NAME:_Solutions_ P221 Background Survey(Spring 2007)
1. What is your major? Lots of Biologists, biochemists and chemists, with a sprinkling of other majors. 2. Did you take a physics course in high school? Only about a quarter of you have seen physics bef
Pull-back of dierential forms
Denition 0.1. Let F : Rn Rm be a smooth map and let be a k-form
on Rm , where k 0. The pull-back F is a k-form on Rn dened as follows:
(1) If k = 0, so that = f : Rm R is a function, we put
F := f F : Rn R.
(2) Let k 1. Then
Orientation on a vector space
1. Orientation on a vector space
Throughout this section let V be a vector space over R of nite dimension
n 1. For two bases B = b1 , . . . , bn and B = b1 , . . . , bn of V , the transition
matrix
TB ,B = (aij )n
i,j=1
from
Differential geometry
Fall 2015 MATH 423, Section B13
MWF, 9am;
Altgeld Hall, rm 145
WWW: http:/www.math.uiuc.edu/~kapovich/423-15/423-15.html
The first midterm exam has been graded and the results have been posted
in ScoreReports. In the meantime, soluti
Basics from linear algebra
Denition. A vector space is a set V with the operations of addition
+ : V V V,
denoted w + v = +(v, w), where v, w V
and multiplication by a scalar
:RV V
denoted rv = (r, v), where r Rv V
such that the following hold:
(1) We hav
Dierential forms
1. Differential forms at a point
1.1. Denition and examples of k-forms.
Denition 1.1. Let p Rn and let k 1 be an integer.
A k-form at p on Rn is a function
: Rn . . . Rn R
p
p
k times
such that satises the following properties:
(1) The m
Math 423
Midterm Exam 1 (Solutions)
Friday, October 2, 2015; 9am-9:50am
Problem 1.[10 points]
For each of the following statements indicate whether it is true or false. You DO
NOT need to justify your answers.
(1) If is an r-form on Rn (where 1 r n) then
H/wk 4. Due Wed, Sept 23 (Solution of Problem 5)
Problem 5. Prove that if F : Rn Rm is a smooth map and if , are smooth
forms on Rm then
F ( ) = F F .
Hint: Youll need to use Proposition-Denition 1.14 of the wedge-product from
the handout on dierential fo
INAME: S GLU'WONS- I
Math 423 Midterm 2
November 5, 2008
0 Answer all questions.
0 Be as clear as possible. Unless otherwise stated, give reasons for your answers.
0 Use both sides of the paper, if necessary.
I Good luck!
'_Questi0n Maximum Your Sco
(c) Show that there are no asymptotic directions at a point where the Gauss
curvature is positive.
(5)
SUM (yo OJ C) M wmcamm
kwo =3 M WMWW Question 6
Let M be a generalized cylinder, i.e. a ruled surface parameterized by
a(u,v) = Btu) + 125
where 5(a)
Question 4
Let f (x, y) be any smooth function of (st, y), dened for all (L y) in an open
set D C R2.
(a) Show that
dirty) = (56,11, aw
is a good patch dening a surface in R3.
(10)
2325
3x 3H3 (b) If f(I,y) = sin("2z) 31M?
) and 0(1):, 3/) is as in part (
DEP 3053 Syllabus, 1/8/2012
DEP 3053 DEVELOPMENTAL PSYCHOLOGY, LIFESPAN, SPRING 2012
Section # 0069
Instructor:
Office Hours:
Office:
Phone:
Email:
ILAN SHRIRA
Wednesday, 3-5pm; also available by appointment
Room 273, Psychology Building
273-0166
[email protected]
Math 423
Midterm Exam 1 (Solutions)
Friday, October 3, 2014; 9am-9:50am
Problem 1.[10 points]
For each of the following statements indicate whether it is true or false.
You DO NOT need to justify your answers.
(1) If is an r-form on Rn (where 1 r n) then
Math 423
H/wk 13
Due Wed, Dec 3, 2014
For this homework you will need to use the handout with a scan of pp.
133-140 from Kuhnel (Covariant differentiation on surfaces) available at the
course webpage. Youll also need to use the formula
kij =
2
X
g km ij,m
Math 423
H/wk 11
Due Wed, Nov 12, 2014
For this homework you will need to use the material for the handout Handout
on orientation and orientability of surfaces available at the class webpage.
1.
Consider the bases B = b1 , b2 , b3 and B 0 = b01 , b02 , b0
Math 423
H/wk 14
Due Wed, Dec 10, 2014
For this homework you will need to use the handout with a scan of pp. 133-140
from Kuhnel (Covariant differentiation on surfaces) and pp. 140-144 from Kuhnel
(geodesics and parallel transport), available at the cours
Math 423
H/wk 12
Due Wed, Nov 19, 2014
1.
Consider the surface S = cfw_(x, y, z) R3 |z = xy R3 . We equip S
with a global coordinate patch : R2 S, (u, v) = (u, v, uv), where
(u, v) R2 .
For the coordinate patch : R2 S and a point (u, v) R2 compute
the fol
Math 423
H/wk 10
Due Wed, Nov 5, 2014
1. Consider the cylinder S = cfw_(x, y, z) R3 |x2 + y 2 = 1.
(a) Verify that : (0, 2) R S, given by (, z) = (cos , sin , z), is
a coordinate patch on S (that is, check that is injective and regular).
(b) For
the coo
Math 423
Midterm Exam 2 (Solutions)
Friday, November 14, 2014; 9am-9:50am
Problem 1.[10 points]
For each of the following statements indicate whether it is true or false. You DO
NOT need to justify your answers.
3
3
(1) Forevery
3 3 matrix
Awith entries
Math 423 Extra Credit problems
Due Monday, December 8, 2014
NOTE: You must provide complete, correct and
maximally detailed solutions in order to get credit
for these problems.
Problem 1.[3 midterm exam points]
Let S R3 be a surface, let q S be a point an
H/wk 5. Due Wed, Sept 30 (partial solutions)
Ch 2.3 Problem 6.
A unit-speed parameterization of a circle in R3 with center c and radius r > 0
may be written as
()
(s) = c + r cos(s/r)e1 + r sin(s/r)e2 ,
where e1 , e2 R3 are such that ei ej = ij .
Let (s)