Homework Solution Differential Geometry Book

-- MTHT 442 (Section Numbers 37035 and 37036) -- Fall 2014


Louis H. Kauffman


533 SEO


(312) 996-3066



Web page:


Office Hours

: 3PM to 4PM on Monday, Wednesday, Friday, or by appointment.

Course Hours

: 1:00PM to 2:00PM on MWF in 300 TH.

This is a course on Differential Geometry and its applications.

The textbook is "Differential Geometry - Curves, Surfaces, Manifolds" by Wolfgang Kuhnel.

A second textbook is "Differential Forms with Applications to the Physical Sciences" by Harley Flanders (Dover Paperback Edition -- see Amazon)

Excerpts from the book "Calculus" by Apostol can be found here: Apostol Curves and Apostol Conics

An article and excercises about the combinatorics and topology of the Gauss Bonnet Theorem can be found here: Gauss Bonnet

A discussion of the Fary-Milnor Theorem can be found here: Fary-Milnor Theorem

Milnor's original paper on curvature of knotted curves can be found here: Milnor

Notes on Inifinitesimal Calculus and Differential Forms by LK are here: Zeroid

Notes on Diagrammatic Matrix Algebra can be found here: Diagrams

Course notes can be found here: Diff Geom Notes 1 Diff Geom Notes 2

First Assignment:

Obtain a copy of the textbook. Read Chapters 1. and Chapter 2. Read the excerpt from Apostol (see above). On page 524 of Apostol, do problems 1-8. On page 538 of Apostol, do problems 2,3,4,5,6. This homework is due on Friday, September 5,2014.

Second Assignment:

Apostol, page 544, problem 4; page 549, problems 10, 11, 22. Read Chapter 2 of Kuhnel. Kunhel, page 49, problems 1,10,11. This homework is due on Monday, September 15,2014.

Third Assignment:

Read Chapter 2 of Kuhnel. Kunhel, page 49, problem 1. Do this problem again and find a good solution! (not to hand in). Kunhel page 49. Problems 9,19,23. And attend to the example on page 31. Show what is claimed there that "one can solve this system of four equations for a,b alpha, beta." Read the "Zeroid" Notes on our website. This homework is due on Monday, September 29,2014.

Fourth Assignment:

Read Kuhnel Chapter 3 and Chapter 4 through page 146. Solve the tollowing two problems: 1. Let A and B be vectors in R^{n} with the usual inner product, denoted here by A*B. Show that (A*A)(B*B) - (A*B)^{2} = Sum_{1<= i < j<= n} (A_{i}B_{j} - A_{j}B_{i}). 2. Using f(u,v) = (u,v, F(u,v)) where F is a diferentiable real-valued function of u and v, work out the three fundamenatal forms,the principal and Gaussian curvatures for this surface element. Choose some specific examples and derive their curvatures as special cases of your work. This homework is due on Monday, November 3, 2014.

Fifth Assignment:

Read Kuhnel Chapter 3 and Chapter 4 through page 166. Solve the following problem: Using f(u_1,...,u_n) = (u_1,...,u_n, F(u)) where F is a differentiable function of u = (u_1,...,u_n),work out the three fundamenatal forms, the principal and Gaussian curvatures for this surface element. Choose some specific examples and derive their curvatures as special cases of your work. (i.e. same problem as before but for hypersurfaces in arbitrary dimensions.) This homework is due on Wednesday, November 12 2014.

Sixth Assignment:

Read Flanders Differential Forms -- Section 4.1 and Section 4.5. Apply Flander's formalism to f(u_1,u_2) = (u_1,u_2, F(u)) where F is a differentiable function of u = (u_1,u_2). Compare with your previous work using the three fundamenatal forms. This is for discussion and not to be handed in. However, please apply our recent formalism for Christoffel symbols and curvature to some examples of your choosing. We will discuss specifics on Monday, December 1. Please hand in some worked examples of your choosing on Wednesday, December 3.


This is a copy of the original website for this course, which I taught at UCLA in 2014. All of the contact information below is out of date.

Math 120A: Differential Geometry

Fall 2014

Instructor: Kristen Hendricks
Office: 6617D Math Sciences Building
Office Hours: M 2:15-2:45, W 10-11, R 4-5
E-Mail: hendricks at math .ucla .edu
TA: Jacob Rooney
Office: MS 2509
Office Hours: T 11-12, R 1-2
E-Mail: jhrooney at ucla .edu

A printable copy of the syllabus is here.

Location and Time

MWF 9-9:50 in Geology 4645. TA discussion section T 9-9:50 in Geology 4645.


This course is an introduction to low-dimensional differential geometry. We will study the geometry of curves in two- and three-dimensional space, touching on curvature and torsion, the Frenet-Serret equations, and the isoperimetric inequality. We will next study surfaces embedded in three-dimensional space, covering the notion of a smooth atlas, various types of surfaces, the first and second fundamental forms, and the Gauss map.


Pressley, Elementary Differential Geometry. Springer (1974). Second Edition. An electronic version of this textbook is available for free through the UCLA network here

We will make use of an appendix of additional exercises published by Pressley, which are here.

You can also find a copy of Classical Differential Geometry by Prof. Peterson here. This may be a helpful supplemental resource.


Math 115A, Math 131A.


Homework will be assigned weekly and due at the beginning of lecture on Friday. There will be nine homeworks. Do not submit homework by e-mail. No late homework will be accepted. However, your lowest homework score will be dropped when computing your grade.

You are encouraged to work in groups on your homework; this is generally beneficial to your understanding and helps you learn how to communicate clearly about mathematics. However, you must write up all solutions yourself. Moreover, since crediting your collaborators is an important element of academic ethics, you should write down with whom you worked at the top of each assignment. You should also cite any other sources other than lecture and the textbook (another book, a blog about analysis, etc) you use.


There will be two in-class midterms on Wednesday, October 29 and Monday, November 24. There will also be a final exam Tuesday December 16, 3:00-6:00 p.m. There will be not be any make-up exams except in extreme and documented circumstances. In particular, note that university policy requires that a student who has an undocumented absence from the final exam be given a failing grade in the course.


Grades will be computed as follows:

  • Homework: 20%
  • Midterms 1 & 2: 20% each
  • Final: 40%

A curve compatible with the department guidelines will be applied to the composite numerical grades. The average will be a B (unless something surprising happens).


We will approximately follow the official schedule of topics here. (In fact the chapter numbers online are for a different textbook, but they correspond almost identically to our book.) This means we will cover most of Chapters 1-8. The exact reading will be posted as the quarter progresses.


Homework 1 (Due October 10th). Solutions.

Homework 2 (Due October 17th). Solutions.

Homework 3 (Due October 24th). Solutions.

Homework 4 (Due October 31th). Solutions.

Homework 5 (Due November 7th). Solutions.

Homework 6 (Due November 14th). Solutions.

Homework 7 (Due November 21st). Solutions.

Homework 8 (Due December 5th). Solutions.

Homework 9 (Due December 12th).


Sample Midterm 1. Solutions.

Midterm 1. Solutions.

Sample Midterm 2. Solutions.

Midterm 2. Solutions.

Study Sheet for the Final.

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