Weekly Agenda

Week 1: Linearity, Mathematical "Nice-ness"

• Meeting 1 (Tue, Apr 1): Introduction to the course, review of three-vectors, what we mean by linear, introduction to basis vectors. 

• Meeting 2 (Thu, Apr 3): Matrix multiplication, index notation, contraction as a linear transformation, summation convention,  "nice" transformations (symmetries), tensors. 

• Discussion (Friday, Apr 4): note: Friday meetings are in Olmsted 1132. The big idea, Q&A. 

Reading 

• Course notes, Chapter 2 on "The Basics" and Chapter 3 on Indexology.

• Optional: review the first five chapters of the 3blue1brown course on linear algebra

• Optional: play with matrix multiplaction in The Matrix Arcade 

• Optional: Chapter 2 of Immersive Linear Algebra

Homework

Due dates listed first, use the hyperlinks to submit homework. Please submit a pdf version of the assignment using the submission links below. Homework due this week:

• Thu Apr 3. Please review the syllabus. No submission link, just take some time to look over the document.

• Thu Apr. 3: Pre-Class Survey. 

• Thu Apr. 3: Short homework 1. Submission link.  

  • The short homework should be quick and something that you can easily after class. It is useful feedback for Prof. Tanedo. 

Homework due next week:

• Tue Apr 8: Please record a 2-5 minute video introducing yourself to the teaching team and the other students. You should start by stating your name. You may also include: where you grew up, your goals for your physics degree, hobbies, anything you are excited about. Submission link. Submission link

  • Prof. Tanedo's welcome video

  • Sasha's welcome video

• Thu Apr. 10: Long homework 1 (updated 4/7). Submission link. 

Week 2: Bases and Linear Transformations

• Meeting 3. (Tue, Apr 8): What is a basis? Meaning of linearity. Tensors as linear transformations.

• Meeting 4 (Thu, Apr 10): Basis of tensors. Basis bras and basis kets, the Kronecker delta. Transformations between bases.

• Meeting 5 (Fri, Apr 11): Indices tell you how tensors transform. Intro to metrics.

Reading

• Course notes (April 10 update), Chapters 4-7. 

  • Update: the material for Thursday's class is in chapters 6-7, please especially make sure you are comfortable with chapter 7.7.

  • This is a large number of chapters for one week. As a guide: you should be following the discussions in class and should be able to do the homework problems. The rest of the course notes are there as support (though I do hope you'll read them all). 

• You can skim David Tong's lecture on tensors from his vector calculus course. Note that that course does not distinguish between upper and lower indices.

• Chapters 2-3 of Jeevakjee's An Introduction to Tensors and Group Theory for Physicists (available free through the UCR library).

Homework

Homework due this week:

• Tue Apr 8: Please record a 2-5 minute video introducing yourself to the teaching team and the other students. You should start by stating your name. You may also include: where you grew up, your goals for your physics degree, hobbies, anything you are excited about. Submission link. Submission link

  • Prof. Tanedo's welcome video

  • Sasha's welcome video

• Thu Apr. 10: Long homework 1 (updated 4/7). Submission link. 

Homework due next week:

• Tue Apr 15: Explainer Video 1. The explainer assignments are available on our internal sheet. The number next to your name is the problem on Long Homework 1 that you should prepare a 5-10 minute video for. You are graded by peers according to our rubric. Submission link.

• Thu Apr 17: Short Homework 2. Inner products and metrics. Submission link.

Extra materials

These references are provided "for culture" and are not part of the main narrative of the course. 

• Extra: there is a cute alternative notation to upper and lower indices called bird tracks. The logo on the cover page of our course notes is an example of bird tracks notation. You can read about it here:

  • Kim et al. "Boosting vector calculus with the graphical notation" American Journal of Physics 89, 200–209 (2021)

  • Alcock-Zeilinger and Weigert, "Simplification rules for birdtrack operators" J. Math. Phys. 58, 051701 (2017)

• Last week we had a brief discussion of birdtracks notation. I reposted a brief discussion of how this is used here. Those who are interested may want to work through the first half of Roger Penrose's Road to Reality.  I say "work through" because you need to do the exercises to appreciate that book. 

  • Here's some general advice about learning anything: when you are given the opportunity to do exercise, do all the exercises. If you are short on time, then only do the exercises that stump you.

