| W 9/8/04 |
-
1.7/10
- 2.1/ Look at exercises 1-11. Do any exercise containing a fact
with which you are unfamiliar or that strikes you as nontrivial.
| F 9/10/04 |
-
Finish reading Section 2.2 (pp. 54-55).
- 2.2/ 1,2,6,8,10,11. What I called
"Groisser's Theorem" in class I really should have called "Groisser's
Conjecture". Problem 1 and the unassigned problems 3 and 4 provide
evidence for this conjecture.
- 2.2/5 is false as stated. Fix it and do the corrected
problem.
- 2.2/9: Required for Computer Science students,
optional for everyone else, but it wouldn't hurt the math students
to locate and use a numerical integrator.
|
| M 9/13/04 |
- 2.1/12. There is a typographical error in this problem; the
upper limit of the integral should be t.
- 2.3/5,7,8
- Do non-book problem 2 as
corrected 9/18/04.
|
| W 9/15/04 |
- Read Corollary 3.5 and Lemma 3.6.
- 2.3/1,6,11
- Do non-book problem 3. This is
intended primarily as a computational problem: you should
compute the matrix exponential, not look for a way to avoid
computing it.
First collection of homework: On Wednesday, Sept. 22 (I
changed the date because of a mistake in non-book problem 2--the
omission of an orthogonal projection--which has now been corrected), I
will collect the following homework problems: 1.5/5; 1.6/1, 4c, 6;
2.3/6; and non-book problems 1,2 (as corrected 9/18/04),3.
The homework you hand in must be neat and easy to read. Please use
8.5''x 11''paper, not torn out from a spiral-bound notebook, and
staple the sheets together. Please do not hand in anything that is
messy, that has been erased and written over, that has shreds of paper
dangling from it, or that has the potential to come apart when I turn
pages.
|
| F 9/17/04 |
No new
homework was assigned. The reason was not that I wanted to give you a
vacation, but that I thought it would be fun for you to do some of the
problems in Section 2.4 involving computer graphics--specifically, 11c
(which is assigned as part of Monday's homework) and 12, 13, 14 (not
assigned, but instructive)--until I tried to do the graphics myself
using Maple, and was unable to do the problems myself in time to decide
whether to assign them. For those of you don't know Maple and want to
learn it, there's an appendix in the textbook that gives an
introduction. However, I found that O'Neill's Maple programs for the
Frenet apparatus did not work well for me, either because I mis-typed
something (or made some similar dumb mistake) or because Maple has
changed so many times since the textbook was written. I eventually
gave up on O'Neill's programs and wrote my own, which worked better
for me but still were not up to the task handling certain curves that
can be handled quickly and easily without a computer, such as the
cycloid in problem 14. But undoubtedly you are all better programmers
than I am, so you may be able to accomplish in minutes what took me
hours, and may want to have some fun with these problems. (Note: if
you choose to use Maple, in the current version you need to use the
"spacecurve" program in the "plots" package in order to plot a curve
in 3D.)
Remark on 2.4/14a: A point on a simple closed plane curve at which
the curvature has a relative extremum is called a vertex. If
you do this problem, you'll see why this name was chosen. It is
obvious that every simple closed plane curve C has at least two
vertices, since the curvature must achieve a maximum and a minimum
value on the compact set C. Much less obvious is the
Four-Vertex Theorem (which we will not prove), which asserts
that every such curve has at least four vertices.
|
| M 9/20/04 |
- See note above concerning the homework for 9/17/04.
- 2.4/3, 11, 16 (as corrected below), 18 (as corrected below), 19
(as modified below). 11(c) and 19(a) are optional if you have no
interest at all in using the computer to do graphs. Note: in #11,
"Ex. 3.6" means Exercise 2.3/6.
- #16: Assume the number a in part (a) satisfies
a2 < 2/3, and use the same interval (-a,a)
in parts (b) and (c).
- #18: (i) The definition given of convexity applies only to
simple (i.e. non-self-intersecting) closed curves. (ii) Replace
"2*pi" by "a positive multiple of 2*pi". To obtain this result, you
don't need to assume that the curve is simple, just that the curvature
is positive (or, less stringently, that the plane curvature--the one
with the tilde over
it--does not change sign). O'Neill's assertion that the total
curvature of a convex simple closed curve is 2*pi is true, but
I don't think it can be proven using only the tools we've developed in
this course. All the proofs I know of require algebraic
topology (specifically, the Jordan Curve Theorem), differential
topology, or complex analysis (the relevant parts of which are
differential topology in disguise); you need to make use of
"intuitively obvious", but not so easy to prove, consequences of the
curve being simple. If you come up with a proof that requires none of
this, let me know! (iii) I'm not sure why O'Neill chose the hint he
gave (for "spherical image", see the last paragraph on p. 71 and first
paragraph on p. 72) instead of the simpler hint "Use problem
2.3/8b". Note: as you probably observed when you did 2.3/8, the kappa
in part (b) should have a tilde over it.
