# Featured What should the Mathematics requirements be for a Physics degree

1. Dec 18, 2017

A good background in Linear Response Theory along with Fourier Transforms is something very important for a well-rounded physics education. I did not have such as course in my undergraduate education, but learned this material in bits and pieces as it came up time and time again in various physics courses including in graduate courses. The convolution theorem for linear response along with the equation in frequency space $\tilde{V}_{out}(\omega)=\tilde{m}(\omega) \tilde{V}_{in} (\omega)$, (where $\tilde{V}_{out}(\omega)$, $\tilde{m}(\omega)$, and $\tilde{V}_{in}(\omega)$ represent Fourier transforms of $V_{out}(t)$, $m(t)$, and $V_{in}(t)$), comes up repeatedly in many places, including electrical circuits, dielectric and magnetic frequency response of a system, and many other places. I think it would be advantageous for this material to be presented to the students in a required mathematics course, rather than needing to pick it up in bits and pieces. It's application is quite universal.

Last edited: Dec 18, 2017
2. Dec 18, 2017

### stefan r

English/writing so that they can write decent papers. Economics/business and maybe political sci so they can run a lab. History/social sci so they know when they are why. Psychology and communications also to run a lab. Some phys ed because brains need food and exercise and also because this is easy to work into a schedule.

If the college/university teaches students to think then they can learn how to do the jobs/careers that they find themselves in. People often find themselves doing things they did not expect to be doing.

3. Dec 18, 2017

### symbolipoint

I mostly agree, that those are all very or at least potentially useful, and sometimes are neglected. My emphasis for the topic is on the MATHEMATICS courses to help educate the Physics students better.

4. Dec 19, 2017

### stefan r

I am saying that you will get better educated physics students if you do not neglect the liberal arts education.

5. Dec 19, 2017

### symbolipoint

The connection is weak. The connection between Mathematics and Physics is much stronger. A bachelor of ARTS degree in Physics might be a degree arrangement encouraging the students to study more of the liberal arts. A bachelor SCIENCE degree in Physics might be a degree arrangement encouraging more technical courses such as Math or Computer Science or other sciences or Engineering course as some possible electives, and still nothing should stop such a student from dipping in to a little bit more of liberal arts.

The ability of the graduate to discuss societel and economics conditions, and write and be able to read is already granted so the focus, as asked initially, is about what set of mathematics courses beyond the usual stated official required ones, should a motivated physics student include. The answers seem to be, anything the student wants if contains topics which his areas of specialty can use the topics, and ultimately, no limit on what more Mathematics course - but one must cut off some because just not enough time when one wants to deal with Physics and not endlessly learn more Math without clear purpose.

6. Dec 19, 2017

### Staff: Mentor

7. Dec 19, 2017

### Dr Transport

These were my UG math courses, Calc I-III, Differential Equations, Linear Algebra, Advanced Multi-variate calculus, PDE's (Fourier series, separation of variables, no Green's functions). That was the required coursework. I also took a course in Special Functions and another special studies course in PDE's.

My department didn't teach a course in math methods (it still doesn't), that was the required advanced multi-variate calculus and PDE's, both of which were taught by a true mathematical physicist who was a professor in the math department. Computational physics wasn't required, neither was numerical methods or complex variables. We did see some complex calculus in the advanced multi-variate course which was essentially an advanced vector calculus, integral theorems (Gauss's law, Stokes theorem and some complex variables and residues).

In graduate school, one place I went to taught a year long course in advanced calculus, special functions and complex in the fall. The spring consisted of linear algebra and group theory. Depending on who taught the group theory, it could have been point and space groups or Lie groups. The other school taught a year long course out of Arfken (yes it was Arfken, not Arfken and Weber, that is how long ago I went to school).

That should shed some light on what math is required for an undergrad physics degree.

8. Dec 19, 2017

### alan2

For what it's worth, I had to take the usual 3 semesters of calculus, linear algebra, ODE's, advanced calculus and one semester each complex variables, vector analysis, and boundary value problems for a BS in physics. But that was 30 years ago and, as I recall, the last three were to satisfy a requirement of three additional math courses at the senior level. I think the experimental people took statistics and other courses instead. I also have degrees in mathematics and, frankly, I see zero use in making a physics major take any course in pure mathematics. Physicists need methods, not formal proof.

9. Dec 19, 2017

### Staff: Mentor

That's quite reasonable.

