Course Grade
A -- (2,10)
A- -- (4,1)
B+ -- (4,0)
B -- (2,1)
B- -- (2,0)
C -- (1,2)
D- -- (1,0)
AB -- (1,0)
This is the home page for Math 423, Differential Geometry, Section E1U, which meets MWF 1:00-1:50 in 152 HENRY. Important news: Math 423 has moved to 241 Altgeld Hall, starting W 9/1.
My intention is to provide, at the very least, an archive for all of the TeX-d handouts in the course and a guide to the semester, class-by-class.
W 8/25 -- Distribution of Class
Organization and How to solve
it guide. We went through selected parts of the first chapter
of the text. More organizational details on Friday.
F 8/27 -- We covered section 2.2. The first assignment, Homework
One, was distributed, due F 9/3. This contains more detailed
instructions about homework than is contained in the class
organization. As per request, yes it is possible to submit two bonus
problems in lieu of two regular problems, but it's not recommended!
M 8/30 -- We finished section 2.2 and began section 2.3. We talked
about frames and the behavior of their derivatives and got ready
to start on Frenet frames for curves parameterized by arclength.
Wednesday's class is the last chance for questions on the first
homework. Bring them on!
W 9/1 -- A full day of Frenet frames. We finished the unit-speed
case and started, barely, on the general case.
F 9/3 -- About 20 minutes on the homework (solutions not linkable) and
then more on Frenet frames with non-unit-speed curves.
The second assignment, Homework
Two, was distributed, due M 9/13. For the record, these were
the webpage corrections to HW1. On #7, yes, I mean the hyperbola lying
in the plane z = 0 (which is consistent with the parameterizations.)
Homework #1 (ungraded) has a botched solution in the book.
M 9/6 -- Labor Day (no class). Last break for 10 weeks! But go to
Lunation
to see an animated version of a month in the life of the moon.
W 9/8 -- "Phalanx o' Formulas" (Frenet Frame fantasia) unleashed
on an unsuspecting class. Implications of zero or constant curvature
and torsion. Bring homework questions for Friday. Don't forget, HW2
is due Monday.
F 9/10 -- Two corrections to the homework, noted above. Homework 1
returned with further comments. Proof of a few short items from
sections 2.3 and 2.4: the minimum curvature of a curve on a sphere
of radius a is 1/a, and if the curvature is always 1/a, then the
curve is part of a great circle; the Taylor series implications of
the Frenet apparatus; the curvature of T, viewed as a curve on the
unit sphere. Lots of good class/instructor interaction. Please continue!
Old Homework Corner on #2: An alert student, Maciej Babinski, points out that the author of the text has an errata list on his webpage: Corrections to EDG/2 In particular, in HW2 #4, that is to say, p.64 #10a, the problem should read sigma = 1/tau, not sigma = 1/r. It is also worth remarking that rho, as it appears in the problem, is *not* the radius of the sphere, even though that is a common notation; rather, rho = 1/kappa. Further, rho' is not some other constant. It is the derivative, with respect to s, of rho; hence rho' = -(1/kappa^2) times d(kappa)/ds. In HW2 #5, the hint is not helpful. As an alternate hint, observe that cosh^2(t) + sinh^2(t) + 1 = 2 cosh^2(t).
M 9/13 -- A small amount of discussion about HW2, then finishing
up on curvature and cylindrical helices. Handwaving about isometries.
This will be finished up on Wendesday, and then, on to surfaces.
The third assignment, Homework
Three, was distributed, due M 9/20.
W 9/15 -- The solution to HW2 #5 was incomplete; a corrected version
was be distributed today. We finished section 3.5 on isometries, to
the extent we'll do it, and then started surfaces. A squirrel attended
for a while, but was frustrated that he missed add day by a few days.
F 9/17 -- Section 4.1 on surfaces, with the book elaborated in some
detail, to the satisfaction of the brave attendees. It takes a long
time for the instructor to learn the difference between a semicircle
and a circle. No squirrels were harmed in the preparation of this
report. HW3 corrections as noted at the top of this page.
M 9/20 -- HW3 is due on W 9/22. Some corrections and hints were given
in class. After that, we continued through 4.1 and started on 4.2.
Parameterizations, cylinders, surfaces of revolution. one point I
didn't make explicitly: in #6, if t_o isn't specified, you can pick
which one you wanted.
