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 = 1 is a regular
surface which is not dieomorphic to a
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 xhel = 1, F hel = xhel xhel = 0,
u
u
u
v
and Ghel = xhe
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 at which the unit tangent vector is equal to (0, 1).
c)
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 is an n n matrix of functions on U satisfying the
equa
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 curvature of S1 , by proposition 3 of 3.2 we have dN1 |(t
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 parameter t
is the angle (t) makes with the y axis from this
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 map
x(u, v ) =
2 v sin
nu
sin u,
2
2 v sin
nu
nu
cos u,
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
curvature kn of is 0, and so N n. Since N t = , then N = b an
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 k is a constant in this case.
Dierentiating the 1st eq
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 and
therefore S is a regular surface. But S cannot be die
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 constant nonzero curvature is a circle (hint: use Frenets
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 corresponding parametrization x : (U R2 open) (V R3 ), then
f x