5/13/13: I am done with writing solutions
to the Sample Final. I have been writing them sporadically all day
long, taking breaks from a Calculus exam that I was giving and grading
today. Good luck for tomorrow's exam!
5/13/13: I am done with writing solutions
to the Sample Final. I have been writing them sporadically all day
long, taking breaks from a Calculus exam that I was giving and grading
today. Good luck for tomorrow's exam!
5/13/13: Coverage for the Tuesday, May 14, 10:30 a.m - 12:30
p.m. Final (in our regular classroom) is basically all we have
studied, except the material on Riemannian geometry covered during the
last class meeting. This is Chapters 5-8 (skipping the Involutes
and Evolutes section from Chapter 5, the Appendix to Chapter 6,
and Chapter 7bis), Chapters 9, 10 (through p. 195),
11 (through the top of p. 210), and the material covered in class on
constant-curvature surfaces, non-Euclidean geometries, the
hyperboloid, the Poincare disk and upper half-plane models of
hyperbolic geometry, smooth manifolds, smooth maps, and
diffeomorphisms. (This roughly corresponds to pp. 218, 224-225, 228,
262-269, and 282-287 from Chapters 12, 14, and 15 in the text, but the
way we dealt with this material was on many occasions quite different
and much simpler.) The best way to get ready for the final would be to
try to solve as many problems as you can on your own (with an open
textbook, if needed, and no timing) from the sample final. For those
problems you cannot solve, it may be a good idea to discuss them with
your friends, i.e., your study group, and with me tomorrow. Come to my
office hours Monday, May 13, 11 a.m. to 12:30 p.m. Try to solve the
real and sample midterm problems once again without consulting your
old solutions. Go over homework problems again in a similar way and
look for problems from similar classes on the web: search for
Differential Geometry. When you know you can get an idea of how you
might approach most of the problems right away, you may feel more sure
that you will do well on our final. I am also working on posting
selected solutions to our sample final, but consult them only after
you have tried hard to solve the problems on your own. Good luck!
5/13/13: I have just posted a solution to
Problem 1 on the Sample Final and plan to post some more, but have
to go to bed now and will have little time, being busy giving an exam
and then grading for another class today.
5/8/13: Sample Final is now posted!
5/8/13: Because the material for Problem Z on the last homework will
be reviewed briefly during the last class, you do not have to turn
that problem in -- just solve it, if you would like to see how well
you understand those things. The material on Riemannian geometry will
not be covered by the Final, either. However, manifolds, smooth maps,
and diffeomorphisms will.
5/8/13: I need to move my office hours on Monday, May 13 ahead by 30
minutes. My office hours will be 11:00-12:30.
5/7/13: Hint to homework Problem U: Use the
equation kg = 0 from the second displayed formula on
p. 188 using a parametrization x = t, y = y(t), unless x = const,
which needs to be treated separately. The equation should then
transform into y y'' + (y')2 = -1 or (yy')' =
-1, which integrates directly as yy' = -t + C, which is a
separable first-order differential equation and is easy to solve.
5/5/13: Some statistics for Exam II: Maximum = 39, Mean = 34.6, Median
= 36, which is 90% of the total score. The exam was probably too easy
for you, folks. On the other hand, it means you have been learning the
material really well!
5/3/13: I plan to post a sample final over the weeekend and later post
solutions to it. I will also hold extra office hours on Monday, May
13: 11:00-12:30.
5/3/13: I have moved the last homework due date to Friday, May 10, but
added one problem on Riemannian geometry to it. Make sure that the
last problem on Homework 12 is Problem Z.
4/26/13: I have just realized that Problem 4 on Sample Test II was too
hard for an exam and removed it from the Sample Test. See comments in
the new, April 26, version of the test. I kept the same numbering of
problems. I apologize about the confusion.
4/25/13: I have added a few problems to Sample Test II and will not be
adding problems to it anymore. There must be 11 problems total.