Homework corner: Two more typos. Problem p.66 10b, the formula should be gamma = alpha + rho N + rho' sigma B; Problem p.75, #6, there is no Example 4.2; however, there is an Example 4.4. Use that one. (Thanks to Moshe Adrian for the second one.)
W 9/22 -- HW3 due, Homework Four distributed, due 9/29. We looked at the example (u+v,u^2+v^2,u^3+v^3), for which there will be a handout on Friday. Pretty much finished 4.2
F 9/24 -- More on ruled surfaces and an introduction to 4.3. Two
longish examples, for which there will be handouts on Monday: the
identification of the hyperboloid of one sheet as a ruled surface
in two ways, and the explication of Cor. 3.3 (p. 146) in case x_1(D_1)
is the Northern Hemisphere and x_2(D_2) is the Western Hemisphere.
(It was unfortunate that 4.3 was omitted from the syllabus. We're
doing it!)
M 9/27 -- Homework 3 returned. A few suggestions on HW 4 (see above).
We nearly finish 4.3 with a review of the chain rule and how it's
applied. We also cover why the double cone is not a surface at the
double point. The two examples are handed out.
Homework notes: In problem #1 (p.132, 4ac) the "patch" is a
"coordinate patch".
W 9/29 -- Finish 4.3, start 5.1. Homework 4 due, Homework
Five distributed, due 10/6. handed out. Test 1 will be on W 10/13, unless
someone can give me a good reason not, and it will cover the first 5
homeworks.
F 10/1 -- Lots of review of various chain rules, and an introduction
to the dreaded shape operator.
M 10/4 -- Examples of the shape operator and how 2 x 2 matrices play
a role in them. Some material is not in the book and will eventually
be in a class handout. No really profound hints on HW 5 (or questions,
either.)
W 10/6 -- More on the shape operator, normal curvature and gaussian
curvature. HW 6 (due 10/20) will be out on 10/8 or 10/11. No more new
material until after the exam, which (I repeat) will cover the first
5 assignments, and so nothing about the shape operator.
F 10/8 -- I reviewed the material to be covered on the exam.
M 10/11 -- I did not prepare any material, but discussed HW 5 and
answered a few questions. I xeroxed Homework
Six, due W 10/20, but forgot to bring it. I'll bring copies
on W 10/13. You do not need to know hyperbolic sine and cosine
for the test, but you should be able to produce the Frenet apparatus
upon request. I will answer questions on the final (and post them
here) up to about 5pm tomorrow.
A student writes: I am assuming a Frenet apparatus problem will be on the test. In the example you did after class on Friday, you evaluated v, a, and a' at zero before you calculated the T, B, and N vectors. However, in most of the other examples you have done in class, you evaluated v, a, and a' at t first. I am assuming you are going to let us know whether to evaluate it at zero or at t before computing the T, B, and N vectors. Am I correct?
My reply: You are correct. There may be a problem where I give you a curve
and ask you to differentiate to determine someinformation, and another
problem where several derivatives are given to you and you have to find
the Frenet apparatus. This is a way of "thinning" the computations.
W 10/13 -- TEST ONE (up through the end of ch. 4.)
F 10/15 -- Getting back up to speed with the shape operator,
normal curvature and the like. A more matrix-oriented approach
than that found in the book.
M 10/18 -- Test 1 returned and discussed. The Gauss map viewed
geometrically. Handout on the shape
operator in an arbitrary basis, and some useful computations
discussed. On the homework, it is sensible to think about the Gauss
map geometrically, and on p.208, #3, I think "normal component" means
with respect to the normal vector on the cylinder, not the normal
vector to the curves.
p. 200 #3. Use #2, and The rank here is the rank of the matrix, or the
dimension of the image.
You may remember the mantra from linear algebra: rank + nullity =
dimension. Look at S(TP(M)) as a vector space, and decide whether it is
2-dimensional, 1-dimensional or 0-dimensional.
p. 200 #4(b). As noted in class, the cone is not a surface if you
include the vertex, so assume z > 0 here.
p. 200 #5, #6 Yes, it's ok to give the answers in parametric form, but
it is much better to give them descriptively or geometrically as well.
p. 208 #3 (Based on an e-mail query) 0^0 only occurs as an
accident of the way we describe differentiation. If g(t) = t = t^1, then
we say that g'(t) = 1 times t^0, leading to uncertainty about g'(0),
but that's an artifact of notation: g'(t) = 1 for
all t, including t = 0.