Week 3: Metric Spaces

• Meeting 6 (Tue, Apr 15): The inner product as a generalization of the dot product, the metric as a symmetric, (0,2) tensor that encodes the inner product.

• Meeting 7 (Thu, Apr 17): The metric raises indices, the inverse metric lower sindices. Isometries are transformations that preserve the inner product. 

  • We had a nice discussion of symmetries: rotations and reflections are isometries. Scale transformation is not an isometry nor is it a symmetry of our universe.

• Meeting 8 (Fri, Apr 18): Special relativity from a diagrammatic perspective. Using Lorentz transformations. Notes from today.

Reading

• Course notes, Chapter 6 and 7.

• Additional course notes on special relativity from a linear algebra perspective from Prof. Tanedo. 

• Optional: "Inside Einstein's head," interactive visualizations of classic relativity thought experiments.

• Optional: Very Special Relativity, by Sander Bais is one of the best introductions to the geometry of special relativity. It's disguised as a little coffee table book for the general public. 

• Optional: David Tong has a set of lecture notes on electrodynamics and relativity. 

Homework

Homework due this week:

• Tue Apr 15: Explainer Video 1. The explainer assignments are available on our internal sheet. The number next to your name is the problem on Long Homework 1 that you should prepare a 5-10 minute video for. You are graded by peers according to our rubric. Submission link.

• Thu Apr 17: Short Homework 2. Inner products and metrics. Submission link.

Homework due next week:

• Tue Apr 22: Peer Review 1. The internal sheet has the peer review assignments. Submission link: Use the same link for each of your three reviews.

• Thu Apr 24: Long Homework 2. (4/22: typos corrected in 1f and 1g, thanks Citali and Emily! /20: typo in an example fixed, thanks Anthony! 4/15: Link corrected, thanks Casey!) Metric spaces, special relativity, Gram-Schmidt. Submission link. 

Additional References

• Jim Branson's Physics 110b course (2012) at UCSD is one of the few places that talks about the transformation of the moment of inertia tensor correctly, see these notes. There is a digression on the moment of inertia tensor in the long homework. 

• For more on the moment of inertia tensor: See also Examples 3.3 (p.55) and 3.14 (p.75) in Jeevakjee's textbook.

Miscellaneous

• One of the exciting parts of putting a class like this together is that I end up spending time learning new things as I ask "how would I have liked to learn this" or "where are the gaps that still exist in my understanding?" Often this ends up sending me off on an illuminating digression that never makes it into our course. In case it is helpful to show what that process is like for me, I attach a set of notes where I was trying to connect the divergence in spherical coordinates to the components of the metric. (The latter part of the notes goes into differential forms, which is unfortunately outside the scope of our class.) 

The image above is from Sander Bais wonderful book Very Special Relativity, which explains relativity in the diagrammatic language we are using in our course. It explains the pole-in-barn paradox. The red lines are the axis of the person running with the pole. The green shaded region is the inside of the barn. The black two-sided arrows are time slices of the pole in our frame, while the red two-sided arrows are the time slices of the pole in the runner's frame.

Week 4: Review Week

This week Prof. Tanedo will be serving on an NSF panel and will not be available on Thursday or Friday. Sasha will lead the class in a problem sesion on Thursday. We will not hold a meeting on Friday. 

Please review the recorded mini-lessons from 2024 on rotations, dual bases, the adjoint, and determinants.

• Meeting A: Tue Apr. 22: Special relativity. Possible other topics: the adjoint, transformation of tensors (why the moment of inertia tensor appears to transform as a matrix), introduction to eigenvalue problems. The following hand written notes have not yet been incorporated into the typed lecture notes.

  • Notes: example of Lorentz transforming velocities.

  • Notes: the adjoint of a matrix, complex spaces

• Meeting B: Thu Apr. 24: Problem session led by Sasha. Please break up into groups to work on any of the problems here: problems for group work. (Corrected 4/22, thanks Daniel B.) I expect that you should be able to get through about 2 problems if working together; pick the problems that you want the most practice on. 

  • Solutions to the problems, these may be helpful as examples for core skills for the second half of the course (Posted 4/29).

• No meeting on Friday, April 25. You are encouraged to meet up with your classmates if you want to review the other problems on the problem sheet. 