Interestingly here in Aus, as I said before, double degrees in applied math and physics or math and engineering for that matter are very popular. They even have a separate strand for it (they have a number of different strands as well eg - stats and decision science). The advanced subjects that must be done for that strand look a lot like physics subjects - they are (from the student handbook of where I attended):

Applied Transport Theory
Applied Transport Theory is the study of the exchange of mass, momentum and energy in physical systems. An understanding of the equations that govern these transport phenomena is fundamental to understanding how the physical world behaves. This unit builds upon knowledge you will have developed in studies of advanced calculus. You will learn how to derive equations from fundamental conservation laws and develop an understanding of the commonality between the equations. Furthermore, in deriving analytical solution techniques for these equations you will develop further skills in calculus and differential equations. Completion of this unit will prepare you for the final semester capstone project.

Partial Differential Equations
Previously you have discovered the power of differential equations for modelling real world processes. In this unit you will extend your capabilities to problems that simultaneously exhibit both spatial and temporal variation. Such problems can be described by partial differential equations. You will learn a variety of analytical solution techniques for these equations, which bring together many of the skills you have learned in earlier study of advanced calculus and ordinary differential equations. You will also learn the techniques of Fourier and complex analysis, which have applications far beyond the realm of differential equations.

Dynamical Systems
Dynamical Systems” is a descriptive term used to represent the analysis of time varying systems. Such systems exhibit a variety of behaviours including exponential approaches to equilibrium states, periodic or oscillatory solutions, or unpredictable chaotic responses to simple inputs. The study of dynamical systems employs topological and function space concepts to provide the analytic structure to systems of nonlinear (and linear) ordinary differential equations, and as such forms the basis for the mathematical interpretation and understanding of numerous real world systems. This unit is an exploration of the more technical aspects of the theory of solutions to systems of ordinary differential equations and as such builds on your prior understanding of such equations while providing the support for the exploration of an exciting area of modern mathematics.

Computational Fluid Dynamics
This capstone unit provides students with the opportunity to apply their knowledge and skills in applied and computational mathematics to simulate complex real-world problems. Students will be presented with several real-world case studies, which will involve model formulation, examining the impact of varying model parameters, and formulating and presenting recommendations for the best course of action to take based on model predictions. Your previous learning in deriving and solving partial differential equations that describe transport phenomena will be extended to include numerical methods of solution. Combined with the computational expertise you have acquired over your degree, you will be able to formulate and solve these complex mathematical models using MATLAB.

Thanks
Bill

10. Dec 20, 2017

### DaveC49

I found having a good grounding in Calculus, ODE's, Linear Algebra, Vector spaces, Group Theory, Numerical method were the maths couses I found most useful for Physics. I also did introductory computer science subjects at first year level and 1st semester 2nd year which proved useful. Like Bhobba, I did a combined Maths- Physics double major degree which I changed to a Physics major at third year level. I only did a very basic statistics course at first year level yet as an applied physicist, I would have benefited from far more depth in statistics. The numerical analysis was however extremely useful in my later career. UQ where I did my undergraduate degree also offered courses in both math and physics at a basic core level and at an advanced level. In maths the core level concentrated on knowing the theorems and fundamantal concepts while the advanced levels required you to learn and understand the proofs of the theorems. Physics also had the same two level structure where the core level subjects gave you a grounding in the concepts and the advanced levels gave you more depth and usually a more rigorous mathematical treatment. Both departments also offered a fourth year to selected students which was pitched largely at a Masters level or Ph.D. course work level in the US generally with prerequisites from the advanced level. One of the difficulties with the combined degree was the mismatch in timing between the relevant maths and physics courses as each department largely arranged their couse structures to meet the requirements of their own graduate programs. Consequently QM in physics preceeded Hilbert spaces in maths by a semester. The mathematics requirement for a mathematical physicist working in areas of string theory or quantum loop gravity or astrophysics are going to be very different from those for someone who ends up working in an accelerator lab . A good solid grounding in core mathematics though gives you a better chance of being able to identify and pick up anything you didn't get as an undergraduate later in your career.

11. Dec 20, 2017

### stefan r

My answer was very much on topic: 3 years calculus and linear algebra. More requirements diminish the "degree".

You are, of course, welcome to disagree. My being wrong does not imply that I am talking about the wrong subject. Natural science professors tend to think their majors should be bigger. Humanities professors usually state the opposite position. This argument really happens at colleges and universities.

The original post also had this question:
Listing off lots of possible math courses does not answer that question at all.

Physics and Mathematics professors at colleges and universities try to add more mathematics requirements. Professors from other departments block those proposals. Reasons are stated more eloquently than I am likely to be capable of.

