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Featured What should the Mathematics requirements be for a Physics degree

  1. Dec 14, 2017 #1

    symbolipoint

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    What should the Mathematics requirements for Physics degree really be?

    The usual official requirement is three semesters of Calculus, and then one more combination course which combines some linear algebra and not-too-complicated differential equations. Often the Physics majors, at least for bachelor's degree, take more than that and find some way of using some.

    If an undergraduate student really wants to be good at Physics, which courses more should or need he do? Why then, are these not listed as the official requirements for the degree?
     
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  3. Dec 14, 2017 #2

    bhobba

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    IMHO distribution theory/generalized functions is a must.

    Just about all applied mathematicians/physicists come up against that damnable Dirac delta function somewhere in their education as well as Fourier transforms. Distribution theory makes Fourier Transforms a snap and the Dirac Delta function no longer a mystery.

    I also want to mention when you study such things.

    Here in Australia we have whats called Math B and C. Math B is a bit below US Calc AB, Math C, which is done in conjunction with Math B, is about Calc BC level with a few slightly different modules a school can choose from such as dynamics or beginning Markov Chains. What some schools are now doing is for good/interested students they do Math B and C in grade 11 and what is usually first year university math in grade 12. That consists of your Calc III, and a subject beginning linear algebra plus differential equations.

    IMHO this is much better preparation when you get to university. You can start on distribution theory immediately, also straight into intermediate QM having studied some linear algebra, differential equations and either are currently studying distribution theory or if leaving it to second semester already studied it. It would allow QM to concentrate on the physics rather than issues that really should have been taught in Math. And for good students you can go straight onto something like Landau - Mechanics rather than the boring general physics you do because you didn't do calculus based basic physics at HS - which IMHO should be done at HS as well.

    Why isn't it done? I think most physics majors who are also not math majors (it's very common here in Aus to do both) are sort of expected to pick it up from their physics texts as they go along. A really inefficient way of going about it IMHO. The physics courses should concentrate on the physics - they should already know the math required for the physics. For example if I remember correctly, Griffiths EM book, which I have a copy of but dusty since I haven't read it for a long time, spends a fair number of pages on the Dirac Delta function giving his personal guarantee coming to grips with it now will greatly enhance their future math/physics education. Really - in an EM book - not the place for it - he is right about its importance but it's in the wrong place - you should already know it - and what it actually is as well which he never does explain. I know most students just want a bit of paper to get a job later and don't really care about issues of real understanding, but some do, some were like me and actually thought about such things. Students like this really do need a proper math course.

    I could even support the idea of not having physics majors alone - but combined physics/applied math majors.

    Thanks
    Bill
     
    Last edited: Dec 14, 2017
  4. Dec 14, 2017 #3
    This is what I would consider to be "typical":
    2-3 semester courses in Differential, Integral, and Multivariable Calculus
    1 course in Ordinary Differential Equations that includes nonlinear 1st order, systems of ODEs (some Matrix Algebra), and stability theory

    This is what it should be:
    1 course in Partial Differential Equations that including the method of characteristics, quasilinear PDEs, and numerical techniques
    1 course in Vector and Tensor Calculus
    1 course in Dynamical systems and Chaos theory preferably covering both recursion relations and continuous models
    2 courses on statistics, data analysis, and Numerical Analysis including Monte Carlo simulations
    1 course on Design of Experiments

    And ideally also:
    1 course in pure Linear Algebra
    1 course in Complex Variables including complex valued vector fields
    1 courses in Mathematical Methods covering Special Functions, Green's functions (scalar, dyadic and tensor), Integral Transforms and Operational Calculus, Integral Equations, and the Calculus of Variations
    1 course in Differential Geometry
    1 course on probability theory
    1 course on stochastic processes
    an assortment of pure math courses in Algebra, Analysis, and Geometry

    They are not required because time is devoted towards learning physics not mathematics. I did as much extra math as I could and also learned more in physics classes. There is only so much material that can be covered in a 4 year undergraduate degree.
     
  5. Dec 14, 2017 #4

    bhobba

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    Agree entirely.

    Here is Australia degrees are 3 and not 4 years - but, while there is some of what I consider we are doing it right, you wrong, discussion about this, we complete what US would consider first year stuff in 11 and 12. We do six subjects at about the UK AS level or IB SL level each of which is considered equivalent to a US single freshman subject. We also have math B and C which is two subjects at about UK A level and equivalent to US Calc BC which I believe gets you 2 credits at most schools. So we have really done 6 subjects before entering uni. Then you do 24 subjects in 3 years at uni to give 30 4 unit subjects or the 120 credits typical US 4 year degrees require - ours is more like the British system than the US. US schools say our 3 year degrees are not equivalent to your 4 year degrees, and we say, for spending a semester here in Australia as part of your US degree, you must be at least second year because we have already done the equivalent of your first year. Its a a bit maddening actually how some don't really understand whats going on and engage in simplistic things like ours are 4 years while yours is 3. Conversely here in Aus we don't understand your AP system and students can easily enter university in the US ready for second year subjects.