Reading

• Course notes (Apr 15), Please review any material that you feel shaky on. Do the exercises in the chapters as practice if you are confused.

• Here are some lectures on rotations, dual bases, the adjoint, and determinants from 2024. I strongly recommend viewing these over the week. 

  • Rotations 1 (13:28)

  • Rotations 2 (17:53)

  • Dual Bases (8:16)

  • Adjoint (17:31)

  • Determinants 1 (23:12) Review of preliminaries.

  • Determinants 2 (10:11) Determinant in 2D

  • Determinants 3 (35:11) Determinant in 3D

Homework

Homework due this week:

• Tue Apr 22: Peer Review 1. Internal sheet with peer review assignments. Submission link: Use the same link for each of your three reviews.

• Thu Apr 24: Long Homework 2. (4/22: typos corrected in 1f and 1g, thanks Citali and Emily! /20: typo in an example fixed, thanks Anthony! 4/15: Link corrected, thanks Casey!) Metric spaces, special relativity, Gram-Schmidt. Submission link. 

Homework due next week:

• Tue Apr 29: Explainer Video 2. Assignments on the internal sheet. Submission link.

• Thu May 1: Short Homework 3. Submission link. 

Additional references

• A nice interactive demonstration of determinants from the Immersive Linear Algebra online, interactive textbook. 

• On Tuesday we discussed the work done "against gravity"

Week 5: Eigenstuff

• Meeting 9 (Tue, Apr 29): Introduction to phase 2 of the class. The postulates of quantum mechanics. Complex vector spaces. The eigenvalue problem and algorithm. 

  • I haven't had a chance to update the typed up lecture notes. Here's a write up of the postulates of quantum mechanics that motivates the second half ot the course. 

  • Examples of eigensystems. (Did you notice that symmetric 2x2 matrices with identical diagonal elements always have the same eigenvectors?)

  • Updated 4/30: there was some confusion (mostly my own) on how I presented a change of basis. In particular, we wanted write the action of a basis bra in one basis acting on a basis ket in another basis. We connected this to the inner product, but there was some justified confusion about the heights of indices. I've written up formally correct revision in the notes, you can find that revision separately here. (Thanks to Lillianne B. and Inchara J. for pressing me to think more carefully about this.)

• Meeting 10 (Thu, May 1): Introduction to nice matrices: self-adjoint (Hermitian) and unitary. Eigenvalues and eigenvectors of Hermitian matrices. 

  • Partial notes: Change of basis, Hermitian matrices

• Meeting 11 (Fri, May 2): Example: A coupled system of springs as an example of an eigenstuff problem.

  • Meeting notes. Not in the notes: using the definition of the adjoint, show that the adjoint of a matrix in Euclidean space is what we normally call the adjoint. 

Reading

• Chapter 15 - 16 of our course notes (Apr 30 version). 

  • We are skipping Chapter 13 on determinants (see the recorded videos in Week 4). Please look over this if you are not familiar with determinants.

  • You can skim Chapter 12, but we will navigate the same material in a different order this year.

  • I recombined some text into a new Chapter 10 on changing basis. Please review that if you are not quite comfortable with changing basis.

• One page summary of the Postulates of QM from Meeting 9. (I haven't had a chance to include these in the course notes yet.)

• Examples from Meeting 9.

• "Eigenvectors and Eigenvalues Explained Visually"

Homework

Homework due this week:

• Tue Apr 29: Explainer Video 2. Assignments on the internal sheet. Submission link.

• Thu May 1: Short Homework 3. Submission link. 

Homework due next week:

• Tue May 6: Peer Review 2. Internal sheet with peer review assignments. Submission link.

• Thu May 8: Long Homework 3. Submission link.

Additional Material

• Solutions to last week's problem sessions The solutions are written to be pedagogical and contain a few digressions, including one long one about representation theory around page 5. 

• One of the unfortunate topics we will not cover in depth are determinants and their connection to the alternating symbol. If you are not familiar at all with determinants, you may want to review these in Chapter 13 of the notes (April 15 version).

  • Determinants 1 (23:12) Review of preliminaries.

  • Determinants 2 (10:11) Determinant in 2D

  • Determinants 3 (35:11) Determinant in 3D

• "Hermitian matrices generate unitary matrices" notes. This is comes up in quantum mechanics. Hermitian matrices are the generators of transformations. This means that a finite transformation (e.g. a rotation about the z axis) is the exponential of a Hermitian matrix.

