PHYS 312 Introduction to Mathematical Physics Syllabus
TENTATIVE AND SUBJECT TO CHANGE UNTIL TERM STARTS

2026 Winter Session, Term 1 (September-Dec 2026)

Calendar Description

PHYS_V 312 (3) Introduction to Mathematical Physics

The application of ordinary and partial differential equations to physical problems; boundary and initial value problems associated with heat, wave and Laplace equations. Fourier analysis; expansions in Bessel and Legendre functions. Credit will be granted for only one of PHYS 312 and MATH 316. [3-0-0] Prerequisite: [MATH215]

Schedule

Lectures: Tue and Thu, 14:00 - 15:30, in Hebb 116.
Office Hours: see Canvas Syllabus Page for current schedule.

Instructors

Professor: Joanna Karczmarek

Email: joanna AT phas.ubc.ca
Homepage
Office: Hennings 406
Outside of lectures and office hours, ask questions on Piazza or reach me via email.

Teaching Assistants (TAs):

TBA
To email the TAs, use Canvas.

Learning sources and References

There is no required textbook for the course. The following is a list of resources you might chose to use to complement, clarify and extend material presented in lectures. When appropriate, lecture notes might refer to one of these resources

Math 257 and 316 cover a lot of the same material as Phys 312. A set of notes for Math 316 is available from Professor Richard Froese's website.
"Elementary Differential Equations with Boundary Value Problems" by William F. Trench is one of the books recommended for Math 316, available for free here.
For a book with a very physics-driven approach, check out "Applied Partial Differential Equations" by J David Logan. I especially like Chapter 1 (The Physical Origins of Partial Differential Equations) for its conservation law based approach. SpringerLink.
Another on-line reference is Notes on Diffy Qs: Differential Equations for Engineers by Jiri Lebl. This book covers ODEs and PDEs; you might have used it for Math 215.
For Linear Algebra, I will mainly use "Linear Algebra Done Right" by Sheldon Axler. You can find there additional discussions, proofs and examples of the definitions we made in class. SpringerLink.
For vector calculus, you can look ahead in your math 317 textbook, CLP-4.
For a more physicsy approach on vector calculus, as related to heat flow, check out The Feynman Lectures on Physics. The complete e-book is available from Caltech. Have a look at Volume II, sections 2-3, 2-4, 2-6, 3-3 and 3-4.
A more advanced book on PDEs is "Introduction to Partial Differential Equations" by Peter J. Olver. In particular, chapter 9 (A General Framework for Linear Partial Differential Equations) fits very well with my philosophy of the course. SpringerLink.
If you are looking for extra practice problems or examples, I suggest books from the Schaum's Outlines series. Relevant to this course are "Fourier Analysis with applications to boundary value problems" and "Differential Equations". The only non-free resource on this list, these are inexpensive paperbacks which contain summary notes of the relevant material, solved problems and supplementary problems with answers.

About SpringerLink: it's an online library that UBC subscribes to. When using UBCSecure, or when logged in through a UBC VPN, you will be able to download full PDF copies of the books mentioned. It's a great resource, and a good place to look for textbooks on many subjects.

Grading scheme and policies

Attendance is mandatory and participation in class is expected. Material might be covered that is not available anywhere else. In-class worksheets will be used extensively. Participation grades will be given for in-class worksheet completion and other activities such as reflections.

If you are unable to attend class, you can still complete the worksheet and hand it in on Canvas. See below for the rules around missed tests.

