Samuel E. Gralla

last updated Februrary 2026

Instructional Materials

Here are some materials I've developed to fill gaps or address deficiencies in standard texts.  Please email me if you end up using them in a course (or if you find a mistake).  I will add more as I make them.

• Taylor series -- three kinds. These notes explain why you shouldn't expect Taylor series to converge.
• Lagrangian mechanics, derived. A video that derives Lagrangian mechanics from Newton's laws.

Videos

For the transition to online learning in Spring 2020, I made a series of videos for undergraduate math methods. The complete playlist may be accessed here.

• Fourier Series (blackboard style)
  • I. Basics
  • II. Example
  • III. Subtleties
  • Fourier Transform and Black Holes
• Partial Differential Equations (blackboard style)
  • I. Types of Equations
  • II. Separation of Variables
  • III. Boundary Value Problems
  • IV. Does it Separate?
• Vector Calculus (with fancy graphics!)
  • I. Basics
  • II. Derivatives
  • III. Integrals
  • IV. The Fundamental Theorem(s)

Courses I teach

Here are the courses I have taught, along with some thoughts.  I am always looking for more ideas (and happy to share my own), so feel free to email me to start a conversation.

• Graduate classical mechanics.  This course is tricky, as I don't think any of the standard books (Landau, Goldstein, Arnold) is really suitable on its own.  I use my own notes that draw from all three and also include other topics that I consider especially interesting or important.  Non-holonomic constraints are a really fun example: Did you know that parallel parking is computing a commutator to show that the car is non-holonomic?  (These notes are not quite ready to be posted publicly, but I am happy to send them to you if you email me.)


• Graduate general relativity.  I closely follow Carroll's book.  This is a rare text that can please students and experts alike.  I think he did a great job of being pedagogical without sacrificing clarity or sophistication.


• Undergraduate electromagnetism I.  I like Griffith's book and follow it closely.  But I do disagree somewhat with the attitude: Griffiths regards point particles as the fundamental building block of the classical theory, with occasional footnotes that this doesn't really make sense because of self-energy and self-force effects.  I would prefer a treatment more in the spirit of modern effective field theory, where you emphasize that different physical models are useful at different scales, with divergences a natural consequence.  I give this viewpoint in lecture, but it would be nice to find a book with this attitude.  (If you know of one, please let me know.)  However, Griffiths has an amazing gift for exposition, and he keeps the students engaged.


• Undergraduate Math Methods.  At Arizona we teach linear algebra as part of a math methods course. I have mixed feelings about this, but it does give us a chance to skip over some boring detail and get to the stuff that is interesting for physics (read: eigenvalues).  I divvy up the course as follows: infinite series (2 weeks), complex numbers (1 week), linear algebra (5 weeks), Fourier Series and PDEs (2 weeks), vector analysis (2 weeks), probability (2 weeks).  I used the book by Boas because of good amazon reviews from students, and because I disliked it less than I disliked other books like Riley and Hobson.  I think many of the students are just going from the lecture, however.

Past Students

Hongji Wei (B.S., Arizona, 2024). Graduate student at Harvard.
Kunal Lobo (Ph.D., Arizona, 2024). Career in actuarial science.
Yangyang Cai (Ph.D., Arizona, 2023). Postdoc in astrophysics.
Will Lockhart (Ph.D., Arizona, 2022). Postdoc in climate science.
Arun Ravinshankar (Ph.D., Arizona, 2021). Career in data science.
Theo Drivas (B.S., Chicago, 2011).  Currently a professor at Stony Brook.
Daniel Brennan (B.S., Maryland, 2014).  Currently a professor at Birmingham.