Week 6: Complex spaces

• Meeting 12 (Tue, May 6): Degenerate eigenvalues, complex metric spaces. Notes.

  • We had a great discussion today about diagonalizing a matrix. I've done my best to summarize them here. 

• Meeting 13 (Thu, May 8): Hermitian conjugate. Eigenstuff of Hermitian matrices.

• Meeting 14 (Fri, May 9): Pauli matrices and spin. Notes from today.

  • During our discussion we talked about the twin paradox in a closed universe. This is a universe that is flat, but if you travel far enough in one direction you return to where you started. This differs from the usual twin paradox because neither twin accelerates to break the symmetry between the two. See Jeffrey Weeks' discussion in The American Mathematical Monthly (JSTOR) in 2001.

Reading

• 2025 Course Notes (Apr 30 version): Chapter 16

Homework

Homework due this week:

• Tue May 6: Peer Review 2. Internal sheet with peer review assignments. Submission link.

• Thu May 8: Long Homework 3. Submission link.

  • Equation (4.1) has a typo: The 4-3 component of F should be Bx so that the tensor is antisymmetric. 

  • It has come to my attention that the relativistic electrodynamics problems are especially difficult for those who have not seen Maxwell's equations in differential form. My apologies. I offer this clip from Star Trek: The Next Generation (S7 E15, "Lower Decks")

Homework due next week:

• Tue May 13: Explainer 3.  Internal sheet with explainer assignments. Submission link. 

• Thu May 15: Short Homework 4, Submission link.

Week 7: Spin Systems, Silly Function Spaces

• Meeting 15 (Tue, May 13): Spin-1/2

• Meeting 16 (Thu, May 15): Histogram space, a silly idea

• Meeting 17 (Fri, May 16): Fourier Series, intro

Reading

• Course notes, Chapter 14 (function spaces), Chapter 13.1-13.5 (Fourier series)

Homework

Homework due this week:

• Tue May 13: Explainer 3.  Internal sheet with explainer assignments. Submission link. 

• Thu May 15: Short Homework 4. Submission link.

Homework due next week:

• Tue May 20: Peer Review 3. Internal sheet with peer review assignments. Submission link.

• Thu May 22: Long Homework 4. Submission link. [See notes below on HW4 errors]

Additional Reading

• Notes on the Dirac δ function from Hitoshi Murayama's Physics 221a class in Berkeley.

• In class we talked about "histogram space" as a template for function spaces. If you look around, nobody else is teaching "histogram space." There's a good reason for it: it's completely silly and breaks down mathematically. The reason why is something of a technicality, but we detail this in Section 14.10 of the Course notes (old link)

Week 8: Catching Up

Professor Tanedo regrets that he is unavailable this week. Sasha will be taking over the class meetings on Tuesday and Thursday. There will be no Friday meeting.

• Meeting C (Tue, May 20): TBA, Fourier Series + Eigenvalue problems

• Meeting D (Thu, May 22): Separation of Variables e.g. / Q&A

Reading

• Course notes, Chapter TBA

Homework

Homework due this week:

• Tue May 20: Peer Review 3. Submission link.

• Thu May 22: Long Homework 4. Submission link. Revised HW4. 

  • Homework 4 has more than its fair share of errors. Thanks to everyone who helped troubleshoot this. 

  • Solution to the neutrino oscillation problem. There were a few points that the original problem glossed over: 

    • The unitary transformation between the mass and eigenbasis is not necessarily a pure rotation. Depending on the order you place the eigenvalues, you may have a pure rotation or a pure rotation times a reflection. Either works; but if you have a reflection then you need to be a bit more careful when definining the mixing angle θ. (See solutions)

    • There was a factor of 2 that we had trouble recovering in the squared sine. This was because I was too sloppy with the relation between energy and mass. In the solutions we carefully Taylor expand the Einstein relation between energy, momentum, and mass.  

  • Partial solution and extra examples for the spring problem.

Homework due next week:

• Thu June 12: Explainer 4 Internal sheet with peer review assignments. Submission link. 

• Thu June 12: Homework 5 (short+long combined). 

