HOMEWORK
MTH 434/534 — Winter 2020


Ground Rules

Suggested Reading
It is to your advantage to skim suggested readings as soon as possible. However, do not expect to master this material the first time around. Don't worry; we'll cover it in class, after which the readings should make more sense. But be warned: we will not always cover the material the same way it is presented in the text.
Warmup Problems
Most assignments will include suggested warmup problems, typically with answers in the back of the text. You should attempt as many of these problems as you feel are necessary, but should not turn them in. If you have questions about any of these problems, ask me, although in most cases further discussion should take place during office hours.
Written Work
It is your job to explain your work to me clearly and completely. Here are some guidelines: You may discuss homework problems with anyone you like, and you may use any reference materials you like, although the use of explicit solutions by others to the same problems is strongly discouraged. However, you must write up the solutions in your own words, and you must indicate what help you used. Late homework will be corrected as a courtesy to you, but can earn at most half credit.
Grading
All written assignments will receive two separate scores, one for content and the other for presentation.

Assignments
Section numbers refer to O'Neill, 2nd revised edition.
(The 2nd edition is nearly identical; the 1st edition differs only in a few problems.)

Week 10
Suggested reading: Read §6.1–§6.2.
Assignment: No assignment this week.
Week 9
Suggested reading: Read §5.3–§5.4; Skim §5.5.
Warmup problems:
§5.3: 1, 6; §5.4: 1, 2, 3;
Assignment (due 3/9/20):
§5.3: 2, 3a; §5.4: 6.
Week 8
Suggested reading: Read §5.1–5.2.
Warmup problems:
§4.3: 4, 5; §5.1: 2, 3; §5.2: 1, 2.
Assignment (due 3/2/20):
  1. Determine the shape operator on a sphere. That is, for your choice of basis tangent vectors $\vf{e}_1$, $\vf{e}_2$, determine $S(\vf{e}_m)=-\nabla_{\vf{e}_m}\Hat{n}$, where $\Hat{n}$ is the (outward) unit normal vector to the sphere.
    (Your basis $\{\vf{e}_1,\vf{e}_2\}$ does not need to be either orthogonal or normalized.)
  2. At any one point on the sphere, compute the normal curvature in any one direction.
  3. MTH 534 only (extra credit for MTH 434):
    Repeat (a) and (b) for the torus given in toroidal coordinates (see HW 5) by setting $\rho=\hbox{constant}$.
    You may skip parts (a) and (b) if you are confident that your solution to (c) is correct, but they are a good warmup.
Week 7
Suggested reading: Read §4.1– §4.4 §4.3.
Warmup problems:
§4.1: 5; §4.2: 1, 2, 3; §4.3: 6, 7.
Assignment (due 2/24/20):
Consider the following two patches on a sphere: \begin{align} \mathbf{x}(\theta,\phi) &= (a\sin\theta\cos\phi,a\sin\theta\sin\phi,a\cos\theta) ,\\ \mathbf{y}(u,v) &= (u,v,\sqrt{a^2-u^2-v^2}) . \end{align}
  1. Compute the adapted bases $\left\{\frac{\partial\mathbf{x}}{\partial\theta}, \frac{\partial\mathbf{x}}{\partial\phi}, \frac{\partial\mathbf{x}}{\partial\theta}\times \frac{\partial\mathbf{x}}{\partial\phi}\right\}$ and $\left\{\frac{\partial\mathbf{y}}{\partial u}, \frac{\partial\mathbf{y}}{\partial v}, \frac{\partial\mathbf{y}}{\partial u}\times \frac{\partial\mathbf{y}}{\partial v}\right\}$.
  2. Determine the curvature of the four (families of) coordinate curves on the sphere with, respectively, $\theta$, $\phi$, $u$, and $v$ held constant. That is, determine the curvature of the curve \begin{equation} \alpha(u)=\mathbf{y}(u,c)=(u,c,\sqrt{a^2-u^2-c^2}) \end{equation} with $c$ constant, and similarly for the other three cases. ("Determine" does not necessarily mean "compute".)
  3. Optional: Discuss the relationship between the relevant basis and the Frenet frame for your curve.
Week 6
Suggested reading: Skim §4.1–§4.2.
Assignment: No assignment this week.
Week 5
Suggested reading: Reread §2.5–§2.7; Read §2.8.
Warmup problems:
§2.8: 1, 2, 3.
Assignment (due 2/10/20):
Toroidal coordinates $(\rho,\psi,\phi)$ can be defined by $$x=(R+\rho\cos\psi)\cos\phi, \qquad y=(R+\rho\cos\psi)\sin\phi, \qquad z=\rho\sin\psi$$ (with suitable domains), where $R$ is a constant.
Perform the following tasks, in any order, using any method.
  1. Draw a picture showing how to interpret these coordinates geometrically.
  2. Construct the frame field $\{\Hat{e}_i\}$ adapted to these coordinates. That is, find an orthonormal basis that points in the directions in which these coordinates increase. (If you wish, you may call your basis $\{\Hat{\rho},\Hat{\psi},\Hat{\phi}\}$.)
  3. Determine the basis of 1-forms $\{\sigma_i\}$ dual to your frame.
  4. Compute the connection 1-forms $\omega_{ij}$ for this frame.
    Your answer should be expressed in terms of the dual basis $\{\sigma_i\}$.
  5. Let $f$ be an unknown function on $\RR^3$ ($f:\RR^3\longmapsto\RR$), and let $\eta$ be an unknown 1-form on $\RR^3$ ($\eta=\sum\eta_i\sigma_i$).
    Find expressions for $df$ and $d\eta$. Again, your answer should be expressed in terms of the dual basis $\{\sigma_i\}$.
  6. Verify that the (first and second) structure equations are satisfied by $\sigma_i$ and $\omega_{ij}$.
Week 4
Suggested reading: Read §2.5–§2.7; Skim §2.8.
Warmup problems:
§2.5: 1 (use any method), 2; §2.6: 1, 2; §2.7; 1.
Assignment (due 2/3/20):
§2.5: 4; §2.6: 3, §2.7: 2, 3, 4.
Week 3
Suggested reading: Read §2.1–§2.4.
Warmup problems:
§2.1: 3, 4, 6; §2.2: 3; §2.3: 5; §2.4: 3a.
Assignment (due 1/27/20):
§2.2: 5; §2.4: 2, 12. (You may, but need not, use a unit-speed curve in #2.)
MTH 534 only:
§2.3: 10a. (The answer in the book may have a typo...)
Week 2
Suggested reading: Read §1.5–§1.6; Skim §1.7–§1.8.
Warmup problems:
§1.6: 1;
MTH 534 only: §1.7: 3
Assignment (due 1/17/20):
§1.6: 3, 4, 7;
MTH 534 only:
Consider the curve $\alpha(u)=(a\cos u,a\sin u,0)$ with $a=\hbox{constant}$ and the vector field $V$ on $\RR^1$ such that $V[f(u)]=\frac{df}{du}$. Choose a function $g(x,y,z)$ on $\RR^3$, and compute $\alpha_*(V)[g]$ as a function of $u$.
(You may instead try the general case, which is §1.7: 7.)
Week 1
Suggested reading: Skim §1.1; Read §1.2–§1.5.
Warmup problems: §1.2: 1a; §1.3: 2, 3ace; §1.4: 3; §1.5: 1.
Assignment (due 1/8/20): Send me an email message at tevian@math.oregonstate.edu.
Please include some information about yourself, such as your math background and your motivation for taking this class.
Assignment (due 1/13/20): §1.4: 4; §1.5: 4, 6;