MATH 113, SPRING 2010
HOMEWORK 1 SOLUTIONS
1: Let (t) be a parametrized curve such that its second derivative (t) is identically zero. What
can you say about ?
Solution: Let t0 be in the domain I of . If = 0, then by the Fundamental Theorem of
Calculus (F
San Jose State University
Math 113, Spring 2010
Midterm 2 Solutions
Assigned on April 28, due on May 5, 2010, by 3 PM
Name:
Granwyth Hulatberi
Score 1 25 2 25 3 25 4 25 Total 100
Explain your work
1. (25 points) Consider a surface patch (u, v ) = (u cos
San Jose State University
Math 113, Spring 2010
Midterm 2
Assigned on April 28, due on May 5, 2010, by 3 PM
Name:
Score 1 2 3 4 Total
Explain your work
1. (25 points) Consider a surface patch (u, v ) = (u cos v, u sin v, log cos v + u), where u R and < v
San Jose State University
Math 113, Spring 2010
Midterm 1 Solutions
Assigned on February 24, due on March 3, 2010, by 3 PM
Name:
Granwyth Hulatberi
Score 1 25 2 25 3 25 4 25 Total 100
Explain your work
1. (25 points) Let (t) = 1 11 11 cos t, + sin t, sin
San Jose State University
Math 113, Spring 2010
Midterm 1
Assigned on February 24, due on March 3, 2010, by 3 PM
Name:
Score 1 2 3 4 Total
Explain your work
1. (25 points) Let (t) = 1 11 11 cos t, + sin t, sin t . 22 22 2
(a) Compute the curvature of . (
MATH 113, SPRING 2010
HOMEWORK 10 SOLUTIONS
11.1 (Pressley): Assume there exists a simple closed geodesic in a surface patch . Then the
interior D of is wholly contained in , hence by Gau-Bonnet
g ds = 2
K dA.
D
The left-hand side equals zero, whereas th
MATH 113, SPRING 2010
HOMEWORK 9 SOLUTIONS
7.6 (Pressley): Recall that the principal curvatures 1 2 of a surface S at a point p are the
minimum and maximum of the normal curvature n (v ) taken over all directions v Tp S . Recall
also that n (v ) = (0) N,
MATH 113, SPRING 2010
HOMEWORK 8 SOLUTIONS
6.5 (Pressley): The coecients of the second fundamental form of the elliptic paraboloid (u, v ) =
(u, v, u2 + v 2 ) (according to ex. 6.1 from Homework 9) are
L=
2
= N,
1 + 4 u 2 + 4v 2
M = 0.
We have (t) = (cos
MATH 113, SPRING 2010
HOMEWORK 7 SOLUTIONS
1. Let S be a smooth surface. Recall that the gradient of a smooth function f : S R is a smooth
map f : S R3 , which assigns to each point p S a vector f (p) Tp S such that
f (p) v = Dp f (v ) =
d
dt
f ( (t),
0
f
MATH 113, SPRING 2010
HOMEWORK 6 SOLUTIONS
1. (Ex. 4.16) Let v be a vector tangent to the surface S at p. Then v = (0), for some curve
: (, ) S such that (0) = p. Since f = 0 on S , we have f ( (t) = 0, for all t. Dierentiating at
t = 0 and using the cha
MATH 113, SPRING 2010
HOMEWORK 5 SOLUTIONS
Ex. 4.2: Let
U = cfw_(u, v ) R2 : a2 < u2 + v 2 < b2
be the open annulus with inner radius a > 0 and outer radius b > 0. Let f : (a, b) R be a
dieomorphism. (We showed in class that such an f exists.) Dene a map
MATH 113, SPRING 2010
HOMEWORK 4 SOLUTIONS
Ex. 3.1: It is enough to show that length and area are invariant relative to translations and
rotations. Let : I R2 be a simple closed curve, L : R2 R2 be a translation or a rotation,
and = L . It is clear that i
MATH 113, SPRING 2010
HOMEWORK 3 SOLUTIONS
Ex. 2.3: Let t(s) be the unit tangent of . Then t(s) = (cos (s), sin (s), for some angle (s).
Since the angle between t and nsg (in that order) is /2, it follows that the signed unit normal
is
nsg (s) = (cos(s) +
MATH 113, SPRING 2010
HOMEWORK 2 SOLUTIONS
1. Let f : R2 R be a C 1 function (i.e., f has continuous partial derivatives) such that the
gradient f (p) of f is nonzero for every p R2 .
(a) Show that every level curve of f is orthogonal to the gradient of f
San Jose State University
Math 113, Spring 2010
Final Exam
Assigned on May 18, 2010
Due on May 25, 2010 by 2 PM
Name:
Score
1
2
3
4
5
XC
Total
You are allowed to consult the literature but not each other.
1. (20 points) The total torsion of a unit speed
Math 310 - Quiz 2 Solutions
Monday, 19 Oct 2009
No calculators, notes or text allowed. Let (sn )n be a sequence dened recursively by s1 = and sn+1 = 2 a. Use induction to prove that sn 1 for all n N. For the base case observe that s1 = > 1. Now, suppose t
Math 310 - Quiz 4 solutions
Wednesday, 4 Nov 2009 (20 pts)
No calculators, notes or text allowed. Each problem is worth 10 points. 1. Write down the rst ve terms of the folowing sequence (sn )n . Find, if possible, a subsequence (snk )k which converges to