12. Dec 20, 2017

### Staff: Mentor

@stefan r, your post #27 was off-topic, and was the reason @berkeman listed your name. The title of this thread is "What should the Mathematics requirements be for a Physics degree" -- (emphasis added).
Also, what does "History/social sci so they know when they are why" mean?

13. Dec 20, 2017

### symbolipoint

Right. berkman does have a point. This point can be made better if done as a separate topic. Some departments and universities seem to like to ADD a separate WRITING course supposedly suited to a particular major field. My belief is that such a separate course should not be needed, since the major field's program should have some significant report-writing as part of more than one of the major-field's courses. A very significant routine lab exercise in some phys sci courses is the writing of formal lab reports. Any student can easily adapt to these. Note that one of the "humanities" areas is the courses which emphasise modern foreign languages; which definitely bring up instruction about both CULTURE & HISTORY, and also the workings of the LANGUAGE. The student will then develop some precise language translation, which is much like what we do mathematically anyway in our physical sciences. When "we" do this, it is between a described or representable physical situation and mathematical expressions and equations.

14. Dec 20, 2017

### Staff: Mentor

Yes - I originally only did the minimum required which believe it or not were 4 subjects - Mathematical Statistics !A, 1B, 2A, 2B. Didnt like it much personally, but the lecturer for the advanced subjects was great - I even remember his name - Dr Ogalvie. I really liked him so, not because I liked stats much, but purely because I liked him , I did 3A and 3B. Turned out to be a very good choice not only for physics but for computing later on - they wanted all these statistical reports etc and it was great to know the background behind them. Not to 3A and 3B level, but it was very useful in understanding QM and it's probability bit such as exactly what does the Heisenberg Uncertainty Principe mean. - you need to know and understand variance for that one.

Thanks
Bill

15. Dec 21, 2017

Besides having courses in advanced calculus, differential equations, and complex variables as an undergraduate physics major, I also had a course in Probability Theory given by the Mathematics Department. One of the most useful things about probability theory is the concept of the distribution function, $F(x)=P(X \leq x)$ compared with probability density function $f(x)=F'(x)$. This concept is very useful in a number of places, including particle scattering, where a section (area ) on the target $\sigma$ maps into a solid angle $\Omega$. (Thereby concepts like the differential scattering cross section $\frac{ d \sigma}{d \Omega}$ are much more readily understood). And I do think in this area, the mathematicians generally have at least a slight edge over most physicists in presenting material such as this. $\\$ The mathematical concepts presented in the Probability course have also been very useful in understanding things such as the Planck spectral function for blackbody radiation, as well as the Maxwell/Gaussian type distribution (density) functions of particle velocities. The Probability course also covered the Standard Normal Distribution function, including how to normalize a function of the Gaussian form. The various probability concepts, including binomial trials, etc., and discrete distributions were also very good things to have learned. I would recommend that anyone who is serious about their physics education have such a course as soon as possible after the basic calculus sequence. Whether such a course should be made a requirement I think would depend somewhat on the academic ranking of the university. I think a really top notch program could do well to have such a course be part of their graduation requirements. $\\$ Editing: And although it might start to be a somewhat heavy load from the math department if the advanced calculus, differential equations, and complex variables courses were required, I think it could be advantageous to have them be required. I still am of the opinion that a course in Linear Response Theory and Fourier transforms should be a requirement as mentioned in post 26. I had one additional course from the mathematics department as an undergraduate, besides those mentioned above, and that was Linear Algebra and Matrices. That, IMO, was the least important of all of these courses. A course in Linear Response Theory and Fourier Transforms would have been much more beneficial than the Linear Algebra and Matrices course.

Last edited: Dec 22, 2017
16. Jan 1, 2018

### Jenab2

Linear algebra, probability and statistics, Fourier and Laplace transforms, vector calculus, and partial differential equations.

17. Jan 1, 2018

### AgentSmith

It depends on how your physics courses are organized. There are bits of math needed here and there that can be fit into a physics course. But you need a minimum of ODE, Linear Algebra, some PDE, some tensor analysis. Its possible to take ODE and linear algebra, then a two semester course in physical mathematics involving PDE, vector analysis. complex analysis, tensors, abstract algebra. A separate and serious sequence in experimental methods & data analysis should be required.
In my opinion, to which I give considerable weight, an applied mathematics degree with a year of intense physics work is also feasible. It just depends. There is no single answer, beyond the minimum most people take anyway.