Additional Reading

• Neutrino oscillations are a bit subtle, even though we like to throw it into quantum mechanics. For a discussion of this, see Jean-Michel Levy's note on the topic: hep-ph/0004221.

  • For those who will continue onto studying particle physics, I also recommend the article "Do charged leptons oscillate?" by Akhmedov, arXiv:0706.1216. 

Week 9: Fourier Series

• Meeting 18 (Tue, May 27): Separation of variables, review

• Meeting 19 (Thu, May 29): Separation of variables, spherical harmonics

• Discussion A (Fri, May 30): No discussion, but the professor is available for office hours on the Barkas patio at this time.

Reading

• Course notes, Chapter TBA

• Fourier Transform Notes from 2024

Homework

Homework due next week:

• Thu June 12: Explainer 4 Internal sheet with peer review assignments. Submission link. 

  • Please see the discussion of Long HW 4 under Week 8. In particular, the following discussion notes from class:

    • Solution to the neutrino oscillation problem.

    • Partial solution and extra examples for the spring problem.

• Thu June 12: Homework 5 (short+long combined). Submission link. 5 June: updated link, corrections to the homework. Thanks Shane and Santosh.

Additional Reading

• "Difficulties with bra-ket notation" a physics.stackexchange post from 2012 that discusses one of the bits of notation that physicists tend to be sloppy about.

Week 10: Wrapping Up

Review of key ideas, highlighting some subtle points, and (if there are no other more pressing discussions), some pictures of how this fits together into more advanced mathematical physics.

• Meeting 20 (Tue, Jun 3): The circular spring problem: subtleties of degenerate eigenvalues, how it works in a typical problem..

• Meeting 21 (Thu, Jun 5): Possible topics: Big picture of Fourier transforms, spring theory (transition to field theory), hints of the complex structure of physics (causality).

Reading

• Course notes, Chapter 13.6-13.9

• Fourier Transform Notes from 2024

Homework

Homework due this week:

• Thu June 12: Explainer 4 Internal sheet with peer review assignments. Submission link. 

  • Please see the discussion of Long HW 4 under Week 8.

• Thu June 12: Homework 5 (short+long combined). Submission link. 5 June: updated link, corrections to the homework. Thanks Shane and Santosh.

• Thu June 12: Course feedback survey. I use this to improve the course and course notes in the future.

• Please complete your course evaluation on Canvas. UCR uses these course evaluations as part of its merits and promotion process for faculty (somewhat controversially) and positive evaluations help support the continued offering of this course. 

Additional Reading

• There's a particle physics textbook called A Standard Model Workbook by Moore that explains the passage from finite to infinite dimensional space in around Box 7.4.  It also does a fantastic job of explaining the factors of 2π when going from Fourier series to Fourier transforms. See around chapter 14: it’s the de Broglie factor of 2π which is, in turn, related to wavelength versus angular wavelength.

• End of the class! Here's the music video for Kendrick's song, Cups, which was featured in Pitch Perfect as "When I'm gone."

• In response to Adrian's question ("what is a Hilbert space?"), this video by Abide by Reason is a pretty good summary for our course. It touches on the idea of Cauchy completeness, which is the reason why the "histogram basis" fails in the continuum limit. 

• "The radiation of a uniformly accelerated charge is beyond the horizon: a simple derivation" (physics/0506049) by Almeida and Saa describes the puzzle of radiation by a uniformly accelerated charge that came up in class on Tuesday. The paper also has a good set of references on the paradox and its solution.

• Thursday's life discussion of scaling laws led me to look up a few things.

  • Kleiber's law states that an animals metabolic rate scales like a three-quarters power of the animal's mass. You can compare this to our estimates to address what could be the limiting factor in this scaling.

  • There are lots of interesting extrapolations of scaling laws. Biologists call this "allometry," and there are some nice papers like "Allometric scaling and accidents at work." The idea of scaling is addressed more thoroughly in Geoffrey West's book Scale.

  • For those who want to dig into the physics side of this, a good starting point is "A hint of renormalization" by Delamotte in the American Journal of Physics (2004).

  • If you can get a hold of a copy, there's a great section on renormalization and scaling in Particle Physics: A Los Alamos Primer (1988), edited by Cooper and West. (See, in particular, the first chapter, "Scale and dimension - from animals to quarks."

Some of my favorite doodles from submitted homework