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Mathematics for the Undergraduate Physics Major
We present below a version of a talk given by Mary L Boas on 28 December 1970 at the meeting of the American Association for the Advancement of Science. It was part of a Symposium on Mathematics in the Undergraduate Program in the Sciences, jointly sponsored by the Committee on the Undergraduate Program in Mathematics. At the time when the talk was delivered Mary L Boas was an Associate Professor of Physics at DePaul University. She had taught both mathematics and physics at several universities. A few years before, in 1966, she had published a textbook on mathematical methods in the physical sciences.
Mathematics for the Undergraduate Physics Major, by Mary L Boas.
I want to talk specifically about the mathematical training of undergraduate physics majors. However, since at this level there is not a very great difference between the needs of chemists and engineers and those of physicists, we may as well think of discussing in general the mathematical training of students in the physical sciences.
There are at least two different schools of thought about the proper mathematical training of an undergraduate physics major.
For many years it has been traditional to have a one or two-semester course for graduate students (or perhaps senior-graduate) in mathematical methods of physics. A similar course at the undergraduate (say, sophomore-junior) level has never become very customary, and I think this is too bad. At the present time the usual physics major "picks up" a great deal of his knowledge of mathematical techniques in his physics courses, where he must use these methods with little or no previous introduction. The few mathematics courses he does take (probably differential equations and perhaps advanced calculus or complex variable or probability - usually not more than one or two of these) go into much greater detail than he finds either interesting or useful. The topics he does not study in mathematics classes he must master from a sketchy introduction in the physics textbook. If the student is not overwhelmed by the new physical ideas he is trying to master, he is likely to be by the combination of new physics and a new mathematical technique presented simultaneously.
Now I am well aware of the physics teacher's counterargument here, that the motivation of the immediate need for a technique is very important and that for this reason the mathematics should not be taught separately. Years of experience have convinced me otherwise.
The Appropriate Mathematics Education.
It is quite possible in a mathematical physics course to tell the students enough about some of the simpler applications of the mathematics topic at hand to keep up interest. Then, in a later physics course, it is merely necessary to review briefly a mathematical concept previously studied and to put the main emphasis on a continuous presentation of the physical ideas rather than to endure long interruptions for the sake of mastering mathematical techniques.
I am also convinced that in the end both the mathematical and the physical ideas are better learned and understood. For example, one mathematical idea may recur in several different physics courses yet the average student may fail to recognise it as the same idea. Worse yet, he may learn what appear to be contradictory ideas in two different courses. For instance, every student of thermodynamics "knows" that the integral of an exact differential around a closed path is zero. But in, say, electricity and magnetism he may run into the integral , which is certainly the integral of an exact differential around a closed path, but is not zero.
Perhaps I should also answer the mathematics teacher who might insist that a physics major should study mathematics as thoroughly and with as much detailed proof as a mathematics major does. My only real objection to this stand is its total impracticality. If the physics student is to take separate courses in linear and abstract algebra, vector and tensor analysis, differential equations (ordinary and partial), probability, advanced calculus, special functions, calculus of variations, etc., he is in the first place taking more mathematics than the average mathematics major; and in the second place, when does he get time to take some physics courses? The alternative which, as I have said, is usually practiced is for the student to take one or two of these courses in the mathematics department, and then "pick up" the rest of the needed techniques from the sketchy outlines in his physics books. I have already indicated why I consider this unsatisfactory.
Now you may ask: What alternative is there? There are so many topics to learn that the student simply cannot study all of these different mathematical topics in a course of reasonable length. After about ten years' experience with such a course at DePaul University and after writing a book [Mathematical Methods in the Physical Sciences, by Mary L Boas, published by John Wiley and Sons, 1966.] which we now use as our text, I think I am in a position to say it can be done. How?
First, such a course must be taught as a scientist's methods course and not as a mathematician's theory course. This approach does not have to mean careless or incorrect teaching. I make a strong point of never teaching anything a student will ever have to unlearn. However, generality can be sacrificed with little loss at this stage. The special cases of theorems that are going to be needed are quite sufficient.
Second, detailed and long proofs can be omitted; unless a short proof is the easiest way of introducing a theorem, I would omit it and substitute a careful statement of the theorem, including a warning about the limits of applicability and a discussion of its meaning and use. With these two methods of condensation (limited generality and few proofs) it is amazing how many techniques can be compressed into a year's course.
When I started teaching this course, some sceptical colleagues in physics believed motivation would be lost, and some in mathematics believed it impossible to cover so much material. By now, many classes have gone through our program and a good many graduates have continued in graduate school and returned with praise for the course; I think its success has been proved.
Development of the Course.