4/24/13: Reminder: the second midterm exam is coming on Wednesday,
May 1. Coverage: Chapters 9, 10 (through p. 195), 11 (through the
top of p. 210), and the material covered in class on
constant-curvature surfaces, non-Euclidean geometries, the
hyperboloid, the Poincare disk and upper half-plane models of
hyperbolic geometry. (This roughly corresponds to pp. 218, 225, 228,
262-269 from Chapters 12 and 14 in the text, but the way we dealt with
this material was quite different and much simpler.) How to get ready:
the more problems you solve before the test, the better you will be
prepared. Solve
the sample
exam, and problems from the past homeworks again (if you feel you
do not remember how to do them). Also, look for similar problems from
Differential Geometry textbooks, such as Differential Geometry and
Its Applications by John Oprea, available in the Math Library,
look for old exams on the web for Math 5378 and similar classes at
other schools. If you know you need only 3 minutes to get an idea on
how to solve at least 50% of random problems you find, then you may be
sure you will do well our test.
4/24/13: I will be out of town, visiting Purdue University the rest of
this week. I will be back on Monday, but I will be passing a
citizenship interview and not be able to teach the class, either. I
have asked my graduate student Denis Bashkirov to substitute for me on
Friday and Monday. I will not hold office hours on Friday, but I will
be here for my office hours on Monday morning. I apologize about the
inconvenience.
4/20/13: Homework 11 is now posted! Sorry
about a delay.
4/9/13: In Problem M on Homework 9, I forgot to mention explicitly
that I was talking about a surface of revolution, even though I used
such notions as parallels and the radius. I have corrected that in the
latest version of the homework.
4/6/13: After you are done with Homework 9, you can add to the
characterization of geodesics below (News of 4/5/13) the
following:
[energy-minimizing curves] <=> [distance-minimizing curves parametrized proportionally to arc length] => [geodesics parametrized proportionally to arc length] <=> [curves satisfying the geodesic equations].
4/5/13: Homework 9 is now posted! It may look
long, because it consists of problems not from the text, but they are
rather straightforward.
4/5/13: Actually, I have made a mistake in that table about geodesics,
as we figured out with a few of the students in an after-class
discussion. Here is a correction:
[distance-minimizing curves] => [geodesics] <= [geodesics parametrized proportionally to arc length] <=> [curves satisfying the geodesic equations].
I was wrong saying that every geodesic was parametrized proportionally to arc length. That confusion came from a different, inequivalent definition of a geodesic that I used to know.
Let me explain why a geodesic parametrized proportionally to arc length will satisfy the geodesic equations. From the determinant formula for geodesic curvature, we see that a regular curve is a geodesic if and only if the projection prN(α''(t)) of α''(t) to the tangent plane to the surface along the unit normal N to the surface is proportional to the tangent vector α'(t). Now, if a curve is parameterized proportionally to arc length, then α''(t) is orthogonal to the tangent vector α'(t). Therefore, prN(α''(t)), being a linear combination of the normals α''(t) to the curve and N to the surface, will also be orthogonal to the tangent vector α'(t). Thus, prN(α''(t)) could be propotional to the tangent vector only when the coefficient of proportionality is zero or prN(α''(t)) = 0, which is equivalent to the geodesic equations.
4/1/13: On Problem 10.8: The tractrix is being rotated about the axis
which it does touch, e.g., the tractrix from p. 84 is rotated
about the y-axis.
3/30/13: One clarification on homework Problem 10.4: the
latitude-longitude parametrization is the same as spherical
coordinates.
3/26/13: I have just learned that there is a free tutor at SMART
Learning Commons who can provide help with Math 5378 Differential
Geometry,
see SMART.
3/26/13: I have to cancel my office hours on Friday, March 29. Sorry!
If you need to see me, please make an appointment.
3/26/13: In Problem 9.7, it should be
u1= √
u2-
a2 , rather than
u1= √
u2+
a2 .
3/14/13: Homework 7 is now posted!
3/11/13: More hints on Problem 9.1: Use Theorem 9.4 and the
computation of the first fundamental form (and thereby the line
element) in the example after Corollary 8.9. If you use the arc length
parametrization of the generating curve in that example, the formula
will be simpler. The rest is to see if you can change your coordinates
on the given surface so as to make the line element change from
Λ(u) (du2 + dv2) to something like
du2 + Λ(u) dv2. See p. 132 on how the
components of the metric change with changes of coordinates.
3/11/13: I have originally misread Problem 9.1, which is on this
week's homework. I have just made corrections to my comments to the
original posting of the problem. You need to prove only that if in a
coordinate chart the line element can be expressed as in the formula,
then this surface is locally isometric to a surface of revolution. And
you can ignore the conformal mapping question.
