical engineering methods. So... if that changes the value of taking the more general numerical methods class above, so be it. Given these latest two pieces of information, what would you say about the matter/recommend?

 

[SteamKing]

Also, by the way you are right to be surprised. Apparently there is a numerical methods class that has specificity to chemical engineering methods. So... if that changes the value of taking the more general numerical methods class above, so be it. Given these latest two pieces of information, what would you say about the matter/recommend?

Not being a ChemEng, I don't know what numerical methods would be specific to this field as opposed to engineering in general. I think any engineer should understand the general algorithms for root finding, interpolation, solving linear systems, solving nonlinear equations, numerical integration, and the solution to ODEs using numerical methods. If your class covered these topics, I'd say you've covered the subject sufficiently.

 

[rhino1000]

So... any recommendations assuming that the chemical engineering applied numerical methods class covers all or the majority of the numerical methods directly relevant to chemical engineering?

By recommendations, I mean recommendations as far as which of these pure math classes would be, in your opinion, most beneficial for me to take. Like, is analysis too abstract to be possibly useful in chemical engineering or to understand chemical engineering? Is it likely to be more or less useful than probability and statistics theory? Do either of them allow for a deeper understanding of the math behind a lot of engineering? etc. Which pure math classes should I take out of the above, if any?

 

[SteamKing]

So... any recommendations assuming that the chemical engineering applied numerical methods class covers all or the majority of the numerical methods directly relevant to chemical engineering?

By recommendations, I mean recommendations as far as which of these pure math classes would be, in your opinion, most beneficial for me to take. Like, is analysis too abstract to be possibly useful in chemical engineering? Is it likely to be more or less useful than probability and statistics theory? Do either of them allow for a deeper understanding of the math behind a lot of engineering? etc. Which pure math classes should I take out of the above, if any?

An analysis course might be beneficial to you if you study PDEs in depth. As to day-to-day engineering calculations, probably not as useful. I would pick ProbStats over it anyway, because you'll run up against this in designing or analyzing experiments, clinical trials, process control, quality control, etc. As far as LA is concerned, if you can solve a set of linear equations using matrix methods, that's probably sufficient. If you need to get into eigenvalues or overdetermined systems, you can pick these techniques up on the fly, as I have done, after getting out of school.

 

[rhino1000]

An analysis course might be beneficial to you if you study PDEs in depth. As to day-to-day engineering calculations, probably not as useful. I would pick ProbStats over it anyway, because you'll run up against this in designing or analyzing experiments, clinical trials, process control, quality control, etc. As far as LA is concerned, if you can solve a set of linear equations using matrix methods, that's probably sufficient. If you need to get into eigenvalues or overdetermined systems, you can pick these techniques up on the fly, as I have done, after getting out of school.

Excellent reply. What do you think about the relative value of probability/stat theory versus partial differential equations, if I had to choose only one? And I doubt if you'd be able to give a good answer for the following question: if analysis would help to understand PDEs, then would that be more valuable than taking another specific chemical engineering class? I understand especially if that last question is too vague and therefore unanswerable.

 

[SteamKing]

Excellent reply. What do you think about the relative value of probability/stat theory versus partial differential equations, if I had to choose only one? And I doubt if you'd be able to give a good answer for the following question: if analysis would help to understand PDEs, then would that be more valuable than taking another specific chemical engineering class? I understand especially if that last question is too vague and therefore unanswerable.

No, it's a good question.

In my line of work, which is naval architecture, I rely on ProbStats quite a bit because certain processes, like analyzing the motion of waves, for example, rely on using methods taught in the usual ProbStats class for figuring out things like the highest wave to expect during a certain period of time. There are plenty of other things which have a probabilistic nature where the methods taught in a ProbStats class would be useful.

PDEs underlie a great many things encountered in engineering, and it's a good thing you desire to study them. I've encountered a number of instances in my career where knowing a little about PDEs helped, but like a said a ways back, most PDE courses are offered at the graduate level. I tried to tackle them on my own, especially learning the methods of their solution, but it took me a number of years of part-time study before I grasped what little I know about them, which is being able to follow what others write in technical papers and such.

I think, based on my experience and training, you should take the course in ProbStats. I believe you'll need to know the basics at some point in your career, especially if you go into a health-related field or even just straight chemical engineering.

 

[rhino1000]

No, it's a good question.

In my line of work, which is naval architecture, I rely on ProbStats quite a bit because certain processes, like analyzing the motion of waves, for example, rely on using methods taught in the usual ProbStats class for figuring out things like the highest wave to expect during a certain period of time. There are plenty of other things which have a probabilistic nature where the methods taught in a ProbStats class would be useful.

PDEs underlie a great many things encountered in engineering, and it's a good thing you desire to study them. I've encountered a number of instances in my career where knowing a little about PDEs helped, but like a said a ways back, most PDE courses are offered at the graduate level. I tried to tackle them on my own, especially learning the methods of their solution, but it took me a number of years of part-time study before I grasped what little I know about them, which is being able to follow what others write in technical papers and such.

I think, based on my experience and training, you should take the course in ProbStats. I believe you'll need to know the basics at some point in your career, especially if you go into a health-related field or even just straight chemical engineering.

Wonderful, thank you!

 

[rhino1000]

Anyone have opinions on real analysis and its worthiness as a class for an engineer/possible graduate student compared to PDE/probability theory? Would it be worth it to use up 3 out of 4 technical electives to take all three of these classes?

 

[ME_student]

Partial differential equations is a class I would recommend as well as linear algebra. I took both, didn't like linear so much, but diff eqs was a lot of fun.