I would now like to support my general philosophy of mathematics education for the scientist by supplying more detail about my background and especially about my course at DePaul and my textbook. I was myself a mathematics major through my master's degree and somewhat beyond but changed to physics for my Ph.D. because I was unable to find direction in mathematics departments for a thesis in mathematical physics. During this time and while I was working for my doctorate in physics at MIT, I had a considerable amount of experience teaching undergraduate mathematics.
During my early years at DePaul, the physics department was small, and I taught at one time or another almost all the different undergraduate theory courses in physics. I gradually began to accumulate a rather complete outline of the mathematics topics which are so often cursorily introduced in undergraduate physics texts in the various fields (of necessity because the students do not get them in their mathematics classes). Over a period of about six or seven years, we gradually changed what I had started as a one-quarter senior-graduate course in mathematical methods to a one-year course which we now give to sophomores and juniors. I think this change and the feeling of the need for it is indicative of how the demand for earlier training in mathematical techniques has been developing in physics teaching and textbooks in recent years. Meanwhile, after trying several existing texts, I finally started supplying dittoed lecture notes to my students; these eventually became a textbook.
Content of the Course.
Perhaps in the remaining time you would find it interesting to know in more explicit detail just what topics we cover in my course at DePaul. We are on the quarter system, and the students with whom I start have had four quarters of calculus and are concurrently taking a course in differential equations.
The first quarter we start by studying probability, which meshes nicely with the thermal physics course that the students are also taking concurrently. Then I give them whatever seems necessary (depending on their calculus background) in: (1) infinite series with special emphasis on obtaining a few terms of a Maclaurin series and an effort to produce some understanding of the use and usefulness of series in approximations in physical problems; (2) complex numbers - with emphasis on using the geometry of the complex plane, e.g., to evaluate by locating it as the point , and a few applications to electric circuits and waves; (3) sufficient knowledge of determinants and matrices to enable them to evaluate a determinant, solve a set of equations by determinants, and multiply two matrices; (4) partial differentiation and multiple integrals review, if necessary (this is covered in fourth quarter calculus, but sometimes students may not have learned cylindrical and spherical coordinates very well or they may not have studied Lagrange multipliers); and (5) vector algebra (by this time, they have had most of this in both mathematics and physics classes but we put some effort into three-dimensional analytic geometry of lines and planes using vectors, e.g., finding the distance between two skew lines).
The second quarter we continue vectors - taking up topics of vector calculus: line integrals, gradient, divergence, curl, and the vector integral theorems. Then we study Fourier series in considerable detail and follow it with two somewhat related topics: Laplace and Fourier transforms, and partial differential equations (studying separation of variables and only those equations which can be solved using Fourier series). Sometimes with a particularly good class we are able also to do calculus of variations in the second quarter - otherwise it is the beginning of the third quarter.
The third quarter is not required for the physics major, and the poorer students have dropped out (although many of them pick it up their senior year); thus, we can work at a little more sophisticated a level. We take up coordinate transformations, learning diagonalization of matrices and the idea of an eigenvalue problem with several applications, and tensor analysis. We then do a rather formal chapter on complex variable theory learning to evaluate contour integrals and a little bit of conformal mapping. With this background, we then go back to Laplace transforms and learn how to find inverse Laplace transforms using the Bromwich integral and the convolution theorem. We discuss the Dirac delta function briefly. Next we take up series solutions of differential equations, studying specifically the Legendre polynomials and Bessel functions. I try to get the students to understand that there are many different orthonormal sets of functions and that they should not be afraid to tackle an unfamiliar differential equation.
(In fact I often give Hermite or Laguerre equations on the final exam, and they cope very well with them.) Finally we complete the work on partial differential equations, reviewing separation of variables and now studying equations whose solutions may involve Legendre polynomials or Bessel functions.
Several general comments are perhaps now in order. This material which I have outlined is intended to supply a mathematical background for the advanced undergraduate physics courses and perhaps for first-year graduate courses. (In fact we use it for a review course for poorly prepared entering graduate students.) However, I do not mean to sell short the more mathematically sophisticated courses offered by the mathematics department. I think that a course such as mine is of tremendous help at about the sophomore-junior level and is all that is necessary for students who stop at the bachelor's degree. However, I always encourage students who are hoping to go on to graduate work to take some mathematics courses at the junior and senior levels. It is important for them to learn to converse with mathematicians and to study some of the more sophisticated "proof"-oriented courses given in mathematics departments. My thesis is not that such courses do not belong in the training of a (graduate-school-bound) physics student, but rather that the techniques course should come first. As a matter of fact, a fair number of mathematics majors take my course because they find the rapid survey of a variety of topics very useful orientation.
Last Updated October 2026