Problem Set # 3
due Th Jan 31, in class
From the textbook:
Sect 2.2: # 7, 11, 12, 15, 16.
Sect 2.3: # 1, 3, 10, 12.
and:
A. Show that the hyperboloid with 2 sheets S = cfw_(x, y, z ) R3 | x2 y 2 + z 2
Math 143 - Homework 5 solutions
May 11, 2015
Extra Problem (1). (a) The helicoid xhel : (u, v) (u cos v, u sin v, v)
has
xhel = (cos v, sin v, 0) and xhel = (u sin v, u cos v, 1),
u
v
so
E hel = xhel
Math 143, Spring 2015
Homework 2 Solutions
1) a) Problem 1 on page 404 of the text.
(a) -3, (b) -4, (c) 1, (d) 0
b) Sketch a regular closed curve in the plane such that there is no point
on the curve
Math 143, Spring 2015
Homework 6, Due Thurs, May 21
1) a) Let U denote the square where |u| < 1 and |v| < 1 in the uv-plane
and let C1 and C2 be n n matrices of smooth functions dened on
U . Suppose M
Math 143 - Homework 4 solutions
May 4, 2015
Section 3.2, Problem 15. Let () be any regular parametrization of
C = S1 S2 , and let N1 , N2 , denote the unit normals to S1 , S2 . Since C is a line
of cu
Math 143 - Homework 1 solutions
April 8, 2015
2
Section 1.3, Problem 4. (a). (t) = (cos t, sin t+csc t) = (cos t, cos tt )
sin
exists for all t (0, ). (Remark: You can recover the fact that the parame
Problem Set # 5
due Th Feb 14, in class
From the textbook: Sect 2.6: # 3, 5 (10 points each)
and:
A (30 points total) For each integer n, consider the n-half twists band Mn given by the image of
the m
PS #6 Solutions
Problem #13/3.2. Assume (s) is an asymptotic curve parametrized by arc length, and consider its Frenet trihedron, plus the unit normal N to S along . Of course K < 0. The normal
curvat
PS #2 Solutions
Problem A. Here we could use the uniqueness part in the Fundamental Theorem, or else solve
the problem by hand as follows:
By Frenets equations, t (s) = k n(s) and n (s) = k t(s) where
PS #1 Solutions
Problem #3/1.2. Integrating twice we get a straight line: (t) = 0 = (t) = c =
(t) = ct + d where c, d are constant vectors. More precisely:
t
(t)dt = (t0 )(t t0 )
(t) = 0 = (t) = (t0
PS #3 Solutions
Problem A. Consider the smooth function F (x, y, z ) = x2 y 2 + z 2 . Its Jacobian Df =
(2x, 2y, 2z ) thus the only critical point of F is 0. In particular 1 is a regular value of F an
Problem Set # 2
due Th Jan 24, in class
From the textbook:
Sect 1.5: # 1(b)(d)(e), #4 and #7(a) (only for arclength parametrization)
Sect 1.7: #2, 5, 6(a)(b), #8
and:
A. Show that a plane curve of con
Math 143 - Homework 3 solutions
April 23, 2015
Section 2.3, Problem 13. () Suppose W R3 is open, contains a point
p on the surface S and f : W R is smooth. If V S is a chart containing
p with correspo