    The answer IMHO is you do a double degree in math and physics where all that stuff is covered. Its possible - but you need better preparation of university entrance both here and in the US - that's the real key.

    Naturally this applies to good, thinking students. Like I said many, possibly even most don't even care - they just want a bit of paper. For them leave things as they are.

    In practical terms here in Aus many do a double degree in math and physics where a lot of what you mention is covered anyway - double degrees here take 4 years eg from where I went:
    https://www.qut.edu.au/study/courses/bachelor-of-science-bachelor-of-mathematics

    I did the math and computer science one:
    https://www.qut.edu.au/study/courses/bachelor-of-information-technology-bachelor-of-mathematics

    But back in my day it was a bit different - you could do it in 3 years - you did about the same number of subjects but many were 3 credits instead of 4. Obviously the 3 credit subjects didn't go as deep into it as the 4 credit ones - or maybe standards have simply slipped.:-p:-p:-p:-p:-p

    Thanks
    Bill
     
    Last edited: Dec 14, 2017
  6. Dec 15, 2017 #5

    Vanadium 50

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    That leaves no time for studying physics. I wouldn't call that ideal.
     
  7. Dec 15, 2017 #6

    Mister T

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    That's impossible to say because there is so much math in physics that a physics major could benefit from lots and lots of math, too much to fit into any degree plan. So a compromise is needed.

    I would say that differential equations is very important. Also important is linear algebra. You you have already acknowledged these, but I think combining them into a single course doesn't do justice to either of them. You really need as much of them as you can learn, so separate courses in them would be better. I found that a course I took in numerical methods was very useful. Differential geometry is also good.

    What you really need to focus on though, is the area of physics you wish to specialize in, and then talk to the professors who teach those physics courses and ask them which math courses they would recommend. The other people who have responded in this thread have made some good recommendations.

    Time and money. Administrators are under pressure to make degrees affordable. They therefore have to come with degree plans that will fit into the time allotted for the degree. The statistics about the average time to complete a degree, for example six years to complete a four-year degree, loom over their heads.

    http://www.politifact.com/wisconsin...e-college-degree-takes-six-years-us-sen-ron-/

    They are being criticized for crippling their graduates with debt that will take years and years to pay off.
     
  8. Dec 15, 2017 #7

    Andy Resnick

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    The question is incomplete, as there are many 'flavors' of physics degrees: BS vs. BA, for example. While many of the responses here seem to fall back on "more math", I take the position that there needs to be more laboratory/computational activities. Statistical analysis of data is a topic that broadly covers all three.

    Let's also remember that there's no simple linkage between earning the degree and subsequent job/career activities, leading to the (still unresolved) question "What does a Physics degree prepare you for?" The reason for electives is a precise response to that question: there is a broad palette of skills that could be asked of a Physics major, who can only become proficient in some subset. Deciding on the subset is the prerogative of the student.
     
  9. Dec 15, 2017 #8

    vela

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    At the schools I attended, undergraduate physics majors were also required to take a year-long course in math methods which covers many of the topics mentioned in this thread. I think most physics majors would have benefited from a course in probability, statistics, and numerical analysis.

    I question the usefulness of having formal mathematics courses in many of the topics mentioned so far. I recall the joke about how mathematicians spend a long time to come up with answers that are 100% correct but still utterly useless. If a student is interested in a particular subjects and wants to see a more rigorous treatment, he or she can choose to take that math course, but to require it of all physics majors seems like overkill. They're majoring in physics, not math.
     
  10. Dec 15, 2017 #9

    bhobba

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    Its the same here - you pick up more advanced topics in the textbooks themselves such as what I mentioned about the Dirac Delta Function in Griffiths.

    But things would be more efficient if the physics courses could concentrate on the physics rather than take a detour to the math.

    When I was doing my Masters and wanted to do QM I spoke to my adviser - he said when he sends people over to the physics department to learn QM they skip introductory courses because the math they have done more than compensated for it. For example in the PDE course you have already solved the Schrodinger Equation for the Hydrogen atom - its a standard PDE they teach. There is no use at all rehashing the same thing again. You are just given a bit of reading to do on physical concepts such as wave-functions etc which I already new and that's it.

