Updates

• Midterm Exam 2 has been posted (Take-home) [2020-04-15 Wed]
• How to upload homework, Video[2020-03-30 Mon]
• Video update message[2020-03-22 Sun]
• Modified Syllabus for Math 481 Updated [2020-03-15 Sun]

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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:http://jpascale.pages.math.illinois.edu/481sp20
• Instructor:James Pascaleff
  • Email:jpascale@illinois.edu; Office: 341B Illini Hall; Phone: (217) 244-7277.
  • Office hours: MWF 2:00pm–2:50pm, Zoom meeting ID: 451 906 965T 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 geometryfocusing on examples, broadly aimed at students in mathematics, thesciences, and engineering. The emphasis on rigorously presentedconcepts, tools, and ideas rather than on proofs. The topics coveredinclude differentiable manifolds, tangent spaces and orientability;vector and tensor fields; differential forms; integration onmanifolds and the generalized Stokes’ Theorem; Riemannian metrics,Riemannian connections and geodesics. Applications to physics willbe discussed.

Policies

• Assessment: Grades will be based on homework (20%), two midtermexams (20% each), and the final exam (40%). The two lowesthomework scores will be dropped. Letter grade cutoffs will not bestricter than 90% for an A-, 80% for a B-, and so on. Individualexams may have more generous cutoffs depending on theirdifficulty. You may view your grades in the ATLAS Gradebook.
• Homework: Homework assignments and their due dates will beposted on this website. Homework is due at the beginning of classon the due date. You are required to submit a paper copy of yourhomework in class. Collaboration on homework is permitted andexpected, but you must write up your solutions individually andunderstand them completely.
• Late homework will not be accepted. However, the lowest twoscores are dropped, so you may miss one or two assignments withoutpenalty. If you are unable to turn in your homework in class onthe due date, please submit it in advance to my mailbox in 250Altgeld.
• Midterm exams: The two midterm exams will held during theregular 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 entirecourse, with some emphasis on material that was covered after thesecond exam.
• Missed exams: If you need to miss an exam for valid reason (suchas illness, accident, or family crisis), please let the instructorknow as soon as possible. Normally, you will be excused from theexam so that it does not count towards your overall grade.
• Cheating: Cheating, that is, an attempt to dishonestly gain anunfair advantage over other students, is taken veryseriously. Penalties for cheating on exams may include a zero onthe exam or an F in the course.
• Disability accommodations: Students who require specialaccommodations should contact the instructor as soon aspossible. Any accommodations on exams must be requested at leastone week in advance and will require a letter from DRES.

Sources of help

• Ask questions in class: Please do not shy away from askingquestions during the lecture. If you are confused by something, itis 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 Ihave reserved for students in this course. You do not need to makean appointment to come see me during this time. If you are unableto meet during office hours, please send me an email and we willmake an appointment.
• Piazza: Piazza is an online discussion forum where you can getyour questions answered by classmates and the instructor. Pleasesign up here. Note that you can use any email toregister for Piazza and can post questions and answers anonymouslyif 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 fromthe book, but I encourage you to read the book, and notnecessarily only the sections that correspond to the material inthe lectures.

  It sometimes helps to have a different perspective on the samematerial. Here are some other books (roughly in increasing orderof difficulty):

  • M. Spivak, Calculus On Manifolds: A Modern Approach ToClassical 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 thetitle suggests, it treats the formalism of tensors verythoroughly.
  • 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 explicitdetail.
  • M. Spivak, A Comprehensive Introduction to Differential Geometry,Volume 1. This is the first volume of a five-volume work. Itcontains 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 differentialgeometry. Some of these problems may refer to concepts we have notdiscussed in the course, and they may use slightly differentterminology, but many of the problems are very relevant to thiscourse. [Note: I am linking here to online PDFs that appear tohave been posted by the authors of the books.]
  • A. T. Fomenko, et al. Selected Problems in Differential Geometry and Topology.If you are looking for the big Russian book of problems from Moscow State University, this appears to be it.
  • W.-H. Steeb, Problems and Solutions in Differential Geometry and Applications.Despite the title, there don’t seem to be solutions included.
  • The book by J. M. Lee listed above also has a good selection of problems.

Homework assignments

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

Drox operative cheat engine code. Update [2020-03-18 Wed]: All remaining homework assignments havebeen posted. The due dates for homework after spring break have beenpushed back. The new dates are as follows.

• Homework 1 (Solutions) due Monday, February 3.
• Homework 2 (Solutions) due Wednesday, February 12.
• Homework 3 (Solutions) due Friday, February 21.
• Homework 4 (Solutions) due Friday, February 28.
• Homework 5 (Solutions) due Wednesday, March 11.
• Homework 6 (Solutions) due Friday, April 3.
• Homework 7 (Solutions) due Friday, April 10.
• Homework 8 due Monday, April 27.
• Homework 9 due Monday, May 4.
• Homework 10 due Wednesday, May 6. This homework is now an extra credit assignment.

Detailed schedule with lecture notes and videos

Update [2020-03-18 Wed]: The schedule has been updated to reflectthat we got a few lectures behind. As a result, the due dates forthe homework assignments have also changed. For all lecturesstarting with lecture 19, a link to a video version of the lecturewill 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