Differential Geometry:
Curves and Surfaces in R^3
Math 421, Spring Semester,
January 8 - April 23, 2025
Physics 119, Mondays and Wednesdays, 10:05 - 11:20 a.m.
Professor
Hubert Bray
bray@math.duke.edu
189 Physics Building
(919)757-8428 (mobile)
Office Hours: After class, or by appointment
Homework is due via Gradescope by 5 p.m. on
Fridays, according to the schedule below. More information about
how to turn in your homework will be emailed to you.
The Midterm will be in class on Wednesday, February 26, 2025.
The Final Exam will be in our usual
classroom on Monday, April 28,
2025, from 9 a.m. - 12 noon.
Here are some practice exams (with solutions!) which you will want
to study to prepare for the midterm and the final exam. Make sure
you are an expert on these exams since problems, solution methods,
and key ideas often get repeated
2018 Midterm Exam
and Solutions
2017
Midterm Exam and Solutions
2016 Midterm Exam and Solutions
2015 Midterm Exam and
Solutions
2014 Midterm Exam and Solutions
2013 Midterm Exam and Solutions (easiest one, so you
might do it first)
2017 Final Exam and Solutions
2016 Final Exam and Solutions
2015 Final Exam and Solutions
2013 Final Exam and Solutions
The lectures of this course were recorded in 2018 and may be
accessed here.
Homework
A few of
the homework problems in the book have typos, sign errors, and
other mistakes in them. There are notes below (next to the
problem) identifying the errors for you, as well as giving you
hints. Extra credit will be awarded for identifying the typos,
sign errors, and other mistakes in the homework problems that are
not already identified below.
Homework #0 (due Friday, January 17, 2025)
Watch Lectures 1
and 2.
Write a 3 page, double spaced paper on any topic you like relating
to either Special Relativity or General Relativity, such as black
holes, the Big Bang, or gravitational waves. Bring good questions to
class on the previous Wednesday to help you finalize the ideas for
your paper.
Homework #1 (due Friday, January 24, 2025)
Watch Lecture 3
and work these problems:
1.1.2
1.1.3
1.1.4
1.1.6
1.1.13
1.1.25
1.1.26
1.1.27
1.2.2
1.2.5
1.3.2
1.3.7
1.3.11
1.3.19
1.3.22
1.3.23
1.3.24
1.3.26
1.3.27
1.3.28
Homework #2 (due
Friday, January 31, 2025)
Watch Lectures 4
and 5
and work these problems:
1.3.30 (after reading example 1.3.29)
1.4.6 (after reading this section, plugging into the
formulas)
1.4.7
2.1.1
2.1.6
2.1.7
2.1.8
2.1.11 (assuming h(u) and the speed of the curve (g(u),h(u))
are both not zero)
2.1.12
2.1.13
2.1.15
2.1.16
2.1.19
2.1.20
2.1.21
2.1.22
2.1.23
2.2.5
2.2.6
2.2.7
2.2.9
2.2.11
2.2.14
2.2.15 (expressions for the shape operator in the book are off by a
sign, for standard U)
2.2.16
Homework #3 (due
Friday, February 7, 2025)
Watch Lectures 6
and 7
and work these problems:
2.3.2
2.3.3
2.3.4
2.3.9
2.3.10
2.3.11
2.4.6
2.4.7
3.1.2
3.1.4
3.1.5
3.1.6
3.1.7
3.1.9
3.1.10
3.1.11
3.2.3
3.2.5
3.2.6
3.2.10
Homework #4 (due
Friday, February 14, 2025)
Watch Lectures 8
and 9
and work these problems:
3.2.12
3.2.13
3.2.14 (note similarity to previous problem)
3.2.15
3.2.16
3.2.18
3.2.19
3.2.26
3.3.1
3.3.3
3.3.4
3.3.5
3.3.6
3.3.7
3.3.8
3.3.10
3.4.2
3.4.4
3.4.5
3.4.6
Homework #5 (due
Friday, February 21, 2025)
Watch Lectures 10,
11,
and 12
and work these problems:
3.4.8
3.5.1
3.5.8
3.5.9
3.5.10
3.6.3
3.6.4
3.6.5
3.6.6 (will delete this problem - don't do it unless you want to)
4.2.3
4.2.4
4.2.5
4.3.2
4.3.5
4.3.6 (will delete this problem too)
The Midterm is in class on Wednesday, February 26, 2025.
Homework #6
(due Friday, March 7, 2025)
Print out a blank midterm exam and submit solutions to each problem,
beautifully written on the exam paper. Obviously you could just copy
the answer key. Please do not do this. I want you to truly
understand the solutions. Go through the solutions, spending the
time to understand them carefully, and then write up your version of
the solutions as you understand them, in your own words. Use
this as an opportunity to solidify your knowledge of the first half
of the course.
(NOTE: Friday, March 15, 2025 is during Spring Break.)
Homework #7 (due
Friday, March 21, 2025)
Watch Lectures 13,
14,
and 15.
Homework #8 (due
Friday, March 28, 2025)
Watch Lectures 16
and 17
and work these problems:
4.4.1
4.4.2
4.4.3
5.1.1
5.1.3
5.1.4
5.1.9
5.1.10 (Assume the curvature of the geodesic is never zero.)
5.1.11
5.1.12
5.1.13
5.2.1
5.2.2
5.2.4
5.2.10
5.4.2 (Conformal to the xy plane, so that E = 1/f^2 = G and F = 0.)
5.4.4
5.4.6
5.4.8
5.4.10
Homework #9 (due
Friday, April 4, 2025)
Watch Lectures 18
and 19
and work these problems:
5.4.11
5.4.14
5.4.15
5.4.19
5.4.20
5.5.6
5.5.7
5.5.10
5.5.12
6.1.1
6.1.2
6.1.4
6.1.5
6.1.9
6.2.1
6.2.2
6.2.3
6.3.1
6.3.3
6.3.4
Homework #10 (due
Friday, April 11, 2025)
Watch Lectures 20
and 21
and work these problems:
6.3.6
6.3.7
6.3.9
6.3.10
6.3.11
6.3.12
6.3.13
6.5.1
6.5.2
6.5.5
6.5.10
6.6.5
6.6.6
6.6.7
Bonus points: Identify the exercise above which has a sign error in
the statement of the problem.
Homework #11 (due
Friday, April 18, 2025)
Watch Lecture 22
and complete the practice final exam from
2013 and turn in your solutions to be graded, mostly for
completion. The solutions are listed above - only look at them as
needed after you've given each problem a good try.
Not tested in course, but you may enjoy: Lectures 23,
24,
and 25.
The Final Exam will be on Monday, April 28,
2025 from 9 a.m. - 12 noon in our usual classroom, Physics 119.