Math 433
Introduction to Dierential Geometry
March 1, 2013
Midterm Exam
Instructions.
1. Two sides of a 3 5 card of notes allowed.
2. Show your work. Explain clearly.
3. There are 5 problems for a total of 60 points through page 7.
Name:
PROBLEM
POINTS
1a
Math 433 HW5 Due April 5, 2013
1. Let v and w be tangent vector elds along a curve : I S . Prove that
d
< v (t), w(t) >=< v, w > + < v, w > .
dt
2. Let (s) be a curve parametrized by arclength s, with nonzero curvature.
parametrized surface
Consider the
Math 433 HW4 Due March 22, 2013
1. Let : R2 R2 be given by (x, y ) = (u(x, y ), v (x, y ), where u and v and dierentiable
functions satisfy the Cauchy-Riemann equations
ux = vy , uy = vx .
Show that is a conformal map from R2 Q into R2 , where Q = cfw_(x,
Math 433 HW4 Due March 22, 2013
1. Let : R2 R2 be given by (x, y ) = (u(x, y ), v (x, y ), where u and v and dierentiable
functions satisfy the Cauchy-Riemann equations
ux = vy , uy = vx .
Show that is a local conformal map from R2 Q into R2 , where Q = c
Math 433 HW3 Due February 22, 2013
1. Compute the rst fundamental forms of the following surfaces
a. (u, v ) = (au cos v, bu sin v, u2 ); elliptic paraboloid.
b. (u, v ) = (au cosh v, bu sinh v, u2 ); hyperbolic parablooid.
Solution:
a.) (a2 cos2 v + b2 s
Math 433 HW3 Due February 22, 2013
1. Compute the rst fundamental forms of the following surfaces
a. (u, v ) = (au cos v, bu sin v, u2 ); elliptic paraboloid.
b. (u, v ) = (au cosh v, bu sinh v, u2 ); hyperbolic parablooid.
2. Calculate the rst fundamenta
Math 433 HW2 Due February 8, 2013
1. Let (s), s [0, l] be a closed convex plane curve positively oriented (i.e. s > 0). The curve
(s) = (s) rn(s), where r is a positive constant, is called a parallel curve of . Show that
a. The length l( ) = l() + 2r.
b.
Math 433 HW2 Due February 8, 2013
1. Let (s), s [0, l] be a closed convex plane curve positively oriented (i.e. s > 0). The curve
(s) = (s) rn(s), where r is a positive constant, is called a parallel curve of . Show that
a. The length l( ) = l() + 2r.
b.
Math 433 HW1 Due January 25, 2013
Without otherwise mentioned, all curves are smooth and regular.
1. A tractrix : (0, ) R2 is given by
t
(t) = (sin t, cos t + ln(tan ),
2
where t is the angle that the y axis makes with the vector (t).
a. Show that is a d
Math 433 HW1 Due January 25, 2013
Without otherwise mentioned, all curves are smooth and regular.
1. A tractrix : (0, ) R2 is given by
t
(t) = (sin t, cos t + ln(tan ),
2
where t is the angle that the y axis makes with the vector (t).
a. Show that is a d