    By doing more math the actual physics becomes more concentrated and you can cover more. Of course it has to be RELEVANT math - not just math for math's sake. I did a number of subjects undergrad totally useless for physics such a Mathematical Economics.

    Thanks
    Bill
     
  11. Dec 16, 2017 #10

    Dr Transport

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    A question then arises, consider Jackson chapters 2 and 3 on electric fields. Ideally a student would have had a semesters course if not a years course in PDE's and separation of variables. What Jackson covers in two chapters is close to what I remember my 1 semester course in Fourier series and PDE's covered (and we didn't really hit upon all that is necessary to deal with Greens functions). How do you divorce the math necessary and just cover the physics?? Those two chapters are just an example of how you need to develop the math at the same time you discuss the physics.
     
  12. Dec 16, 2017 #11

    jtbell

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    At some or many schools in the US, physics majors learn topics like Fourier analysis and PDEs not in a course taught in the math department, but instead in a course taught in the physics department under a name like “Mathematical Methods”.
     
  13. Dec 16, 2017 #12
    It's easier to define a body of mathematical knowledge that should be required, bodies that may be recommended for various specializations, etc. than it is to specify how that knowledge needs to be imparted (math courses vs. physics courses). I managed to get by with Calc 1, Calc 2, Calc 3, Diff Eq, Linear Algebra, and Numerical Analysis as the formal math courses I took. But we also has a physics course called, "Mathematical Methods in the Physical Science" (used Boas) that was essential. I also picked up along the way that instructors in physics courses just taught on the fly (Calculus of Variations, PDE, Runge-Kutta, Fourier Analysis, Complex Analysis, etc.)

    Most of the better Physics departments (top 100) in the US seem to know what they are doing as it relates to how to ensure their majors have the math they need one way or the other. I tend to get more suspicious when the Calc 3 courses are powder puff and the Physics majors never get the math they need to properly understand Maxwell's Equations or the Hydrogen atom at the more advanced undergraduate levels. These shell games are much more common at universities ranked below 75 or so, and I'd be hesitant to recommend schools ranked below 100 for students aspiring to PhD programs without having a hard look at the specific school to know whether the math (and the Physics that depends on it) is getting short shrift.

    But course names on a transcript may also not be meaningful. Lots of Calc 3 and Diff Eq credit is being gifted, and lots of slacker physics students are passing E&M 1 and 2, and QM 1 and 2 who are woefully under prepared for graduate school. The grade gifting is not even giving these students Cs any more to keep them out of graduate school, lots are being gifted As and Bs.
     
  14. Dec 17, 2017 #13
    The usual offerings at a decent university are good enough. Physics students should not waste a minute of their time in philosophy courses from the math department that might cover such irrelevant topics as generalized functions/distributions; the Dirac delta function is obvious to people who do not get their jollies off estimating the number of angels who fit on a pin head, and a good physicist can pick up the mathematics if needed from a strong foundation (Calculus through vector calculus, applied linear algebra, ordinary and partial differential equations, probability theory,and more recently, computational methods)
     
  15. Dec 17, 2017 #14

    symbolipoint

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    I do not believe that; which is why I asked the question. I had only found hints from some physics students many so many years ago, that more advanced and alternative Mathematics courses were getting into play for some of these students academic studies.
     
  16. Dec 17, 2017 #15

    atyy

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    You can just learn the maths with the physics. If one is interested, one can take the maths courses one likes. These are not official requirements, since there are many types of physicists. If you make too many requirements, you leave less time for people to explore their own interests.

    Also, the maths in some parts of physics was still not rigorous even 15 years ago - eg. renormalization, Landau damping, the KPZ equation etc.
     
  17. Dec 17, 2017 #16

    bhobba

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    Really? Irrelevant? Are Rigged Hilbert Spaces irrelevant to QM? If you say yes then for anyone that actually thinks a bit the usual formalism is an utter mess. Ballentine knows this and explains, not in detail mind you, but enough so someone that actually thinks, gets the gist of whats happening, and why its important. And learning Fourier transforms - try that one without distribution theory - you soon become bogged down in issues like how the hell does one have the Fourier transform of say e^ix which crops up all over the place. You can look it up in a table of such things - but anyone with even a very basic knowledge of integration is lost - you cant do it.

    Of course you may think its just philosophy - that's OK - but I don't classify things that create confusion as just philosophy - it IMHO is critical.

