**Spring 2019**

**Textbook**

Calculus on Manifolds (5^{th}
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 StokesÕ

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 StokesÕ theorem.

**Instructor**

Hang Xu

Email: hxu@math.jhu.edu

Office Location: Krieger Hall Rm 220

Office Hours: Tuesday 4:30-5:30 pm

**Course
Assistant**

Cheng Zhang

Email: czhang67@math.jhu.edu

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

**Lecture**

TuTh 3:00-4:15 at Krieger 300

**Grading**

**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.

**Take-Home Final** due by the end of May 15,
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 **

**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, studentdisabilityservices@jhu.edu. 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 ÒAcademic Ethics for UndergraduatesÓ and the Ethics Board Web site (http://e-catalog.jhu.edu/undergrad-students/student-life-policies/#UAEB) 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.

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

We meet at **Bloomberg
178** on 2/5.

Homework 1 due.

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

Homework 2 due.

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

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: chapter 2, problems 26, 34, 35, 37 (a), 38, 39, 41 (a)(b),.

Meet at **Bloomberg
178** on 3/5.

Midterm in ThursdayÕs lecture.

No homework due.

**Week 7 (3/12 & 3/14): Integrable
Functions, FubiniÕs Theorem**

Homework 5 due.

Homework 6: chapter 3, problems 2, 3, 6, 9, 10, 12, 14, 15, 16.

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

Homework 6 due.

Homework 7: chapter 3, problems 7, 18, 20, 21, 23, 26, 31, 32.

Homework 7 due.

Homework 8: chapter 3, problems 22(you can use the conclusion in 21), 35, 36, 37, 38, 39, 40, 41.

Homework 8 due.

Homework 9: chapter 4, problems 1, 2, 3, 4, 5, 10, 11.

Homework 9 due.

Homework 10: chapter 4, problems 13, 14, 16, 17, 18, 19, 20, 21.

Homework 10 due.

Homework 11: chapter 4, problems 22, 23, 25, 26, 27(you can use 4-24), 29, 30, 31.

Homework 11 due.