W 10/20 -- We went through a computation of the shape operator for the surface z = xy at a general point, and there was a handout. A revised, extended and corrected handout will be distributed on 10/25. Homework 6 solutions were distributed and Homework Seven, due W 10/27, was distributed. There is a 2-point homework problem based on replicating the handout for a different surface, and this will be discussed in class on 10/25.
F 10/22 -- We went through section 5.4 and computed H and K for a
large class of surfaces. There will be a handout eventually.
M 10/25 -- Returned homework 6 and discussed some variations of the
proofs. Three handouts: 10/20 Handout
verstion 2.0, a hand-written handout on curvature for surfaces
of revolution and a stacked container of patches of surfaces parallel
to z = xy; viz. a can of Pringles. HW 7 is due Wednesday.
W 10/27 -- Homework 7 solutions were distributed and Homework Eight, due W 11/3, was passed out. I went through section 5.5 in some detail.
F 10/29 -- I covered section 5.6 (the part on principal curves), and worked
out, using Monday's handout on surfaces of revolution, an analytic proof
that meridians and parallels are principal curves.
M 11/1 -- Some elaborations on the solutions to HW 7. More
on principal curves and an introduction to asymptotic curves.
In homework 8, problem #1, the surface has a Monge patch of the form
x(u,v) = (u,v,g(u)+h(v)), and in #3 (p.223, 36), you have to compute
the curvature for two different surfaces. In #5, the definition of
principal direction is on p. 204, and I interpret this problem as
3 questions asked of each of two different curves. And since we didn't
really talk about geodesics in class yet, let me point out the
definition: alpha is a
geodesic curve iff alpha'' is normal to M.
W 11/3 -- More on asymptotic curves and geodesics. Worked out detail with the surface z = xy, more on that Friday. Homework 8 solutions were distributed and Homework Nine, due W 11/10, was passed out.
F 11/5 -- Deeper into 5.7 and surfaces of revolution. A discussion of
the principal curves on the surface z = xy. (There will be a handout
on this eventually.) Also, a botched discussion of the catenary and
the brachistochrone problem. Rather than kill some trees for a black
and white version of these, I'll refer you to two websites:
Brachistochrone
Problem and
Catenary.
M 11/8 -- Discussion of homeworks 8 and 9 (see tip at top of webpage).
Finish of section 5.7. The Third version of
z = xy handout was distributed; although with a 4th page that
isn't on the .pdf.
Here is a way to set up a couple of the problems on HW 9 involving
determinants. We know that there are two orthonormal principal
directions at every point -- call them u1 and u2, and assume that
S(u1) = k1 u1 and S(u2) = k2 u2. (If k1 = k2, then any two orthonormal
directions will do; otherwise, they are uniquely determined up to
sign.) Now, at the point P of the problem, suppose xu = a u1 + b u2
and xv = c u1 + d u2, so that S(xu) = a k1 u1 + b k2 u^2, etc. One can
calculate E, F, G, L, M, N in terms of a, b, c, d, k1, k2, and, so,
figure out the quadratic equation given by the determinant. You have
to distinguish two cases: if k1 = k2, then every vector is in a
principal direction; if k1 \neq k2, then v1 xu + v2 x2 = v1(a u1 + b
u2) + v2(c u1 + d u2) is principal if and only if it is a multiple of
u1 or u2 alone; that is, if and only if v1 a + v2 c = 0 or
v1 b + v2 d = 0. This should be enough to let you do the problem in
a reasonable amount of time.
A student writes: After your hints at the first homework problem in
class, I have encountered another
problem. You said "Let Xu=au1 + bu2 and Xv=cu1 + du2". Now, in computing
E,F,G,L,M, and N, I have found that the Unit Vector U=0 (specifically
the cross
product of Xu and Xv), since u1 and u2 are orthonormal vectors. And the
multiplication of two orthonormal vectors = 0. If this is the case,
then L, M, and
N are also 0. I have a feeling this is incorrect. Maybe I am looking
at this the wrong way.
My reply: If xu = a u1 + b u2 and xv = c u1 + d u2, then
xu x xv = ac u1 x u1 + ad u1 x u2 + bc u2 x u1 + bd u2 x u2 =
(ad - bc) u1 x u2.
Remember that i and j are orthonormal vectors and i x j = k.
