University of Minnesota
School of Physics & Astronomy

Phys 5041.001

Mathematical Methods for Physics

modified 17-Jan-2017 at 9:29AM by Oriol T. Valls


I- Objectives of the course:

The main objective of the course is to help students develop the level of
understanding, skill, and ease in the use of mathematical techniques for
problem solving which is required
of professional physicists. The purpose is to give students
not only the abilities they need,
but also to instill in them the equally necessary
confidence in their ability to handle advanced mathematics.

The course will emphasize "how to get it done", not mathematical rigor.
It will also emphasize the blending of analytic, symbolic and (occasionally)
numerical techniques, as used by practicing physicists in contemporary research,
rather than their separate aspects.

II- Prerequisites:

This course is intended for Physics graduate students. Many advanced
undergraduates take it also. It should be useful as well
to students majoring in other
scientific or engineering subjects.

Prospective students unsure about their preparation should contact the
instructor. Permission will be liberally given to all genuinely interested

The course is not mandatory for anybody, and therefore it is rather informal.

III- Class meetings and Office hours:

The course is currently scheduled to have four 50 minute
meetings on MTWF at 8:00 in room 110 PAN. There are no recitations but
the Wednesday lectures will begin with an extended period of
student questions and discussion. This is not intended in any way to discourage
students from asking questions at any other time during the lectures, but only
to allow for the deeper questions (often requiring longer answers) which arise
during studying and problem solving attempts.

Instructor office hours (in McNamara 1480-2, accessible via
Suite 160) will be on Th at 4:40 (after the colloquium)
and Fr at 1:25. The TA will also hold office hours, in PAN 442,
on Th at 12:35.

IV- Textbooks:

No particular book will be followed in the lectures, but students should expect
to need to do a fair amount of book reading to supplement their class notes.
Many textbooks cover the subjects that will be discussed. Two books are
recommended: for general purposes, Arfken & Weber "Methods of Mathematical
Physics" contains an encyclopedic assortment of topics (many of
which will be covered in class, not necessarily in the same way).
Nearly all standard mathematical
methods that physicists are expected to know how to use are covered.
It is available for FREE in electronic form at: d=0&renderMode=poppedOut&displayMode=full&frbrVersion=&fctN=facet_frbrgroupid&dscnt=0&vl%282404505UI0%29=any&fromSitemap=1&onCampus=false&query=any%2Ccontains%2Cmathematical%20methods%20for%20physicists&frbg=502847&fctV=502847&loc=local%2Cscope%3A%28tcsearch%29&dym=true&dstmp=1370356334995&highlight=true&lang=eng&cs=frb&vl%281UIStartWith0%29=contains&group=GUEST&vl%28freeText0%29=mathematical%20methods%20for%20physicists&vid=TWINCITIES&institution=TWINCITIES

The second book, Bender & Orszag "Advanced Mathematical Methods for Scientists
and Engineers", is shorter and more advanced, but very useful. Buy it if
you can. The Springer edition is identical (except for a slightly different
title) to the earlier McGraw-Hill edition. Save money and buy it used.
The book is a "keeper", which should remain in the students' bookshelf for a long
time after the course is over. Bender & Orszag covers
methods with a reputation for being "advanced" or "difficult", in such a way as
to make these topics very accessible. According to Garrison Keillor, this is a
very well-written scientific book, see .

There are many other books with roughly the same scope: if you
already own one of them, you may not need anything else.

Other materials:

Students need to have access to a symbolic mathematical package such as
Mathematica or Mathlab. All Physics students have such access through the
Physics Department servers. Other students should contact the instructor.
Students will be expected to develop
familiarity with symbolic software packages during the course, and some previous
experience, while not required, is a plus.

V- Major topics that will be discussed include:

a.- Applications of Complex Analysis to integration and series summation.
b.- Asymptotic series. Their generation. Numerical summation. Borel summation.
c- Divergent series. Pade approximants.
d.- Fourier series. FFT.
e.- Integral Transforms (Fourier & Laplace).
f.- Multiple delta functions; response functions.
g.- Operational methods: Green functions. Boundary Conditions in PDE
h.- Integral equations.
i - Differential equations. Series solutions.
j.- Asymptotic behavior of solutions. Singular solutions
to differential equations.

