Math 5041 – Geometry I: Calculus on Manifolds (Fall 2018)

Section information

  • Class time and location: Tuesdays and Thursdays from 2:30PM to 4:00PM in Capples II L001
  • Tentative office hours: Tuesdays 4PM-5PM (immediately after class); Wed and Fri 12:00PM-2:00PM


This is an introduction to differential and integral calculus on manifolds. It may be regarded as a more mature version of Calc III, in that it will cover topics that are familiar from multivariate calculus such as directional derivatives, vector fields, the inverse and implicit function theorems, integration (of various things) on surfaces (of arbitrary dimensions), Stokes’s theorem, etc., but done in a more sophisticated way and in much greater generality.


I plan to use a few different texts for somewhat different purposes, the first of which (see below) will be our designated “official” textbook. Important! Do not buy them. They are available in electronic form through the Olin Library web site. (Paper copies may also be purchased under Springer MyCopy plan, and can be had for $15.00 plus taxes, also through the Library’s web site.)

  • An Introduction to Manifolds by Loring W. Tu, Second Edition, Springer 2011. Library link here. Tu’s book will be the course’s “official” text. This means that I will try to follow its overall plan, and will define the content of the course by it. See below for more details on topics we intend to cover.

  • Introduction to Smooth Manifolds by John M. Lee. Second Edition, Springer 2013. Library link here. Lee’s book is very polished and has a greater variety of problems than Tu’s. It may also be more detailed; for this reason, you may find it helpful to have it close to hand, should the explanations of the official text (or my own!) not be sufficient. But I don’t intend to refer to it in class, unless I use it as the occasional source of a homework problem. (I’ve adopted this text before and find it excellent, except for it being so lengthy. Tu’s has a more manageable size, although I haven’t tested it in classes yet. I’ll appreciate your feedback about it, and on everything else, as the course progresses.)

  • Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers by Pedro M. Gadea, Jaime M. Masqué, Ihor V. Mykytyuk. Second Edition, Springer 2013. Library link here. This text contains a large number of fully solved problems. I may take homework problems from it, but you won’t be asked to submit your answers to them. The point of occasionally assigning problems with available solutions, for which you won’t have to submit your own, is that those additional problems become part of the pool from which I can derive exam problems. I’ll have more to say about this point in class.

Topics we hope to cover. (Chapters refer to text by Tu)

  • Extended review of multivariate calculus - Chapter 1

  • Manifolds: generalities - Chapter 2

  • Smooth functions, submanifolds, vector fields, flows, etc. - Chapter 3

  • Lie groups and Lie algebras (mostly covered in assignments) - Chapter 4

  • Exterior calculus - Chapter 5

  • Integration - Chapter 6.

Coursework, exams and grades

There will be weekly homework assignments, a mid-term exam, and a final (and qual.) exam. The grade breakdown is: Homework 40%, MT 20%, Final 40%. The mid-term exam will be October 30 (Tuesday) during regular class time. The final is set for December 13 from 11:30AM to 2:30PM. Letter grades will be given according to the following scale: A+ = (97, 100], A= (93,97], A-=[90,93]. Similar for B, C, D; and F=[0,60).

Homework assignments