 

[rhino1000]

Partial differential equations is a class I would recommend as well as linear algebra. I took both, didn't like linear so much, but diff eqs was a lot of fun.

Just making sure you know that I am talking about partial differential equations (not the typical introductory differential equations) and upper level linear algebra (not the typical introductory linear algebra). The reason I say this is that I know the normal versions are required in all of the major engineering majors across the country, and the way that you word it ("I've taken linear" and referring to PDE as "diff eqs"); I've taken a combined Diff eq./Linear algebra course already. Like I said, just making sure you caught that distinction. Thanks for the reply! As of now I am planning on taking the probability and statistics class next semester and PDE the semester/a semester following that!

Other thoughts/replies still appreciated! I'm guessing not many people have experience with real analysis which is why I'm not getting too much thoughts on its relevance to engineering.

 

[Sashwat Tanay]

As per your branch, the most useful ones should be
1. probability and statistics
2. numerical methods

 

[Math10]

I highly recommend Numerical Methods and Applied Linear Algebra.

 

[Mark Harder]

I'm thinking here mostly of grad-level as well as college level courses, but these are among the choices you may consider for your senior year:
Since you brought up PDEs and Numerical Analysis, you might need both of these, although I think of mechanical analysis of solid structures when I say this. I don't know enough CE to say whether hydrodynamics might be involved. If it is, you'll need PDEs and systems of PDEs. But these systems will be solved numerically, through finite element analysis and related methods. So if you do in fact need to know theoretic PDE analysis, you will need to know the numerical analysis required. This means you need numerical analysis course that includes finite element analysis. This is a pretty advanced topic; I took a 2 quarter course in NA that included solutions of ODEs and never came close to PDE methods. To cover all the bases, this will probably require a full-year course in NA. NB! Do not mistake numerical METHODS for numerical ANALYSIS. The latter includes analyses of errors and stabilities in the methods presented. If you are going to do any statistics in your profession, you will absolutely need both probability and statistics, more than 1 quarter each. Linear algebra is involved in all of the above. It's as fundamental as calculus, actually. You may want some theoretical LA as well as applied. Linear algebra of infinite dimensional systems plays an essential part of quantum mechanics. I don't know how much QM you will need in CE. Courses covering analysis of higher dimensional spaces, R^n and C^n, are really quite theoretical. I would skip these unless you are advised/required to know the subjects.

Some more comments. Often textbooks and courses intended for engineers are clearer than those for fundamental sciences. Look for these. Also, physical chemistry is essential for understanding features of chemical processes such as kinetics (systems of ODEs by the way). In a 1-yr course, thermodynamics will be covered as will quantum chemistry and statistical mechanics. The SM is essential for understanding irreversible and non-equilibrium processes. It is possible, however, that thermodynamics and chemical kinetics for engineers will be offered by your CE dept. Good luck!

 

[rhino1000]

As per your branch, the most useful ones should be
1. probability and statistics
2. numerical methods

I highly recommend Numerical Methods and Applied Linear Algebra.

Your quote isn't appearing, probably because you put your reply itself into a quote. :D

Hey guys, thanks for the replies! I have locked in probability and statistics theory class for next semester. I see that you both recommend numerical methods; I forgot to mention this in OP, but I am already required to take a numerical methods class that has specificity to chemical engineering. I am guessing this subtracts from the value of the numerical methods elective course, so would I be right to say that your replies did not take this into consideration (admittedly by my fault)? Also, @Mark Harder : I don't think I was considering numerical analysis; I was considering real analysis, which is essentially advanced proof-based calculus. It's possible I accidentally did say that though.

 

[rhino1000]

Every time I come onto this thread, I really hope to see someone recommend the real analysis class :/.

 

[Mark Harder]

Rhino. I wasn't confusing Numerical Analysis with Real Analysis. The word analysis is used in a different sense in the two titles. I believe I recommended Numerical ANALYSIS over Numerical METHODS because the former includes the analysis of errors, stability, speed, etc., which a course in methods alone might not. You can copy algorithms and download subroutines that function just fine as numerical methods. But how will you evaluate them? I admit I sometimes did this in my work, and it's worked for me because the methods offered were judicious. One example is the use of the usual formula for the roots of a quadratic. You might encounter cases in which this method of solution is way, way off base because of rounding errors. Pocket calculators use more memory than Fortran in their arithmetic, so they will give the correct answers. If you're writing a program that involves root-finding, you would be better off using one of the many approximation algorithms, which do a far better job than a formula that SEEMS to be exact. I recommend the book "Numerical Methods that Work" for an introduction full of pithy remarks by an author with definite opinions. It might provide you with a flavor of the sort of thing I'm saying.

 

[rhino1000]

Rhino. I wasn't confusing Numerical Analysis with Real Analysis. The word analysis is used in a different sense in the two titles. I believe I recommended Numerical ANALYSIS over Numerical METHODS because the former includes the analysis of errors, stability, speed, etc., which a course in methods alone might not. You can copy algorithms and download subroutines that function just fine as numerical methods. But how will you evaluate them? I admit I sometimes did this in my work, and it's worked for me because the methods offered were judicious. One example is the use of the usual formula for the roots of a quadratic. You might encounter cases in which this method of solution is way, way off base because of rounding errors. Pocket calculators use more memory than Fortran in their arithmetic, so they will give the correct answers. If you're writing a program that involves root-finding, you would be better off using one of the many approximation algorithms, which do a far better job than a formula that SEEMS to be exact. I recommend the book "Numerical Methods that Work" for an introduction full of pithy remarks by an author with definite opinions. It might provide you with a flavor of the sort of thing I'm saying.

I see. Thanks for the clarification! Numerical analysis, huh.