- #19: (i) Remark: a knot is a simple closed curve in
R3 (this includes curves such as a circle that
aren't knotted in the conventional sense; in knot theory these
"un-knots" are considered to be special cases of knots). (ii) Remark:
"tau" in this problem is just O'Neill's name for this curve; it has
nothing to do with torsion. (iii) After doing part (a), also plot the
projected curve given in part (b); you may find that this helps you
interpret the picture you got in part (a). (iv) In part (b) you'll
get to use the famous "tan(theta/2) substitution" that we (sometimes)
teach in Calculus 2, which reduces any integral of a rational function
of sine and cosine to a rational algebraic function. (At least, that's
how I did the integral; maybe you will come up with another way.)
(v) In part (c), in place of what O'Neill says to do, prove
that the curve in part (b) can be deformed in R3 to
a knot with total curvature arbitrarily close to 4*pi. (Hint: interpolate
between the curve in part (a) and the curve in part (b). Note that you
will have to show that your deformed curve is simple.) Optional:
write a program to compute curvature (or use O'Neill's if it works for you),
numerically compute the total curvature of the trefoil knot in (a),
and compare the answer to 4*pi.
- 2.5/1b, 2ab, 4.
- 2.7/2.
|
| W 9/22/04 |
- Read the book's proof of the Second Structural Equation (I've
decided just to let you read this instead of spending class time on it
on Friday).
- Do non-book problems 4 and 5.
|
| F 9/24/04 |
3.2/3,4.
|
| W 9/29/04 |
3.3/4,5. Sorry for not getting these posted till Thursday afternoon.
|
| F 10/1/04 |
3.4/5.
Clarification of non-book problem 4. Part (c), in the sentence
"Apply part (a) ..." what I meant to say was: First, figure out
dy in terms of dx, then use this result, plus part (a), to
write the triple with partial-with-respect-to-y's in term of
the triple with partial-with-respect-to-x's.
|
| M 10/4/04 |
3.5/3,7,8.
|
| F 10/8/04 |
- 4.1/4,5,6,8,9,10,11
- 4.2/2, 5abc (part (d) optional), 6a (part (b) optional)
|
| W 10/13/04 |
4.3/1, 3b (notation as on p. 149), 4, 6, 13, 14, 15. At first it may
strike you that #14 contradicts the existence of non-proper patches,
of which I showed (pictorially, not with a formula) an example in
class. The reason there is no contradiction is that in #14 one has
the additional assumption that the image of the patch lies in a
smooth surface. The example given in class does not have this
property.
|
| F 10/15/04 |
- No new homework.
- Midterm. I am tentatively planning to hand out the
take-home midterm on Fri. Oct. 29 and collect it on Mon. Nov. 1.
These dates could change, but the earliest I can conceive of handing
out (or making available) the exam is Tues. Oct. 26; the latest is
Fri. Nov. 5. (The dates would be the same for everyone in the class.)
Please let me know, as soon as possible, any potential problems with
dates in this range.
I have not yet started to prepare the exam, so I don't know how
long it will be. As far as content goes, my general rule is that
anything covered in class, in homework, or in the accompanying reading
is fair game (including problems you have never seen before but that
can be done by synthesizing material from class, homework, textbook,
and prerequisite classes).
|
| M 10/18/04 |
4.4/1, 3
|
| W 10/20/04 |
4.5/1, 3, 4, 5, 13. In #3 there's very little to show, and it can be
done in a line or two (but remember that "mapping" means
smooth mapping); I'm assigning it because it's used in #4.
In #13, it is more conventional to say that G
descends to a (smooth) map F from M to N, or
that the formula in the book determines a well-defined function
F, than to say that the formula is consistent; however,
there is nothing wrong with O'Neill's terminology. The issue
underlying this problem is that in general
G may not be one-to-one, so that unless the consistency
condition is met there will not be a unique map
from
the set M to the set N--here by "map" I just mean
function in the set-theoretic sense, not in the differentiable (or
even continuous) sense--such that {F composed
with
G}=F-tilde. In fact the consistency condition is
equivalent to the uniqueness of such a set-map (existence of such a
set-map is guaranteed because G is assumed to be onto). In
essence, what you are showing in part (a) of this problem is that,
given smooth maps G and
F-tilde as in the problem (with G surjective),
uniqueness of a set-map F from M to
N such that {F composed with
G}=F-tilde implies smoothness of that map.
|
| F 10/22/04 |
- No new homework.
- Correction to statement in lecture. The following is true.
