Differential Geometry:
Curves and Surfaces in R^3
Math 421, Spring Semester,
January 10 - April 24, 2024
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 28, 2024.
The Final Exam will be in our usual
classroom on Monday, April 29,
2024, 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 19, 2024)
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 26, 2024)
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, February 2, 2024)
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 9, 2024)
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 16, 2024)
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 23, 2024)
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
4.2.3
4.2.4
4.2.5
4.3.2
4.3.5
4.3.6
The Midterm is in class on Wednesday, February 28, 2024.
Homework #6 (due
Friday, March 8, 2024)
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, 2024 is during Spring Break.)
Homework #7 (due
Friday, March 22, 2024)
Watch Lectures 13,
14,
and 15.
Homework #8 (due
Friday, March 29, 2024)
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 5, 2024)
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 12, 2024)
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 19, 2024)
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 29,
2024 from 9 a.m. - 12 noon in our usual classroom, Physics 119.