If you follow my hints (now on the webpage), there is no need to follow
the book's hint.
W 11/10 -- The day was spent going over the homework and enjoying
some highly carbohydrated and sweetened tori, in partial instructoral
penance for the long and surprisingly crunchy homework. The next
topics are in section 6.4 -- isometries. Oh, and Homework
Ten, due W 11/17, was passed out. This is the last homework on
which the second test will be based. There's one more homework, due
after Thanksgiving, that will be covered on the Final, but not the
second exam.
F 11/12 -- We began to discuss isometries and local isometries,
pretty much following 6.4.
M 11/15 -- Minor comments on HW 10. More detailed discussion of
the solutions to HW 9, a more elaborate presentation of the
helicoid/catenoid isometry than in the book and a beginning to
the proof of Gauss' Theorem Egregium, for which there will eventually
be handouts.
When I said, a "numerical (rather than symbolic)" answer, I didn't
phrase my intent very well. My point was to emphasize that the
desired answers are at a specific point, and should not be given
in terms of u and v. If you need to say "the square root of 2", e.g.,
there are some circumstances under which that is considered a symbolic
answer, whereas 1.414... is a numerical one. My point is that I want
"the square root of 2", not "the square root of u^2 + v^2, evaluated
at (u,v) = (1,1).
A student writes: I had a question on #2. I seemed to be able to do everything
fine except for finding the principal vectors. H, K, and the
principal curvatures were not a problem. I assume that there
is some way to use these to get the principal vectors, but I
couldn't find it in the notes or in the book. If you could
toss a hint, that would be most appreciated.
My reply: Well -- If you know S(x_u) and S(x_v) in terms of x_u and x_v,
finding the principal vectors is essentially finding the eigenvectors
of the operator, and you already know the eigenvalues... -- BR
In response to a similar question, I wanted to add that it is
perfectly ok to use problems we've already done in the homework, such
as 5.4 -- #2, 9, to simplify the computation. And there is a useful
hinto to p.224 -- #11 in the back of the book.
In response to yet another query, it appears that problem 7 isn't
quite right as it stands. Do your best with it, or use google, dare I say?
W 11/17 -- The completion of the proof of the Theorem Egregium, with
notes to follow after break. Also, Homework
Eleven, due W 12/1, was passed out. A few comments are at the head
of this webpage; a few more comments will surely be made over break
and on Monday. This material will only be covered on the final.
F 11/19 -- A few people showed up and I talked about Gauss and Christoffel.
Thanksgiving
M 11/29 -- My narrative review for test.
W 12/1 -- Answering of your questions, both as a group, and as you
gradually left the room, individually.
Don't forget: I've already announced that the second test will be on
Friday Dec. 3, covering Chapter 5 of the test and homeworks 6 through
10.
In order to help you focus on the second exam, I have decided to delay the due date for HW 11 to Monday, Dec. 6. This should guarantee to the skeptical that it's the last homework of the semester!
I have put corrections onto a Homework Eleven Tip-sheet. This was distributed in class and will also be available at the test on Friday.
F 12/3 -- Test 2, on chapter 5
M 12/6 -- Review of Test 2 and homework 11.
W 12/8 -- Three or four topics not covered in the course, but
on the final, leading up to the Gauss-Bonnet theorem.
F 12/10 -- Final review and questions and ICES.
S 12/11 -- Office Hours on Reading Day, Sat. Dec. 11 -- 1:30 -> 4:00,
327 Altgeld, actually, we'll meet in 441 Altgeld, just around the
corner and up the stairs.Reading Day (Extended Office Hours)
A student writes:
I have what I think is a simple question about the final
concerning isometries. I know that you said isometries would
be in the final and I am wondering what sort of problems I
should know about them:
Should I know simply how to prove two surfaces are isometric,
or should I actually know how to demonstrate the isometry
analytically as in the last homework (the problem that was
quite hard and involved using some terms from complex
variables)?
My reply: What you need to know are; (i) the formal definition of a local isometry; (ii) the theorem that local isometries preserve E, F, G; (iii) the content of Theorem Egregium; (iv) the mechanism of the map between two locally isometric surfaces: ie, pull back from M_1 to D and then map to M_2. You don't need to know the proofs of (ii) or (iii). Lots of stuff on the homework was harder than would be on an exam!
Tu 12/14 -- Final Exam, 1:30-4:30 PM. This time is fixed in stone.