Vector and Tensor Analysis (Math 481, Spring 2020)


Basic information

  • Lectures: Video lectures are posted below and in the Kaltura/Mediaspace channel: Math 481 Spring 2020. MWF 2:00pm–2:50pm in 145 Altgeld Hall.
    • This represents Sections F13 (CRN 38087) and F14 (CRN 38089).
  • Course website:
  • Instructor: James Pascaleff
    • Email:; Office: 341B Illini Hall; Phone: (217) 244-7277.
    • Office hours: MWF 2:00pm–2:50pm, Zoom meeting ID: 451 906 965 T 4:00–5:00pm, W 3:00–4:00pm, F 11:00am–Noon.
  • Prerequisites: Multivariable calculus (Math 241 or equivalent) and linear algebra (Math 415 or Math 416 or equivalent).
  • Textbook: Theodore Frankel, The Geometry of Physics: An Introduction, Third Edition, Cambridge University Press, 2012.

Course description

This is an introductory course in modern differential geometry focusing on examples, broadly aimed at students in mathematics, the sciences, and engineering. The emphasis on rigorously presented concepts, tools, and ideas rather than on proofs. The topics covered include differentiable manifolds, tangent spaces and orientability; vector and tensor fields; differential forms; integration on manifolds and the generalized Stokes’ Theorem; Riemannian metrics, Riemannian connections and geodesics. Applications to physics will be discussed.


  • Assessment: Grades will be based on homework (20%), two midterm exams (20% each), and the final exam (40%). The two lowest homework scores will be dropped. Letter grade cutoffs will not be stricter than 90% for an A-, 80% for a B-, and so on. Individual exams may have more generous cutoffs depending on their difficulty. You may view your grades in the ATLAS Gradebook.
  • Homework: Homework assignments and their due dates will be posted on this website. Homework is due at the beginning of class on the due date. You are required to submit a paper copy of your homework in class. Collaboration on homework is permitted and expected, but you must write up your solutions individually and understand them completely.
  • Late homework will not be accepted. However, the lowest two scores are dropped, so you may miss one or two assignments without penalty. If you are unable to turn in your homework in class on the due date, please submit it in advance to my mailbox in 250 Altgeld.
  • Midterm exams: The two midterm exams will held during the regular class periods on Monday, March 2 and Wednesday, April 15. The second midterm will be a take home-exam.
  • Final exam: The final exam will be a take-home exam. It will cover the entire course, with some emphasis on material that was covered after the second exam.
  • Missed exams: If you need to miss an exam for valid reason (such as illness, accident, or family crisis), please let the instructor know as soon as possible. Normally, you will be excused from the exam so that it does not count towards your overall grade.
  • Cheating: Cheating, that is, an attempt to dishonestly gain an unfair advantage over other students, is taken very seriously. Penalties for cheating on exams may include a zero on the exam or an F in the course.
  • Disability accommodations: Students who require special accommodations should contact the instructor as soon as possible. Any accommodations on exams must be requested at least one week in advance and will require a letter from DRES.

Sources of help

  • Ask questions in class: Please do not shy away from asking questions during the lecture. If you are confused by something, it is likely that others are as well.
  • Come to office hours: I have office hours Tuesday 4:00–5:00pm, Wednesday 3:00–4:00pm, and Friday 11:00am–Noon. This is a time I have reserved for students in this course. You do not need to make an appointment to come see me during this time. If you are unable to meet during office hours, please send me an email and we will make an appointment.
  • Piazza: Piazza is an online discussion forum where you can get your questions answered by classmates and the instructor. Please sign up here. Note that you can use any email to register for Piazza and can post questions and answers anonymously if you prefer.
  • Books: The lectures are intended to go with the official textbook:

    • T. Frankel, The Geometry of Physics: An Introduction.

    The exact order of topics in the lectures may be different from the book, but I encourage you to read the book, and not necessarily only the sections that correspond to the material in the lectures.

    It sometimes helps to have a different perspective on the same material. Here are some other books (roughly in increasing order of difficulty):

    • M. Spivak, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus. Full text PDF available via UIUC Library. This text is primarily concerned with differential forms and the integrals thereof.
    • R. Bishop and S. I. Goldberg, Tensor Analysis on Manifolds. This book is available in a low-price Dover edition. As the title suggests, it treats the formalism of tensors very thoroughly.
    • J. M. Lee, Introduction to Smooth Manifolds. Full text PDF available via UIUC Library. This is a graduate-level textbook that covers many examples in explicit detail.
    • M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1. This is the first volume of a five-volume work. It contains a wealth of examples and scholarly remarks.
  • Extra Problems: If you are looking for extra practice problems, here are a couple of books of problems on differential geometry. Some of these problems may refer to concepts we have not discussed in the course, and they may use slightly different terminology, but many of the problems are very relevant to this course. [Note: I am linking here to online PDFs that appear to have been posted by the authors of the books.]