    Thanks
    Bill
     
  18. Dec 18, 2017 #17

    Mister T

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    It depends a lot on what it is you plan on doing with the physics you learn, but also what you like to learn. Your interests may lean more toward the practical or more towards a deeper understanding of things. Be careful, though, because too much of either one may turn out to be a bad thing.
     
  19. Dec 18, 2017 #18

    bhobba

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    Sure - but IMHO things that even a smidgen of thought show is nonsense should have at least an overview of whats going on. Some students don't care about such hings - even if it doesn't really make sense, as long as it works who cares. Others get quite confused by this and you loose them. Yes it depends on the student and purpose, but we need at least a bit of an aside explaining something like - if you are on the ball you recognize its nonsense but rest assured it can be fixed. If interested they can then go away an look into it.

    It just isn't generalized functions but crops up all over the place eg the Hall Effect. The usual explanation is its absences of electrons, 'holes', moving in the other direction. This then proceeds onto explaining semiconductors and what not. But a little though shows it utter hogwash - holes moving one way are exactly the same as electrons moving in the usual direction - it explains nothing. Many don't care - but a few thinks a bit and realizes it's a big porky. Whats really going on - well the holes are actually quasi particles of positive charge and conductors contain both electrons and holes moving about. It doesn't take much to explain it - good students can investigate it a bit further (although that will involve a lot of advanced QM) - but at least are not confused.

    The one thing we cant do to students is make them think things are not quite 'cosha'. We don't need to tell them the full detail - but we do need students to know these questions do have answers.

    Thanks
    Bill
     
  20. Dec 18, 2017 #19
    Formal logic and set theory.

    Why generalized functions/distributions are irrelevant ?
     
    Last edited: Dec 18, 2017
  21. Dec 18, 2017 #20

    bhobba

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    Of course they are not irrelevant. Its just some students simply do not care. That's fine - but some do and dismissing those means possibly loosing them - they are turned off. That simply IMHO is unacceptable - we must do better.

    Its not confined to students either - read the opening pages of Von-Neumann's - Mathematical Foundations of QM - Dirac - yes the great Dirac - get's a big serve - correctly. Of course Von-Neumann can do that from his exalted position, but it leaves a big problem to be solved. It was solved - but it took a while. IMHO we are doing a disservice to students not at least outlining the solution.

    Thanks
    Bill
     
  22. Dec 18, 2017 #21

    George Jones

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    I agree that the amount of math required for a typical North American physics degree is about right.

    While the amount of required math is about right, I think it would be sad if no physics students took pure math courses like point-set topology or abstract algebra, just as it would be be sad if no physics students took non-required biology courses or non-required chemistry courses.

    Science, including pure mathematics and non-applied fundamental physics is part of who we are as a species. Fundamental science is as much part of our culture as musics, art, and literature. If we lose the desire and ability to ask fundamental "Why?" questions in science, we have failed as humans.
     
  23. Dec 18, 2017 #22
    In the US, most BS degrees top out at 120-128 credit hours, and this limit on credits is set by the institution, not by the Physics department.

    The institutions also have core requirements (humanities, foreign language, etc.) so that the major requirements represent a fixed number of hours much lower than 120. In most cases, the requirements for a BS in Physics already has maxxed out the number of credit hours in Physics + Math, so you cannot simply add more. Each addition needs to remove something else, so that the total number of credit hours in Physics + Math is constant.

    Folks are full of great ideas for what to add, but few have realistic proposals for what should be removed to keep the number of credit hours in Physics + Math constant while still representing an improvement with their proposed additions.

    Students do have a small number of electives that they can use for additional math or science courses. They tend to make these based on combinations of perceived need, interest, and keeping their perceived workloads manageable. I am mentoring a couple outstanding science majors who are having a hard look at minoring in math and using electives to fulfill the requirements for a math minor. If they complete this proposed path, they'll be better prepared for possible theory in graduate school, but not really at much advantage should they choose experimental paths. Programming courses would suit them better in those cases.

    On the whole Physics (and other science) majors need more opportunities, not more requirements: opportunities for research, opportunities for programming coursework, opportunities for advanced math coursework, opportunities for graduate coursework (possibly with instructor approval). One student I mentor was denied enrollment in a graduate course due to an administrative rule, even though the instructor wanted her in the course and even invited and strongly encouraged her.
     
  24. Dec 18, 2017 #23

    Dr Transport

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    Really, in 30+ years since I got my degree I have not had the need for either...... I've used group theory and that is about the most advanced math I've used.
     
  25. Dec 18, 2017 #24
    I am sorry, it was a bad joke.
     
  26. Dec 18, 2017 #25

    ZapperZ

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