Math 210C (Mathematical Methods in Physics and Engineering III)

Instructor: Prof. Michael Holst (5739 AP&M,
Term: Spring 2016
Lecture: 1:00p-1:50p MWF, TBA
Office Hours: 2:00p-2:50p M, 5739 AP&M
Class webpage:

  • Electronic copy of Book Used for Second Part of Course [Frank12] Theodore Frankel. The Geometry of Physics: An Introduction. Cambridge University Press, New York, NY, Second(Third) Edition, 2004(2012).
  • Required [Boot10] William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, San Diego, CA, Revised Second Edition, 2003.
  • Recommended [Isha10] Chris J. Isham. Modern Differential Geometry for Physicists. World Scientific, River Edge, NJ, 2001.
  • Recommended [Hass06] S. Hassani. Mathematical Physics: A Modern Introduction to Its Foundations. Springer, NY, NY, 2006.
Printable Syllabus: Can be found [ here ].

COURSE DESCRIPTION: The following is a brief description of the course. Note that the UCSD catalog currently has a very dated description of this course that was written a couple of decades ago. Our goal for the course remain the same as for the original course: provide graduate researchers in applied mathematics, physics, science, and engineering with a more sophisticated and powerful set of mathematical tools for performing their research. However, we will update some of the topics to reflect changes in the mathematical tools needed for modern science and engineering research; obvious examples being the growing importance of simulation algorithms over the last two decades, and the more recent emergence of data science techniques in all areas of science and engineering research. The prerequisites for the course remain the same as for the older description of the course.

210C. Mathematical Methods in Physics and Engineering (4)
Prerequisites: Math 210B or consent of instructor.

  1. Topology
  2. Differentiable Manifolds
    • HW1, covering portions of Topics 1 and 2, is posted [ here ]
      (NOTE 6/7/16: Fixed typos are noted in red on the newly posted HW1.)
    • Lecture notes (pretty rough) on implicitly defined manifolds, optimization, mechanics are [ here ]
  3. Vector Fields, Tensor Fields, Vector Bundles
    • HW2, covering portions of Topics 2 and 3, is posted [ here ]
      (NOTE 6/7/16: Fixed typos are noted in red on the newly posted HW2.)
  4. Exterior Derivative, Lie Derivative
  5. Riemannian Manifolds, Covariant Derivative, Curvature
    • HW3, covering portions of Topics 3, 4, and 5, is posted [ here ]
Applications appearing throughout the year: Linear and Nonlinear Ordinary Differential Equations (ODE), Partial Differential Equations (PDE), Integral Equations, Wavelets and related tools.

GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Course information, such as the planned lecture topics for the week, any homework assignments, and so forth, will be maintained on this class webpage. Note that I sometimes make minor changes to both the lectures and homework assignments as the quarter progresses, based on how much I am able to cover in the lectures, and which directions we go based partially on the interests of the students. Therefore, CHECK THE WEBPAGE FREQUENTLY.

The course will be graded on attending most of the lectures, doing some homework assignments, and a final "take home" examination, according to the following guidelines:

Participation in Class (I.e., coming to most of the lectures): 25% of grade
Homeworks (3-5 written homeworks, each covering a major part of the course): 25% of grade
Final Exam (take home exam) 50% of grade

The "participation in class" part of the course is that I simply want you to try to come to most of the lectures; that will give you full marks on that part of the grade. (I will not actually take attendence; you are all adults.)

The "homeworks" will be based on the lectures, and will just give you the opportunity to use some of the tools we go over in the lecture. They will not be particularly time-consuming; they are for your benefit. Similar to the participation metric, if you make a good attempt on the homework then I will give you full marks. (I do not plan to formally grade the homeworks, but will try to give you feedback if you feel you need it.)

The final will be a "take home" exam, which will simply be some problems from the list of homework problems (that you may have already worked on during the quarter). During the actual scheduled time of the final exam for 210C, I will be in my office and can answer any last-minute questions about problems on the take home final. (I will also have some coffee available for anyone who comes by...)

Here are some important dates:

First lecture: MON 3/28
Last lecture: FRI 6/3
Finals week: MON-FRI, 6/6-6/10
Office Hours During Finals week: TUE/WED/FRI: 10a-12p, THU: 11:30a-2:30p
Final Exam Period (Office Hours in 5739): THU 11:30a-2:20p
Take Home Final Exam Due Date: FRI 6/10 (by 5pm in my office or under my door, OR scan it and email it to me by midnight)

TAKE HOME EXAM (Announced in Class Friday 06/03/16): This quarter the take-home final exam in 210C consists of working out a total of eight of the assigned homework problems that were listed above as HW1-3. You are completely free to select your eight problems from the list of problems I assigned in the Homeworks, EXCEPT that you must choose three from each of HW1 and HW2 (six of the eight problems), and two from HW3. That way you will be challenged to learn a little bit about each of the major sections of the material that we covered this quarter. HW3 are problems on exterior calculus, which we have only barely begun in the lectures. But, all of this material is very standard, and can be found in all four of the books listed for the course, and also in my lecture notes (I will post a scanned version as soon as I can). The best resource is actually the book by Ted Frankel (a now retired UCSD professor); I have a link to the electronic version (free to all UCSD students) at the top of this webpage. By pushing you to look at the exterior calculus problems just a little bit, my hope is to get you interested enough to pursue reading on your own this summer.

Some comments on the final exam problems:
  • As requested in the last lecture, if you choose as one of your final problems the long multi-part problem on classical mechanics (problem 1.9 in HW1), then I will count this as two of the three problems you are to turn in from HW1.
  • Some of the problems on all three homeworks are just definitions; please do not pick those problems for the final exam problems :)
  • Some of the problems in the homeworks, particular HW3, are not easy (I actually do not know how to do some of them yet). Just make a good attempt; that is what I am looking for here.
UPDATE ON FINAL (06/09/16, 9am): Unfortunately, we do not have a room in APM today for me to give an extra 1-hour lecture during our scheduled final (Thu 11:30a-2:30p), so I will just be in my office (with coffee to share, maybe even some snacks) during that period.

Early in the quarter the department asked all instructors not giving formal final exams to give up their assigned final exam room to the department for use as OSD exam spaces, and our room is being used for this. The department is really short of space for these exams during finals, together with hosting large review sections for lower division courses, so all rooms are actually scheduled the entire day.

If anyone has not had enough of tensors and p-forms by the end of this week, I will offer to you a personal 1-hour lecture any afternoon next week :).