Additional topics, depending on student interest, may be covered if time allows.

VI- Homework:

The objectives of the course can only be attained through practice. Therefore
homework is a very important part of the course. A set of problems will be
handed out every Monday and solutions will be due one week later. No late
homework will be accepted. If illness or other valid reason prevent a
student from doing a set, an adjustment will be made in the denominator of the
homework percentage. Making an intelligent effort to solve the homework is
mandatory: this is enforced as explained below. Getting the answers right the
first time is of course not what is mandatory: trying to get answers is.

Sample solutions will be posted at the course web site shorty after the
deadline for handing in the solutions. Sets will be graded: Each problem will be
assigned, in addition to the standard grade based on how correct the solution
is, a second "attempt" grade on a binary scale: a "1" if a serious attempt was
made to solve the problem (even if the attempt was unsuccessful) and "0"

It is quite acceptable, and for most people a good idea, for students to get
together in groups to do the homework, but each student should hand in his or
her own version of the work and be prepared to defend it. When doing
homework in a group make sure you contribute your fair share and do any
freeloaders a favor by ruthlessly expelling them from the group.

VII- Exams and grades:

There will be a final exam on Friday, May 12, 1:30-4:30 pm, a date and time
determined by the University and the Department, which cannot be changed. There will be a
one one-hour midterm tentatively scheduled for March 10. This date could be
changed. There could be instead (if a clear majority of the students prefer it)
two one-hour midterms. The exam grades will be based on each individual
student successful solution of the problems that will be posed.

The grades will be determined by two factors:

The regular portion of the grade, R, will be composed of: Successful homework
solutions (20%), the midterm (30%, or 20% each if there are two) and the final
(50% or 40% weight depending on number of midterms). R is expressed as a

The second, participation, P, grade will be computed as
follows: 80% from the number of homework problems seriously
attempted (seriously attempted means a solution handed in showing substantial
work, even if it was partly or totally incorrect), and
20% from class participation (asking questions etc) as judged by the
instructor. Any diligent student should get a P near 100%.

The overall grade T is determined by the square root of P times R.
T=sqrt(R*P). This means for example, that a student getting P=1 (100%), which
is quite doable, and a regular "exams and homework" grade of 64% (a C
according to the formula below) would have the grade transformed into 80%
(a B/B+).

Letter grades will be based on overall grade T with 5% intervals corresponding
to +/- increments, that is 15% increments corresponding to every letter.
Thus, one needs 45% to get a D, 55% to get C-, 70% to get B-, 85% to get A-.
This scheme awards A+ to students getting 95% or higher. The University, for
some bizarre reason, does not recognize the A+ grade, students earning one will
have a plain A in their official transcripts, but will receive an email from
the Instructor (which they can frame if they wish) informing them. The
University does not recognize the D- grade either, and any such would
become and F. Students taking the course on an S/F basis must earn at least a
C-, a D level grade is not satisfactory.


Every student is expected to behave professionally and honestly. (See for the
University conduct code).

No cheating or other unprofessional behavior will be tolerated.
Handing in solutions copied from another person, or found in the web, would
be, besides cheating, evidence of stupidity, of lack of interest
in learning, and of inability to keep minimum professional standards.
The minimum penalty for cheating is an automatic F for the course. All cases
will be considered for whatever maximum the Supreme Court allows.

Web site:

To find this web site go to the Physics department main page at and follow the link to class pages and then to 5041.
Updates of this syllabus and other announcements will be found here.
Students should periodically check the site, as they are responsible
for knowing the course announcements and other information posted.
Brief solutions to each homework set will also be posted at the web site
in .pdf format.

Legal stuff:

For anything not covered above, all relevant University Policies will be
followed. A very comprehensive index of such policies is at