3/10/13: Some statistics for Exam I: Maximum = 40, Mean = 28.5, Median
= 28.5, which is 71.25% of the total score. Good job!
3/7/13: Homework 6 is now posted!
3/3/13: You do not need to turn in any homework this week. Concentrate
on getting ready to the test.
3/2/13: I have added one problem (Make sure you have a copy of the
test with 7 problems!) to the sample test and moved it to the Homework
page.
3/2/13: Reminder: the first midterm exam is coming on Wednesday,
March 6. Coverage: Chapters 5-8, skipping the Involutes and
Evolutes section from Chapter 5, the Appendix to Chapter 6, and
Chapter 7bis. How to get ready: the more problems
you solve before the test, the better you will be prepared. Solve
the sample
exam, and problems from the past homeworks again (if you feel you
do not remember how to do them). Also, look for similar problems from
Differential Geometry textbooks, such as Differential Geometry and
Its Applications by John Oprea, available in the Math Library,
look for old exams on the web for Math 5378 and similar classes at
other schools. If you know you need only 3 minutes to get an idea on
how to solve at least 50% of random problems you find, then you may be
sure you will do well our test.
2/27/13: I have posted a sample test before the real one which will be
in a week, on March 6.
2/25/13: I have added the following clarification (and simplification)
to the homework problem 8.4: Find the principal curvatures just at the
north pole of the ellipsoid ax2 + by2 +
cz2 = 1, a, b, c > 0, and at the origin for the cylinder z
= ky2, k > 0, and the saddle z = ax2 -
by2, a, b > 0.
2/21/13: Yesterday late night, I made up HW5, but posted HW4 again
instead. Sorry. The flaw is now corrected. Homework 5 is short, as we
are getting close to the first test, and I plan to post a sample one
soon.
2/18/13: Also added the following clarification to Problem 7.9 on
HW4. The graph is revolved about the u-axis. The function is assumed
to be positive. The problem refers to Example (4) on page 119.
Finding (x-1 ◦ xbar) means solving the equation
x(u,v)=xbar(ubar,vbar) for u and v. This is an equation on vector
functions x and xbar, so it is actually a system of three equations,
one for each coordinate. A solution should produce u and v as
functions of ubar and vbar. Beware of the domains of these functions,
as you expect them to be the intersection of the two charts expressed
in the ubar and vbar parametrization, i.e., xbar-1 of the
overlap(s).
The equations x = xbar look awfully complicated, but you may notice
some structure. In the first two coordinates, they look like a sin u =
b cos ubar, a cos u = - b sin ubar. They have a simple solution, but
it is not obvious how to find it. Your life would have been simpler,
if the equations were between two sines and two cosines. But this is
easy to achieve: sin u = cos (u-π/2) and cos u = - sin
(u-π/2). Thus, your equations are
a cos (u-π/2) = b cos ubar, a sin (u-π/2) = b sin ubar.
Since cos2 + sin2 = 1, these imply that a = + or
- b. Recall what a and b are and realize that only one of these
options is possible: a = b > 0. Thus we get the polar coordinates of
one and the same point in the plane. Therefore, they must be equal, up
to a multiple of 2π in the angle:
u-π/2 = ubar + 2π k.
We know that both u and ubar must be between 0 and 2 π. This leaves
two options: k = 0 and k = -1, which corresponds to two overlaps
between those two charts. Indeed the Moebius band is just a piece of
band after all, glued in a funny way at the ends. The two charts you
can imagine cover half of the glued band and a bit more, and you
expect them to overlap over two separate regions. Think of covering a
circle with two intervals!
Now, you have to recall what a and b are and add to the equation a = b
the equation which comes from making the z's equal in x (u,v) = xbar
(ubar, vbar):
v cos u/2 = vbar cos (π/4 + ubar/2),
which we have ignored so far. You may also wish to plug in the two
solutions for u through ubar that you have found above. The two
solutions correspond to two different overlaps, with which you will
have to work separately. For each overlap this gives a system of two
equations, very similar to what we had above, but with u already being
expressed through ubar, considerably simpler. Solve it for v, and you
will get u and v expressed through ubar and vbar in the two
overlaps. Now work with the partials and see whether the Jacobians are
positive.