Participation	10%	Each lecture will have an in-class work and discussion component, usually centered around a worksheet. This work will be graded mostly on completion rather than correctness, unless otherwise indicated. The grades assigned will be either 1 (full class participation), 0.5 (partial class participation) or 0 (missing class participation). Late participation work will not be accepted; it will be due at 11:59pm on the day of the lecture. Your participation grade will be computed as follows: 1.25*(sum of your lecture participation grades)/(number of lectures with participation element). In other words, you have to participate in 80% of the lectures for full participation marks.
Weekly Homework	15%	The weekly Homework will be posted by Thursday and due the following Tuesday, after class (ie, you will have at least 5 days to complete it). Two lowest homework grades will be dropped from the final course grade. For long term absences, when further accomodation is warranted, the weight of those HWs might be moved to the final exam. Homework solutions should be uploaded via Canvas (in PDF format). If you experience technical problems, you can email the solutions to me instead. If you miss the deadline, but I receive your work before the solutions are posted, I will accept it. Homework that is late (even by an hour) might not be accepted for credit, since solutions will be posted immediately after deadline. Group discussion of Homework is encouraged, but the solutions you hand in must be your own work. This means you should not be looking at anybody else's notes or assignment while writing up your solutions. If asked questions, you may share your thinking with classmates, but not your completed work. Both copying from another student's completed work and sharing your completed work with another student will be considered academic misconduct. Similarly, using generative AI to create complete homework solutions is not permitted. A detailed policy on AI usage is given in this document.
Tests and Final exam	75%	There will be five tests, administered during class time, each worth each worth one 'exam unit'. The final exam will be worth three 'exam units' (each assigned the same grade). Questions on these tests will be very similar to those on Homework. Of these eight units, your best six will count toward your final grade. Tests and final exam will constitute 75% of the course grade, so each unit is worth 75/6 = 12.5%. In other words, if your grade on the final exam is better than your two lowest test grades, those test grades will be dropped and the exam will be worth 37.5%. If your grade on the exam is better than just the lowest test grade, this lowest grade will be dropped and the exam will be worth 25%. Finally, if your grade on the exam is lower than all your test grades, all the tests will count and the exam will contribute just 12.5% to your course grade. Tests will be held in class and cannot be taken online. Test dates: Sept 24, Oct 8, Oct 22, Nov 12, Nov 26. The tests will be up to one hour long and we will hold class in the remaining time. If you require extra time accomodations (through CfA), please try to start early rather than end late so you are not missing class. If you miss a test, the weight of the test will be moved to the exam, and exam will be worth one more unit. No Doctor's note or similar is needed for this accommodation. If you are sick on a final exam day, do not attend the exam. You must apply for deferred standing (an academic concession) through Science Advising no later than 48 hours after the missed final exam/assignment. Students who are granted deferred standing write the final exam/assignment at a later date. Learn more and find the application online: https://science.ubc.ca/students/advising/concession. Physics and Astronomy usually holds an SD exam session on Friday afternoon of the first week of classes in January. For additional information about academic concessions, see the UBC policy here: http://www.calendar.ubc.ca/vancouver/index.cfm?tree=3,329,0,0.
Total	100%

Top level learning goals

In this course, you will learn to:

Recognize separable equations and apply the method of separation of variables to reduce partial differential equations to systems of ordinary differential equations/eigenvalue problems.
Think about linear differential equations and their solutions using the framework of Linear Algebra: vector spaces, subspaces, homogeneous and inhomogeneous linear equations, linear operators/maps and their kernels and images, bases, inner products, symmetric/Hermitian operators; quote and sometimes prove relevant theorems.
Construct formal and explicit solutions to a variety of linear differential equations with and without boundary and/or initial conditions.
Use series expansions (Fourier as well as special function expansions) to write solutions to differential equations.
Identify common differential equations (Laplace's, wave, heat, Schrodinger) with and without sources; connect those differential equations as well as various boundary conditions to physical situatons; intepret solutions of such equations in a physical manner.
Use Fourier transforms to find and study solutions to differntial equations.
Discretize differential equations in a variety of bases to allow them to be solved numericaly and implement such numerical solutions (for example, in Matlab).

UBC provides resources to support student learning and to maintain healthy lifestyles but recognizes that sometimes crises arise and so there are additional resources to access including those for survivors of sexual violence. UBC values respect for the person and ideas of all members of the academic community. Harassment and discrimination are not tolerated nor is suppression of academic freedom. UBC provides appropriate accommodation for students with disabilities and for religious, spiritual and cultural observances. UBC values academic honesty and students ae expected to acknowledge the ideas generated by others and to uphold the highest academic standards in all of their actions. Details of the policies and how to access support are available here.

UBC takes academic misconduct (this includes copying of homework, cheating on exams and plagiarism) very seriously, and the penalties are stiff. See sections on Academic Honesty and Standards and Academic Misconduct in the UBC Academic Calendar.

Copyright notice: all material in this course is copyrighted by Joanna Karczmarek (the instructor). It is provided online to you, the students registered in the course. You may not post, share, publish or upload (including to gen AI tools) any of the course materials without explicit permission from the instructor (Joanna Karczmarek). This applies even to those portions of the course materials that are shared with the public by the instructor or other authorized agents. Moreover, sharing exam, test and/or quiz questions, or solutions to any questions posed in the course (including those in worsheets, homework, quizzes, tests, exams and any practice materials) might constitute academic misconduct.

Further policies applicable to PHYS 312 are available in this document.