Homework assignments

Update [2020-04-18 Sat]: The due date for homework 8 has been pushed back a week.

Update [2020-03-18 Wed]: All remaining homework assignments have been posted. The due dates for homework after spring break have been pushed back. The new dates are as follows.

Detailed schedule with lecture notes and videos

Update [2020-03-18 Wed]: The schedule has been updated to reflect that we got a few lectures behind. As a result, the due dates for the homework assignments have also changed. For all lectures starting with lecture 19, a link to a video version of the lecture will be posted below.

Update [2020-03-22 Sun]: The lecture videos are posted on the Kaltura/Mediaspace Channel: Math 481 Spring 2020

I have set up a recurring Zoom meeting for MWF 2:00pm-2:50pm. The meeting ID is 451 906 965

Date Lecture Remarks
[2020-01-20 Mon] Martin Luther King, Jr. Day No class.
[2020-01-22 Wed] 1. Motivation, smooth functions. First class.
[2020-01-24 Fri] 2. Coordinate charts and atlases.  
[2020-01-27 Mon] 3. First examples of manifolds.  
[2020-01-29 Wed] 4. The precise definition of a manifold.  
[2020-01-31 Fri] 5. Tangent spaces.  
[2020-02-03 Mon] 6. More on tangent spaces. Homework 1 due. 10th day add/drop deadline.
[2020-02-05 Wed] 7. Examples of tangent vectors.  
[2020-02-07 Fri] 8. Smooth maps between manifolds.  
[2020-02-10 Mon] 9. Inverse function theorem.  
[2020-02-12 Wed] Catch-up lecture. Homework 2 due.
[2020-02-14 Fri] 10. Submanifolds.  
[2020-02-17 Mon] 11. Regular values of maps and Sard’s theorem.  
[2020-02-19 Wed] 12. Vector fields.  
[2020-02-21 Fri] 13. Ordinary differential equations on manifolds. Homework 3 due.
[2020-02-24 Mon] 14. The tangent bundle.  
[2020-02-26 Wed] 15. Examples of vector fields and flows.  
[2020-02-28 Fri] 16. The cotangent bundle. Homework 4 due.
[2020-03-02 Mon] Exam 1 on Lectures 1–14. Solutions.  
[2020-03-04 Wed] 17. Multilinear algebra and tensors.  
[2020-03-06 Fri] 18. Examples of tensors on manifolds.  
[2020-03-09 Mon] Catch-up lecture.  
[2020-03-11 Wed] Catch-up lecture. Homework 5 due.
[2020-03-13 Fri] 19. Riemannian metrics. Video  
[2020-03-16 Mon] Spring Break  
[2020-03-18 Wed] Spring Break  
[2020-03-20 Fri] Spring Break  
[2020-03-23 Mon] 20. The Riemannian distance function. Video  
[2020-03-25 Wed] 21. Alternating tensors. Video  
[2020-03-27 Fri] 22. Differential forms. Video  
[2020-03-30 Mon] 23. The exterior differential. Video  
[2020-04-01 Wed] 24. Integrating differential forms. Video  
[2020-04-03 Fri] 25. Orientations and orientability. Video Homework 6 due.
[2020-04-06 Mon] 26. Integration examples. Video  
[2020-04-08 Wed] 27. Manifolds with boundary. Video  
[2020-04-10 Fri] 28. Boundary orientations. Video Homework 7 due.
[2020-04-13 Mon] 29. Proof of Stokes’ theorem. Video  
[2020-04-15 Wed] Exam 2 on Lectures 15–27. Solutions.  
[2020-04-17 Fri] 30. Topological applications of Stokes’ theorem. Video  
[2020-04-20 Mon] 31. Closed and exact forms, de Rham cohomology. Video  
[2020-04-22 Wed] 32. Properties of de Rham cohomology. Video  
[2020-04-24 Fri] 33. Lie brackets and Lie derivative. Video  
[2020-04-27 Mon] 34. More on Lie brackets. Video Homework 8 due.
[2020-04-29 Wed] 35. Connections. Video  
[2020-05-01 Fri] 36. Riemannian (Levi-Civita) connections. Addendum. Video  
[2020-05-04 Mon] 37. Surfaces of revolution. Video Homework 9 due.
[2020-05-06 Wed] 38. The Riemann curvature tensor. Video Homework 10 due.
[Optional] 39. The Gauss-Bonnet theorem. Video Optional lecture.
[2020-05-11 Mon] Final exam Take home exam due [2020-05-15 Fri 23:59]