κ (α; t) = κ (β; s) = det
(β'(s)t β''(s)t) = (dt/ds)3
det (α'(t)t α''(t)t) = 1 /||α'
(t)||3 det (α'(t)t
α''(t)t).
Last modified: (2014-01-20 16:34:56 CST)
2/18/13: I have added the following clarifying and simplifying remark
to Problem I on HW4. Use any of your favorite charts for the sphere
and assume the point (x0, y0, z0
2/18/13: Correction for the textbook: The top formula on page 133
should have the bars on the left-hand side, rather than on the
right-hand side.
2/17/13: Hint to Problem K on Homework 4: You may assume that (u,v)
↦ xu × xv/||xu ×
xv|| is a smooth function from the surface to
S2.
2/16/13: Here is a solution of Problem H from Homework 3. WLOG, you
can assume that the first two rows of the Jacobi matrix for
parametrization y are linearly independent at and near a point on the
surface S2. By the Implicit Function Theorem, it means that
the surface S2 near that point may be given as the graph of
a function x3 = F(x1,x2). This means
that y(u,v) = (y1(u,v), y2(u,v),
F(y1(u,v),y2(u,v))), and by the Inverse Function
Theorem, the map (y1(u,v), y2(u,v)) has a smooth
(i.e., differentiable) inverse (u(x1,x2),
v(x1,x2)). The map y-1 on a point
(x1,x2,x3) of the surface
S2 will then be given by the formula y-1
(x1,x2,x3) =
(u(x1,x2), v(x1,x2)). The
composition y-1 ◦ f ◦ x will then be
(u(f1(x), f2(x)),
v(f1(x),f2(x))) and thereby a composition of
smooth functions. Another solution may be noticing that in the proof
of Proposition 7.5, one might use any smooth, not necessarily regular
function in lieu of x, in this case the function f ◦ x. Of
course, you have to repeat the proof in your own words, if you write
this as a solution of the problem.
2/14/13: I have just posted homework for next
Wednesday. Reminder: At the beginning of each homework, I give
a reading assignment for that homework. Not later than shortly after
each class, I outline what has been covered in class and give a
reading assignment on
the
Class Outline page. Some important announcements, including
homework hints and study advice are made on this Class News page.
2/12/13: Here is a hint to Problem 7.5. Do not try to find
x-1, find (x-1 ◦ xbar) without knowing an
explicit inverse. I would have hard time looking for x-1,
too. I am not sure I would be able to do it with formulas in a
reasonable amount of time.
2/7/13: Folks, I was too sleepy last night to post the homework and
too busy in the morning today. Sorry. I will do that today early
afternoon.
2/6/13: If you try to use google or other search engines to arrive at
our class pages, make sure that you are looking at Math 5378 Spring
2013. The thing is that I also taught this class in 2002, and you
probably do not want to do homework from 2002.
1/31/13 (after midnight): Homework 2 is now posted!
1/28/13: I have made an error computing β''(s), when β(s) =
α(t(s)) is the reparametrization of the curve α(t) by its
arc length s = s(t), today in class: it should be β"(s) =
α" (t(s)) (dt/ds)2 + α'(t(s))
(d2t/ds2), rather than β"(s) = α"
(t(s)) (dt/ds) + α'(t(s))
(d2t/ds2). Sorry! This should yield the
required computation of the curvature in terms of α(t) as
follows:
1/28/13: A little clarification on the textbook Problem 6 from Chapter
5. Let us reword it as follows: Take any circle of radius R=a with
center on the y-axis. Show that the lower half of the circle meets the
tractrix at a right angle.
1/25/13: You should be able to do all the homework problems, except
#'s 4 and 6, over the weekend. We will study the tractrix and oriented
(a.k.a. directed) curvature on Monday.
1/22/13: I have to move my office hours just for one day, Wednesday,
January 23, from 11:15-12:05 to 12:20-1:10, because of a conflict with
a special seminar.
1/21/13: I recommend the following way to study. Attend each class,
take notes, participate in class actively. After each class review
your notes and study the corresponding part of the text. You can find
out which part of the text at
the
Class Outlines page. Then do relevant
homework problems.
1/21/13: If you need to register for this class, please, send a
message to Ms. Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
1/21/13:
Homework will be collected before each class, starting next
Wednesday, January 30, and a small selection of problems will be
graded. ("Small," because the grader will allocate only 3 hours per
week for our class.)