Math 441: Calculus on Manifolds

Spring 2019


Calculus on Manifolds (5th edition), Michael Spivak,

Cambridge University Press, 2000.


Course Description

The purpose of this course is essentially a semester-long study of the modern version of Stokes’

theorem and the mathematics needed to build it up. Major topics to be addressed are likely to include:

_       Calculus in higher dimensional Euclidean spaces.

_       Tensors, differential forms and singular chains.

_       Calculus on manifolds, the Stokes’ theorem.


Hang Xu


Office Location: Krieger Hall Rm 220

Office Hours: Tuesday 4:30-5:30 pm 

Course Assistant

Cheng Zhang


Office Hours: Monday 9:00-11:00 am at Krieger 213


TuTh 3:00-4:15 at Krieger 300



is based on 350 points distributed as follows:

11 homework problem sets, worth 10 points each, due in the lecture on the following Thursday. The lowest score will be dropped.

Midterm on March 7 in the lecture, worth 100 points.

Final TBA, worth 150 points.


Course Policies

No late homework is accepted. Staple your problem sets! Study groups are encouraged, but homework has to be written down independently.

No makeup exam in this course. If you have to miss an exam for a documented, legitimate reason, please inform me as early in the semester as possible.

You are responsible for lecture notes, any course material handed out, and attendance in class. The lectures will be conducted as if you have already read the material and attempted some homework problems. In this manner, you can focus mainly on those parts of the lectures that cover the areas of your reading you found difficulty to understand.

If you have any math questions, please feel free to ask me anytime. You can find me after lectures, in office hours, or you can reach me by email.


Help Room

Krieger Hall 213. The hours are 9am – 9pm on Monday through Thursday, and 9am – 5pm on Friday. This free service is a very valuable way to get one-on-one help on the current material of a class from other students outside the course. It is staffed by graduate students and advanced undergraduates. Outside of me and the Grader for the course, definitely take your questions to the Help Room. This course is simply an analysis course directed toward particular maps and differential equations. Most graduate students should be able to "see" through the many problems stated in this course. And your attempts to help guide them will be of huge benefit to you also. 


Students with disabilities

Students with documented disabilities or other special needs who require accommodation must obtain an accommodation letter from Student Disability Services, 385 Garland, (410)516-4720, After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.


JHU ethics statement

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of student affairs and/or the chairman of the Ethics Board beforehand. See the guide on “Academic Ethics for Undergraduates” and the Ethics Board Web site ( for more information.


Tentative Schedule (will be updated as the course progresses)

Homework will be posted by every Wednesday and due in the lecture on the next Thursday.

Week 1 (1/29 & 1/31): Norms and Inner Product, Subsets of Euclidean Space

Homework 1: chapter 1, problems: 5, 7, 10, 13, 14, 15, 16, 18.

Week 2 (2/5 & 2/7): Functions and Continuity, Differentiation

We meet at Bloomberg 178 on 2/5.

Homework 1 due.

Homework 2: chapter 1, problems 1, 20, 22, 23, 24, 25, 27, 30.

Week 3 (2/12 & 2/14): Chain Rule, Partial Derivatives

Homework 2 due.

Homework 3: chapter 2, problems 1, 4, 6, 7, 8, 12, 13.

Week 4 (2/19 & 2/21): Derivatives, Inverse Functions

Homework 3 due.

Homework 4: chapter 2, problems 21, 22, 23, 24, 25, 29, 30, 32.


Week 5 (2/26 & 2/28): Implicit Functions, Integration

Homework 4 due.

Homework 5:


Week 6 (3/5 & 3/7): Measure Zero and Content Zero, Midterm

Meet at Bloomberg 178 on 3/5.

Midterm in Thursday’s lecture.

No homework due.


Week 7 (3/12 & 3/14): Integrable Functions, Fubini’s Theorem

Homework 5 due.

Homework 6: 


Week 8 (3/19 & 3/21): Spring Break


Week 9 (3/26 & 3/28): Partitions of Unity, Change of Variable

Homework 6 due.

Homework 7:


Week 10 (4/2 & 4/4): Multilinear Algebra, Fields and Forms

Homework 7 due.

Homework 8:


Week 11 (4/9 & 4/11): Singular Chains, the Fundamental Theorem of Calculus

Homework 8 due.

Homework 9:

Week 12 (4/16 & 4/18) Manifolds, Fields and Forms on Manifolds

Week 13 (4/23 & 4/25): Stokes’ Theorem

Homework 9 due.

Homework 10:


Week 14 (4/30 & 5/2): The Volume Element, the Classical Theorems

Homework 10 due.

Homework 11:


Final Exam