Let M, N be surfaces, with M compact and N
connected. If F is a one-to-one map from M to N
whose derivative is an isomorphism at each point, then F is a
diffeomorphism. In class, I omitted the compactness hypothesis,
without which the statement is false. (For the non-math students: a
subset of Rn is compact if and only if it is
closed and bounded. That's not the definition of compactness;
it's a theorem--the Heine-Borel Theorem--that makes the
concept of compactness much easier to understand.) A counterexample if
"M compact" is omitted: take M to be the open unit disk
in R2, take N=R2, and take
F to be the inclusion map. The mistake I made in class was the
ridiculous statement that, under a continuous map, the image of a
closed set is always closed. However, it is true that, under a
continuous map, the image of a compact set is always compact,
hence closed. The rest of the argument I gave is correct.
Another true statement related to the one above is this:
Let M, N be surfaces, and let F be a one-to-one map from
M to N whose derivative is an isomorphism at each
point. Then F is a diffeomorphism onto its image.
(This is true whether or not M is compact and whether or not
N is connected. However, the price you pay by weakening the
hypotheses this way is that the conclusion is weaker; you cannot
conclude that
F is onto.)
|
| M 10/25/04 |
|
| F 10/29/04 |
For the problems below, assume Stokes' Theorem (which we'll prove Monday).
- 4.6/3, 7 (easy corollary of facts proven in class).
|
| M 11/1/04 |
- 4.7/3, 8
- Reminder about midterm exam: As announced in
class, this take-home exam will be handed out on Wednesday 11/3/04 and
will be due at the beginning of class on Monday 11/8/04.
Warning: I will be ending class a few minutes early on Friday
to get to the airport to attend a weekend conference. This means that
you will have no opportunity to ask me questions about the midterm,
not even by email, after Thursday mid-afternoon; consider 3:00
p.m. Thursday the cutoff time. Email about the exam received after
that time will not be answered.
|
| W 11/3/04 |
In view of the midterm, the due-date for the problems below is Wed. 11/10/04.
|
| (posted 11/4/04) |
One more typographical error on midterm: in problem 3,
equation (1), the "g" on the right-hand side should be "Q".
|
| M 11/8/04 |
5.2/1
|
| W 11/10/04 |
- 5.3/1,3,7. Clarification of #7:
- In the first line of
part (a), note that the formula given for
F*(v) expresses only the "vector" part of
this tangent vector; the basepoint is F(p), not
p. (S, of course, is the shape operator of M
at p).
- A related comment applies to the first displayed
equation (the one involving cross-products) in part (a): the left-hand
side of the equation gives only the vector part of a tangent
vector in TF(p)R3, while
the right-hand side is the vector part of a tangent vector based at
p. (In the second displayed equation, H, K, and
k1, k2 are, of course, the mean,
Gaussian, and principal curvatures of M.)
- A similar comment applies again to part (b): given a point
p in M, (i) the
statement about unit normals should really be that the vector part of
the unit normal to \bar{M} (M with a bar over it) at
F(p) equals the vector part of U at p, and
(ii) in the equation relating the two shape operators, the left-hand side
denotes the vector part of a tangent vector based at
F(p), while the right-hand side
denotes the vector part of a tangent vector based at p. If
you do not take this into account, you will end up doing a formal
calculation that makes no sense and yields the wrong answer.
- In part (c) "M" should be
"\bar{M}".
- Also in part (c), in the formulas
that you are asked to derive, "\bar{K}(F)" should be
what would be written in LaTeX as $\bar{K}\circ F$, the composition of
\bar{K} with F. A similar comment applies to
"\bar{H}(F)".
- 5.4/2,3,6,7
|
| M 11/15/04 |
- See clarification of 5.3/7 on previous assignment. Today's lecture
should make this problem easier to do than it was when originally
assigned.
- 5.3/8 (note that this has parts (a) and (b); (b) is on the next
page)
- 5.4/12,13,15
|
| W 11/17/04 |
- Do non-book problems 6 and 7. (The version of #7 posted Wed. night
was incomplete--my apologies--but I've now completed it and also made
some minor modifications to wording and notation.)
- 5.6/2 (see pp. 233-234 for "asymptotic curve), 3, 9. The
situation in #9 is related to non-book problem 7c, case (i): the
monkey saddle is preserved by an isometry of R3 of
order 3 that fixes the origin.
|
| W 11/24/04 |
|
| W 12/1/04
(not posted till 12/2/04) |
- 6.3/1,2,4,5. An additional hint for #2 and #5: Theorem 2.6, p. 262.
|
| F 12/3/04
|
|
| M 12/6/04 (posted 12/7/04)
|
- 6.4/1,2
- 6.5/1,2,4
- Read section 6.6 and do problems 6.6/1,2
|
|